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Students need rigorous training to familiarize themselves to the style of the exams they

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CONTENTS

Mathematics --·----e~·e-e ---·--

1. Integers .......................................... 9 -13

2. Fractions and Decimals .................. 14- 20

3. Data Handling ............................... 22- 26

4. Simple Equations ........................... 27- 31

5. Lines and Angles ............................ 33- 40

6. Triangles ......................................... 41-46

7. Congruence of Triangles ................. 48-54

8. Comparing Quantities .................... 55- 59

9. Rational Numbers .......................... 61- 66

10. Practical Geometry ........................ 67- 71

11. Perimeter and Area ....................... 73- 77

12. Algebraic Expressions .................... 78-82

13. Exponents and Powers ................... 84-87

14. Symmetry ...................................... 88- 93

15. Visualising Solid Shapes ................. 95 - 101

Questions@stimulating-minds ........ 103- 104

Model Test Paper ......................... 105 - 106

Explanatory Answers ................... 107 - 134


CHAPTER

1 Integers

+ Natural numbers (N): Counting numbers 1, 2, 3, 0000000 etc., are called natural numbers.

N = {1 I 2, 3, 4, 0000000}

Representation of natural numbers on a number line: To represent natural numbers on a number line we should draw a line and write the numbers at equal distances on it as shown.

I I I I I I I 1 2 3 4 5 6 7

I I I I I • l I 8 9 10 11 12 ....... :

+ Whole Numbers (W): The set of natural numbers together with zero is known as the set of whole numbers.

w: {0, 1 1 2, 3, oooooooo}

+ Integers (Z): The set containing negatives of natural numbers along with whole numbers is called the set of integers.

Z = [ ,-4, ~ 3, - 2, - 1, 0, 1, 2, 31" uouo)

Negatives of natural Whole numbers numbers

1, 2, 3, 4, 000000 etc., are called positive integers and are denoted by z+. :. z+ = {1, 2, 3, 4, 0000000}

- 1, -2, -3, -4, 000000 etc., are called negative integers and are denoted by z-. :. r = {.0000000000000000 - 3, - 2. - 1}

Note: 1. Usually, negative numbers are placed in brackets to avoid confusion arising due to two signs in evaluations. e.g.,3+ (-5) =-2

2. 0 is not included in either z• or r. Hence, it is non-negative.

(i) To represent quantities like profit, income, increase, rise, high, north, east, above, depositing, climbing and so on, positive numbers are used.

1. Integers II

Jl BMA's Talent & Olympiad Exams Resource Book Class VII - Mathematics

(ii) To represent quantities like loss, expenditure, decrease, fall, low, south, west, below, withdrawing, sliding and so on, negative numbers are used.

Note: 1. 0 is neither positive nor negative. 2. The +sign is not written before a positive number.

3. ! and 0.3 are not integers as they are not whole numbers.

+ Representation of integers on a number line: Integers are represented on the number line as shown.

[ ~ .... -1-j I I I I I -6 -5 -4 -3 -2 -1

On a number line when we

I 0

(i) add a positive integer, we move to the right. (ii) add a negative integer, we move to the left.

I 1

(iii) subtract a positive integer, we move to the left. (iv) subtract a negative integer, we move to the right.

+ Properties of integers:

I 2

I 3

I 4

l I Ill 5 ..... .

(i) Closure property: Closure property is satisfied with respect to addition, subtraction and multiplication in the set of integers. For a, b e Z, a + b e Z, a - b e Z and a x b e Z.

(ii) Commutative property: Commutative property is satisfied with respect to addition and multiplication in the set of integers. If a, b e Z, then a + b = b + a and a x b = b x a.

(iii) Associative property: Associative property is satisfied with respect to addition and multiplication in the set of integers. If a, b, c e Z, then a + (b + c) = (a + b) + c = c + (b + a) and a x (b x c) = (a x b) x c = c x (b x a).

(iv) Distributive property: Multiplication is distributed over addition and subtraction in the set of integers. For a, band c e Z, a (b + c) = ab + ac and a (b - c) = ab - ac.

(v) Identity element: 0 is the identity under addition and 1 is the identity under multiplication. For a e Z, a + 0 = a = 0 + a and a x 1 = a = 1 x a.

(vi) Multiplication by zero: For any integer a, a x 0 = 0 x a = 0.

II 1. Integers

BMA's Talent & Olympiad Exams Resource Book Class VII - Mathematics

0

• • 0

• • • • •

What do we call the set of negative numbers and whole numbers? (A) Natural numbers (B) Integers (C) Positive numbers (D) The set of whole numbers. Which of the following is the smal lest positive integer? (A) 0 (B) 100 (C) 1 (D) 9 Where are the negative numbers located on a horizontal number line? (A) On the right of 0 (B) On the left of 0 (C) Above 0 (D) BelowO

What is the opposite of earning ~ 100? (A) + ~ 1 00 (B) Profit of ~ 1 00 (C) Gain of ~ 1 00 (D) Spending ~ 1 00 How is the withdrawal of ~ 200 represented? (A) Depositing ~ 200 (B) - ~ 200 (C) ~ 200 (D) - 200 Which of the following is true with respect to -28 and -32? (A) -28 < -32 (B) -28 = -32 (C) -32 > - 28 (D) -28 > - 32 Where do we place the positive numbers on a vertical number line with respect to 0? (A) Above (B) On its left side (C) On its right side (D) Below What is the representation of 30 km towards the west? (A) 30 km east (C) 30 km

(B) -30 km (D) 30

What is the nature of the product of a negative integer by itself, odd number of times? (A) Positive (B) Negative (C) Non negative (D) Cannot be determined

1. Integers

• CD

What is the nature of the product of a negative number by itself even number of times? (A) Negative (B) 0 (C) Positive (D) Non-negative

Calculate (-32) x (-4) x (-3) x 0 x (-6) .

(A) 27648 (B) 276480 (C) 0 (D) -27648 CD If the dividend and the divisor have like signs, what is the sign of the quotient? (A) Positive (B) Negative (C) Zero (D) Indeterminate

• If the dividend and divisor have unlike signs, what is the sign of the quotient? (A) Positive (B) Negative (C) Zero (D) Indeterminate

• Match the following .

(i)

(ii)

(iii)

(iv)

Column - 1 Column - II

(132) 7 (-12) ( ) (a)

(-144) 7 (+16) ( ) (b)

(-32) 7 (-4) ( ) (c)

{196) 7 (4) ( ) (d)

(A) (i)-(b), (ii)-(a), (iii)-(c), (iv)-(d) (B) (i)-(a), (ii)-(b), (iii)-(d), (iv)-(c) (C) (i)-(d), (ii)-(c), (iii)-(b), (iv)-(a) (D) (i)-(c), (ii)-(d), (iii)-(a), (iv)-(b)

49

8

(-9)

(-11)

• With respect to which of the following operations is closure property satisfied by the set of integers? (A) +,x (B) +.~.x (C)+,x,- (D) +. -.~

4Z) What is the additive identity for the set of integers?

(A) 0 (B) (-1) (C) 1 (D) + 10

II

Jl BMA's Talent & Olympiad Exams Resource Book

G Which of the following is the multiplicative identity in the set of integers? (A) 1 (B) (-1) (C) 0 (D) (-1 0)

• Whatisthevalueof124 x 4-3 + 118 7 2? (A) 552 (B) 496 (C) 553 (D) -553

• Which of the following orders is used while evaluating an expression?

•

•

(A)[ 1. (), {} (B) {}. (), [ 1 (C) (), {}. [ 1 (D) (), [ 1. {} If a negative sign precedes a bracket, what happens to the terms inside it? (A) Their signs are changed. (B) The terms are reciprocated . (C) The signs remain the same. (D) The terms are doubled. If a positive sign precedes a bracket, what happens to the terms inside it? (A) Signs of the terms will be changed. (B) Every term is reciprocated. (C) Every term will become zero. (D) No change occurs in any of the terms. What is the value of the expression 7- [13 - {-2- 6 (6 of -5)}]? (A) -172 (B) 180 (C) 172 (D) 0

• • •

Class VII · Mathematics

What is the sign of the product obtained when a positive integer is multiplied by -1?

(A) Positive (C) 0

(B) Negative (D) Non negative

The sum of two integers is 62. If one of the integers is -48, what is the other?

(A) 14 (C) -110

(B) - 14 (D) 110

The product of two integers is -48. If one of the integers is -6, what is the value of the other?

(A) 1 (B) 288 (C) 0 (D) 8

A man walked 3 km towards North then 8 km towards South. What is his final position with respect to his initial position?

(A) 5 km towards East (B) 3 km towards South (C) 8 km towards North (D) 5 km towards South

What is the smallest negative integer?

(A) -1 (C) 0

(B) - 10 (D) Does not exist

What is the sign of the product of two integers with like signs? (A) Negative (B) Positive

~Previous Contest Questions~

•

(C) 0 (D) Cannot be determined What is the sign of the product of two integers with unlike signs? (A) Negative (B) 0 (C) Positive (D) Cannot be determined Which of the following operations on integers satisfy the commutative property? (A)-. + (B)-, X (C) +, - (D)+, X

Over which of the following operations is multiplication distributed in the set of integers? (A) -. + (B) -,X (C) +, - (D) x, 7

II

0

• In a quiz, positive marks were given for correct answers and negative marks for incorrect answers. If Guru's scores in five successive rounds were 35, -10, -15, 20 and 5, what is his total score at the end?

(A) 25 (B) 35 (C) 45 (D) 55

A deep well has steps inside it. A monkey is sitting on the topmost step (i.e., the first step). The water level is at the ninth step. If the monkey jumps 3 steps down and then jumps back 2 steps up, how many jumps does it have to make to reach the water level?

(A) 11 (B) 9 (C) 7 (D) 5

1. Integers


• A certain freezing process requires that room temperature be lowered from 4oC at the rate of soc every hour. What is the room temperature after 10 hours? (A) 0 oc (B) -5 oc (C) - 1 0 oc (D) -15°C

• In a class test containing 10 questions, 3 marks are awarded for every correct answer and (-1) mark is awarded for every incorrect answer and 0 for the questions not attempted. Srinu gets two correct and six incorrect answers out of eight questions he attempts. What is his total score? (A) 0 (B) 2 (C) - 2 (D) 6

• What should be multiplied by (-12) in order to get 180?

(A) 15 (B) -15 (C) 16 (D) - 16 0 A lift descends into an underground floor at the rate of 6 metres per minute. If the descent starts from 10 metres above the ground level, how much time will it take to descend 350 metres? (A) 30 minutes (B) 50 minutes (C) 1 hour (D) 1 hour 30 minutes

1. Integers

8 The temperature at 12 noon was 1 ooc above zero. If it decreases at the rate of 2 oc per hour unti l midnight, what would be the temperature at 9 p.m.?

(A) -8°C (B) -6°C (C) 8°C (D) 6°C 0 What is the identity element with respect to subtraction in integers?

(A) 0 (B) 1 (C) - 1 (D) Does not exist 0 Which of the following statements holds correct?

(A) NeWe Z (C) We NeZ

(B) Ze NeW (D) ze weN

The quotient of two numbers is (-17). If one of the numbers is (-340), what is the other number?

(A) 20 (B) 17 (C) (-20) (D) ( -30)

II

CHAPTER

• •

2 Fractions and Decimals

A fraction is a part of a whole . p

A number of the form - ,where p and q are whole numbers and q * 0 is known as a q fraction.

+ In the fraction £., p is called the numerator and q is called the denominator. q + The numerator tells us how many parts are considered of the whole.

+ The denominator tells us how many equal parts the whole is divided into.

Note: Usually fractions are written in their lowest terms. The numerator and the denominator of a fraction in its lowest terms are coprime. That is, their H.C.F. is 1.

+ Types of fractions:

(i) Simple fraction: A fraction in its lowest terms is known as a simple fraction.

12 5 4 e.g., 25, 7, · 3 etc.,

(ii) Decimal fraction: A fraction whose denominator is 10, 100, 1000 etc., is called a decimal fraction.

3 7 24 131 e.g., 10' 1 00 '1 000' 1 000 etc.,

(iii) Vulgar fraction: A fraction whose denominator is a whole number other than 1 0, 100, 1000, etc., is called a vulgar fraction.

2 4 11 27 e.g., 9' 13' 20' 109 etc.,

(iv) Proper fraction: A fraction whose numerator is less than its denominator is called a proper fraction.

3 5 23 73 e.g., 7' 11' 40 ' 100 etc.,

(v) Improper fraction: A fraction whose numerator is greater than or equal to its denominator is called an improper fraction.

11 25 41 53 eg -----etc . ., 7 ' 12 ' 36 ' 53 .,

II 2. Fractions and Decimals


(vi) Mixed fraction: A number which can be expressed as the sum of a natural number and a proper fraction is called a mixed fraction.

3 5 9 6 e.g., 14 . 4 7 . 7 D. 1 2 s etc.,

(vii) Like fractions: Fractions having the same denominator but different numerators are called like fractions.

5 9 11 e.g., 14.14. 14 etc.,

(viii) Unlike fractions: Fractions having different denominators are called unlike fractions.

2 5 9 e.g., 5. 7. D etc.,

+ An important property: If the numerator and denominator of a fraction are both multiplied by the same non zero number, its value is not changed.

3 3x2 3x3 3x4 Thus, - = -- = -- = -- etc.,

4 4x2 4x3 4 x 4 + Equivalent fractions: A given fraction and the fraction obtained by multiplying (or dividing)

its numerator and denominator by the same non-zero number, are called equivalent fractions.

9 3 6 12 e.g., Equivalent fractions of 12 are 4, 8. 16 etc.,

+ Method of changing unlike fractions to like fractions: Step 1: Find the L.C.M. of the denominators of all the given fractions.

Step 2: Change each of the given fractions into an equivalent fraction having denominator equal to the L.C.M. of the denominators of the given fractions.

5 7 11 e.g., Convert the fractions 6. 9 and 12 into like fractions.

L.C.M. of 6, 9 and 12 = 3 x 2 x 3 x 2 = 36

Now, ~ = 5 x 6 = 30 ; !._ = 7 x 4 = 28 and ~ = ~ = ~ . 6 6 X 6 36 9 9 X 4 36 12 12 X 3 36 30 28 33

Clearly, 36' 36 and 36 are like fractions.

a + Irreducible fractions: A fraction b is said to be irreducible or in lowest terms, if the H.C.F

of a and b is 1. They are also called simple fractions. a

If H.C.F. of a and b is not 1, then b is said to be reducible.

+ Comparing fractions: Let ~and~ be two given fractions. Then, b d

a c (i) - > - ~ ad> be b d

a c (ii) - = - ~ ad = be

b d

2. Fractions and Decimals

a c (iii) - < - ~ ad < be

b d

II

BMA's Talent & Olympiad Exams Resource Book r l Class VII - Mathematics

+ Method of comparing more than two fractions: Step 1: Find the L.C.M. of the denominators of the given fractions. Let it be m. Step 2: Convert all the given fractions into like fractions, each having mas denominator. Step 3: Now, if we compare any two of these like fractions, then the one having larger

numerator is larger. + Addition and subtraction of fractions:

(i) Add/Subtract like fractions: To add/subtract like fractions, add/subtract the numerators and place the sum/ difference on the same denominator as that of the given fractions.

e.g., (1) 2 3

Add 7 and 7 .

2 3 2 + 3 5 -+-=--=-7 7 7 7

4 6 e.g., (2) Subtract 7 from 7 .

6 4 6 -4 2 7 7 7 7

(ii) Add/Subtract unlike fractions: To add/subtract unlike fractions, first convert them into like fractions and proceed as in (i).

1 2 3 e.g., (1) Add 3. S and 7 .

L.C.M of 3, 5 and 7 is 1 05.

: . ~ + ~ + ~ = ~ + ~ + ~ = 35 + 42 + 45 = 122 = 1_22_ 3 5 7 105 105 105 105 105 105

e.g., (2) 3 7

Subtract 4 from 12 . 7 3 7 -9 -2 - 1 ---=--=- = -12 4 12 12 6

+ Multiplication of fractions: (i) Multiplying a whole number with a proper or an improper fraction: To multiply a

whole number with a proper or an improper fraction, we multiply the whole number with the numerator of the fraction, keeping the denominator same.

4 12 4 e.g., 3 x 9 = 9 = 3

(ii) Multiplying a whole number by a mixed fraction: To multiply a whole number by a mixed fraction, first convert the mixed fraction to an improper fraction and then multiply.

eg 14 x 2 ~= 1 4x .:!2 =34 . . , 7 7

II 2. Fractions and Decimals


(iii) Multiplying a fraction by a fraction: The product of two or more fractions is the product of their numerators divided by the product of their denominators expressed

in lowest terms i.e., if ~ and % are the fractions then their product is :~ expressed

in the lowest terms.

+ Calculating fractional part of a quantity:

To know the fractional part of a quantity, the fraction and the quantity are multiplied.

1 1 e.g., 3 of ~ 90 = ~ 3 x 9 0 = ~ 30

+ Reciprocal of a fraction:

Two fractions are said to be the reciprocal of each other, if their product is 1.

e.g., i and ~ are the reciprocals of each other, since ( i x ~) = 1.

In general, if ~ is a non-zero fraction, then its reciprocal is .; .

Note: Reciprocal of 0 does not exist.

+ Division of fractions:

(i) Division of a whole number by any fraction: To divide a whole number by a fraction, we have to multiply the whole number by the reciprocal of the given fraction.

e.g.,

(ii) Division of a fraction by a whole number: To divide a fraction by a whole number, we have to multiply the given fraction by the reciprocal of the whole number.

2 22 1 2 e.g., 4 5 .;- 11 = S x 11 = 5 While dividing a mixed fraction by a whole number, convert the mixed fraction into an improper fraction and then divide.

(iii) Division of a fraction by another fraction: To divide a fraction by another fraction, we have to multiply the first fraction by the reciprocal of the second.

e.g., 3 .:!_ + 2..!. = ~ + .!2 = .ll x 2. = ~ = 11.! 4 5 4 5 4 11 44 44

2. Fractions and Decimals II


0

• • • • • • •

Which of the follow ing is true with

respect to ~ and .:!2? 16 5

(A) ~ > 13 16 5

(B) 9 = 13 16 5

(C) ~ < 1 3 (D) 1 3 < ~ 16 5 5 16

Which of the following is the improper 1

fraction form of 12 5 ?

(A) !.!:. 5

(B)~ 5

(C) 1 08 (D) 85 5 5

3 7 3 What is the sum of 4 ' 6 and 15 ?

(A) 3_!_ 2

331 213 412 (B) 60 (C) 60 (D) 120

13 11 What is the difference of 15 and 35 ?

58 5 8 26 23 (A) 105 (B) 105 (C) 30 (D) 15 Which of the following is a proper fraction?

(A) ~ (B) 3_!_ 1 5 23

(C) ~ 7

(D) 34 3

What is the product of 1..:!.. 3 .2. and !.._ ? 3 4 8

(A) 3~ (B) 2~ (C) 3~ (D) 2~ 24 24 24 24

3 Guru reads

5 of a book. He finds that

there are still 80 pages left to be read. What is the total number of pages in the book? (A) 100 (B) 200 (C) 300 (D) 400

1 What is '7 of 49 litres?

(A) 1 l (B) 5 l (C) 7 l (D) 6!

II

• Indian cricket team won 4 more matches than it lost with New Zealand. If it won 3 5 of its matches, how many matches

did India play?

(A) 8 (B) 12 (C) 16 (D) 20

What is the product of a fractional number and its reciprocal? (A) 0 (B) same number (C) 1 (D) undefined

21 . 26 What should be added to

27 to make 1t

27?

(A) ~~ (B) 267 (C) 257 (D) :7 What are fractions with different denominators called? (A) Like (C) Proper

(B) Unlike (D) Improper

G) What is the equivalent fraction of 181

having the numerator 40? 40 44 40 10

(A) 11 (B) 40 (C) 55 (D) 40

• What do you call the fractions which have one as the numerator?

(A) Like fractions (B) Unlike fractions (C) Unit fractions (D) Equal fractions

In which of the following fractions is the numerator greater than the denominator?

(A) Like (C) Proper

1

(B) Improper (D) Mixed

9 of a number is 5. What is the number?

(A) 5 (B) 9 (C) 14 (D) 45

Given that ~ = t. which of these is true?

(A) pq = st (B) ps = qt p s (C) pt = sq (D) - =-t q



•

1 What should be added to 3 to get 3- ? 5

3 1 2 2 (A) 5 (B) 5 (C) S (D) 35

7 X If 4- = - , what is the value of x?

11 11 (A) 51 (B) 7 (C) 44 (D) 28

12 What is the lowest form of 30 ?

1 1 2 6 (A) 5 (B) 2 (C) S (D) 15 In wh ich of the following does the shaded part represent one third of its whole?

(A) tE (C) A

(B) u (D)

How many parts should be shaded in the figure B to make it represent the same fraction as the unshaded part of the figure A?

M ffiHHHE Figure A Figure B

(A) 3 (B) 15 (C) 8 (D) 12

Madhavi eats one full bar of chocolate . Then she divides another one into 5 equal parts and eats 3 of them. What is the total number of chocolates that she has eaten?

4 3 (A) - (B) -

5 5 8

(C) -5

8 (D) 1Q

How many one-fourths need to be 1

added to 24 to make 5?

(A) 3 (B) 4 (C) 5 (D) 11

11 Between which two numbers does 4 1ie?

(A) 1 and 2 (B) 2 and 3 (C) 3 and 4 (D) 11 and 12


(26-27): Look at the alphabets given.

•

•

PRASAD What fraction of alphabets are made of 3 straight lines?

1 (A) -

3 1

(B) -6

2 (C) -

5 5

(D) -6

What fraction of alphabets are made of semicircles and straight lines ?

1 (A) -2

(B) ~ 5

(C) ~ 6

4 (D) -5

Which of the following is an improper fraction? (A)® (C) ~

(B) ITIIIIJ (D) ITIIIIIJ

What does the shaded part of the following strip represent?

2 1 2 3 (A) 7 - 7 + 7 = 7 (B)

2 1 2 1 - +- - - = -7 7 7 7

2 1 3 (C) - + - =-7 7 7

1 2 2 5 (D) - + - + - = -7 7 7 7

Which of the following statements is true?

(A) Fractions with the same numerator are called like fractions.

(B) Fractions with the same deno-minator are called unlike fractions.

(C) Difference of two like fractions

difference of numerators comm on denominator

(D) A fraction with the numerator greater than or equal to the deno-minator is called a proper fraction.

II


1 Which figure shows - shaded?

10

(A) I I I I I I I I I I I (B) I I I I I I I I I I

~;::::;::::;;::::;::::;;:::::;:::;~~ (C) I I I I I I I I I I ~~~~~

(D) I I I I I I I I I I How is the decimal number 9.6 read?

(A) Nine point six (B) Ninety six (C) Nine six (D) Ninety and six What is the decimal equivalent of

4 9 - +0+-- ? 10 1000 °

(A) 0.049 (C) 0.409

(B) 409 (D) 0.490

What is the place value of 3 in 9.365?

(A) 3 (B) 30 3 3

(C) 10 (D) 1 000

How can 8 hundredths be written? (A) 0.008 (C) 0.800

(B) 0.08 (D) 800

What is the decimal fraction shown by the shaded part of the figure given?

I I I I I I I I I I I (A) 2.5 (B) 0.5 (C) 0.6 (D) 0.4

• Previous Contest Questions ..o111111111111

0 Find the value of 3 i. x 2 3.. x 1 ~. 7 5 4

• (A) 1 (B) 5 (C) 10 (D) 15 A book consists of 216 pages. During last

3 week, Suresh read 4 of the book. How

many pages did he read? (A)126 (B)162 (C)116 (D)161

II

• • • • 0

0

0

Class VII - Mathematics

2 If the cost of 55 litres of milk is

~ 1 01~ , what is its cost per litre?

(A) ~ 1 8 ~ (B) ~ 17 ~ 4 4

(C) ~ 1 6 ~ (D) ~ 1 5 ~ 4 4

The product of two numbers is 15~ . If 6

one of the numbers is 6 3.. , what is the other number?

3

(A) I (B) 1~ (C) 2~ (D) 3~ 8 8 8 8

The product of two decimals is 1.5008. If one of them is 0.56, what is the other? (A) 2.86 (C) 2.68

(B) 26.8 (D) 0.268

A badminton player won 6 games and lost 4. What fraction of the games did he win?

(A) ~ 4

4 (B) -

6 6 5

(C) 10 (D) 10 What is the value of

3_2_ - [1~ + {22-(12-2)l]? 12 4 2 2 3 f 0

(A) 0.5 (B) 2 (C) 1 (D) 0

5 2 Ravi had 6 of a cake. He ate 3 of it.

What part of the cake did he eat?

5 (A) -

9 (B)~

12 10 10

(C) - (D) -6 3

How is the fractional number for 5 out of 7 written?

(A) ~ 7

1

7 (B) -

5 (C) ~

7 4

(D) -5

If 40- 5 x P = 0, then what is the value of P?

(A) 0 (B) 2. (C) 199

(D) 200 5 5



Data Handling

+ Representing data with the help of bars or rectangles of uniform w idth in a diagram is called a bar graph or a bar diagram.

+ Each bar represents only one value of the data and hence there are as many bars as there are values in the data.

+ The length of the bar indicates the value of the item. The width of the bar does not indicate anything.

+ All bars should rest on the same line called the base. + The bars may be drawn horizontally or vertically. + A double bar graph helps us to compare two collections of data at a glance. Collection and Tabulation of Data: + The word data means information in the form of numerical figures or a set of g iven information. + Data obtained in the original form is cal led a raw data. + Arrang ing the numerical figures of a data in ascending or descending order is called an array. + Arranging the data in a systematic tabular form is called tabulation or presentation of the data.

Tabulated data is easy to understand and interpret. + Each numerical figure in a data is cal led an observation. + The number oftimes a particular observation occurs is called its frequency. + The difference between the highest and the lowest values of the observations in a given data is

called its range. + When the number of observations is large, we make use of tal ly marks to find the frequencies. + Tallies are usually marked in a bunch of five for the ease of counting. Mean, mode and median: + Mean in statistics is the same as average in arithmetic. Average is a number that shows the central

tendency of a group of observations.

For a raw data,

M Sum ofobservat ions Sumofthe n umbers

ean = Mean of 'n' numbers = --------Number of observat ions Number of addends

+ Mode: The observation which occurs for a maximum number oftimes is cal led the mode of the given data.

+ Median: After arranging data in ascending or descending order of magnitudes the value of the middle term is called the med ian of the data. (i) When the number of observations is odd, there w ill be only one middle term and this term is the

median. (ii) When the number of observations is even, there will be two middle terms. The average of

these two midd le terms is the median of the data. + Some situations in our li fe happen certainly. Some are impossible and some that may or may not

happen. This is called chance or probability of an event to occur.

II 3. Data Handling


0

• • 0

• • •

•

What is the arithmetic mean of 1, 2, .... , 9 and 10? (A) 5.5 (B) 6 (C) 7.5 (D) 10 In a data, 10 numbers are arranged in increasing order. If the 71h entry is increased by 4, by how much does the median increase?

(A) Zero (B) 4 (C) 6 (D) 5 What is the mean of x, x + 3, x + 6, x + 9 and x + 12? (A) X + 3 (B) X + 6 (C) X + 9 (D) X + 12 The daily sales of kerosene (in litres) in a ration shop for six days is given in the box.

[ 75, 120, 12, 50, 70.5, 140.5 )

What is the average daily sale?

(A) 150 l (B) 1 0 l (C) 142 l (D) 78l The mean of five numbers is 27. If one of the numbers is excluded, the mean gets reduced by 2. What is the excluded number?

(A) 35 (B) 27 (C) 25 (D) 40 What is the median of the data 46, 64, 87, 41,58,77,35,90,55,33,92?

(A) 87 (B) 77 (C) 58 (D) 60.2 Which of the following is true about mean?

(A) It occurs most frequently.

(B) It d ivides observations into two equal parts.

(C) It is representative of the whole group.

(D) It is the sum of observations. If each entry of a data is increased by 5, how does the mean change? (A) Remains the same . (B) Increases by 5.

3. Data Handling

• (C) Decreases by 5. (D) Becomes half. The arithmetic mean of five given numbers is 85. What is their sum? (A) 425 (B) 85 (C) A number between 85 and 425. (D) A number greater than 500. The average weight of a sample of 10 apples is 52 g. Later it was found that the weighing machine had shown the weight of each apple 10 g less. What is the correct average weight of an apple?

(A) 62 g (B) 54 g (C) 56 g (D) 52 g

The mean of6,y, 7,xand 14 is 8.Which of the following is true?

(A) X + y = 13 (B) X - y = 13 (C) 2x + 3y = 13 (D) x2 + y2 = 15

Which of the following is correct about mode?

(A) It is central. (B) It occurs most frequently. (C) It lies between the maximum and

minimum observations .

(D) It is the average of the two middle terms. Rajani has a box with 6 marbles numbered from 1 to 6 on each of them. She picks a marble from it without seeing. What is the probability that the marble picked has the number 3 on it?

1 2 3 (A) - (B) - (C) -

6 3 4 (D) 1

4 (14-18): The heights of six mountains are 8200

m, 6000 m, 8600 m, 7500 m, 8800 m and 6500 m . Based on this information, answer the questions given.

• What is the approximate average height of the mountains?

(A) 7657 m (C) 7756 m

(B) 7600 m (D) 7765 m

II


• Find the median height of the mountains.

(A) 7850 m (C) 8750 m

(B) 7580 m (D) 5780 m 0 What is the mode of the heights?

(A) 6000 (B) 8800 (C) Does not exist (D) 7 500 4D Which of the following statements is true?

(A) The mean height of the mountains is greater than their median height.

0

(B) The mean height of the mountains is less than their mode.

(C) The median height of the mountains is less than their mode.

(D) The median height of the mountains is greater than their mean height.

Rakesh and Sanjay planned to go trekking on any of these mountains. They wrote the heights on chits of paper, shuffled them and picked one. What is the probability that the height picked is the maximum?

1 (A) -3

2 (B) -3

1 (C) -6

(D) _!_ 4

(19-21): The ages (in years) of some teachers of a school are given in the box.

26, 32, 38, 41, 26, 31, 35, 33, 26, 37 l Based on this information, answer the following questions. What is the range of the ages of the teachers?

(A) 15 years (C) 41 years

(B) 26 years (D) 32 years

What is the mean age of the teachers?

(A) 23.5 years (B) 32.5 years (C) 35 years (D) 38 years

What is the mode of the given data?

(A) 32 years (B) 41 years (C) 26 years (D) 31 years

II

(22-25): The heights of 10 students, measured in em are as follows: 143,132,150, 139,128,135, 151, 146, 141, 149 Based on this information, answer the following questions.

fZ) What is the height of the shortest girl?

(A) 128 em (B) 141 em (C) 151 em (D) 150cm

• What is the range of the data?

(A) 35 em (B) 31 em (C) 23 em (D) 28 em

What is the mean height of the students?

(A) 141.4 em (B) 141 em (C) 142cm (D) 151 em

The height of how many students is greater than the mean height?

(A) 4 (B) 3 (C) 5 (D) 2

In which of these situations is a double bar graph useful?

(i) Enrolment of students in class VII in 2009 and2010.

(ii) Marks obtained in Term I and Term II examinations.

(iii) Marks obtained in all subjects of a term examination.

(A) (i) and (ii) (C) (iii) and (i)

(B) (ii) and (iii) (D) (i) only

Which of these is certain to happen?

(A) You look younger today than yesterday. (B) You look older today than yesterday. (C) A tossed coin will land heads up. (D) Tomorrow will be a cloudy day.

Which of these is impossible to happen ?

(A) A tossed coin lands with heads up. (B) A tossed die lands up with 4 on top. (C) The next traffic light is red. (D) A die thrown lands up with 7 on top.

3. Data Handling


•

Which of these can happen but not certainly?

(A) A tossed coin lands with heads up. (B) The sun rises in the east. (C) A die thrown lands with 8 on the top. (D) The Earth revolves around the Sun.

Which of these is impossible?

(A) The next traffic light seen is red. (B) A tossed die lands up with 8 on top. (C) A flipped coin lands up with head on top. (D) It rains tommorow.

Find the range of the data.

128, 139, 148, 132, 152, 154,140,143,146,149,142

(A) 9 (B) 26 (C) 5 (D) 24

The table shows the number of hours Pavan studies on different days of a week.

Mon Tue Wed Thurs Fri Sat

3 4 2 5 4 3

How many hours per day does he study on an average?

(A) 3.5 hours (C) 4 hours

(B) 3 hours (D) 4.5 hours

(33-35): The bar graph shows the sales of fruits and vegetables in a store in 4 hours on a certain evening.

5 (}

(}

(}

(}

(}

0

Sales of fruits and vegetables in a store

D fruit s etables 45 D veg -

~ ~ 35 -

~ 26 ,....-

2020 --

5 p.m 6 p.m 7 p.m 8 p.m

3. Data Handling

•

How many kilograms of fruits were sold during the four hours?

(A) 120 kg (B) 124 kg (C) 126 kg (D) 144 kg

When was the sale of fruits lesser than that of vegetables?

(A) 7 p.m (B) 6 p.m (C) 8 p.m (D) 5 p.m

During the four hours, which of the following is true about the sale of fruits and vegetables?

(A) The sale of vegetables was lesser than that of fruits.

(B) The sale of vegetables is 126 kg. (C) The sale of fruits is 11 8 kg.

(D) The sale of vegetables is greater than that of fruits .

The number of chapatis needed for 30 students of a class are given in the box.

2, 1, 2, 3, 3, 2, 2, 2, 4, 4 3, 2, 3, 3, 2, 2, 2, 2, 3, 3 2, 3, 2, 2, 2, 3, 3, 3, 3, 4

Calculate the mode of the data.

(A) 3 (B) 2 (C) 1 (D) 4

• Previous Contest Questions~

0 Which of the following is true?

•

(A) The mean of the first 5 natural numbers is the same as their median.

(B) The mean of the first 5 natural numbers is the same as the mean of the first 5 whole numbers.

(C) The median of the first 5 whole numbers is the same as the mean of the first 5 natural numbers.

(D) The mode of first 5 natural numbers is 5 .

What is the arithmetic mean of first five prime numbers?

(A) 6.2 (B) 5.2 (C) 6.5 (D) 5.6

II


• A comet passed by the Earth in the year 1835. It passes by the Earth every 60 years. Based on this information, in which of the fo llowing years can the comet be expected to pass by the Earth ? (A) 2035 (B) 2060 (C) 2075 (D) 2080

(4-6): The bargraph shows the marks obtained by four students in quarterly examination.

•

y

600

~ 500 ... nl E 400 0 ... 300

Q) .JJ E 200 :s z 100

0

r-

I ~ s:: ~

~ II)

r-

r-

s:: s:: nl

"C r- ~ s:: nl

nl ~ z a. .s::; Q) II) Q) 0

X Students

Whose perforamnce was the best? (A) Nandan (B) Shravan (C) Sharanya (D) Deepa

II

• •

Which two students secured equal marks? (A) Shravan and Deepa (B) Sharanya and Shravan (C) Deepa and Nandan (D) Sharanya and Deepa

Whose performance was the worst? (A) Sharanya (B) Deepa (C) Shravan (D) Nandan

(7 -8): The pie-chart depicts the results of a survey conducted to identify the favourite juice of some students.

• • Pineapple

How many students like other juices if the total number of students is 360 ? (A) 70 (B) 80 (C) 65 (D) 60

How many students like orange juice, if the strength of the school is 720 ? (A) 150 (D) 360 (C) 180 (D) 240

3. Data Handling

Simple Equations

+ Variable: A symbol which takes various values is known as a variable. Normally it is denoted by letters x, y etc.

+ Constant: A symbol having a fixed numerical value is called a constant.

Sometimes, 'c', 'k' etc., are used as symbols to denote a constant.

+ Coefficient: In the product of a variable and a constant, each is called the coefficient of the other.

Sometimes, symbols like a, b,l, m etc., are used to denote the coefficients.

+ Expression: An expression can be defined as a combination of constants, variables and coefficients by some or all of the four fundamental mathematical operations ( +, - . x and + ).

e.g., 3y - 14 Here, 3 is the coefficient of 'y'. 'y' is the variable and - 14 is the constant.

+ Equation: A statement of equality of two algebraic expressions involving a variable is called an equation.

+ Simple linear equation: An equation which contains only one variable of degree 1 is called a simple linear equation.

e.g., (i) 3x - 2 = 5 - 4x (ii) 2(t- 4) = 6 (iii) 2y + 5 = r - 2 6

p - 1 2p (iv) --+- = 3

6 7 + Solution of an equation: The value of the variable, which when substituted in the g iven equation,

makes the two sides L.H.S. (Left Hand Side) and R.H.S. (Right Hand Side) of the equation equal is called the solution or root of that equation.

e.g., 3x + 4 = 1 0 ~ 3x = 1 0 - 4 = 6 ~ x = 2 Verification: Substituting x = 2, we have L.H.S. = 3x + 4 = 3 (2) + 4 = 6 + 4 = 10 = R.H.S.

x = 2 is a solution of the given equation 3x + 4 = 10.

+ Rules for solving an equation: (i) Same number can be added to both sides of an equation.

(ii) Same number can be subtracted from both sides of an equation.

(iii) Both sides of an equation can be multiplied by the same non-zero number.

(iv) Both sides of an equation can be divided by the same non-zero number.

(v) Transposition: Any term of an equation may be taken to the other side w ith the sign changed. This process is called transposition.

e.g., 4x- 5 = 3x + 5 ~ 4x = 3x + 5 + 5 [Transposing '-5' to R.H.S.]

~ 4x- 3x = 10

~X= 10

[Transposing '3x' to L.H.S.]

(vi) Cross multiplication: If ax +db = £..then q(ax + b) = p(cx + d). This process is called cross ex+ q

multiplication.

4. Simple Equations II


0

• •

0

•

What is the va lue of 'x' in

3x - 1 _ 1+x =3 - x - 1? 5 2 2 .

(A) 5 (B) -7 (C) 7 (D) -5

If 0.2(2x -1)- 0.5(3x - 1) = 0.4, what is the value of 'x'?

1 (A) 11 1 3 3

(B) - 11 (C) 11 (D) - 11 In each of these figures the solution of an equation is given in brackets. Which of them is correct?

(A) fj;Jif] (B) 1\ ~

(C) I (I::) I ~ (D)~

Sunil wrote an equation as ~ = 4.

Ravi wrote a statement for Sunil's equation. Which of these is the statement of Ravi if he has written correctly?

(A) One-fifth of'm' is 4. (B) One-fifth of a number is 5. (C) One-fourth of 'm' is 4. (D) One-fourth of a number is 4. In which of the following cases does an equality NOT hold?

(A) Adding the same number on both the sides.

(B) Not performing the same operation on both the sides.

(C) Subtracting the same number from both the sides.

(D) Multiplying both the sides by the same non-zero number.

II

• 0

0


What are the two steps involved in solving the equation 15x + 4 = 26?

(A) Subtracting 4 from both the sides and then dividing both sides by 15.

(B) Adding 4 on both sides & then multiplying both sides by 15.

(C) Subtracting 4 on the L.H.S. and multiplying by 15 on the R. H.S.

(D) Adding 4 on the L.H.S. and dividing by 15 on the R.H.S.

Which of the following equations can be constructed with x = 2?

(A) 3x + 4 = 8 (C) 3x + 4 = 2

(B) 3x- 4 = 2 (D) 3x - 4 = 8

5 7 subtracted from 2 of a number results

in 23. What is the number?

(A) - 10 (C) 12

(B) 10 (D) - 12

In a coconut grove, (x + 2) trees yield 60 coconuts per year, x trees yield 120 coconuts per year and (x - 2) trees yield 180 coconuts per year. If the average yield per year per tree is 1 00, find x. (A) 4 (B) 3 (C) 2 (D) 1

4 is added to a number and the sum is multiplied by 5. If 20 is subtracted from the product and the difference is divided by 8, the result is equal to 10. Find the number.

(A) 16 (C) 8

(B) 12 (D) 20

A number is 31ess than two times the other. If their sum is increased by 7, the result is 37. Find the numbers.

(A) 9, 11 (C) 11 ,19

(B) 11,13 (D) 9,13

4. Simple Equations

BMA's Talent & Olympiad Exams Resource Book

CD ~ is subtracted from a number and the

difference is multiplied by4.1f25 is added to the product and the sum is divided by 3, the result is equal to 10. Find the number.

•

• • •

3 (A) -

5 7

(B) -4

(C) .§. 7

2 (D) -3

The present age of A is twice that of B. 30 years from now, age of A will be 1 Y2 times that of B. Find the present ages (in years) of A and B respectively.

(A) 60, 30 (B) 30, 60 (C) 40, 50 (D) 50, 40

5th

A person travelled 8 of the distance by

fh train, - by bus and the remaining15

4 km by boat. Find the total distance travelled by him.

(A) 90 km (C) 150 km

(B) 120 km (D) 180km

The total cost of three prizes is 3 th < 2550.1fthe value of second prize is 4

1 of the first and the value of 3'd prize is 2 of the second prize, find the value of first prize. (A) < 900 (C) < 1200

(B) < 1500 (D) < 450

Which of the following is an equation?

(A) 4x + 5 = 65 (B) 4x + 5 < 65 (C) 4x + 5 > 65 (D) 4x + 5 ::;; 65

Which of the following is an algebraic expression for the statement "The sum of 3x and 11 is 32."?

(A) 11x + 3 = 32 (C) 3x + 32 = 11

(B) 3x + 11 = 32 (D) 11x + 32 = 11

Choose the statement that best 1

describes the equation 4 m = 1 0.

4. Simple Equations

•

•


(A) One - fourth of 10 is m. (B) One - fourth of m is 3 more than 3. (C) One - fourth of m is 1 0. (D) Four times m is 1 0. Vinay's father is 44 years o ld. If he is 5 years older than thrice Vinay's age, which of these equations gives, the age of Vi nay's father?

(A) 3x + 5 = 44 (B) 44 + 5x = 3x (C) 44-3y= 5+3y (D) 3x-5 = 44

Which of the following statements is false?

(A) The solution of 4x = 60 is 12. (B) y = 7 satisfies the equation y + 0 = 7.

5 (C) p = 2 is the solution of 12p- 5 = 25.

3 (D) m = 2 is the solution of 4(m + 3) = 18.

Which of the following does not affect the given equation? (A) Adding 0 on the L.H.S. and 1 on the

R. H. S. (B) Adding 1 on the L.H.S. and (-1) on

the R.H.S. (C) Adding the same number on both

sides of the equation. (D) Adding 0 on the R. H. S. and 1 on the

L.H.S . P is a linear equation. How many solutions does P have? (A) 1 (B) 0 (C) 3 (D) Infinitely many Ramesh got 5 marks more than Sonu in a test. If the total marks secured by them is 15, how many marks did Ramesh get?

(A) 25 (B) 5 (C) 15 (D) 10

In a test match, Sachin scored twice as many runs as Sehwag. Together, their runs fell two short of a double century. How many runs did Sachin score?

(A) 66 (C) 198

(B) 132 (D) 200

II


• In a math test, the highest marks obtained by a student in the class is twice the lowest marks plus 7. If the highest score is 87, what is the lowest score?

(A) 42 (C) 40

(B) 39 (D) 44

A teacher asks the students of her class to write an equation for the statement 'Ten times a number pis 100." Three students wrote the following equations. Which is correct?

(A) (i) only (C) (iii) only

(i) 10 p = 100

(ii) 10 = 100 p

( ... ) lOp 100 Ill p =

(B) (ii) only (D) Both (B) and (C).

On transposing terms from one side of the equation to the other, which of these changes takes place ?

(A) Addition becomes subtraction. (B) Multiplication becomes addition. (C) Addition becomes multipl ication. (D) Multiplication becomes subtraction.

2x + 5 3x - 2 M = -- and N = --. What value 7 4 of x makes M = N?

- 17 (A) 3

34 (C) 13

- 34 (B) 13

17 (D) 3

• Given A= P(1 + rt), what is the value of 'r'

when A = 27, P = 18 and t = 5?

(B) 1

(A) - -2 5

27 1 (C) - (D) -

5 10

II

• 1 1 1

Given - +- = -f, find the value of 'v' u v when f = 20 and u = 30.

(A) -20

(C) 60

(B) -60

(D) -30

The sum of three consecutive integers

is 75. Which is the largest among them?

(A) 26

(C) 24

(D) 25

(D) 23

C!) The lengths of the sides of a triangle are

(2a + 1) em, (3a + 2) em and (4a -1) em.

For what value of 'a' is the perimeter of

the triangle 92 em?

(A) 5 (C) 8

(B) 9 (D) 10

A father is 26 years older than his son. In 3 years' time, the son's age will be one-third his father's age. What is the present age of the son?



Pankaj has 96 marbles and Arun has 63 marbles. How many marbles should Arun give Pankaj so that Pankaj will have twice as many marbles as Arun?

(A) 9 (B) 12 (C) 7 (D) 10

~Previous Contest Questions~

• If 3P + 2 _ 4P - 3 + P - 1 = 4, find the 5 7 35

value of p.

(A) 65 (B) 63 (C) 36 (D) 56 • The sum of five times a number and 13 is 48. What is the number? (A) 3 (B) 5 (C) 7 (D) 9

4. Simple Equations


• • • •

•

Guru is 20 years o lder than his son. If the sum of their ages is 50 years, how old is his son?



If one-fourth of a number decreased by 12 is 30, what is the number?

(A) 168 (C) 148

(B) 186 (D) 184

If C = ~ (F - 32) , what is the value F? 9

(A) 5C- 32 (B) 9C- 32 9 5

(C) 9C + 32 5

(D) 5C + 32 9

The scale shown is balanced. Each cube on the left side weighs the same amount.

How much does one of the cubes weigh?

(A) 1 gram (B) 4 grams (C) 5 grams (D) 20 grams

The figure shows the selling price of pens in a shop. Vimal paid ~ 50 for the purchase of a few pens and received ~ 0.50 in change.

5 for ~11.25

How many pens did Vimal buy?

(A) 26 (C) 19

(B) 22 (D) 16

4. Simple Equations

0

0


The sum of two-thirds of a number and one-fifth of the same number is 13. Find the number.

(A) 15 (C) 13

(B) 3 (D) 5

x - 4 2x + 1 5x + 1 Evaluate -

3- - -

6- = -

2- .

3 (A) -5

5 (C) 6

4 (B) 5

-4 (D) 5

The denominator of a fraction is 3 more than its numerator. If 2 is added to both the numerator and the denominator, the

2 new fraction is equivalent to 3 What is the original fraction?

(A) ~ 7

(B) i 7

2 (C) -3

3 (D) -5

144 beads were shared equally among some children. If there were 3 children fewer, each child would have 16 beads each. How many children were there? (A) 8 (B) 9 (C) 12 (D) 11

II


CHAPTER

5 Lines and Angles

+ Point: A point is a geometrical representation of a location. It is represented by a dot. + Line: A geometrical line is a set of points that extends endlessly in both the directions i.e., -a line has no end points. A line AB is represented as AB .

[ ~ : : ~ ) + Line segment: A line segment is a part of a line. A line segment has two end points. A line

segment AB is represented as AB . ,....-------...... [ : : l

+ Ray: A ray is a part of the line which has one end point (namely its starting point).

[ ~ : ~ ) 0 -A ray OP IS denoted as OP.

+ Angle: An angle is the union of two rays with a common initial point.

The symbol of angle is L . An angle is measured in degrees (0).

--+ --+ The angle formed by the two rays AB and AC is denoted by L.BAC or L.CAB .

The two rays AB and Al are called the arms and the common initial point 'A' is called the vertex of the angle ABC.

+ Types of Angles: (i) Right angle: An angle whose measure is equal to 90° is called a right angle.

I :~J (ii) Acute angle: An angle whose measure is less than 90o is called an acute angle.

5. Lines and Angles II


(iii) Obtuse angle: An angle whose measure is greater than go• but less than 180. is called an obtuse angle.

·---~ A C (iv) Straight angle: An angle whose measure is equal to 180. is called a straight angle.

[•a "i'tso·c·] (v) Complete angle: An angle whose measure is exactly equal to 360. is called a complete

angle.

(vi) Reflex angle: An angle which is greater than 180• but less than 360• is called a reflex angle.

[·-------,~ (vii) Zero angle: An angle whose measure is o· is called a zero angle.

[ . ! . l - - -Note: In a zero angle, the rays OA and 08 coincide without any rotation of 08. That is no angle is formed between the two rays.

+ Related Angles: (i) Complementary angles: Two angles are said to be complementary if the sum of

their measures is equal to go•.

Here Lx + Ly =go·, therefore L x and L y are complementary angles.

(ii) Supplementary angles: Two angles are said to be supplementary if the sum of their measures is equal to 180·.

[,...--• -~ -vd-x 8-.A -----,.1

Here, Lx + Ly = 1so·, therefore L x and L y are supplementary angles.

II 5. Lines and Angles


+ Adjacent angles: Angles having a common vertex, a common arm and the non-common arms lying on either side of the common arm are called adjacent angles.

IAI --+ In the given figure, LAOB and LCOB have a common vertex '0 ', a common arm OB and

--+ --+ --+ OA and oc are on opposite sides of OB . So they are adjacent angles. + Linear pair of angles: Two adjacent angles make a linear pair of angles, if the non-common

arms of these angles form two opposite rays (with same end point). In the figure given, the angles BAC and CAD form a linear pair of angles because the non-common arms AB and AD of the two angles are the opposite rays, with the same vertex A.

[. D 4 • B

Moreover, LBAC + LDAC = 180° .

Note: 1. A linear pair is always supplementary. 2. A linear pair is always adjacent, while a pair of adjacent angles need not be a

linear pair.

+ Vertically opposite angles: Two angles having the same vertex are said to form a pair of vertically opposite angles, if their arms form two pairs of opposite rays.

In the figure given, LBOD and LAOC are a pair of vertically opposite angles because they have common vertex at 0 and also OB, OA; OC, OD are two pairs of opposite rays. Vertically opposite angles are formed when two lines intersect.

1~1 Similarly, we find that LBOC and LAOD is another pair of vertically opposite angles.

Note: If two lines intersect each other, the vertically opposite angles formed are equal.

Pair of Lines: + Intersecting lines: Two lines which are distinct and have a common point are called intersecting

lines. The common point is called the point of intersection of the two lines. + Perpendicular lines: If two lines l and m intersect at right angles, they are called

perpendicular lines, denoted as z .l m, read as l is perpendicular to m. + Parallel lines: Two lines l and m are said to be parallel, if they lie in the same plane and do

not intersect when produced however far on either side and is written as l I I m read as l is parallel to m.

5. Lines and Angles II


+ Transversal: A line which intersects two or more lines at distinct points is called a transversal.

In the given figure, p is a transversal to the lines land m. + Angles made by a transversal:

In the figure given, lines l and m are cut by the transversal p. The eight angles marked 1 to 8 have names given in the table.

Interior angles L3, L4, LS, L6

Exterior angles Ll, L2, L1, L8

Pairs of corresponding angles Ll and LS, L2 and L6, L4 and L8, L3and L7

Pairs of alternate interior angles L3 and LS, L4 and L6

Pairs of alternate exterior angles L l and 0, L2 and L8

Pairs of interior angles on the L4 and LS, L3 and L6 same side of the transversal

+ If two parallel lines are cut by a transversal, then (i) each pair of corresponding angles is equal. (ii) each pair of alternate interior angles is equal. (iii) each pair of interior angles on the same side of the transversal is supplementary. (iv) each pair of alternate exterior angles is equal. (v) each pair of exterior angles on the same side of the transversal is supplementary.

Note: (i) The F-shape stands for corresponding angles.

(ii) The Z-shape stands for alternate angles.

+ Two lines are said to be parallel, when a transversal cuts these lines such that pairs of (i) corresponding angles are equal. (ii) alternate interior angles are equal. (iii) interior angles on the same side of the transversal are supplementary.

II 5. Lines and Angles


• • • 0

• • • •

Which of the following are the units of an angle?

(A) Seconds (C) Degrees

(B) Kilograms (D) Kilometres

What do we call an angle which exactly measures go·?

(A) An obtuse angle (B) An acute angle (C) A right angle (D) A reflex angle

What do we call an angle whose measurement is exactly equal to 0"? (A) An obtuse angle (B) A straight angle (C) A zero angle (D) A right angle

What is an angle which measures exactly 180. called?

(A) A zero angle (B) A right angle (C) A straight angle (D) An acute angle

Which instrument is used to measure or construct angles?

(A) Compasses (C) Protractor

(B) Scale (D) Set squares

How many rays can be drawn from a given point? (A) 2 (C) 8

(B) 5 (D) Infinitely many

What do we call a 169. angle?

(A) An obtuse angle (B) An acute angle (C) A right angle (D) A zero angle

What happens to the measurment of an angle after the extension of its arms? (A) Doubles (B) Triples (C) Remains the same (D) Cannot be said

5. Lines and Angles

•

CD

In L ROP. what is the vertex?

(A) R (B) p (C) 0 (D) PR

What are the two arms of L DEF?

--(A) ED and EF - -(B) DE and EF --(C) FE and FD - -(D) DE and FD

When two line segments meet at a point forming right angles, what type of segments are they called?

(A) Parallel segments (B) Perpendicular segments (C) Equal segments (D) Bisecting segments - -How is " AB is perpendicular to CD " written symbolically? --(A) AB ..LCD --(B) AB II CD --(C) AB :t CD --(D) AB = CD

• 00 .l PR. What is the measure of

LQOR?

[,.-r-•. [ (A) 180° (B) 45° (C) 90° (D) 120°

• A line AB is parallel to the line CD. How is this symbolically written? -- --(A) AB :t CD (B) AB = CD -- --(C) AB ..l CD (D) AB // CD

II


CD What are the lines which lie on the same plane and do not intersect at any point called?

(A) Perpendicular lines (B) Intersecting lines (C) Parallellines (D) Collinear lines 0 When two lines are parallel, what is the distance between them?

• (A) Remains equal. (B) Does not remain equal. (C) Increases on the right. (D) Decreases on the right.

What is the number of pairs of parallel lines in the given figure?

p Q

CJ s

(A) Z (B) 1 R

(C) 4 (D) 3

What is the complementary angle of zoo?

(A) 70° (B) 180° (C) goo (D) 150°

What is the supplementary angle of 1ZOO?

(A) zoo (B) goo (C) 60° (D) 180°

What is the measure of a comple-mentary angle of an angle greater than 45°?

(A) Less than 45° (B) Equal to 45° (C) Greater than 45° (D) Equal to goo

Which of the following is true?

(A) Two acute angles are supplementary. (B) Two obtuse angles are supplementary. (C) Two right angles are supplementary . (D) Two reflex angles are supplementary.

Find the angle which is a complement of itself.

(A) 30° (B) 45° (C) goo (D) 180°

Which of the following ang les is a supplement of itself?

(A) goo (B) 180° (C) 45° (D) 11 oo

II

(24- 30}: Observe the given figure in which l II m and n is the transversal and answer the questions that follow.

• What type of angles are 'a' and 'p'?

(A) Corresponding angles (B) Alternate angles (C) Vertically opposite angles

(D) Interior angle on the same side of the transversal

What type of angles are 'c' and 'p'?

(A) Corresponding angles

(B) Alternate angles

(C) Vertically opposite angles

(D) Interior angles on the same side of the transversal

Which of the following is a pair of corresponding angles?

(A) d and c (B) s and r (C) c and r (D) p and q

Which of the following is a pair of vertically opposite angles?

(A) a and b (B) a and p (C) s and r (D) p and r If the measure of 'c' is 11 0°, what is the measure of 's' ? (A) 45° (B) 11 oo (C) 70° (D) 180°

G) If the measure of b = 70°, what is the measure of s ?

• (A) 11 oo (B) 70° (C) goo (D) 180°

If b = 70°, what is the measure of b + p? (A) 180° (B) 11 oo (C) goo (D) 70° G The angle between the two blades of a scissors is 1g4o. What type of an angle is it?

(A) straight ang le (B) reflex ang le (C) obtuse angle (D) complete angle

5. Lines and Angles


• •

Observe the figure given. . s

Which of the following PQ = RS?

is true if

(A) PQ + QR = RS (C) PQ + QS = RS

(B) PR = QS (D) PQ-RS = QR

PQR is a straight line.

~ n12nooo /t T 2~

~ ~

p Q R

What is the value of x? (A) 20° (B) 25° (C) 15° (D) 30° In the given figure, what is the measure of x?

(A) 32° (B) 148° (C) 64° (D) 180° In the figure, AB, CD and U are three straight lines that interesect at 0. If y is thrice x, find the value of y. E*D A~ B

C F

(A) g7.5° (B) 35° (C) 32.5° (D) gao

In the figure, AB II CD and XV is the transversal.

Which of the following is incorrect? (A) p = 115° (B) q = 115° (C) q = 65° (D) r = 115°

5. Lines and Angles

Find the angle x in the given figure, if AB II CD .

c

(A) 75° (B) 55° (C) 160° (D) 145° Through the vertex A of ~ABC , a line XV is drawn parallel to BC.

Which of the following is correct? (A) b = y (B) c = X (C) a= b (D) a+b +C=X +a+y

• Find the unknown angle x in the figure.

c~ __ o..,.........,.--::-- E

F

(A) 45° (B) 125° (C) goo (D) 80°

• ' Observe the figure given. r---------------~

~D ~ A B C

Compute the sum of x, y and z. (A) 180° (B) 70° (C) 1goo (D) 80° If the angles (2a - 1 0)' and (a - 11 )' are complementary, what is the value of 'a'? (A) 37' (B) 27' (C) 1 r (D) r -- +-+ If OP is a ray standing on a line QR

such that LPOQ = LPOR , what is the measure LPOQ ? (A) 45o (B) 60° (C) 7 5o (D) go•

II


Previous Contest Questions~

0 In the figure, ACE, BCF and DCG are straight lines and AB II HC.

F

Find the angles p, q, rand s.

(A)

(B)

(C)

(D)

p

45°

45°

45°

45°

q r

50° 65°

65° 50°

65° 50°

50° 65°

s

50°

50°

65°

65°

• In the figure given, what is the value of Lt?

• p q

(A) 3ao (B) 4ao (C) sao (D) 6ao

In lhe figure given, if AB II CD, what are the respective values of 'p' and 'q ' ?

F

(A) 7S0 and 2ao (C) 2S0 and 1ao

E

(B) 2ao and 7S0

(D) 1ao and 2S0

II

0 In the figure given, a II b and c II d. If L1 = 7 so , what is the measure of L3?

•

•

(A) 1aso (B) 1Sao (C) 7So (D)1aao

In the figure given, LXOZ and LYOZ form a linear pair. If p- q = sao, what are the respective values of p and q?

I·~ P~·l (A) sao and 13ao (B) 13ao and sao (C) 12ao and 6ao (D) 6ao and 12ao

In the figure given, what is the value of 't'?

(A) so (B) 1 ao (C) 1So (D) 2ao

0 Given ABE is a straight line.

• a A B E

Find angle y. (A) 6ao (B) 1200 (C) 1Sao (D) 3ao

In the given figure, what is the measure ofq?

(A) 48° (B) 900 (C) 42° (D) 11 ao

5. Lines and Angles

CHAPTER

6 Triangles

+ A triangle is a simple closed figure bounded by three line segments. It has three vertices three sides and three angles. The three sides and three angles of a triangle are called its six elements. It is denoted by the symbol fl.

lnfl ABC, Sides: AB, BC and CA ;Angles: LBAC, LABC and LBCA ;Vertices: A.BandC

+ A triangle is said to be

(a) an acute angled triangle, if each one of its angles measures less than go·. (b) a right angled triangle, if any one of its angles measures goo. (c) an obtuse angled triangle, if any one of its angles measures more than goo.

Note: A triangle cannot have more than one right angle. A triangle cannot have more than one obtuse angle. In a right triangle, the sum of the acute angles is go•.

+ Angle sum property: The sum of the angles of a triangle is 180°.

+ Properties of sides: (i) The sum of any two sides of a triangle is greater than the third side.

(ii) The difference of any two sides is less than the third side.

+ Property of exterior angles: If a side of a triangle is produced, the exterior angle so formed is equal to the sum of interior opposite angles.

A

B

e.g., Exterior angle, X0 = LA + LB = 70° + 40° = 110°

+ A triangle is said to be

(a) an equilateral triangle, if all of its sides are equal.

(b) an isosceles triangle, if any two of its sides are equal.

(c) a scalene triangle, if all of its sides are of different lengths.

6. Triangles II


• The medians of a triangle are the line segments joining the vertices of the triangle to the midpoints of the opposite sides.

A Here AD, BE and CF are medians of f:J. ABC.

+ The medians of a triangle are concurrent.

• The centroid of a triangle is the point of concurrence of its medians. The centroid is denoted by G.

The centroid of a triangle divides the medians in the ratio 2 : 1 .

The medians of an equilateral triangle are equal.

The medians to the equal sides of an isosceles triangle are equal.

The centroid of a triangle always lies in the interior of the triangle .

• • • • • Altitudes of triangle are the perpendiculars drawn from the vertices of a triangle to the opposite sides.

A Here AL, BM and CN are the altitudes of t:J. ABC.

• The altitudes of a triangle are concurrent.

• The orthocentre is the point of concurrence of the altitudes of a triangle. Orthocentre is denoted by H.

• The orthocentre of an acute angled triangle lies in the interior B c of the triangle.

+ The orthocentre of a right angled triangle is the vertex containing the right angle.

+ The orthocentre of an obtuse angled triangle lies in the exterior of the triangle.

Properties: (i) The altitudes drawn on equal sides of an isosceles triangle are equal. (ii) The altitude bisects the base of an isosceles triangle.

(iii) The altitudes of an equilateral triangle are equal. (iv) The centroid of an equilateral triangle coincides with its orthocentre.

+ In a right angled triangle, the side opposite to the right angle is called the hypotenuse and the other two sides are known as its legs.

+ Pythagoras'Theorem: In a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the remaining two sides.

In the right angled triangle ABC, AC2 = AB2 + BC2 •

A

~ c B + In a right angled triangle, the hypotenuse is the longest side.

II 6. Triangles


• • • 0

• • •

In a tJ. ABC, if AB2 = BC2 + AC2, at which vertex is the right angle?

(A) A (C) c

(B) B (D) Either A or B

Which type of triangle is formed by BC = 7.2 em, AC = 6 em and L c = 120<>?

(A) An acute angled triangle. (B) An obtuse angled triangle . (C) A right angled triangle. (D) An isosceles triangle.

Which triangle is formed by AB = 3 em, BC = 4 em and AC = 8 em?

(A) A scalene triangle. (B) An isosceles triangle. (C) An equilateral triangle. (D) No triangle is formed.

If two angles in a triangle are 65° and 85°, what is the third angle?

(A) 30° (B) 45° (C) 60° (D) goo

If one angle is the average of the other two angles and the difference between the greatest and least angles is 60°, which triangle is formed?

(A) An isosceles triangle. (B) An equilateral triangle. (C) A right angled triangle. (D) A right angled isosceles triangle.

In tJ. ABC, if AB = BC and LB = 80°, what is the measure of L C ?

(A) 50° (C) 130°

(B) 100° (D) 180°

Which triangle is formed by BC = AC = 7.2 em and L C = goo?

(A) A right angled triangle. (B) An isosceles triangle. (C) A right angled isosceles triangle. (D) No triangle is formed.

6. Triangles

•

•


Which of the following statements is false? (A) The sum of two sides of a triangle is

greater than the third side. (B) In a right angled triangle, hypotenuse

is the longest side . (C) A, B and C are collinear if

AB+ BC =AC. (D) The sum of angles in a triangle is

less than 180°. What is the length of the hypotenuse of a right angled triangle whose two legs measure 12 em and 0.35 m? (A) 37 em (B) 3.72 em (C) 0.372 em (D) 37 m If the two legs of a right angled triangle are equal and the square of the hypotenuse is 100 sq units, what is the length of each leg?

(A) 1 0 units (B) 5.J2 units (C) 1 o.J2 units (D) 15 units

In a tJ. PQR. PQ = PR and L Q is twice that

of LP . What is the measure of L Q? (A) 72° (B) 36° (C) 144° (D) 1 08° If two sides of an isosceles triangle are 3 em and 8 em, what is the length of the third side? (A) 3 em (B) 8 em (C) 3 em or 8 em (D) Neither (A) nor (B)

If in a tJ. ABC, L A = 60° and AB = AC, of what type is tJ. ABC? (A) An isosceles triangle. (B) A right angled triangle. (C) An isosceles right angled triangle . (D) An equilateral triangle. In a tJ. ABC, if AB + BC = 1 0 em, BC + CA = 12 em, CA + AB = 16 em, what is the sum of the lengths of its sides? (A) 1g em (B) 17 em (C) 38 em (D) 30 em

II

Jl BMA's Talent & Olympiad Exams Resource Book Class VII · Mathematics

• Which of the following are the angles in a right ang led triangle other than the right angle?

(A) Acute angles (B) Obtuse angles (C) Right angles (D) Reflex angles

In a D. ABC, if LA = LB + LC , what is the

measure of L A

(A) 60° (B) 45° (C) 900 (D) 135°

If a, b and c are the sides of a triangle, which of the following is correct? (A) a - b > c (B) c > a + b (C) c = a + b (D) b < c + a If the angles of a triangle are in the ratio 1 : 2: 7, what type of a triangle is it?

(A) An acute angled triangle. (B) An obtuse angled triangle. (C) A right angled triangle. (D) A right angled isosceles triangle.

A triangle always has

(A) exactly one acute angle. (B) exactly two acute angles. (C) at least two acute angles. (D) exactly 2 right angles.

How many independent measurements are required to construct a triangle? (A) 3 (B) 4 (C) 2 (D) 5

In a !J. ABC, if LB is an obtuse angle, which is the longest side?

(A) AB (C) AC

(B) BC (D) Either (A) or (B)

If P : An isosceles triangle is right angled . Q: LA = LB = 45° and LC = 90°, which of the following statements is true?

(A) P is true and Q is the correct explanation of P.

(B) P is true and Q is not the correct explanation of P.

(C) P is false.

(D) P is the correct explanation of Q.

II

An isosceles triangle can be obtuse angled. (A) Always false. (B) Always true. (C) Cannot be determined. (D) Sometimes false. Which of the following statements is correct? (A) The difference of any two sides is less

than the third side. (B) A triangle cannot have two obtuse

angles. (C) A triangle cannot have an obtuse

angle and a right angle. All the above. A (D)

W Two chimneys 18m and 13m high stand upright on a ground. If their feet are 12m apart, what is the distance between their tops? (A) 5 m (B) 31 m (C) 13 m (D) 18 m The top of a broken tree touches the ground at a distance of 15 m from its base. If the tree is broken at a height of 8 m from the ground, what is the actual height of the tree? (A) 20 m (B) 25 m (C) 30 m (D) 17 m

• What is the ratio in which the centroid of a triangle divides the medians ?

• •

(A) 1 : 2 (B) 1 : 3 (C) 2: 1 (D) 3 : 1 The centroid of a triangle is the point of concurrence of which of these? (A) Angle bisectors (B) Perpendicular bisectors (C) Altitudes (D) Medians Which of the following statements is true? (A) The centroid of an acute angled

triangle lies in the interior of the triangle.

(B) The orthocentre of an acute angled triangle lies in the interior of the triangle.

(C) The medians of a triangle are concurrent.

(D) All the above.

6. Triangles


•

•

•

In .6.ABC, Dis the midpoint of BC and G is the centroid of the triangle. If GO = 2 em, what is the length of AD ? (A) 4 em (B) 6 em (C) 2 em (D) 8 em

In a .6. ABC, E is the midpoint of AC and G is the centroid of the triangle. What is BE: GE? (A) 1 : 2 (B) 2 : 1 (C) 3 : 1 (D) 1 : 3

Which of the following statements is true? (A) The orthocentre of a right angled

triangle is the vertex containing the right angle.

(B) The median of a triangle joins a vertex to the midpoint of the opposite side.

(C) The centroid of a right angled triangle lies in the interior of the triangle.

(D) All the above. In a scalene triangle ABC, X is the midpoint of BC. What is AX?

A

~ B

(A) Median (C) Centroid

X c (B) Altitude (D) Orthocentre

In APQR, PQ = PR ; M is a point on QR

and PM l_ QR . What do you call PM?

p

ffi Q

(A) Centroid (C) Altitude

M R

(B) Median (D) Vertex

In .6. XVZ , XP is the median.Which of the following is correct? (A) XP = XV (B) VP = PZ (C) XP = XZ (D) XV= XZ

6. Triangles

•

•

From the figure given, which of the following statements is true ?

A EA ~

o"' c B

(A) L x + Ly = L1 + L 3 (B) Ly = L2 (C) L x = L1 (D) Lx + Ly = L1 + L 2

The exterior angle of a triangle is 110". If one of the interior opposite angles is 55", what is the measure of the other ? (A) 45" (B) 65" (C) 55" (D) 35" 0 In .6. PQR, LQ =go·. Which of the following is the longest side?

• (A) RQ (B) PQ (C) PR (D) Either (A) or (B) Which of the following statements is NOT true?

(A) A triangle can have three 60° angles. (B) A triangle can have a right angle. (C) A triangle can have two right angles. (D) A triangle can have all three angles

equal. From the following figure, what are the respective values of x and y?

(A) 80°,60° (C) 60°, 800

(B) 600,400 (D) 400,600

• Previous Contest Questions~ 1 Angles of a triangle are (x + 10") ,

(x + 40") and (2x- 30") . What is the value of x? (A) 30" (B) 40" (C) 20" (D) 1 o·

II


•

•

• •

• •

In the figure given, what are the values of Lb, Lc and La respectively?

A

~ D B C E

(A) 18°, 70° and 92o(B) 92°, 70° and 18°

(C) 70°, 92° and 18o(D) 70°, 18° and 92°

In ~ABC, AB = AC, LA = 40° . 0 is a point inside ~ABC such that LOBC = LOCA . Find the measure of LBOC.

A

sL----~c

(A) 11 oo (B) 35° (C) 140° (D) 155°

The triangle XYZ is right angled at X. Which is the longest side of !1XYZ ? (A) XV (B) XZ (C) ZX (D) YZ Which of the following statements is correct? (A) The exterior angle of a triangle is

equal to its interior adjacent angle. (B) The median of a triangle joins its ver-

tex to the midpoint of its opposite side. (C) The altitude of a triangle is drawn

from a vertex to the midpoint of the opposite side.

(D) All the above . What is the number of medians in a triangle ? (A) 0 (B) 2 (C) 3 (D) 1 Find the measure of yin the given figure.

p

(B) 60° (D) 120°

II

0 lnthefigure,AB = ACand AD II BC Find the respective measures of x, y and z.

E

B C

(A) X = 70°, y = 70°, z = 55° (B) X = 70°, y = 55°, z = 55° (C) X = 55°, y = 55°, Z = 55° (D) X = 70°, y = 55°, z = 70° 0 Find the angles x andy respectively.

D

C B A (A) X= 47°, y = 25° (B) X= 27°, y = 45° (C) x = 45°, y = 27° (D) x = 25°, y = 4 7°

Find the value of x.

(A) 106° (C) 26°

(B) 53° (D) 52°

B

Find the measure of x in the following figure .

l:i:1 W X

(A) 112° (B) 56° (C) 68° (D) 106°

6. Triangles


CHAPTER

7 Congruence of Triangles

• Two figures having exactly the same shape and size are said to be congruent.

• Two triangles are said to be congruent, if pairs of corresponding sides and corresponding angles are equal.

Note: The symbol ~ is used for 'is congruent to' relation.

• Two line segments are congruent, if they have the same length. AB = CD is read as line

segment A B is congruent to the line segment CD .

• Two angles are congruent, if they have the same measure. "LA is congruent to LB " is written symbolically as L A = L B or LA = LB .

• S.S.S. congruence condition: If the three sides of a triangle are equa l to the three corresponding sides of another triangle, then the two triangles are congruent.

e.g., A D

7cm 7cm

In the given figure, L\ABC = L\DEF by S.S.S. congruence condition.

• S.A.S. congruence condition: If two sides and the included angle of a triangle are respectively equal to the two corresponding sides and the included angle of another triangle, then the two triangles are congruent.

e.g., A D

B c E F

In the given figure, L\ABC = L\D EF by S.A.S. congruence condition.

II 7. Congruence of Triangles


+ A.S.A. congruence condition: If two angles and an included side of one triangle are re-spectively equal to the two correspond ing angles and the corresponding included side of another triangle, then the two triangles are congruent.

e.g., A D

F

In the given figure, ~ABC = ~DEF by A.S.A. congruence condition.

+ R.H.S. congruence condition: If the hypotenuse and a side of a right angled triangle are equal to the hypotenuse and the corresponding side of another rightangled triangle, then the two triangles are congruent.

e.g.,

B c E F

In the given figure, ~ABC = ~DEF by R.H.S. congruence condition.

+ In congruent triangles, the congruent angles are opposite to equal sides, and the congruent sides are opposite to equal angles.

+ There is no A.A.A. congruence cond ition for congruence of triangles. Two triangles with equal corresponding angles need not be congruent. In such case, one of the triangles can be an enlarged copy of the other.

+ The order of the letters in the names of congruent triangles displays the corresponding relationships.

Thus,if ~ABC = ~E DF , AliesonE,BonDandConF.Aiso AB liesalong ED, BC along - - -DF and AC along E F .

7. Congruence of Triangles II


0

• •

0

•

In 11 ABC, AB = AC and AD is perpendicular to BC. State the property by which 11 ADB =: 11 ADC.

(A) S.A.S. property (B) S.S.S. property (C) R.H.S. property (D) A.S.A. property

Two students drew a line segment each. What is the condition for them to be congruent?

(A) They should be drawn with a scale.

(B) They should be drawn on the same sheet of paper.

(C) They should have different lengths.

(D) They should have the same length.

In the given figure, if AD = BC and ADIIBC, which of the following is true?

D c

I I A B

(A) AB =AD (B) AB= DC (C) BC=CD (D) AC =AD

In 11 PQR and 11 XYZ, L P = 50°, XV = PQ, and XZ = PR. By which property are

1::::. XYZ and 11 PQR congruent?

(A) S.S.S. property (B) S.A.S. property (C) A.S.A. property (D) R.H.S. property

A: Two triangles are said to be congruent if two sides and an angle of one triangle are respectively equal to the two sides and an angle of the other.

R : Two triangles are congruent if two sides and the included angle of one are equal to the corresponding two sides and included angle of the other.

II

0

• •

•


Given A & R, which of the following statements is correct?

(A) A is false and R is the correct explanation of A.

(B) A is true and R is the correct explanation of A.

(C) A is true and R is false.

(D) A is false and R is true. Two triangles, 11 P Q R and 11 DE F are of the same size and shape. What can we conclude about them?

(A) 11 PQR is smaller than 11 DEF. (B) 11 PQR is larger than 11 DEF. (C) 11 PQR is congruent to 11 DEF . (D) 11 PQR is not congruent to 11 DEF .

Which of the following examines the congruence of plane figures?

(A) Trial and error method (B) Superposition method (C) Substitution method (D) Transposition method

Which of the following is a pair of congruent figures?

(A) A regular pentagon and a regular hexagon.

(B) A rhombus and a square.

(C) Two equilateral triangles of the same length of their sides.

(D) A quadrilateral and a rectangle.

In 11ABC and 11PQR, AB = x em, BC = y em and CA = z em. What are the measures of sides PQ, QR and RP of 11PQR if 11ABC =: 11ABC =: 11PQR?

(A) (B) (C) (D)

PQ QR RP xcm ycm z em ycm xcm zcm xcm zcm ycm zcm xcm ycm

7. Congruence of Triangles


(10-11): In the given figure, AD = CD and AB = CB.

c G What are the three pairs of equal parts?

CD

(A) LADB = LCDB, LABD = LCBD; BD = BD

(B) AD = AB, DC = CB, BD = BD

(C) AB = CD, AD = BC, BD = BD

(D) LADB = LCDB, LABD = LCBD;

LDAB = LDBC Which of the following is the correct conclusion?

(A) 8ABC and 8CBD are isosceles triangles.

(B) BD bisects LADC . (C) BD bisects LBAD . (D) 8ABC and 8CBD are equilateral

triangles.

In the given figure, AB = AC and AD is the bisector of LBAC .

A

& B 0 C

Which among the following statements is true?

(A) 8 ADB :: ilABC (B) ilADC :: ilABC (C) L B = L C (D) L ABC = L CAB

In 8 ABC and 8 D E F , AC = OF, AB = DE and BC = EF. By which property are ilABC and ilDEF congruent?

(A) R.H.S. property (B) S.S.S. property (C) S.A.S. property (D) A.S.A. property


• For the triangles 8ART and 8PE N given, if S.A.S. criterion should be used given LT = LN , what are the respective measures of PN and RT?

•

•

T R N E

(A) TR and PE

(C) AT and EN

(B) AR and PE

(D) AR and PN

Which of the following is an example of A.S.A. criterion of congruency for two triangles L ADB and L DEF?

(A) A B = EF, L B = L E and L C = L F

(B) BC = EF,LB = LEandLC=LF

(C) AC = EF, LB = LD and LC = LF

(D) AC = DE, LB = LD and LC = LF

In the figure given, AC = BD and LBAC = LCDB = 90o.

B

If ~ABC :: ~ DCB by R.H.S. property which of the following is required?

(A) The measure of AB. (B) The measure of CD. (C) The measure of BC. (D) AC = BD

Which of the following criterion does not exist?

(A) A.S.A. criterion (B) R.H.S. criterion (C) A.A.A. criterion (D) S.S.S. criterion

II


• In two triangles, the three angles of one triangle are correspond ingly equa l to three angles of another triangle. Which of the following is a correct statement?

(A) One triangle is an enlarged copy of other.

(B) The two triangles are necessarily congruent.

(C) The two triangles are congruent by A.A.A. congruency criterion.

(D) All of the above.

(19-20): MBC is congruent to ~PQR under thecorrespondence ABC ~ RQP.

• Which part corresponds to PQ ?

(A) C B (B) A C (C) Q R (D) A B

G) Which part of .llABC corresponds to

RP ?

(A) AB (B) AC (C) CA (D) BC

.llABC is congruent to ~XYZ. Find the

measures of Lx and L y respectively.

A

il B Scm C

(A) 800, 60° (C) 800, 40°

''\)' X

(B) 60°,40° (D) 60°,80°

~ABC :: ~ DEF,and AB = 3 em, EF = 8 em and OF = 1 0 em. What are the respective lengths of AC and DE in em? (A) 10,3 (B) 10,8 (C) 8,3 (D) 3,10

MBC =~FOE What is the measure of

L F? A E D

~ B C

v F

(A) 700 (B) 50° (C) 130° (D) 600

II


In the figure, PQ = PS and QR = SR. If .llPQR is congruent to .llPSR, which of the following is correct?

(A) .LQPR = L PRS (B) L RPS = L RQP (C) L QRP = L SRP (D) PR = RS

AB and AC are two chords of a circle with centre 0 as shown.

A

If .llAOB is congruent to .llAOC, which of the following is correct?

(A) LOBA = L OCA (B) LAOC = LOCA (C) AO = AC (D) AB = OC

A regular hexagon is divided into four triangles.

A B Which of the following is correct?

(A) .llABF:: .llEDF (B) .llFDC:: .llFCB (C) .llABF:: ilFBC (D) .llFDC :: .llBCF

MBC and .llDEF are as shown. A D

~~ B C E F

By which condition of congruence is .llABC:: .llDEF?

(A) S.S.S. condition (B) S.A.S. condition (C) R.H.S. condition (D) A.S.A. condition.



• Given AB II CD and AB =CD, which of the following is correct?

A B

C D (A) L\AOB :: L\DOC (B) L\AOB :: L\ODC (C) L\BOA::: L\DOC (D) L\BAO::: L\COD

• MBC is isosceles, AB = AC and AD .1 BC . B D C w

A

Which of the following is correct?

(A) L\ADC ::: L\ADB (B) MDB i MDB (C) L\ADB :: L\ABC (D) MBC ::MDC

G) Observe the triangles given in the figure.

A

2~ Q 4cm R

B 4cm C

2cv.: State the condition L\ABC ::: L\PQR.

p

under which

(A) A.S.A. (C) S.A.S.

(B) S.S.S. (D) R.H.S.

• In the given figure, AB = AC and AD = AE. C E D B v

A

Which of the following statements is not true?

(A) L CAE = L BAD (B) L\ACE ::: L\ABD

(C) L\AEC :: L\ABD (D) BE = DC


•

•

ABCD is a parallelogram in which AB = DC and AD = BC.

lSl A B

By which condition of congruence of

triangles is L\ABD = L\CDB?

(A) R.H.S. (C) S.S.S .

(B) S.A.S. (D) A.S.A.

Given L\XYZ and L\LMN.

~ tz' Y Z N

Choose the correct statement.

(A) L\XYZ ::: L\LMN (C) XV= MN

(B) YZ = LM (D) XZ = LM

Identify the incorrect statement.

(A) The corresponding sides of congruent triangles are equal.

(B) The corresponding angles of congruent triangles are equal.

(C) Two triangles cannot be congruent. (D) There are four congruency conditions

for congruence of triangles.

f4i.? Previous Contest Questions~

0 By which congruency property, are the two triangles PQS and PRS given in the following figure congruent ?

p

Q R (A) S.S.S. property (B) S.A.S. property (C) A.S.A. property (D) R.H.S. property

II


• •

0

• •

In the following figure, two triangles ABC and EDC are such that AC = EC, BC = DC, LE = 60°, LDCE = 30o and LB = goo . By which property is .!lABC :: .!lEDC?

(A) S.S.S. property (B) S.A.S. property (C) A.S.A. property (D) R.H.S. property

For the congruence of .!lABC and .!lPQR which one of the following sets of conditions is not sufficient?

66 A c B P Q

(A) L ABC = L PQR a = p, c = r (B) L CAB = L RPQ L ABC = L PQR c = r (C) b = q, L CAB = L RPQ, a = p (D) a = p, c = r, L ABC = L PQR

Which of the following are congruent?

(A) Two~ 1 coins (B) A ( 1 coin and a ( 2 coin (C) A ~ 2 coin and a ~ 5 coin (D) A~ 5 coin and a~ 10 coin

Which of the following is important in congruence of triangles?

(A) Naming the angles of the triangles using capital letters.

(B) Measures of angles in degrees.

(C) The order of letters of the triangles.

(D) Exact length of the sides of the triangles .

In the figure given, which of the following statements is true?

p

Q s

R

(A) .!lQPR:: .!lSPR (B) .!lPSR :: .!lRQP

(C) .!lPRS :: .!lQPR (D) .!lQRP :: .!lPSR

II

• 0

•


In two triangles, pairs of corresponding sides and the corres-ponding angles are equal. What can be concluded from this?

(A) The triangles are small. (B) The triangles are congruent. (C) The triangles are equilateral. (D) The triangles are equiangular. Which of the following are measure-ments of two triangles by which they are congruent under S.A.S condition?

(A) .!lABC : LB = 50°, BC = 5 em, and AB = 7 em .

.!lDEF : LE = 50°, EF = 7 em, and DE= 5 em.

(B) .!lABC : BC = 6 em, AC = 4 em and LB = 35° .!lDEF : DF = 4 em, EF = 6 em and LE = 35°

(C) .!lABC : AB = 4.5 em, AC = 4 em and LA = 60° .!lDEF : DE = 4 em, FD = 4.5 em and LD = 55°

(D) Either (B) or (C) .!lABC is isosceles with AB = AC, AD .1 BC ; which of the following is a correct statement?

A

.~, D

(A) MBC :: MDC (B) MBD :: MCD

(C) .!lADB:: .!lACD (D) .!lADC :: .!lABD

Are triangles in the given figure congruent by R.H.S. condition?

3 ~Scm ~Scm cm~3cm~

A

(A) Yes (C) No

B p Q (B) Insufficient data (D) Either (B) or (C)


CHAPTER

8 Comparing Quantities

+ Ratio is a method of comparing two quantities of the same kind by division.

+ The symbol used to write a ratio is':' and is read as 'is to'.

+ A ratio can be expressed as a fraction.

+ A ratio is always expressed in its simplest form. + A ratio does not have any unit, it is only a numerical value.

+ A ratio consists of two terms. The f irst term is called the antecedent and second term is called the consequent.

+ A ratio can be written in its simplest form by dividing the antecedent and the consequent by their H.C.F.

+ The antecedent and the consequent of a ratio cannot be interchanged. + To express two terms in a ratio, they should be in the same units of measurement.

+ When two ratios are equal they are said to be in proportion. The symbol for proportion is:: and is read as 'as to'.

+ The two terms in the middle of a proportion are called means and the first and the last terms are cal led extremes.

+ If two ratios are to be equal or to be in proportion, their product of means should be equal to the product of extremes.

+ If a : b:: c: d then the statement ad = be holds good. + If a: band b: care in proportion such that b2 = ac then b is called the mean proportional of a: band

b:c.

+ Multiplying or dividing the terms of the ratio by the same number gives equivalent ratios.

+ Unitary method: To find the value of many quantities when the value of one is given, the operation is multiplication {x).To find out value of one when the value of many is given, the operation is division (~).To f ind out value of many when the value of many is given, unitary method can be used.

+ X Many --~) One One Many

{Division) {Multiplication)

+ Another way of compar ing quantities is percentage. The word percent means per hundred. Thus 12% means 12 parts out of 100 parts.

+ Fractions can be converted into percentages and vice-versa.

e.g., (i) ~ = ~ x 1 00% = 40% 5 5

8. Comparing Quantities

(ii) 25% = ~ = .:!_ 100 4

II

Jl •

• •

• • • • • • • •


Decimals can be converted into percentages and vice-versa . e.g . (i) 0.36 = 0.36 x 100% = 36%

( .. ) 43 II 43% = - = 0 .43

100


If a number is increased by a% and then decreased by a% or is decreased by a% and then increased az

by a%, then the original number decreases by 1 00

% .

A number can be split into two parts such that one part is P% of the other. Then the two parts are

100 p 100 + p x number and 1 00 + p x number.

If the circumference of a circle is increased (or) decreased by P% then the rad ius of a circle in-creases (or) decreases by P%. Gain = Sell ing Price (S.P.)- Cost Price (C.P.) Loss = C.P. - S.P .

Gain% = gain x 100% C.P.

Loss% = loss x 1 00% C.P.

= (1 00 + ga in%) x C.P. = (1 00 -loss%) S.P. 100 100 X C.P .

c p = . X S.P. = X S.P . ( 100 ) ( 100 )

· · 100 + gam% 100 -loss%

PTR S.l. = 1 0 0

S.l. =Simple Interest P = Principal T=Time R = Rate percent per annum

+ Amount (A)= Principal+ Interest

PT R ( TR ) = p + 1 0 0 = p 1 + 1 00

• R X T = 100 (N -1) where R = rate percent

T =time N = The number of times the sum gets multiplied (i.e., doubled, tripled .... etc.)

+ S.l. is calculated uniformly on the original principal throughout the time period.

II 8. Comparing Quantities


• •

• • • • • •

In an office the working hours are 10:30 a.m. to 5:30p.m. and in between 30 minutes are spent for lunch. Find the ratio of office hours to the time spent for lunch. (A) 7 : 30 (B) 1 : 14 (C) 14 : 1 (D) 30: 7 Which of the following is the condition for two ratios to be equal? (A) Product of means is equal to

antecedents. (B) Product of extremes is equal to con-

sequents. (C) Antecedents are equal to conse-

quents. (D) Product of means is equal to

product of extremes . If 20: 5 : : p : 1, what is the value of p?

(A) i (B) 4 (C) 5 (D) 1

What is the mean proportion of 25 : 20 :: 20: 16? (A) 25 (B) 20 (C) 16 (D) 100 A boy has enough money to buy 20 books. If each book costs 25 paise less, he could buy two more books and still have 70 paise left. How much money does the boy have? (A) ~ 16 (B) ~ 24 (C) ~ 48 (D) ~ 36 When a number is reduced by 4, it becomes 80% of itself. Find the number. (A) 20 (B) 30 (C) 40 (D) 50

1 If 2 2 % of a number is 0.2, what will be

120% of it? (A) 10.8 (B) 4.8 (C) 9.6 (D) 12.4 The number of seats for admission is increased by 1 0% every year. If the number of seats in 2001 was 400, what was the number of seats in 2003? (A) 824 (B) 484 (C) 500 (D) 480


•

CD


In an election between two candidates, the candidate who gets 30% of the votes polled is defeated by 15000 votes. What is the number of votes polled for the winning candidate? (A) 37 500 (B) 30000 (C) 26250 (D) 11250 In 2003, Indian cricket team played 60 games and won 30% of the games played. After a phenomenal winning streak, this team raised its average to 50%. How many games did the team win in a row to attain this average? (A) 36 (B) 24 (C) 48 (D) 12

A pudding is made of 200 g sugar, 800 g eggs, 600 g flour and 200 g dry fruits. What percent of sugar is present in the whole pudding?

(A) 11i%

(C) 6 .:!_ % 4

2 (B) 163%

(D) 32_% 2

Which of the following statements is wrong? (A) Jill + 625 = 80% of 1200-320 (B) 5Y2 of 240 = 150% of 880 (C) 25% of 50 = 0.125 (D) 150 g is 20% of a kg

35% population of a town are men and 40% are women. If the number of children is 20,000, what is the number of women?

(A) 3200 (C) 32000

(B) 30,050 (D) 31500

A student has to secure 40% marks to pass. He got 40 marks and failed by 40 marks. What is the maximum number of marks?

(A) 400 (C) 200

(B) 300 (D) 100

II


•

•

After a deduction of 5% from a certain sum and then 10% from the remainder, a sum of ~ 171 is left. What was the original sum?

(A) ~ 200 (B) ~ 250 (C) ~ 150 (D) ~ 300

The C.P. of 25 articles is equal to the S.P. of 20 articles. What is the gain%? (A) 25% (B) 20% (C) 30% (D) 50%

The value of a machine depreciates every year by 5%. If the present value of the machine is~ 100000, what will be its value after two years?

(A) ~ 100050 (C) ~ 99250

(B) ~ 90250 (D) ~ 96150

By selling an article for ~ 600 a man loses 20%. At what price should he sell it in order to gain 25%?

(A) ~ 800 (B) ~ 750 (C) ~937.50 (D) ~ 1 000

What profit percent is made by selling an article at a certain price if by selling at

t !h of that price results in a loss of 10%?

(A) 20% (B) 15% (C) 25% (D) 1 0%

11 oranges are bought for ~ 10 and 10 oranges are sold for ~ 11. Find the gain (or) loss percent.

(A) 21% loss (C) 21% gain

(B) 11% gain (D) 11 % loss

A man buys a radio for~ 600 and sells it at a gain of 25%. What is the S.P of the radio? (A) ~ 700 (C) ~ 900

(B) ~ 750 (D) < 1000

A shopkeeper sold two watches for < 425 each, gaining 10% on one and losing 10% on the other. Which of the following is true? (A) He gains 1%. (B) He loses 1 %. (C) Either (A) or (B) (D) Neither (A) nor (B)

II

•

•


A man purchased a bag of rice containing 70 kg for < 17 5. He sold it at the rate of~ 2.75 per kg. Find the profit or loss%. (A) 12% loss (C) 12% gain

(B) 10% gain (D) 10% loss

Sneha bought a purse for ~ 480. She sold

it to Neha at a gain of 6-;}% and Neha sold it to Devi at a gain of 10%. How much did Devi pay for it? (A) ~ 561 (B) ~ 560 (C) ~ 550 (D) ~ 525 On what sum of money lent at 9% per annum for 6 years does the S.l. amount to~ 810? (A) ~ 1000 (C) < 1200

(B) ~ 1500 (D) < 1600

At what rate of interest per annum will a sum double itself in 8 years?

(A) 25%

(C) 12 ~ %

(B) 6 ~% 2

(D) 3s%

The S.l. on a certain sum is 16 over 25 of the sum. Find rate percent and t ime, if both are equal. (A) 8%, 8 years (B) 16%, 16 years (C) 10%, 1 0 years (D) 12%, 12 years

A milkman borrowed ~ 2500 from two money lenders. For one loan, he paid 5% p.a. and for the other, he paid 7% p.a. The total interest paid after two years was ~ 275. Calculate the money he borrowed at 7% interest. (A) < 1875 (C) ~ 625

(B) < 1000 (D) ~ 1200

Of a certain sum, j rd is invested at 3%,

ilh at 6% and the rest at 8%. If the S.l. for 2 years from all these investments amounts to~ 600, what was the original sum? (A) ~ 2000 (C) < 4000

(B) ~ 3000 (D) < 5000



•

•

A certain sum of money lent out at a certain rate of interest per annum, doubles itself in 10 years. In how many years will it triple itself? (A) 20 years (B) 16 years (C) 12 years (D) 10 years

A sum was invested at S.l at a certain rate for 4 years. Had it been invested at 2% higher rate, it would have fetched ~ 56 more. Find the sum. (A) ~ 500 (B) ~ 600 (C) ~ 700 (D) ~ 800

1th The S.l on a sum of money is 9 of the principal and the number of years is equal to the rate percent per annum. Find the rate percent.

1 1 1 1 (A) 2 3 % (B) 3 3 % (C) 4 2 % (D) 3 2 %

If ~ 85 amounts to ~ 95 in 3 years, what will ~ 102 amount to in 5 years at the same rate? (A)~ 120 (B)~ 104 (C)~ 116(D) ~ 122

~WMt.l1fii·'·1M1·"M1i'·'rff I 0 If 2A = 3B = 4C, what is the value of A: B:C?

• • •

(A) 2 : 3 : 4 (B) 4 : 3 : 2 (C) 3 : 4 : 6 (D) 6 : 4 : 3

A certain sum of money amounts to t of itse l f in 5 years. What is the rate percent per annum? (A) 5% (B) 7% (C) 9% (D) 12% The ratio of copper and zinc in an alloy is 7: 8.1fthe weight of zinc is 9.6 kg, what is the weight of copper in the alloy? (A) 9.4 kg (B) 8.8 kg (C) 8.4 kg (D) 4.8 kg In an examination, 96%ofthecandidates passed and 50 failed. What is the number of candidates who appeared for the examination? (A) 1520 (C) 1530

(B) 1250 (D) 1350


• 0

• 0

0


Ashok sells a room heater for f 322, 1th

gaining G of its cost price. What is his

gain percent ?

(A) 13~% (B) 15~% (C) 16~% (D) 17~% 3 3 3 3

A sum of money at simple interest doubles itself in 8 years 4 months. What is the time for which it triples itself? (A) 18 years 6 months (B) 16 years 8 months (C) 15 years 8 months (D) 16 years 1 0 months Guru and Chiru borrowed ( 2250 and ~ 2500 respectively at the same rate of simple interest for 3 years. If the interest paid by Chiru is ( 45 more than that paid by Guru, what is the rate of interest per annum? (A) 3% p.a. (C) 5% p.a.

(B) 4% p.a. (D) 6% p.a.

How many times will the sum get multiplied in 10 years at 20% per annum rate of interest? (A) 2 times (C) 4 times

(B) 3 times (D) 5 times

In how many years does a certain sum amount to three times the principal at

2 the rate of 163%?

(A) 12 years (B) 8 years (C) 4 years (D) 16 years Divide ( 8000 into two parts so that the S.l on the first part for 5 years at 12% per annum is equal to S.l on the second part for 2 years at 18% per annum. (A) ( 2000, ( 6000 (B) ( 5000, ~ 3000 (C) ~ 4000, ( 4000 (D) ( 3000, ~ 5000 Find the principal if S.l. is f 31.50, time

1 period is 1

4 years and rate percent per

1 annum is 54 %.

(A) ( 460 (C) ~ 480

(B) ( 430 (D) ~ 4800

II


Rational Numbers

+ Natural numbers (N): 1, 2, 3 .4 ... etc., are called natural numbers. + Whole numbers (W): 0,1, 2, 3, ..... etc., are called whole numbers. + Integers (Z): ...... , -3, -2, -1, 0, 1, 2, 3, ....... etc., are called integers, denoted by I or Z.

1, 2, 3, 4, ... etc., are called positive integers denoted by z•. -1, -2, -3, -4, ...... etc., are called negative integers denoted by z-.

Note: 0 is neither positive nor negative.

+ Fractions: X

The numbers of the form y, where x and y are natural numbers, are known as fractions.

3 2 1 e.g., 5' 1' 125 · ......... etc.

+ Rational numbers (Q): p

A number of the form q ( q =f:. 0 ), where p and q are integers is called a rational number.

-3 5 10 -11 e.g., 17' - 19'1' - 23, ...... etc.

r-------------------------------~

Note : 0 is a rational number, since 0 = i. + A rational number !: is positive if p and q are either both positive or both negative. q

3 - 2 e.g., -5 '-- 7

+ A rational number !: is negative if either of p and q is positive and the other is negative. q -5 7

e.g., 9' - 23

Note: 0 is neither a positive nor a negative rational number.

+ Representation of Rational numbers on a number line: We can mark rational numbers on a number line just as we do integers. The negative rational numbers are marked to the left of 0 and the positive rational num· bers are marked to the right of 0.

1 1 Thus, 3 and- 3 would be at an equal distance from 0 but on its either side.

9. Rational Numbers II


Similarly, other rational numbers with different denominators can also be represented on the number line. Thus, in general, any rational number is either of the following two types.

m (i) - where m < n

n 1 3 5

m (ii) - where m > n

n

e.g., 2 , 4 , 6 etc., 7 3 15 e.g., _, _ , _ etc., 6 2 6

Representation of : on the number line where m < n:

5 The rational number 6 (5 < 6) is represented on the number line as shown.

4 I 11111!1 -1 0 :1 2 3 .

5 6

Representation of : on the number line where m > n:

17 Consider the rational number 5 .

17

.. 4

First convert the rational number 5 into a mixed fraction and then mark it on the number

17 2 line. i.e., 5 = 3 5

[ • _j 0 1 2 II I I I II 3 : 4 . 3~

5

5

+ Standard form of a rational number:

A rational number.!: is said to be in standard form if q is a positive integer and the integers q p and q have no common factor other than 1.

+ Comparison of two rational numbers: Step 1 : Express each of the two given rational numbers with a positive denominator. Step 2: Take the L.C.M. of these positive denominators. Step 3: Express each rational number with this L.C.M. as common denominator. Step 4 : The number having a greater numerator is greater.

+ Rational numbers between two rational numbers: There exist infinitely many rational numbers between any two rationa l numbers. So, we can insert any number of rational numbers between any two given rational numbers.

II 9. Rational Numbers


+ Operations on rational numbers: (i) Addition: To add two rational numbers with the same denominator, we simply add

their numerators and divide by the common denominator.

-5 1 - 5 + 1 -4 e.g., 7 + 7 = - 7- = 7 When denominators of given rational numbers are different f ind their L.C.M and express each one of the given rational numbers with this L.C.M as the common denominator. Then add as usual. Additive Inverse:

~ p p ~ q is the additive inverse of q and q is the additive inverse of q .

e.g., ~2 + ~ = 0 = ~ + ( ~2). The sum of a number and its additive inverse is 0 (the additive identity).

(ii) Subtraction: While subtracting two rational numbers, we add the additive inverse of the rational number to be subtracted to the other rational number.

7 2 7 2 7 (-2) 21+(-16) 5 Thus, - - - = - +additive inverse of-=-+-- = = -

8 3 8 3 8 3 24 24

(iii) Multiplication: To multiply two rational numbers, we multiply their numerators and Product of t he numerators

denominators separately and write the product as d f h d . · Pro uct o t e enommators

a c (a c) (axe) Thus, for any two rational numbers b and d , their product is b x d = (b x d) .

Reciprocal of a rational number: If the product of two rational numbers is 1 then each rational number is called the reciprocal of the other.

Thus, the reciprocal of ~ is .; and we write, ( ~ r = ~ Clearly, (a) reciprocal of 0 does not exist.

(b) reciprocal of 1 does not exist. (c) reciprocal of - 1 is -1.

(iv) Division: To divide one rational number by another non-zero rational number, we multiply the rational number by the reciprocal of the other.

a c c Thus, if b and d are two rational numbers such that d =t. 0 ,

then ( ~+~) = ~ x ( reciprocal of~) = ( ~x~}

-7 4 -7 ( . 4) -7 3 -21 e.g., 2 + 3 = 2 x rec1procal of 3 = 2 x 4 =a

9. Rational Numbers II


0 p: Every fraction is a rational number. 0 Which of the following statements is true?

• •

• • •

q: Every rational number is a fraction. Which of the following is correct? (A) p is true and q is false. (B) p is false and q is true. (C) Both p and q are true. (D) Both p and q are false.

Which of the following is not a rational number(s)?

-2 (A)-9

(B) _i_ - 7

(C)~ (D) J2 - 17 3

p: All integers are rational numbers.

q : Every rational number is an integer. Which of the following statements is correct?

(A) p is false and q is true. (B) p is true and q is false. (C) Both p and q are true. (D) Both p and q are false.

-3 What type of a number is 0 ?

(A) A positive rational number. (B) A negative rational number. (C) Either a positive or a negative

rational number. (D) Neither a positive nor a negative

rational number.

Which among the following is a rational 5

number equivalent to 3?

- 20 25 25 (A) ""15 (B) - 15 (C) 15 15

(D) 25

0 What type of a numerator does 7 have?

(A) Positive (B) Negative (C) Either positive or negative (D) Neither positive nor negative

II

0 0

3 -12 (A) ->--8 32

3 -12 (B) -8 = 32

3 -12 (C) - < --8 32

3 4 (D) ->-5 3

-4 -32 If - = -, what is the value of x? 7 X

(A) -56 (B) 56 (C) 46 (D) -46

Which is the greatest?

(A) -5 11

5 -5 (B) -12 (C) 17

-5 (D) 13

Which is the correct descending order of

-2 _i_ - 11 ~? '-5 ' 20 ' 4 °

3 - 11 4 (A) - >-2> - > -4 20 -5 3 -11 -4

(B) - > - >- > - 2 4 20 5 3 4 -11 (C) - >->-2>-4 - 5 20 3 4 - 11 (D) ->->-> - 2 4 - 5 20

What is the average of the two middle 4 1 2 5

rational numbers if 7. 3. 5 and 9 are arranged in ascending order?

86 86 43 43 (A) 90 (B) 45 (C) 45 (D) 90

What is the percentage of the least number in the greatest number of 3 9 1 7 - -, - and-? 5' 5 5 5 °

1 (A) 119% (B) 10% (C) 20% (D) 25%

What is the difference between the greatest 5 1 11

and least numbers of 9. 9 and 9?

2 4 10 2 (A) 9 (B) g (C) 9 (D) 3

9. Rational Numbers


• Which of the following pairs represent the same rational number ?

• - 7 1 (A) -and-21 3 - 5 5 (C) - and --9 -9

1 - 1 (B) -and-

3 9

(D) .i_ and - 24 - 5 15

P :The quotient of two integers is always a rational number.

1 Q : 0 is not rational. Which of the following statements is true? (A) P is true and Q is the cor rect

explanation of P. (B) P is fa lse and Q is the correct

explanation of P. (C) P is true and Q is false. (D) Both P and Q are false.

• What is the result of 2- ~~ + 256

?

(A) 149 (B) 149 (C) 149 (D) 149 39 78 76 98

A Which is the equivalent of - 143 ? ., 21

(A) -6 + ~ (B) 6 + ( -l7 ) 21 21

(C) (-6) + ( ~17 ) (D) - 6

• Of which property is

- 7 +(__3_+ - 13 )=(-7 +-2_)+ - 13 5 - 11 25 5 - 11 ) 25

an example? (A) Closure property (B) Commutative property (C) Associative property (D) Identity property G) Which of the following statements is correct? (A) 0 is called the additive identity for

rational numbers. (B) 1 is called the multiplicative identity

for rational numbers. (C) The additive inverse of 0 is zero

itself. (D) All the above.

9. Rational Numbers

•


The sum of two rational numbers is -3. If -7

one of the numbers is 5· find the other number.

(A) - 8 5

(B) ~ 5

(C) -6 5

(D)~ 5 - 5

What number should be added to 6 to

3 get 2?

- 7 1 8 (A)- (B) 2- (C) -

3 3 3 (D) - 8

3 Which of the following statements is true?

(A) The reciprocals of 1 and -1 are themselves.

(B) Zero has no reciprocal.

(C) The product of two rational numbers is a rational number.

(D) All the above.

Name the property of multiplication illustrated by

-4 x(-6 +~)=(-4 X -6 )+(-4 X~,~. 3 5 7 3 5 ) 3 7"

(A) Associative property (B) Commutative property (C) Distributive property (D) Closure property

What is the product of a rational number and its reciprocal? (A) 0 (B) 1 (C) - 1 (D) 2

The product of two rational numbers is - 9 - 4 "1"6· If one of the numbers is 3• what is the other number?

(A) 36 (B) 25 (C) 27 (D) 3!_ 48 64 49 64

- 8 By what rational number should

39 be

multiplied to obtain 26?

(A) 507 (B) - 507 (C) 407 (D) _ 407 4 4 4 4

II


• How many pieces of equa l size can be cut from a rope of 30 metres long, each

measuring 3~ metres?

(A) 8 (B) 10 (C) 6 (D) 12

Evaluate p + q given p = ( -2~1 and

q = (-1i). (A) 1 _!_

1 5

(C) ( -2 185)

(B) ( - 3185 )

(D) 3~ 15

fJ Compute 1± + ( ~8 ) - ( ~5 j.

•

•

(A)- 31 13 15 36 (B) 36 (C) 36

Find the value of - 6 ~ x ~. 3 5

(D) - 29 36

(A) 2 3_ (B) 2 .2_ (C) -2 ~ (D) - 2 2 3 5 3 3

For what value of 'x' is

~ -(- ~1) + X = 3 274 ?

- 1 1 2 (A) 3 (B) 3 (C) 3 (D) - 3

Determine 'y' so that Y- 2i = ( - 3172 }

1 1 1 1 (A) - 14 (B) 1'4 (C) - 112 (D) 112 Find the simplest form of ( -~ + i + i) ( ~2 j.

- 2 2 (A) 5 (B) 0 (C) 1 (D) 5 Rohit, Peter and Santosh walk around a

1 2 5 circularpark.Theytake 3h. Shand 12hto

complete one round. What is the total time taken by them to complete a round in minutes? (A) 79 minutes (C) 60 minutes

(B) 50 minutes (D) 69 minutes

II

• Find a rational number between 3 9 -and-. 4 11

(A) .2_ (B) ~ 2 11

(C) 69 (D) .2_ 88 4

• Previous Contest Questions ..o111111111111

• • • • •

- 7 4 What should be added to 8 to get 9?

72 (B) 95 - 72 - 95 (A) 95 (C) 95 (D) n 72

-2 What should be subtracted from 3 to

5 get 6?

(A)~ 2

(B) -3 2

(C) ~ 3

(D) -2 3

What is the reciprocal of - 8? 1 (C)~ (A) 8 (B) - (D) -8 8 8

What is the sum of the rational numbers 2 - 3 -and -? 5 7 °

1 1 (A) 0 (B) 1 (C) 35 (D) - -

35 - 33

By what number should S be divided

- 11 to get -

2- ?

- 4 4 (A) 3 (B) 3 (C) ~

4 (D) - 3

4

• Express [~1 - (~1 )]~[i+(~1 )] inthe

simplest form.

- 2 (A) -3

(B) ]_ 2

2 (C) -

3 (D) - 1

2

G Given a= 1%. b = ±·c = ~ andd = ( -1~). evaluate a(b - c) ~ d.

-4 (A)-

21 - 6

(B)-23

- 5 (C)-

27 4

(D)-21

9. Rational Numbers

Practical Geometry

+ A ruler and compasses are used for constructions. + Given a line l and a point not on it, a line parallel to l can be drawn using the idea of 'equal

alternate angles' or 'equal corresponding angles'. + Three independent measurements are required to construct a triangle. + A rough sketch is drawn with the given measurements before actually constructing the

triangle. + The sum of lengths of any two sides of a triangle is greater than its third side. + The difference of lengths of any two sides of a triang le is lesser than its third side. + The sum of angles in a triangle is 180°. + The exterior angle of a triangle is equal in measure to the sum of interior opposite angles. + The following cases of congruence of triangles are used to construct a triangle.

(i) S.S.S: A triangle can be drawn given the lengths of its three sides. (ii) S.A.S: A triangle can be drawn given the lengths of any two sides and the measure

of the angle between them. (iii) A.S.A: A triangle can be drawn given the measures of two angles and the length of

the side included between them. (iv) R.H.S: A triangle can be drawn given the length of hypotenuse of a right angled

triangle and the length of one of its legs. + A triangle is said to be,

(a) an equilateral triangle, if all of its sides are equal. (b) an isosceles triangle, if any two of its sides are equal. (c) a scalene triangle, if all of its sides are of different lengths.

+ A triangle is said to be, (a) an acute angled triangle, if each one of its angles measures less than goo. (b) a right angled triangle, if any one of its angles measures goo. (c) an obtuse angled triangle, if any one of its angles measures more than goo.

+ Pythagoras' theorem: In a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the remaining two sides.

A

~ Here, AC' = AB' + BC' . CL..JB 10. Practical Geometry II


0

•

• •

•

Which among the following is used to construct a triangle? (A) The lengths of the three sides. (B) The perimeter of the triangle. (C) The measures of three angles. (D) The names of three vertices. In the given figure, find the measure of LRQT , if PQ = QR and LQPR = 60°.

p p . T

(A) 60° (C) 120°

(B) 140° (D) 100°

A triangular sign board is isosceles. If the unequal side is 7 em and one of the equal sides is 6 em, what is the measure of the third side? (A) 5 em (C) 7 em

(B) 6 em (D) Either (A) or (C)

Which of the following statements is incorrect? (A) The sum of angles in a triangle is 2

right angles. (B) The exterior angle of a triangle is equal

to the interior angle of the triangle. (C) The hypotenuse is the longest side

of a right angled t riangle. (D) All the above . Identify the true statement. (A) A triangle with 3 equal sides is

isosceles. (B) A triangle with a 11 oo angle is right

angled. (C) A triangle with 3 acute angles is

acute angled. (D) A triangle with 2 equal sides is

equilateral.

II

• •

In which of the following cases is the construction of a triangle not possible?

(A) Measures of 3 sides are given.

(B) Measures of 2 sides and an included angle are given.

(C) Measures of 2 angles and a side are given.

(D) Measures of 3 angles are given .

Choose the correct option in which a triangle CANNOT be constructed with the given lengths of sides.

(A) 3 em, 4 em, 5 em (B) 7 em, 6 em, 5 em (C) 1 0 em, 7 em, 2 em (D) 12 em, 8 em, 6 em

( 8-1 0): David folds a sheet of paper. The dotted lines as shown in the figure are the creases formed, which are

• •

named as l, m and n. n . .

---------1·-------- l

········r······· m

Which of the following is true? (A) l I I m (B) l I I n (C) n I I m (D) Either (B) or (C) What can you say about lines l and n?

(A) l II n (B) l.l n

(C) l is the same line as n (D) Neither (A) nor (B)

What do you call the linen with respect to the lines l and m? (A) n is a line parallel to l and m. (B) n is a line parallel to l only. (C) n is a transversal. (D) n is a line parallel to m only.

10. Practical Geometry


•

Which of the following is used to draw a line parallel to a given line?

(A) A protractor (B) A set square (C) A ruler (D) A ruler and compasses

How many parallel lines can be drawn passing through a point not on the given line?

(A} 2 (D) 3

(B) 1 (D) 0

Based on the sides of a triangle, which of the following is a classification of triangles?

(A) A right angled triangle (B) An acute angled triangle (C) An obtuse angled triangle (D) An isosceles triangle

Which type of triangle is in the classification based on angles?

(A) An equilateral triangle (B) An isosceles triangle (C) A right angled triangle (D) A scalene triangle

In which of the following cases can a triangle be constructed?

(A) Measures of three sides are given.

(B) Measures of two sides and an included angle are given.

(C) Measures of two angles and the side between them are given.

(D) All the above.

0 The measurements of 6 DEF are EF = 8.4 em, L E = 103• and L F =as·. Which of the following is correct?

(A) !!.. DE F can be constructed.

(B) !!.. DE F is an obtuse angled triangle.

(C) !!.. DE F cannot be constructed.

(D) !!.. DE F is an acute angled triangle.


• In !!.. XYZ , x, y and z denote the three sides. Which of the following is incorrect? (A) X - y > z (B) X + z > y (C) X - y < z (D) X + y > z 0 Which of the following can be used to construct a 30• angle?

(A) Construct a so· angle using compasses and bisect it.

(B) Construct a perpendicular bisector of a line segment.

(C) Construct the bisector of any angle. (D) Construct an angle congruent to

any given angle.

Study the steps of construction given.

Step 1: Draw a ray OA.

Step2: With 0 as centre and any convenient radius drawanarcMNtocut OA atM.

Step3: With Mas centre and the same radius draw an arctocutMNatP.

Step4: With Pas centre and the same radius, draw an arctocutMNatQ.

Step 5: Draw OQand produce it to D. An angle AOD is constructed.

What is the measure of L AOD ?

(A} so· (C) 120°

(B) 30• (D) 45•

Satish followed the steps given in the box.

Step 1: Construct an angle of90°.

Step 2: Bisect t he go• angle.

Step 3: Bisect one of the angles obtained in step 2.

Which steps is not required to construct a 45• angle?

(A) Step 1 (B) Step 2 (C) Step 3 (D) Steps 2 and 3

II


• Which of the following is NOT constructed (C) The difference of any two sides is using a ruler and a set square? lesser than the third side.

(A) A perpendicular to a line from a (D) All the above. point not on it. • How many perpendicular lines can be

(B) A perpendicular bisector of a line drawn to a line from a point not on it? segment. (A) 1 (B) 2 (C) 0 (D) Infinite

(C) A perpendicular to a line at a point • In flXYZ, XV > YZ > ZX Which of the on the line. following is the smallest angle?

(D) A line parallel to a given line through (A) X (B) Z (C) y (D) X = Y = Z

• a given point. • t:,.PQR is constructed with all its angles Given PQ = 4 em, QR = 3.5 em and measuring 60° each. Which of the RP = 4.5 em, what type of a triangle can following is correct? be constructed? (A) fl PQR is an equilateral triangle. (A) An acute angled triangle (B) fl PQR is isosceles triangle. (B) An obtuse angled triangle (C) fl PQR is a scalene triangle. (C) An equilateral triangle (D) fl PQR is a right angled triangle. (D) A right angled triangle • A triangle is constructed as shown in the • t:,. PQR isconstructedsuchthatPQ = 5 em, figure. PR = 5 em and LRPQ = 50° Identify the

E~F type of triangle constructed. (A) An isosceles triangle Which of the following is not correct (B) An acute angled triangle about fl DEF? (C) An obtuse angled triangle (A) fl DEF has all its sides equal. • (D) Both (A) and (B) (B) fl DEF is an acute angled triangle . Identify the condition when a triangle (C) fl DEF is a scalene triangle. can be constructed? (D) fl DEF is not an equilateral triangle . (A) All three acute angles are given. • An isosceles triangle is constructed as (B) A side and an acute angles are given. shown in the figure. (C) Two obtuse angles are given. .6, • (D) All given sides are equal. Identify the false statement. (A) A triangle with three equal sides is Which of the given statements is incorrect?

called an equilateral triangle. (A) PR is the hypotenuse of flPQR. (B) A triangle with a right angle is called (B) flPQR is an equilateral triangle.

a right angled triangle. (C) t:,.PQR is a right angled triangle. (C) A triangle with two equal sides is (D) If right angled t:,.PQR has its equal

called a scalene triangle. angles measuring 45° each.

(D) A right angled triangle has two • In fl ABC, LC = 50° and LA = LB. What

• acute angles and a right angle. is the measure of LA? Identify the condition to be checked (A) 75° (B) 80° (C) 65° (D) 45° before constructing a triangle. • Which vertex of fl ABC is right ~led if (A) Sum of the three angles is 180°. - -

(B) The sum of any two of the sides is AB =8cm,AC =6cm,and BC =10cm?

greater than the third side. (A) LC (B) LA (C) LB (D) A or C

II 10. Practical Geometry


• (A)

{B)

(C)

(D) • •

A triangle PQR is constructed w ith PQ = 1 Ocm, PR = 8 em and L P = goo.ldentify the correct classification of 11 PQR.

Based on sides Based on angles

Scalene Right angled Isosceles Acute angled

Scalene Acute angled

Isosceles Right angled

11PQR is such that L P = LQ = L R = 60• which of the following is true?

(A) 11 PQR is equilateral. (B) 11 PQR is acute angled. (C) Both (A) and (B) (D) Neither (A) nor (B) Which of the following are the measures of a triangle that can be constructed using the S.S.S. criterion?

(A) X y = 6 em, LX = 40• , LV = 10• (B) DE = 8 em, EF = 7 em, FD = gem

(C) PQ = 4 em, QR = 6 em, L Q = ao•

(D) AB = 5 em, BC = 4 em, LC = go•

~;t§'·''f'·'·'MI·"'iiiH:fP I • A line p and a point X not on it are given.

• •

Which of the following is used to draw a line parallel to p through X? (A) Equal corresponding angles. (B) Congruent triangles. (C) Angle sum property of triangles. (D) Pythagoras' theorem . To which of these triangles is the Pythagoras' property related? (A) A scalene triangle. (B) An acute angled triangle. (C) A right angled triangle. (D) An obtuse angled triangle. Given AB = 3 em, AC = 5 em, and L.B = 30•, 11 ABC cannot be uniquely constructed, with AC as base, why?


0

• 0

• 0

•


(A) Two sides and included angle are given.

(B) The other two angles are not given. (C) The vertex B cannot be uniquely

located. (D) The vertex A coincides with the

vertex C.

A triangle 11 PQR with L.Q = go·, QR = 8 em and PR = 10 em is constructed. What would be the measure of PQ? (A) 3 em (B) 4 em (C) 10 em (D) 6 em The idea of equal alternate angles is used to construct which of the following? (A) A line parallel to a given line (B) A triangle (C) A square (D) Two triangles

In 11 ABC , the measures of two sides are given and L A is a right angle. Which of these properties is used to construct the triangle?

(A) S.S.S. property (B) R.H.S. property (C) S.A.S. property (D) A.S.A. property

Identify the criterion of construction of the equilateral triangle LMN given LM = 6 em.

(A) S.A.S. criterion (B) R.H.S. criterion (C) A.S.A. criterion (D) S.S.S. criterion A right triangle DEF is constructed with DE = 5 em, L F = go· and DF = 4 em. Choose the correct statement from the following.

(A) DE is the hypotenuse of 11DEF . (B) LE + LD = go• (C) EF = 3 em (D) All the above.

In D.RST, RS = 5 em, L.SRT = 45• and L RST = 45•. Which criterion can be used to construct 11RST ? (A) A.S.A. criterion (B) S.A.S. criterion (C) S.S.S. criterion (D) R.H.S. criterion

II


CHAPTER

11 Perimeter and Area

+ Perimeter is the distance around a closed figure. + Area is the part of plane occupied by the closed figure. + (a) Perimeter of a square = 4 x side units

(b) Perimeter of a rectangle = 2 x (length + breadth) units (c) Area of a square = side x side sq. units (d) Area of a rectangle = length x breadth sq. units

+ Area of a parallelogram = base x height sq. units

+ Area of a triangle = 1 (area of the parallelogram generated from it)

= ~ x base x height sq. units

+ Area of a trapezium = t (a+ b) h sq. units, where 'a' and 'b' are lengths of parallel sides

and 'h' is the height. + A circle is a closed curve in a plane drawn in such a way that every point on it is at a

constant distance (r units) from a fixed point 0 inside it. The fixed point 0 is called the centre of the circle and the constant distance r is called the radius of the circle.

+ Circumference of a Circle: The perimeter of a circle is called its circumference.

Circumference = 2m = nd, where r = radius and d = diameter.

Here n (Pi) is a constant, equal to 3.14 approximately.

+ Area of a Circle: Area of a circle with radius r units is equal to 7tr2 sq units.

+ Area of a Ring:

The region enclosed between two concentric circles of different radii is called a ring.

Area of path formed between two concentric circular regions = ( nR2 - nr2

) sq. units

= 7t (R2 - r2

) square units

= 1t (R + r) (R - r) square units

11 . Perimeter and Area II


0

• • • • • 0

•

The circumference of a circle is 44 m. What is its area?

(A) 6084.5 m2 (B) 276.5 m2

(C) 154m2 (D) 44m2

If the area of a circle is 2464 m2, find its diameter.

(A) 56 m (C) 176m

(B) 154m (D) 206m

The circular grass lawn of radius 35 m has a path of width 7 m around it on the outside. What is the area of the path? (A) 1496 m2 (B) 1450 m2

(C) 1576m2 (D) 1694 m2

The difference between the circumference and radius of a circle is 37 m. What is the circumference of the circle?

(A) 7 m (C) 154m

(B) 44 m (D) 88 m

How many plants (approximately) will be there in a circular bed whose outer edge measures 30 em allowing 4 cm2 for each plant?

(A) 18 (D) 24

(B) 750 (D) 120

When the circumference and area of a circle are numerically equal, what is the diameter numerically equal to?

(A) Area (B) Circumference (C) 21t (D) 4 If the ratio of circumferences of two circles is 4 : 9, what is the ratio of their areas?

(A) 9:4 (C) 4:9

(B) 16:81 (D) 2:3

If the area of a circle is A, radius is r and circumference is C, which of the following is true?

(A) rC = 2A (B) c A 2

(C) r2

(D) A AC =- - = C

4 r

II

e G)

What is the breadth of a rectangular sheet of perimeter 100 em and length 35 em?

(A) 35cm (B) 20cm (C) 15cm (D) 50 em

A wall hanging is in the shape given in the figure. Find its perimeter.

(A) 176 em (C) 44 em

(B) 146 em (D) 88 em

• A wire bent in the form of a circle of radius 42 em is again bent in the form of a square. What is the ratio of the regions enclosed by the circle and the square?

(A) 11 : 12 (B) 21 : 33 (C) 22 : 33 (D) 14 : 11

41) The length and the breadth of a rectangular piece of land are 400 m and 250 m respectively. What is the cost of the land at ( 1000 per square metre?

(A) ( 10 lakhs (B) ( 1 crore (C) ( 1 0 crores (D) ( 10 thousands

• In the given figure, ABCD is a parallelogram. DL _l AB and DM _l BC. If AB = 18cm, BC = 12 em and DM = 1 0 em, find DL.

C ,....-------==--"'- D ----------------- I B

(A) 6 ! em

(C) 6 f cm

(B) 6 em

(D) 6 t cm

11. Perimeter and Area


• In the fo llowing f igure, ABCD is a parallelogram. DL _i AB and AB = 13 em = AD. If the area of paralle logram is 156 cm2, find AL.

c ,.....---------,

B L A (A) 5 em (B) 6 em (C) 7 em (D) 8 em

The length and breadth of a rectangular hall in a model are 0.4 m and 30 em respectively. What is the distance between the opposite corners of the wall in the model?

(A) 34.16 m (C) 34.16 em

(B) 50 m (D) 50 em

In the figure given, PR = 22 em, SN = MQ = 3 em. What is the area of PQRS?

p

Q

(A) 66 sq. em (C) 34 sq. em

s

R

(B) 63 sq. em (D) 198 sq. em

Find the area of a verandah 2.25 m wide constructed outside a room 5.5 m long and 4 m wide.

(A) 36 sq. m (C) 64 sq. m

(B) 63 sq. m (D) 84 sq. m

The longer side of a parallelogram is 81 em and the corresponding altitude is 16 em. If the length of shorter side is 24 em, what is the altitude corresponding to the shorter side?

(A) 36 em (C) 54 em

(B) 48 em (D) 24 em

11 . Perimeter and Area


One side of a parallelogram is 2.4 dam and its area is 576 m2• Find the corre-sponding altitude. (A) 24 m (B) 24 em (C) 24 dm (D) 24 dam A field is in the form of a parallelogram whose base is 420 m and altitude is 3.6 dam. Find the cost of watering the field at 1 0 paise per sq m. (A) ~ 15120 (B) ~ 1 512 (C) ~ 151.20 (D) ~ 1512.28 Find the area of the triangle whose base is 14 dam and height is 650 em. (A) 455 dm2 (B) 455 dam2

(C) 455 cm2 (D) 455 m2

Find the altitude of a triangle whose base is 24 em and area is 672 cm2.

(A) 48 em (B) 56 em (C) 46 m (D) 58 em Two sides of a right triangle containing the right angle are 100 em and 8.6 em. Find its area. (A) 430 sq. em (C) 430 sq. m

(B) 43 sq. m (D) 430 em

If ABCD is a rectangle having length 30 em and breadth 20 em, E, F and G are midpoints of AB, CD and AD respectively, find the area of the unshaded part.

D F C GD A

(A) 400 cm2

(C) 375 cm2

E B (B) 450 cm2

(D) 500 cm2

What is the difference of circumferences of the circles shown?

(A) 22 em (C) 24 em

(B) 20 em (D) 26 em

II


• In a parallelogram ABCD, the diagonal AC measures 34 m and the per-pendicular distance of AC from either of the vertices Band D is 12 m. Find area of parallelogram.

(A) 204 sq. m (C) 816 sq. m

(B) 408 sq. m (D) 806 sq. m

ABCD is a rectangle having length 30 em and breadth 25 em. P. Q, R and S are midpoints of AB, BC, CD and AD respectively. What is the area of the shaded part?

D R C

s

A (A) 375m2

(C) 475m2

p

Q

B (B) 375 cm2

(D) 425 cm2

If ABCD is a parallelogram, what is the ratio of areas of parallelogram ABCD and !:! ABC? (A) 1 : 2 (C) 3:2

(B) 2: 1 (D) 2:3

What is the area of the shaded part in the figure given that the side of square ABCD is 90 m and the side of square PQRS is 100m?

R S

D Q~o-----' P

(A) 1900 sq m (B) 2000 sq m (C) 1 000 sq m (D) 2500 sq m

A farmer had a rectangular plot measuring 500 m by 1 00 m. If he fences the plot 4 times with barbed wire, what length of wire was used?

(A) 4800 m (B) 1 000 m (C) 300 m (D) 1200 m

II

•

The playground of school is as shown.

~lOOm----.

What is the perimeter of the playground?

(A) 420 m (B) 200 m (C) 220 m (D) 840 m

The hour hand of a clock is 4.5 em long. What distance does its tip cover in 12 hours?

(A) 28 m (B) 336 em (C) 28.28 em (D) 12 em

The figure shows a parallelogram PQRS. S R

L.~cm I P 20cm Q

What is its area ?

(A) 100 cm2 (B) 1 m2 (C) 100 em (D) 1 m

The measurements in the given figure are in em.

L

2(x + 3) x+S

h 6x+9

What is its area?

(A) 200 cm2

(C) 210 cm2 (B) 186 cm2

(D) 196 cm2

~ _f> Previous Contest Questions~ 1

• If the length of the diagonal of a square is 12 J2 em, what is its perimeter?

(A) 38 em (C) 48cm

(B) 44 em (D) 54 em

If the perimeter of a semicircle is 144 em, what is its area?

(A) 1232 cm2 (B) 1223 cm2

(C) 1322 cm2 (D) 1323 cm2



• • • • •

The diameter of a wheel of a cycle is 70 em. It moves slowly along a road. What distance will it cover in 24 complete revolutions?

(A) 5820 em (B) 5280 em (C) 5028 em (D) 5082 em

If the perimeter of a square increases by 25%, what is the increase in its area?

(A) 245 % 4

(C) 225 % 4

(B) 235% 4

(D) 215 % 4

How many square centimetres make 1 square metre?

(A) 100 (C) 1000

(B) 10000 (D) 100000

Find the circumference of the semicircu-lar region w ith diameter 10 em.

25 (A) -em

7 5 (C) 27- em 7

(B) 25~ em 7

(D) 25 em

If the radius of a circle is increased by 1 unit, what is the ratio of circumference and the diameter of circle so formed?

(A) 1t : 1 (B) 1 : 1t (C) 1t : 3 (D) 3 : 1t


•

•


Find the difference between the perimeters of the square and circle in the figure given.

(A) 57.2 em (C) 72.8 em

(B) 15.6 em (D) 52.7 em

The length around a rectangular mat is 21 m. If its length is 5 m 60 em, what is its width?

(A) 4 m 90cm (C) 49 m

(B) 490 m (D) 49 em

The perimeter of rectangular piece of paper is 56 em. If the length is three times its width, what is its width?

(A) 8 em (C) 7 em

(B) 56 em (D) 21 em

The floor of a room measures 12 m by 1 0 m. A carpet is placed on the floor from wall to wall. What is the area of the carpet?

(A) 120 m2 (B) 1200 m (C) 120m (D) 1200 m2

II

Algebraic Expressions

+ Algebra: It is a branch of mathematics in which we use literal numbers and statements symbolically. Literal numbers can be positive or negative. They are variables.

+ Variable: A symbol which takes various values is known as a variable. Normally it is denoted by x, y, z etc.

+ Constant: A symbol having a fixed numerical value is called a constant.

Sometimes, 'c', 'k', etc., are used as symbols to denote a constant.

+ Coefficient: In a term of an algebraic expression any of the factors with the sign of the term is called the coefficient of the product of the other factors in that term.

Sometimes, symbols like a, b, l, m etc., are used to denote the coefficients. Coefficients that are numbers are called numerical coefficients.

+ Algebraic expression: A combination of constants and variables connected by some or all of the four fundamental operations +, -, x and 7 is called an algebraic expression.

e.g., -5p + 12 is an algebraic expression.

Here -5 is the coefficient of the variable 'p' and 12 is the constant.

+ Terms of an algebraic expression: The different parts of the algebraic expression separated by the sign + or -, are called the terms of the expression.

e.g., 3x - 5 + 4xy is an algebraic expression containing 3 terms -3x, -5 and 4xy. + Like and unlike terms: In a given algebraic expression, the terms having the same literal

factors are called like or similar terms, otherwise they are called unlike terms.

e.g., 3xy and -4xy are like terms while 6xy and -4x are unlike terms.

+ Factors: Each term of an algebraic expression consists of a product of constants and variables.

A constant factor is called a numerical factor, while a variable factor is known as a literal factor.

+ Various types of algebraic expressions: (i) Monomial: An algebraic expression which contains only one term, is called a

monomial.

Thus, Sx, 2xy, -3a2b, -7, etc., are all monomials.

(ii) Binomial: An algebraic expression containing two terms is called a binomial.

Thus, (2a + 3b), (8- 3x), (x2 - 4xy2), etc., are all binomials.

(iii) Trinomial: An algebraic expression containing three terms is called a trinomial. Thus, (a + 2b + 5c), (x + 2y- 3z), (x3 - y3- z3), etc., are all trinomials.

II 12. Algebraic Expressions


(iv) Polynomial: An expression containing two or more terms is called a polynomial.

+ Addition of Algebraic Expressions: While adding algebraic expressions, we collect the like terms and add them. The sum of several like terms is another like term whose coefficient is the sum of the coefficients of those like terms. The like terms are added and the unlike terms are left as they are.

+ Subtraction of Algebraic Expressions: The difference of two like terms is a like term whose coefficient is the difference of the numerical coefficients of the two like terms.

e.g., 4x2 - 6x2 = (4- 6)x2 =- 2x2

Rule for subtraction: Change the sign of each term of the expression to be subtracted and then add.

+ Value of an expression: The value of an algebraic expression depends on the values of the variables forming the expression.

+ Using algebraic expressions- Formulae and Rules:

0 0

• 0

Rules and formulae in mathematics are written in a concise and general form using algebraic expressions.

Thus, the area of a rectangle = lb, where l is the length and b is the breadth of the rectangle.

The general (nth) term of a number pattern (or a sequence) is an expression in 'n'.

Thus, the nth term of the number pattern 11, 21, 31, 41, .... is (1 On + 1 ).

If x + y = S, y + z = 7 and z + x = 12, what is the value of x + y + z? (A) 12 (B) 2 (C) S (D) 24

How many auxiliary formulae can be formed from the expression in the box?

[ A =th(a+b) ]

(A) 2 (B) 3 (C) 4 (D) 1

What is the difference between 3a + 2b and -2a- Sb?

(A) Sa + 7b (C) Sa- 7b

(B) -Sa- 7b (D) a- 3b

The length and breadth of a rectangular plot are l and b. Two rectangular paths each of width 'w' run inside the plot one parallel to the length and the other

• •

parallel to the breadth. What is the total area of the paths?

(A) (Z+w)(b+w)-Zb (B) l b- (l - w) (b- w) (C) (l + b - w) w (D) l b- (l - 2w) (b- 2w)

In a two digit number, the units digit is x and tens digit is (x + 3). What is the sum of the digits in the number?

(A) 11 X + 3 (B) 2x + 3 (C) 3 + x (D) 11x + 30 A and B are polynomials and each is the additive inverse of the other. What does it mean? (A) A = B (B) A + B is a zero polynomial. (C) A - B is a zero polynomial. (D) A- B = B- A

12. Algebraic Expressions II


8 When a certain number, 'm' is divided by 5 and added to 8, the result is equal to thrice the number subtracted from 4. What is the value of 'm'?

(A) 2 (B) ~ (C) ~1 (D) ~5

• 5 added to thrice a number is equal to 12 added to twice the number. What is the number?

•

• (i)

(ii)

(ii i)

(iv)

(A) F49 X~ 7

(C) 7

(B) ~343 X!._ 7

(D) Both (B) and (C)

In the figure given what is the perimeter, in em, of the triangle ?

p

(3x - 2) em (A) (8y + 4x - 3) em (B) (8y - 4x + 3) em (C) (14x- 2y - 3) em (D) (12xy- 3) em

If c = x - a , find the value of x . x - b

(A) bC - a (B)

C - a -- --c - 1 C - b

(C) C +a

(D) 1- c --

C+b a - bC

Match the following.

Column - 1 Column - II

4m2p, 4mp2 ( ) (a) Binomial

5 - 3t ( ) (b) Unlike terms

5 ( ) (c) -7x - x Trinomial

I 2

1 +x+x 2 ( ) (d) Like terms

(A) (i)-(a), (ii)-(b), (iii)-(c), (iv)-(d) (B) (i)-(b), (ii)-(a), (iii)-(d), (iv)-(c)

II

(C) (i)-(d), (ii)-(c), (iii)-(b), (iv)-(a) (D) (i)-(b), (ii)-(c), (iii)-(a), (iv)-(d) CD Which of the following is true? (A) The product of numbers p and q

subtracted from 7 is 7 + pq.

(B) y- y 3 is a monomial.

(C) The coefficient of y2 in 2x2y + 7y2 is 7.

(D) 100z3 is a binomial.

• If x = 3, a = (-1) and b = (-2), what is the value of 2- 6x + 4a- 3b?

(A) -22 (B) - 14 (C) 12 (D) 14 CD For what value of 'm' is 9 - 5m = (-1 )?

(A) -1 (B) -2 (C) 2 (D) 1

• Simplify x2yl- 1.5 x2f + 1.4x2yl.

(A) 0.9x2yl (B) -0.9x2yl (C) 0.9 (D) - 0.9 0 What is the value of (a3 - 2a2+ 4a- 5)- (-a3 - 8a + 2a2+ 5)?

(A) 2a3 + 7a2+ 6a - 10 (B) 2a3 +7a2+12a-10 (C) 2a3-4a2+12a-10 (D) 2a3-4a2+ 6a-10

By how much is x 4 - 4x2y2 + y4 less than x4 + 8x2y2 + y4? (A) -12x2y2 (C) -12xy

(B) 12xly2 (D) 12xy

a2 What is the sum of 2 2a2 3b3 4c3

2 J J -+-- - and a +b +c?

3 4 5 .

(A) ~a + ~b3 - -

1 c3

6 12 20

(B) ~a2- _1 bJ + ~cJ 6 20 12

25 1 + -a3 + - cJ

12 20

12. Algebraic Expressions


G Match the following.

Column-/ Column - II

(i) a' - b .. when

( ) (a) 0 a= 3 and b = 2

(i i) z3

- 3 (z -10) ( ) (b) -12

when z= 10

(i ii ) i + 2x + 1

( ) (c) 1000 whenx=-1

(iv) Sp - 2 when

( ) (d) 1 p=-2

•

(A) (i)-(d), (ii)-(a), (iii)-(b), (iv)-(c)

(B) (i)-(d), (ii)-(c), (iii)-(b), (iv)-(a)

(C) (i)-(a), (ii)-(b), (iii) -(c), (iv)-(d)

(D) (i)-(d), (ii)-(c), (iii)-(a), (iv)-(b)

Simplify the following expression. x (y - z) - y (z - x) - z (x - y)

(A) 2x (y- z) (B) 2y (z- x) (C) 2x (z- y) (D) 2z(x- y)

What is the 41h term of a pattern described by the expression n2 + 1?

(A) 18 (B) 17 (C) 24 (D) 16

The third term of the series 7 n + 20 is 41. What is the 1Oth term ?

(A) 90 (B) 56 (C) 63 (D) 87

What is the expression related to the pattern 5, 8, 11, ....... ?

(A) 2n - 1 (B) 3n + 2 (C) 4n + 1 (D) n2 - 1

Which expression gives the predecessor of a natural number 'n' ?

(A) 2n-1 (C) n-1

(B) n+ 1 (D) 2n+ 1

If 'n' denotes a natural number, what does '2n' denote?

(A) A prime number (B) An even number (C) An odd number (D) A composite number


• • • •


For any natural number n, what does 2n + 1 denote?

(A) An even number (B) An odd number (C) A composite number (D) A prime number

Identify the like terms in 21 p - 32 - 7p + 20p.

(A) 21 p, - 32 and 20p (B) -32, -7p and 20p (C) 21p, -7pand20p (D) -7p, 21 p, and 32

What is the symbolic form of "one-fourth of the product of m and n"?

(A) ~mn 4 (B) t (m + n)

(C) 1 (D) j_m 4 (m-n) 4 n

What do we call the algebraic terms with same literal coefficients?

(A) Equivalent (C) Constants

(B) Unlike terms (D) Like terms

What is the coefficient of 'y ' in the expression 3xy - 13?

(A) 3x (B) 3 (C) -13 (D) Either (A) or (B)

S. l'f 3 2 1 1 d 1mp 1 y - x - - ax - y + - ax -- x an 4 5 3 8 find its value when a = 3, x = (-2) and y = (-6).

(A) 5 230 (B) 3 230 (C) ~~ (D) 52~ A rectangle is 3p em long and 2p em wide. Find the perimeter of the rectangle when p = 12.

(A) 102 em (C) 210 em

(B) 120 em (D) 10p em

From the sum of 7x - 2y - 3z and 3x + 5y - 8z, take away x - 3z.

(A) 9x- 3y + 8z (B) 9x + 3y- 8z (C) 9x + 3y + 8z (D) 9x - 3y- 8z

II


The angles of a quadrilateral are (p+25)0 ,

2p0, (2p - 1St and (p + 20)0

• What is the value of the smallest angle?

(A) 105° (B) 65° (C) 115° (D) 65°

The sides of a right angled triangle are 2a em, (2a + 2) em and (4a- 2) em long. What is the length of the shortest side of the triangle if its perimeter is 24 em?

(A) 8 em (B) 6 em (C) 10 em (D) 3 em

® ;'§Jt.JfWMM'·!f'i1it.!ifP I 0 What is the sum of 3y2 + Syz,

-2/ - 2yz - z2 and -yz + 2z2?

• • • •

(A) y2 - 2yz + z2

(C) l- 2yz - z2 (B) y2 + 2yz + z2

(D) -y2 + 2yz- z2

Evaluate the expression. p-(p-q)-q-(q-p)

(A) p - q (C) p + q

(B) - p + q (D) - (p + q)

What is the value of the expression 2x 2y + xy 2 + xy for x = 1 andy = - 2?

(A) -2 (B) -3 (C) -4 (D) -5 5

If P + 4 (4- 2P) = -4, find P.

(A) -6 (B) 6 (C) 18 (D) -18

How is "4 is less than half of x " written in symbolic form ?

(A) 4 > ~ 2

(C) .: + 4 2

(B) x- 4 2

(D) 4 < _: 2

II

• • • • •


A basket has 23 oranges and bananas. How many bananas are there in the basket if there are 'p' oranges in it?

(A) 23 p (B) 23 - p (C) 23 + p (D) p - 23

'x' packets of 6 sweets each are divided equally among 10 children. How many sweets does each child get ?

(A) 6x (B) 6x- 1 0

(C) 3; (D) 3x - 5

The length of a rectangle is 2(x + 6) em, and its width is half its length. What is its perimeter?

(A) 6(x - 3) em (B) 6(x - 6) em (C) 3(x + 6) em (D) (6x + 36) em

Simplify~ (66x + 44) + ~(33x - 33) 11 11 .

(A) 33x + 7 (B) 33x - 7 (C) 33x - 7x (D) 33 + 7x Express in the simplest form . r-----------------~

9 5 - (30 + St) + - (1St -12) 10 6

(B) 17t - 39 2

(D) 17t + ~ 2



Exponents and Powers

+ Exponential form is the short form of repeated multiplication. A number written in exponential form contains a base and an exponent.

105 is the exponential form of 1 ,00,000, since 1,00,000 = 10 x 10 x 10 x 10 x 10.

In 105, 1 0 is the base and 5 is the exponent or index or power.

+ Base denotes the number to be multiplied and the power denotes the number of times the base is to be multiplied.

a x a = a2 (read as 'a squared ' or 'a raised to the power 2')

a x a x a = a3 ( read as 'a cubed' or 'a raised to the power 3')

a x a x a x a = a4 (read as 'a raised to the power 4' or 4th power of a)

a x a x a .... (n factors) = an (read as 'a raised to the power n' or nth power of a)

• (i) When a negative number is raised to an even power the value is always positive.

e.g., (- 5)4 = (- 5) X (-5) X (-5) X (-5) = + 625

(ii) When a negative number is raised to an odd power, the value is always negative.

e.g., (-3)5 = (-3) X (-3) X (-3) X (-3) X (-3) = (-243)

+ Laws of Exponents:

Note: (a) (-1)oddnumber = _1

(b) (-1)evennumber = +1

For any non-zero integers 'a' and 'b' and whole numbers 'm' and 'n:

(i) a X a X a X ...... X a (m faCtOrS) = am

(ii) am X an= am +n (v) (ab)m = ambm

(iii) am

(vi) (~J am

a" = am-n' if m > n bm

= 1, ifm = n (vii) ao = 1

ifm < n

(iv) (am)n = amn

+ Any number can be expressed as a decimal number between 1.0 and 1 0.0 including 1.0 multiplied by a power of 10. Such a form of a number is called its standard form.

II 13. Exponents and Powers


• • • • • • • • •

What is the value of ( '! -4° ) x 52?

(A) 25 (B) 0 (C) - 25 (D) 1

(16 )

2

What is the value of 81 ?

(A)~ 2

2 (B) -

9 (2 J8

27 (C) 3 (D) 8 Which of the following is true?

(A) 10 X 1 011 = 1 0011 (B) 23 X 32 = 65

(C) 23 > 32 (D) p0 = 1 000°

What is the value of (12 x 3°-8 x 5°)?

(B) 2 (C) 4

Find the value of 2° + 3° + 4°. (A) 3 (B) 234 (C) 1 (D) 24

What is the value of 27 x 53?

(A) 7500 (B) 16000 (C) 11200 (D) 14000

Which of the following is the least?

[ (-1f, (-10)3, (1)5 and (-1)4 J

(A) 15 (B) (-1 0)3 (C) (-1 )4 (D) (-1 )3

Write a X a X a X C X C X C X C X d X d in exponential form.

(A) a3c3d3 (B) a3c3d (C) a3c3d2 (D) a3c4d2

(-11 )2 x (- 11)4 =(-11)x. What is the value of x? (A) 2 (B) 4 (C) 6 (D) - 2

Which of the following statements is correct?

(A) (23)2 and (32)4 are not the same. (B) (23)2 and (34)2 are the same. (C) (750)2 = 7502 (D) (57)3 = 573

13. Exponents and Powers

CD Evaluate ( i J. (A) ( ~ J (B) (56)5 (C) ~: (D) (55)6

4D Which of the following values are equal?

(i) 14 (ii) 4° (iii) 04 (iv) 41

(A) (i) and (ii) (C) (i) and (iii)

(B) (ii) and (iii) (D) (i) and (iv)

What is the sum of the powers of the prime factors in 108x 192? (A) 5 (B) 7 (C) 8 (D) 12

• Express ( - 2)3 .;. m3 in the form ( ~ J

- 23 (A) m3 (B) (-! J

(- 2)3 (C) m3 (

2)(-21 (D)- m ~)

G) What is the value of [ ( ~ J -( ~ J] x 26?

(A) 1 (B) 2 (C) 3 (D) 4

A 2x 34 x 25

W What is the result of 9 x 42 ?

(A) 36 (B) 42 (C) 46 (D) 48

CD 729 X 64 . Express as a product of pnme 270

factors in exponential form.

• How is 65950 expressed in standard form?

(A) 6.59 X 106 (B) 6.595 X 105

(C) 6.595 X 1 04 (D) 6.59 X 1 05

II


• Express 15625 as a power of 5.

(A) 56 (B) 54 (C) 58 (D) 57

3 - 1 Find the value of 164 x 4 2 .

(A) 8 (B) 1 (C) 4 (D) 0

- 3

Find the numerical value of (4096f4 .

1 (A) 1024 (B) 512 (C) 256 (D) 512

1 [ n ]1

'

3

(B) 5 mz 61o

[

2 ] 1/3 (C) i m~-1o [

2 ] 2/3 (D) i mn61o

G Solve 94x .;- 32x = 2187.

•

•

6 (A) x = -

7 7 (C) X= -6

-7 (B) X= -

6 -6

(D) X= -7

Simplify and leave the answer in exponent form.

(A) ~

- 6 (C) 7

(~J-2

x (I. r 7 6J

(B) 1

(D) ~ 7

43 24 -23 Simplify P q x P r x q r .

p -3q3r3

(A) p3q4r7

(C) p-sq2r-4

(B) p3q4r1o

(D) psq-2r4

II

Find the numerical value of 256°·75.

(A) 64 (B) 256 (C) 1 (D) 4-3

(a-n)m X a-2m Simplify amn X a2m

(A) a2m(n-2l (C) a-2m(n+2)

(B) a2m(2n-4) (D) a2m(n-4)

1 - 1

(16a2)2 x (36 a4 )2 Simplify 1 3 9

2al X 5al X 8a4

25 (A) 120 a 4

-25 (C) 24 a 4

(B) 25 120 a 4

25 (D) 24 a 4

Express 62x 7-4 x (8-2)2 x 63 x 72 x 84 in the simplest exponential form.

65 (A) 65 72 (B) J2 (C) 6-5 72

G) Solve 92x = 729.

3 (A) x =-4 5 (C) X= -6

G Find the value of x.

(D) 6-5 7-2

2 (B) x =-

3 3 (D) x =-2

[ (53)"' X (Ss)x = sn ] 6 (A) X= 6-11

(C) X= 6

(B) x = 9

(D) X= 8

Simplify -1b-s 9 1ob11 -12 13b-14 15 . a c xa c xa c

(A) a11b1sc-15 (B) a11b1sc15 (C) a-11b-1sc15 (D) a-11b1sc15



~Previous Contest Questions~ • • • •

The mass of the Earth is 5,976,000,000, 000,000,000,000,000 kg. How is the mass of Earth expressed in standard form?

(A) 5.976 X 1024 kg (B) 5.976 X 1021 kg (C) 5976 X 1021 kg (D) 59.76x 1023 kg

What is the simplified form of (-5xy)4 + (-5xy)2?

(A) (-5xy)6

(C) (-5xy)4 (B) (-5xy)8

(D) (-5xy)2

Which of the following equals (-1)?

(A) (3 - 2)235 (B) (-1 )47 (C) Both (A) and (B) (D) Neither (A) nor (B)

What is the simplified form of the product given?

( (-3p)4 (6q)5 (3r)6

(A) (-3)15 25 p4 q5 rs (B) p4 q5 rs (C) 315 25 p4 q5 rs (D) ( -3)15 25 p4 q4 r4


• • • •


The speed of light in vacuum is 300, 000, 000 ms-1. Express it in standard form .

(A) 3.Q X 108 mS-1

(B) 3.0 x 1010 ms-1

(C) 3.0 x 106 ms-1

(D) 3.0 x 1012 ms-1

Which is greater 23 or 52?

(A) Both are equal. (B) 23 (C) 52 (D) Cannot be determined.

Solve (95Y = (94 r .;. 92 .

(A) x = 4 (C) X=- 3

(A) 2-l• (C) 2lt

(B) x = - 2 (D) x = 2

II

Symmetry

+ Linear symmetry : If a line divides a given figure into two coinciding parts, we say that the figure is symmetrical about the line and the line is called the axis of symmetry or line of symmetry.

0

0 + A line of symmetry is also called a mirror line.

+ A figure may have no line of symmetry, only one line of symmetry, two lines of symmetry or multiple lines of symmetry.

+ Regular polygons have equa l sides and equal ang les. They have multiple lines of symmetry.

+ Each regular polygon has as many lines of symmetry as its sides.

+ A scalene triangle has no line of symmetry.

+ A parallelogram has no line of symmetry.

+ A line segment is symmetrical about its perpendicular bisector.

+ An angle with equal arms has one line of symmetry.

+ An isosceles triangle has one line of symmetry.

+ An isosceles trapezium has one line of symmetry.

+ A semicircle has one line of symmetry.

+ A kite has one line of symmetry.

+ A rectangle has two lines of symmetry.

+ A rhombus has two lines of symmetry.

+ An equilateral triangle has three lines of symmetry.

+ A square has four lines of symmetry.

+ A circle has an infinite number of lines of symmetry.

+ In English alphabet, the letters A, B, C, D, E, K, M, T. U, V, W andY have one line of symmetry and the letters H, I, X have two lines of symmetry.

• II 14. Symmetry


+ In English alphabet, the letters F, G, J, L, N, P, 0. R. Sand Z have no line of symmetry. The letter 0 has many lines of symmetry.

+ The line symmetry is closely related to mirror reflection. When dealing with mirror reflec-t ion, we have to take into account the left H right changes in orientation.

+ Point symmetry: A figure is said to be symmetric about a point 0 , called the centre of symmetry, if corresponding to each point P on the figure, there exists a point P' on the other side of the centre, which is exactly opposite to the point P and lies on the figure.

Note: A figure that possesses a point symmetry, regains its original shape even after being rotated through 180".

(i) (ii) {iii)

p p A p B

Sh s IXl D p' c p' P'

Letters of the English Une of symmetry alphabet

A, M, T, U, V, W andY Vertical B, C, D, E and K Horizontal H, I and X Both vertical and horizontal F, G, J, L, N, P, Q, R, Sand Z None 0 Infin itely many

+ Rotational symmetry: A f igure is said to have rotational symmetry if it fits onto itself more than once during a complete rotation.

+ The number of t imes a figure fits onto itself in one complete rotation is called the order of rotational symmetry.

+ A line segment AB possesses a rotational symmetry of order 2 about the midpoint 0 of the line segment.

+ An equilateral triangle ABC possesses a rotational symmetry of order 3 about the point of intersection 0 of the bisectors of the interior angles.

+ A square ABCD possesses a rotational symmetry of order 4 about the point of intersection 0 of its diagonals.

+ A rhombus ABCD possesses a rotat iona l symmetry of order 2 about the po int of intersection 0 of its diagonals.

+ A rectangle ABCD possesses a rotational symmetry of order 2 about the point of intersection 0 of its diagonals.

+ A parallelogram ABCD possesses a rotational symmetry of order 2 about the point of intersection 0 of its diagonals.

14. Symmetry II


• A regular pentagon possesses a rotational symmetry of order 5 about the point of intersection 0 of the perpendicular bisectors of the sides of the pentagon.

• A regular hexagon possesses a rotational symmetry of order 6 about the centre 0 of the hexagon.

• A circle with centre 0 possesses a rotationa l symmetry of an infinite order about the centre 0.

• The following letters of the English alphabet have rotational symmetry about the point marked on them.

HIN0SXZ

0

• • 0

• • •

Which of the following alphabets has a vertical line of symmetry?

(A) M (B) B (C) Q (D) E Which of the following alphabets has a horizontal line of symmetry?

(B) K (A) C (C) D (D) All the above

Which of the following alphabets has no line of symmetry?

(A) A (B) B (C) p (D) 0

Which of the following alphabets has many lines of symmetry?

(A) I (B) 0 (C) P (D) F

Which of the following triangles has no line of symmetry?

(A) An equilateral triangle (B) An isosceles triangle (C) A scalene triangle (D) All of the above

What is the mirror image of B in a horizontal mirror?

(A) B (B) a (C) cc (D) a~ Which of the following figures has only two lines of symmetry?

II

•

•

(A)? (B)~ (C)¢=::> (D) D Which of the following figures has rotational symmetry of order more than 1?

(A) ffi (B) 6. (C) + (D) All of these

What is the order of rotational symmetry for the given figure?

(A) 4 (C) 2

(B) 3 (D) 1

14. Symmetry


0 What is the order of rotational symmetry for the given figure?

(A) 3 (B) 4 (C) 6 (D) 12 • Which of the following figures has only

one line of symmetry?

(A)~ (B) t:J

(A) 2 (B) 4 (C) 1 (D) 3 • In the given figure, the dotted line is the

line of symmetry. Which figure is formed if the given figure is reflected in the dotted line?

(A) A square (B) A rhombus (C) A triangle (D) A pentagon

• What is the other name for a line of symmetry of a circle? (A) An arc (B) A sector (C) A diameter (D) A radius

• In 1}. XVZ , XV = XZ and XM _i VZ and ZP _i XV . About which of the following is the triangle symmetrical? (A) XM (B) VN (C) ZP (D) XZ

14. Symmetry

0 What is the order of rotational symmetry of the English alphabet Z? (A) 0 (B) 1 (C) 2 (D) 3 Which of these quadrilaterals have both line and rotational symmetries of order more than 3? (A) A triangle (B) A square (C) A kite (D) A rectangle

• Which of these letters has only rotational symmetry? (A) S (B) E (C) B (D) P

• A square has a rotational symmetry of order 4 about its centre. What is the angle of rotation? (A) 45° (B) 90° (C) 180° (D) 270° What is the order of rotational symmetry of the figure given?

(A) 2 (B) 1 (C) 4 (D) 3 (21-28): Find the number of lines of

symmetry of each of the figures given.

(A) 0 (B) 2

·v (A) 4 (B) 2 •a (A) 2 (C) 1

(C) 8 (D) 4

(C) 8 (D) 6


II


(A) 4 (C) 2

(A) 4 (B) 2

(A) 0 (B) 1


(C) 6 (D) 8

(C) 2 (D) 3

Which of the given figures has an order 4 rotational symmetry?

(A) @ (B) 0 (C) 0 (D)©

Find the order of rotational symmetry of the given figure.

+ (A) 0 (B) 2 (C) 4 (D) 8

Which of the English alphabets has a rotational symmetry of order 0?

(A) H (B) A (C) I (D) N

II

Identify the figure with 5 lines of symmetry. (A)o (C) 0

(B)* (D)()

What is the order of rotational symmetry of the figure given ?

(A) 3 (B) 1 (C) 5 (D) Infinitely many G What is the order of rotational symmetry of the given figure?

(A) 0 (B) 1 (C) 3 (D) 4

• Which figure completes the figure given about the line l, the line of symmetry.

[ ·~-+l] (A) ~ +-L ______ \._. z (B) ·~-+l

(C) •~-+l (D) +-~•l

14. Symmetry


•

What is the order of rotational symmetry of given figure?

(A) 3 (C) 0

(B) 6 (D) Many

Which of the following figures has both linear symmetry and rotational symmetry?

(A) An isosceles triangle (B) A scalene triangle (C) A parallelogram (D) A square

Which of the following is the odd one out?

(A) A pentagon (B) A scalene triangle (C) A semicircle (D) An isosceles triangle

Which of the following statements is correct?

(A) An equilateral triangle has three lines of symmetry.

(B) A rectangle has four lines of symmetry.

(C) A circle has only one line of symmetry.

(D) A parallelogram has two lines of symmetry.

'Iii Previous Contest Questions .....

• •

How many lines of symmetry does a regular polygon have?

(A) Infinitely many (B) As many as its sides (C) Only one (D) Zero

Which of these has 3 lines of symmetry?

(A) Any triangle. (B) An isosceles triangle. (C) An equilateral triangle. (D) A right angled triangle.

14. Symmetry

• • • • •

(A)

(B)

(C)

(D)

•


Which of these letters of the English alphabet has reflectional symmetry about a vertical mirror?

(A) U (C) p

(B) B (D) F

p = No. of lines of symmetry of a square,

q = No. of lines of symmetry of a rectangle.

Which of the following is true? (A) p < q (B) p = q (C) q > p (D) p > q A figure looks exactly the same as its original position after a 60° rotation about its centre. At which other angle does this repeat?

(A) 600 (C) 270°

(B) 120° (D) 45°

Which of these letters of the English alphabet has both multiple line and rotational symmetries? (A) 0 (B) S (C) H (D) L

Which of the following is matched incorrectly?

Lines of Order of Figure symmetry rotational

symmetry

An isosceles 1 0 triangle

A rhombus 2 2

A parallelo-2 0 gram

A circle Infinitely Infinitely many many

Find the order of rotational symmetry of the following figure.

(A) 0 (C) 2

(B) 1 (D) 3

II


Visualising Solid Shapes

+ Descript ion of Some Basic Shapes: (i) Square

Side

Corner ~..__~t--J

It has four sides and four corners.

All its sides are of the same length.

(ii) Rectangle

....____.____r ~: ~ ,;,, Corner~-- r

It has four sides and four corners.

The opposite sides of a rectangle are of the same length.

(iii) Triangle

Side

It has three sides and three verti-ces.

(iv) Cuboid

Face Edge

L...---------1'--~ Vertex

It has 6 flat faces, 12 straight edges and 8 vertices.

15. Visualising Solid Shapes

(v) Cube

Face

It has 6 flat faces, 8 vertices and 12 straight edges.

(vi) Cylinder

Curved face

Curved edge

It has 3 faces ---7 1 curved face and 2 flat faces. It has 2 curved edges.

(viQ Cone

Curved face

Curved edge

Flat face It has 2 faces ---7 1 curved face and

1 flat face.

It has 1 curved edge.

II


+ Three dimensional shapes have length, breadth and height or depth. + Two-dimensional shapes have only length and breadth. + Three-dimensional (or 3-D) shapes can be visualised on a two-dimensional (or 2-D)

surface. + A net is a skeleton-outline in 2-D which when folded results in a 3-D shape. The same solid

can have several types of nets. + Dice are cubes with dots on each face. Opposite faces of a die always have a total of seven

dots on them. Some dice have number, 1 to 6 on their faces. + A solid can be sketched in two ways.

(a) An oblique sketch which does not have proportional lengths, but conveys all important aspects of the appearance of the solid.

(b) An isometric sketch, drawn on an isometric dot paper, which has proportional measurements of the solid.

+ Different sections of a solid can be viewed in many ways:

•

(a) Slicing the shape results in the cross-section of the solid. (b) Observing a 2-D shadow of a 3-D shape. (c) Looking at the shape from different angles, i.e., the front-view, the side-view and the

top-view. Description of few more solid shapes

S.No.

1.

2.

3.

4.

Name of the figure

Triangular pr ism

Tr iangular pyramid or Tetra-hedron

Square pyramid

Sphere

Figure

1\ \

4J

Description

A triangular prism resembles a kale idoscope. It has triangular bases.

It has a triangular base.

It has a square as its base.

No flat face. It has only a spherical face.

Components

Faces : 5 Edges : 9 Ver tices : 6

Faces : 4 Edges: 6 Ver tices :4

Faces : 5 Edges: 8 Ver tices: 5

Faces : 1 Edges : 0 Ver tices : 0

II 15. Visualising Solid Shapes


0

0

•

•

How many edges does the following • Which of the following pair of shapes, figure have? when joined together (by placing them

4 edge to edge) can form a rectangle?

(A) nn(B) ~~ ~~ (D)DV (A) 12 (B) 8 (C)

(C) 6 (D) 4 Which of the following figures has six • Which of the following is different from faces? the other three?

(B) Cl-P (A) Q (B) L I )- - (C) c=J (D) D

(D) L 17 8 A solid object when seen from one side, looks like this .

Which shape is divided into two EQUAL parts by the dotted line?

(A) b. (B) -0-(C) \11 (D) l//r

I / I

How many triangles can be seen in this figure?

(A) 3 (C) 6

A X

(B) 5 (D) 7

D The same solid, when viewed from top, looks like this.

Which of these shapes could it be?

(A)® (B) 0 (C) 7 (D) _0. u ~

15. Visualising Solid Shapes II


• •

CD

(i)

(ii)

(iii)

(iv)

How many corners does the shape given have?

(A) 6 (C) 12

(B) 10 (D) 13

Identify the correct statement from the following.

(A) A triangle has 3 sides and 4 vertices.

(B) A cylinder has 3 faces.

(C) All sides of the rectangle are equal.

(D) A cuboid has 4 flat faces and 12 straight edges.

Rakesh has 10 one rupee coins of similar kind. He puts them exactly one on the other. What shape will he get finally?

(A) Circle (B) Cylinder (C) Cube (D) Cone

Identify the false statement from the following.

(A) A cuboid has 3 pairs of opposite faces. (B) The number of vertices of a cube is 6. (C) All sides of a square are equal. (D) A cuboid is a three dimensional figure.

Match the following.

Column -I Column -II

(d ( ) (a) Q 6 ( ) (b) oW CllJ ( ) (c) ~ ,.

4 ( ) (d) cQJ

II

(A) (i)-(a), (ii)-(b), (iii)-(c), (iv)-(d) (B) (i)-(d), (ii)-(a), (iii)-(b), (iv)-(c) (C) (i)-(c), (ii)-(b), (iiii)-(a), (iv)-(d) (D) (i)-(d), (ii)-(a), (iii)-(c), (iv)-(b)

• Wh ich of the following is an oblique sketch of a cube of edge 4 em?

lem (A) ~

lem

(B) 4 em ./ c em

"' 2em

LTI '3C'~ ~ 4em 4cm

(C) (D) (;~

4em ~

• What is the total on the face opposite to 4 + 3 on the dice given?

(A) 3 (B) 5 (C) 12 (D) 7

41) What is the number on the face opposite to 5 on a die?

(A) 1 (B) 2 (C) 3 (D) 6

0 The oblique sketch of 3 dice each with 2 em edge placed side by side is given. What are its respective length, breadth and height?

(A) 6 em, 2 em, 2 em (B) 6 em, 4 em, 2 em (C) 4 em, 6 em, 2 em (D) 2 em, 4 em, 6 em



A die is cut horizontally.What is the cross-section obtained?

(A) A triangle (B) A rectangle (C) A square (D) A cube

Cl) The following figure shows a source of light (L), a solid (S) and its shadow (P) on a screen (M).

® @ What is the shape of P?

(A) A rectangle (B) A sphere (C) A circle (D) A triangle

The shadow of a 3-D object is given.

D Which of following could the object be?

(A) A book (B) A ball (C) A die (D) A pipe

The following is the shadow of a 3-D object.

0 Which of these objects does the shadow belong to?

(A) (2) (B)[] (C) - (D) )

• The front view of a solid is ~. Which of these is the solid?



(A) A die (C) A pyramid

(B) A match box (D) A ball

The front, side and top views of an object is as shown.

DOD Identify the object.

(A)

(C)~ (B) -(D)®

Which of the following is a part of the front view of a honey comb?

(A) ~.-1 --.1' (B) 0 (C) ~ (Dl/ I Identify the front view of the object given.

--{0 (Al u (Bl .__I ---J

Observe the object given.

1\ 3-What is its side view indicated by the arrow?

(A) I (B) \ \ (C) D (D) ~

II


• Which of the following has on ly one f lat surface?

(A) b (B) .__( __ -JIC1

(C) u (D) 1\ \ • Which of the given objects has exactly

two flat faces?

•

(A) 8 (B) () )

(D)(() (C) 1\ \ Identify the number of plane faces of the given solid.

(A) 4 (C) 5

(B) 3 (D) 2

Identify the number of vertices of the given solid.

(A) 8 (C) 12

f-f-~ (B) 6 (D) 10

Identify the cross-section of the given solid at the dotted line.

4 I

' () I ) I I

I

I I

If

(A) 0 (B) D (C) 0 (D) I I

II

•

•

A hollow pipe is viewed from the side indicated by the arrow. In which shape is it?

(A) A ring (C) A cylinder

(B) A circle (D) An ellipse

The following arrangement of cubes is painted blue on all sides.

How many square faces are painted blue?

(A) 16 (C) 18

(B) 9 (D) 12

Identify the solid which has the following views.

Side view Top view Front view (B) A cuboid (A) A cube

(C) A cone (D) A sphere

• Previous Contest Questions~

0

• What is the shape of the given figure?

cJ (A) A rectangle (B) A square (C) A quadrilateral (D) A parallelogram

Which of these is the difference between a cube and a cuboid?

(A) There is no difference between a cube and cuboid.

(B) A cube has equal length, breadth and height where as a cuboid has different measures for length, breadth and height.

(C) A cube has unequal length, breadth and height and a cuboid has equal length breadth and height.

(D) Either (A) or (C).



• • •

• • •

Which of the given geometric solids has the maximum number of vertices?

(A) Cone (C) Cuboid

(B) Cylinder (D) Pyramid

If two cubes of dimensions 3 em by 3 em by 3 em are placed side by side, what would the dimensions of the resulting cuboid be?

(A) 6 em x 6 em x 6 em (B) 12 em x 12 em x 12 em (C) 9 em x 6 em x 3 em (D) 6 em x 3 em x 3 em

Which of the following is not the net of a cube?

(A) ct (B) rrr§ (C) Etta (D) ~ What solid do you get when you give a vertical cut of a brick of dimensions

5 em x 5 em x 10 em along 10 em side?

(A) A cuboid (C) A cube

(B) A cylinder (D) A triangle

What is the shape formed by rotating a right triangle about its height?

(A) A sphere (B) A cylinder (C) A cone (D) A cuboid

What is the total length of edges of the following cube?

(A) 24 em (C) 32 em

2cm

(B) 30 em (D) 36 em


0 Identify the correct statement.

(A) A cone has 2 vertices. (B) A cube has 8 vertices. (C) A cylinder has 1 vertex. (D) A cuboid has 1 0 faces.

G) The top view and the front view of a solid are given in the figure.

oo Top view Front view

Identify the solid.

(A) A cylinder (C) A prism

(B) A cone (D) A pyramid

Identify the net of the given solid.

~ (A) ? (B) 'Y!1 (C) (J (D) ¢

Identify the solid with 3 rectangular and 2 triangular faces, from the following.

(A) & (B) (J

(C)£::) (D)~ Ill



1. 3

Shruti and Fida want to buy the same book. Shruti has 4 of the money needed to buy the

book and Fida has half of the money needed to buy the book. If the book was < 3 cheaper, then together they would have exactly enough money to buy 2 copies of the book. What is the original price of the book?

2. A quiz has three questions, with each question worth one mark. If 20% of the students got 0 questions correct, 5% got 1 question correct, 40% got 2 questions correct, and 35% got all 3 questions correct, find the overall class mean (average) mark.

3. In square ABCD, P is the midpoint of DC and Q is the midpoint of AD. If the area of the quadrilateral QBCP is 15, what is the area of square ABCD?

4. A star is made by overlapping two identical equilateral triangles, as shown. The entire star has an area of 36. What is the area of the shaded region?

5. Lines PS, QT and RU intersect at a common point 0, as shown. P is joined to Q, R to S, and T to U, to form triangles. Q Find the value of L P + L Q + L R + L S + L T + L U.

Q

p

A B

D p c

u

6. If x and y are positive integers with x > y and x + xy = 391, what is the value of x + y?

7. Four congruent rectangles and a square are assembled without overlapping to form a large square, as shown. Each of the rectangles has a perimeter of 40 em. Find the total area of the large square.

Questions@stimulating-minds IIIII


8. Suppose that x and y are positive numbers with

1 xy = 9

What is the value of (x + 1 )(y + 1 )?

7 X (y + 1) = 9

y(x +1) = 158

9. The price of each item at the Gauss Gadget Store has been reduced by 20% from its original price. An MP3 player has a sale price of~ 1120. What would be the cost of same MP3 player if it was on sale for 30% off of its original price?

10. If x is a positive integer less than 100, how many values of x make ,.}1 + 2 + 3 + 4 + x an integer?

11. Three different numbers are chosen such that when each of the numbers is added to the average of the remaining two, the numbers 65, 69 and 76 result. Find the average of the three original numbers.

12. Let N be the smallest positive integer whose digits have a product of 2000. What is the sum of the digits of N.

13. The rectangular flag shown is divided into seven stripes of equal height. The height of the flag is h and the length of the flag is twice its height. The total area of the four shaded regions is 1400 cm2• What is the height of the flag?

2h

14. If a = 7 and b = 13, what is the number of even positive integers less than ab?

1 25 15. If p, q and rare positive integers and p + --1 = 1'9. find the value of q.

q+-r

l h

l

Ill Questions@stimulating-minds


Score 0

Model Test Paper [;{)

0

•

•

• •

1th fh S of a flag pole is black, 4 is wh ite and

the rema ining three metres is pa inted yellow. Find the length of the flag pole.

5 60 (A) 511 m (B) 11 em

6 (C) 5 km (D) 5- em

11 The given rationa l numbers are 1 4 - 7 2' _5 , and 8 . If these numbers are ar-

ranged in ascending order or descending order, which is the middle number?

1 (A) 2

4 (C) - 5

(B) -7 8 1 7

(D) - and - -2 8

MBC is cong ruent to !lXYZ. Find the measures of Lx and L.y .

~ZVV B Scm C X (A) x = 80°, y = 60° (B) x = 60°, y = 40° (C) X = 80°, y = 40° (D) X = 60°, y = 80°

If 24-carat gold is 100% pure gold, what percentage of pure go ld is in 22 -carat gold?

(A) 61~% (B) 71~%

(D) 91~% The sum of 10 observations is 250. If one observation, 25, is deleted, what is the new mean?

(A) 25 (B) 20 (C) 28 (D) 22

Model Test Paper

• 0

• • •

A man sold 10 eggs for 5 rupees and gained 20%. How many eggs d id he buy for 5 rupees?

(A) 12 25

(B) 12 (C) 25 (D) 20

If the ratio of areas of two circles is 16 : 25, what is the ratio of their circum-ferences?

(A) 25: 16 (C) 4:5

(B) 5: 4 (D) 3:5

The angles of a triangle are X0, (2x + 5t and

(3x - 5t, Find the largest angle.

(A) 180° (B) 65° (C) 85° (D) 30°

1 1 1 Given - +- = -f, find the value ofVwhen u v f = 20 and u = 30.

(A) - 20 (B) - 60 (C) 60 (D) - 30

If !lPOR = flXYZ , which ofthefollowing is

correct?

(A) YZ =OR (C) XZ =OR

(B) XV= PR (D) YZ =PO

The dai ly earnings (in rupees) of 10 workers in a factory are 8, 16, 19, 8, 16, 19, 16, 8, 19, 16. What is their median wage?

(A) < 17.50 (C) < 19.00

(B) < 8.00 (D) < 16.00

A school has 560 students. The number of

2 girls is 14 7 % of the number of boys. How

many girls are there in the school?

(A) 320 (B) 80 (C) 490 (D) 70

The difference of one-fifth of a number and 4 is 3. Which of the following is the number? (A) 7 (B) 35 (C) 5 (D) -5

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rl BMA's Talent & Olympiad Exams Resource Book

•

• • • •

The m inute hand of a circular clock is 14 em long. How far does its tip move in 1 hour?

(A) 94.2 em (B) 88 em (C) 196 em (D) 28 em

lnthefiguregiven, LXOZ and LYOZ form a linear pair. If p - q = so·. what are the respective values of p and q?

II( •

X

(A) so· and 130• (C) 120· and 60•

0 y

(B) 130• and so· (D) 60. and 120•

What fraction of the f igure g iven is unshaded?

1 (A) -

4 1

(B) -3 2

(C) -3 3

(D)-4

What is the value of 2(5a - 1) + 2 (6 + 7b) where a = 1 and b = 2a?

(A) 58 (B) 48 (C) 56 (D) 44

The area of a triangular field is 1.5 hectares. If its altitude is 300m, find the correspond-ing base.

(A) 100 em (C) 100m

(B) 10m (D) 10 em

The maximum daily temperature (in °C} of a city during a week are 24.6, 28.7, 22.7, 27.5, 26.5, 25.8 and 26.9. What is the mean tem-perature (in °C}?

(A) 26.7 (C) 27.1

(B) 27.7 (D) 26.1

Key

• •


- 8 The product of two rational numbers is

9 .

-4 If one of the numbers is ..,..-s , what is the

other number?

3 10 10 (A)-3 (B) 10 (C) 9

9 (D) 10

Madhavi glued 4 white cubes together as shown. Then she painted the entire object red.

I I I I ~ I I I I

I I I I How many faces of the 4 cubes were painted red? (A) 4 (B) 9 (C) 18 (D) 24 Which of the following values of x satisfies

3 4 7 the equation -- + -- = -?

2x-1 2x + 1 2x

7 (A) -

2 7 2

(B) -- (C) -2 7

2 (D)--7

5 out of 2250 parts of the Earth is Sulphur. What is the percentage of Sulphur in the Earth?

2 1 2 3 (A) - % (B) - % (C) - % (D) - %

9 9 5 7 How many kilometres does a bicycle wheel of rad ius 30 em cover in 70 revolutions? (A) 0.0132 km (B) 1.32 km (C) 13.2 km (D) 0.132 km AB and CD intersect at 0 .

Find the value of x. (A) 40° (B) 48° (C) 32° (D) 36°

1 2 3 4 5 6 7 8 9 1 0 11 12 13 14 15 16 17 18 19 20 A c A D A c CDC ADD 8B 8 B 8 CD A

21 22 23 24 25 c A A D D

II Model Test Paper


~ 1 1 [ Explanatory Answers

I 1. Integers I ~ Multiple Choice Questions 1. (B) The set of negative numbers and

whole numbers is called integers. z = {' .... -2, -1, 0, 1, 2, ...... }

2. (C) 1 is the smallest positive integer. 3. (B) All the other positve numbers to

its right are greater than, the negative numbers are located on the leftside of 0.

4. (D)

5. (B) 8. (B)

9. (B)

10. (C)

11. (C)

12. (A)

16. (A)

17. (A)

18. (A)

I I I I I -2 -1 0 1, 2 The required opposite is spending ~ 100.

6. (D) 7. (A)

West is represented by a negative integer. (-1) x (-1) x (-1) .... .. odd times = (-1) which is negative. (-1) x (-1) x (-1) ...... even times = 1, which is positive. (-32) X (-4) X (-3) X 0 X (-6) = 0 aS the product of any number and 0 is 0.

13. (B) 14. (C) 15. (C)

0 is identity element with respect to addition of integers. e.g., (-5) + 0 = + 0 + (-5) = (-5).

1 is the identity element with respect to multiplication. Use BODMAS rule & simplify.

124 x 4 _ 3 + 118+2 J, J,

= 496 - 3 + 59 = 552 19. (C) ( ), { }, [ ] is the correct order of

evaluation of brackets. 20. (A) If a negative sign precedes a bracket,

the signs of the terms inside the bracket are changed.

21. (D) If a positive sign precedes a bracket, the signs of the terms inside the bracket are not changed.

Explanatory Answers

22. (C)

23. (B)

24. (A)

27. (B)

28. (D)

29. (D)

30. (D)

31. (D)

J Use of the order of removal of brackets and simplify. 7- [13- {-2 -6 (6 of -5)}] = 7 - [13 - {-2 -6 X -30}) = 172 The sign of the product of two like integers is positive. e.g., 5 x 4 = 20 and- 2 x -4 = 8

25. (D) 26. (C)

The product of a positive number and a negative number is negative. e.g., 6 X -1 =-6 which is negative. x + ( - 48) = 62, where x is the unknown integer. ~ X = 62 - ( -48)

= 62 + 48 = 110 P x - 6 = - 48, where P is the unknown integer. ~ p = -48 7 - 6 = 8

3 km Initial position

S km

Final position

The smallest negative integer does not exist.

~ Previous Contest Questions 1. (B) 35 + (- 10) + (- 15) + 20 + 5

= 35 - 10 - 15 + 20 + 5 = 35 - 25 + 25 = 35

2. (A) Here jumping downwards is taken as positive and jumping upwards is taken as negative. Also given that the monkey is sitting on the first step. 1 + (+3) + (- 2) + (+3) + (- 2) + (+3) + (-2) + (+3) + (-2) + (+3)+(-2) + (+3) = 1+3- 2+3- 2+3- 2+3- 2+3 - 2+3 is 11 steps.

Ill


3. (C)

4. (A)

5. (B)

6. (C)

Initial temperature of room= 40 oc Given that the temperature lowers at the rate of 5 oc every hour. For 10 hours --7 10 X (-5 °C) =-50 oc . . Room temperature 10 hours after the process begins = 40 oc - 50 oc = - 10 oc Of the 10 questions in the test, 2 are correct and 6 are incorrect 2 are not attempted. So, the total score =2 x(+3) +6 x(- 1)+2 x (0) =6 - 6+0=0 Let the unknown number be x . Then X X (- 12) = 180

180 180 => X= - = - - = - 15 -12 12

The lift has to descend 360 m so as to reach-350m, as it is 10m above the ground level. Given that the lift descends at the rate of 6 m/minute. 6 m --7 1 minute

360 1 60 . 360 m --7 -x = mmutes 6

= 1 hour 7. (A) Given the temperature at 12 noon

is 10 oc.

8.

I

The temperature decreases at the rate of 2 oc per hour until mid night. From 12 noon to 9 p.m. it is 9 hours.

~ Temperature decrease =9x (-2°C) =-18 °C

(A)

Temperature at 9 p.m = 10 oc + (-18 °C) = -8 oc

9. (A) 10. (A)

2. Fractions and Decimals I [i' Multiple Choice Questions

1. (C) L.C.M. is 16 x 5 = 80 9 45

16 = 80' 13 208 5 80

II

2.

3.

4.

7.

8.

9.

(B)

(B)

(A )

(B)

(C)

(D)

10. (C)

15. (B)

16. (D)

17. (C)

18. (B)


45 208 => - < -80 80

N D

WNx D +N D

9 13 .. - < -16 5

where W is the

whole number, N is the numerator and D is the denominator.

12 1 _12x5+1 = 60 +1 = 61 So, 5- 5 5 5

3 7 3 To find the sum of -+-+1-4 6 5' find the L.C.M. of the denominators and then add them. L.C.M. of 4, 5 and 6 is 60.

3 7 3 3 7 8 31 .. - + - + 1- = - + - + - =3-4 6 5 4 6 5 60

5. (B) 6. (C)

No . pages left =(1 -~} total pages :. Total number of pages

5 =BOx- =200 2

..!. x 49 litres and 7 litres. 7 Let the number of matches lost be x. The number of matches won = x + 4. Total matches played = x + x + 4

3 We have, x + 4 = -(2x + 4) 5

~ x = B :. Total matches played = 20

11. (C) 12. (B) 13. (C) 14. (C)

Let 'x' be the number. 1 - of x = 5 => x = 5x9 = 45 9 p s - =- => pt=sq q t

(By cross multiplication.)

1 3- = 3+x 5

16 3 16-15 1 .. x=---=--=-5 1 5 5

Explanatory Answers


4 _2_= 4x11+7 =51=~ 11 11 11 11· 19. (A)

So, x = 51.

12 2 20. (C) 30 = 5 is the required lowest

21. (D)

form.

Total parts = 3; Shaded parts = 1 Shaded part in the figure in

1 option (D) represents 3 of the whole.

1 22. (B) Figure A represents 4 shaded part

3 and 4 unshaded part.

3 So, 4 x 20 = 15 parts should be

shaded in figure B. 23. (C) Total number of chocolates eaten

24. (D )

25. (B)

27. (A)

3 5+3 8 =1+ - =-- = -5 5 5

The required number of one fourth

=5-2.!. = 5 - ~ = 11 = 11x .!. 4 4 4 4

26. (A)

P,R,D

36. (B) 5 parts out of 10 are shaded. 5 - = 0.5

10 l@f Previous Contest Questions

4 2 3 25 12 7 1. (D) 3-x2-xl-=-x-x-= 15

7 5 4 7 5 4 2. (B) Total number of pages in the book

= 216 No. of pages read by Suresh

=(~of 216 )=( 216x~j = 162

Hence, Suresh read 162 pages during last week.

2 1 3. (A) Cost of 55 litres of milk= ~ 1014

~ Cost of 1 litre of milk

_ (405+27) - (405x~) - ~ 4 5 - ~ 4 27

Hence, the cost of milk is 3

U8 4 per litre.

4. (C) Let the other number be x. 2 5 Then 6- xx = 15 -3 6 95 3 19 3 => x= - X- = - =2-6 20 8 8

The fraction of alphabets made of 5. (C) Product of given decimals= 1.5008 One decimal = 0.56 semicircles and straight lines is

3 1 - = - The other decimal = 1.5008 + 0.56 6 2

28. (B) The fraction that represents the 6

= (1.5008 X 100) = 150.08 = 2_68 0.56 100 56

figure is 6. It is an improper 6. (C) Games won= 6 fraction.

29. (D) : . The total shaded part 1 2 2 5 =-+-+-=-7 7 7 7

30. (C) Difference of two like fractions

Difference of numerators Common denominator

31. (C) 32. (A ) 33. (C) 34. (C)

35. (B) 8 8 hundredths = - = 0.08 100

Explanatory Answers

Total games = 6 + 4 = 10

: . The required fraction = ~ 10

7. (D) 3 112 -[ 1~ + {2~-( 1~-~ )}]

= 37 - [2. + ~] = 37- 37 = 0 12 4 6 12 12

8. (A) Part of the cake eaten

5 2 10 5 = -X- = - = -6 3 18 9

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9. (A) 10. (D)

I 3. Data Handling I [i' Multiple Choice Questions

1. (A) By definition,

A.M.= 1+ 2 +3+ .. ... +10 = 5.5 10

10+1 By shortcut, A.M. = -2

- = 5.5

2. (A) For 10 observations, the median would be the average of the 5th and 6th observations. Since they are unaffected by increase in 7th entry, the median will be unchanged.

3. (B) By definition,

Average = x + (x + 3) + (x + 6) + (x + 9) + (x + 12) 5

= 5x +30 = x+ 6 5

4. (D) By definition of average, the average daily sale

= c 5+ 120+12+~0 +70.5 +140.5}

= 78l However, without calculating we can say that the answer is D since t h e average lies between t h e maximum and the minimum.

5. (A) Let the sum of four numbers bey and the excluded number be x .

y + x y Then-- -27and - -25orx+y 5 - 4 -= 135 and y = 100 ~ x = 35

6. (C) Arranging the given data in ascending order, we have, 33, 35, 41, 46, 55, 58, 64, 77, 87, 90 and 92. The sixth entry is 58. : . Median is 58.

7. (C) Mean is representative of whole group.

8. (B) Let the observations be xl' x2, ...... X0

•

After the increase they are x1 + 5, x2 + 5, .... ,X

0 + 5.

II

A.M. = X1 + 5 + x2 + 5 + .... + X0 + 5 n

=A.M. before increase+ 5. 9. (A) 10. (A) 11. (A) 12. (B) 13. (A) Each of the 6 marbles has an equal

chance of being picked. So, the probability that the marble picked

1 is 3 will be 1 out of 6 i.e., 6 ·

14. (B) Average height of the mountains 8200 + 6000 + 8600 + 7500 + 8800 + 6500 --------------~-------------m 6

= 45600 m = 7600 m 6

15. (A) Arranging the heights in ascending order, we have 6000, 6500, 7500, 8200, 8600, 8800.

7500+ 8200 :. Median height = 2

= 7850 m 16. (C) 17. (D) 18. (C) 19. (A)

sum of observations 20. (B) Mean = number of observations

325 = 10 = 32.5 years

21. (C) Mode of a given data is the value that occurs most frequently. So, mode = 26 years.

22. (A) The height of the shortest girl is 128 em.

23. (C) The range of the data = maximum value minimum value = (151- 128) em = 23 em

Sum of the heights 24. (A) Mean height= No. of students

1414 = ---w- = 141.4 em

25. (C) Mean height of the students is 141.4 em. No. of students whose height is greater than 141.4 em is 5.

26. (A) Double bar graphs help to compare X' two sets of data at a glance.

27. (B) Our age increases day by day.

Explanatory Answers


28. (D) A die has 6 faces numbered from 1 to 6. So, it will not land up with 7 on top.

29. (A) A tossed coin may land with heads or tails up.

30. (B) The 6 faces of a die are numbered as 1 to 6.

31. (B) The range of a given data is the difference between its highest and lowest values.

32. (A)

From the given data, range = 154 - 128 = 26.

A 3+4 +2 +5 + 4+3 verage = 6

21 =- = 3.5 hours 6

33. (C) No. of kilograms of fruits sold during the four hours = 35 + 26 + 45 + 20 = 126.

34. (B) 35. (A) 36. (B)

~ Previous Contest Questions 1. (A) Mean of first 5 natural numbers

= 1 + 2 + 3 + 4 + 5 = 15 = 3 5 3

Median of 1, 2, @, 4, 5, = 3 2. (D) The first five prime numbers are 2,

3, 5, 7 and 11.

. Th. 2+3 + 5 +7 +11 . . e1r mean = ------5

= 5.6 3. (C) Since the comet passes by the

Earth every 60 years, find the year from the options that is a multiple of 60 added to 1835. 2075 = 1835 + 4 X 60 :. The comet can be expected to pass by the Earth in the year 2075.

4. (A) 5. (B) 6. (B)

7. (D) The angle representing students who like other juices is

360° - (90° + 75° + 135°) = 60• Total number of students = 360 : . Required number of students

60° =--x360=60 360°

Explanatory Answers

8. (A) The strength of the school is 720. So, the number of students who like

75° orange juice is --x720 = 150. 360°

I 4. Simple Equations I ~ Multiple Choice Questions

(c) 3x - 1 _ 1 +X + X - 1 = 3 1. 5 2 2

~ 6x = 42 ~ x = 7

2. (B) 0.2(2x - 1) - 0.5(3x - 1) = 0.4 - 1

~ X= ll 3. (C) Transposing 3 to the R.H.S. gives

the value of m. 4. (A) 5. (B) 6. (A) 7. (B)

8. (C) Let the number be x . 5x 5x => - - 7 = 23 => - = 23 + 7 2 2

2 => x=30x- =12 5

9. (A) The average yield par tree per year

= (x+2)x60+xx120+(x-2)x180 = 100 (X + 2 + X + X - 2)

~ 60x = 240 ~ x = 4 10. (A) Let the number hex. Then according

to the problem, (x+4)x5 - 20 = 10

8 ~ X = 16

11. (C) 12. (B) 13. (A) Let the present ages of A and B be

2x years and x years. After 30 years, their ages will be A= (2x + 30) years B = (x + 30) years

1 So,2x+30= 12 (x+30)

~ x = 30 :. Present age of B = 30 years ~ Present age of A = 2 x 30

= 60 years

II


14. (B) Let the total distance be x km.

:. x-(5; +~ )= 15 (Given)

7x ~ X - 8 = 15 ~ X = 120

15. (C) Let the first prize be ~ x . 3

· Second ptize = ~ - x . . 4

1 3x 3x Third prize=~ 2x4 = ~ 8

.'. ~ (X+ 3: + 3:) = ~ 2550

~ X= ~ 1200 16. (A) 17. (B) 18. (C) 19. (A) 20. (A) 21. (C)

22. (A) A linear equation has only one variable of degree 1. So it has only one solution.

23. (D ) Let the marks of Sonu be x . Then Ramesh's marks = x + 5. Total marks= 15

~ 2x + 5 = 15 ~ x = 5

: . Marks got by Ramesh = x + 5 = 10 24. (B) Let the number of runs scored by

Sehwag be 'r '.

25. (C)

29. (D)

30. (C)

II

According to the problem, the number of runs scored by Sachin = 2r Also r + 2r = (2 x 100- 2)

r = 66 ~ Sachin's score

= 2r = 2 x 66 = 132 26. (A) 27. (A) 28. (C)

A = P(1 + rt)

~ 27 = 18(1 + 5r)

27-18 1 1 1 1 ~ r=---X-=-X- = -

18 5 2 5 10 1 1 1 -+-=-u v f

1 1 1 3- 2 1 ~ - =---=--=-

v 20 30 60 60

~ v= 60

31. (A) Let the smallest integer be x- 1. Then the three consecutive integers are x- 1, x and x + 1. Their sum = 75

~ (x - 1) + x + (x + 1) = 7 5

~x= 25

:. The largest of the numbers =X+ 1 = 26

32. (D ) Perimeter = (2a + 1) + (3a + 2) + (4a- 1) = 92

92-2 90 ~a=--=-= 10

9 9 33. (A) Let the son's present age be x years .

34. (D)

Then the father's age is (26 + x) years. In 3 years' time, son's age = (x + 3) years and father's age= (26 + x + 3) years = (x + 29) years.

1 ~ x + 3 =- (x + 29)

3 ~x= 10

:. The present age of the son is 10 years.

Let the number of marbles that Arun should give Pankaj be 'x'. Then according to the problem, (96 + x) = 2(63 - x)

~ 3x=30 ~ x= 10

~ Previous Contest Questions

1. (D) Given 3p+2- 4p- 3 + p-1 =4 5 7 35

~ 2p + 28 = 140 ~ p = 56 2. (C) Let the number be 'x'.

Given 5x + 13 = 48 ~ x = 7 3. (C) Let the age of the son be 'x' years.

So the age of Guru is x + 20 years. Given sum of the ages of Guru and his son = 50 years ~ X + X + 20 =50 ~ 2x = 50 - 20 ~ X = 15

4. (A) 5. (C)

6. (C) Let x g be the weight of each cube. Then 4x g = 20 g ~ x = 5 grams

Explanatory Answers


7. (B)

8. (A)

9. (D)

Given that the cost of 5 pens is ~ 11.25, the number of pens purchased by

. l _ ~ (50-0.50) x5 V1ma - ~ 11.25

= ~ 49.50 x5 =22 ~ 11.25

Let the number be 'n'.

2n n Then - + - =13 3 5

=> n= 13x15 = 15 13

x-4 2x+1 5x+1 --- --=--

3 6 2 -24 -4

=>x=-=-30 5

10. (B) Let the numerator of the original fraction be x. Then its denominator is x + 3.

X ~ Original fraction = --3 · x+

x+2 2 => --=-(Given) => x = 4

x+5 3 4

· Ori<rinal fraction =-. . .,. 7

11. (C) Let the number of children be x. Then (x - 3)16 = 144

144 => X = - + 3 = 9 + 3 = 12 16

I 5. Lines and Angles I ~ Multiple Choice Questions 1. (C) A common unit of measurement of

angles is degrees. 2. (C) An angle which exactly measures

90° is called a right angle.

:r 3. CC) 4. CC) 5. CC) 6. (D) An infinite number of rays can be

drawn from a point.

Explanatory Answers

7.

8.

9.

(A) An angle which lies between 90° and 180° is called as an obtuse angle. So, 169° is an obtuse angle.

(C) Extending the arms of an angle does not affect the angle between them.

A _s~--!3-l'

L_ __ .,. __ . .,. 0 B D F

(C) The vertex of an angle is the common point of the rays that form the arms of an angle. Here, it is 0.

10. (A) - -ED and EF are the two arms of

11. (B)

12. (A)

16. (A)

LDEF. Perpendicular segments meet at a point forming right angles.

13. (C) 14. (D ) 15. (C) The distance between the parallel

( I ; ) l lines is the same. ! m : m <· :) m

17. (A) There are two pairs of parallel lines as the opposite sides of a rectangle are parallel.

18. (A) Complementary angles add up to

90°. LPQR=20° => Its comple-mentary angle is 90° - 20° = 70°

19. (C) The sum of supplementary angles is 180°. L ABC = 120° => Its supplementary angles add up to 180°.

20. (A) 21. (C) 22. (B) 23. (A)

24. (A) l II m, n is the transversal. 'a' is an exterior angle and 'p' is an interior angle both on the same side of n. So, 'a' and 'p' are corresponding angles.

25. (B) l II m, n is the transversal. 'c' and 'p' are both interior angles, but on different sides of n. So, 'c' and 'p' are alternate angles.

II


26 (C) c and rare corresponding angles as 'c' interior and 'r' is exterior angle both on the same side of 'n'.

27. (D) p and r are vertically opposite angles formed at the intersection ofn and m.

2S. (C)

2g. (B)

34. (B)

35. (A)

c and s are interior angles on the same side of transversal, which are supplementary. So, if c = 110•, 's' measures 1SO· - no· = 70•.

30. (A) 31. (B) 32. (B) 33. (A) PR is a straight line and SO, X = 1S0°- 32° = 14S0

•

From the figure, LAOC=50° (Vertically opposite angles) Given y is thrice x, we have x + so• + y = 1so• (Angle on a striaght line) ~ x + 50° + 3x = 1S0°

130° ~ x=--=32.5°

4 :. y = 3x = 3(32.5) = g7.5•

36. (B) 115• and q are interior angles on the same side of the transversal. So, 115° + q = 1so• ~ q = 1so• - n5• = 65°

37. (D) Draw PQ II AB and CD.

From the figure, x = 20° + (1S0° - 55°) as PQ II AB II CD.

~X= 20° + 125° =145° 3S. (D) a + b + c = 1SO•

(Sum of angles in a triangle) Also, x + a + y = 1SO• (Angle on a straight line)

:.a+b+c = x+a+y

II

3g. (B) F

DF II CH II BG ::} a+ b = x and a= 45° (Corresponding angles)

b = 1so• -1oo• = so• (Angles on the same side of transversal.) ~ X =a+ b = 45° + S0° = 125°

40. (C) From the figure, 150 - X + 70 - X + X = 1S0° ~ X = 220° -1S0° = 40°

Since AE II BD, y = x as they are alternate angles. In llBCD, LBDC = x (Alternate angles)

70 - x+x+z = 1S0° ~ z = 110°

:. The required sum = x+y+z = 40°+40°+ no• = 1goo

41. (A) Given, the angles (2a- 10)0 and (a - 11) 0 are complementary

angles. : . (2a- 10)• + (a - n )• = go• :. a =37°

42. (D) Given OP is a ray on line QR. Also LPOQ = LPOR.

Q 0 R

LPOQ= LPOR ..... (1)

LPOQ + L POR = 1SO• ..... (2) From (1) and (2), we have

2LPOQ =1SO•

~ L POQ = 1SO• = go• 2

Explanatory Answers



1. (A) In t..ABC, r = 180°-50°-65° = 65•

2. (C)

HC II AB => q = 50° . (Alternate angles) s = q (Vertically opposite angles) Hence, s = 50° Since BCF is a straight line,

p + 20° + q + r = 180° ~ p = 45" . . p = 45°, q = 50°, r = 65° and s = 50° are the required values.

p 0

In the given figure, t + 10° + t + t + 20° = 180° ~ t=50°

q

3. (D) The lines AB and EF intersect at G. LEGB =LAGF

(Vertically opposite angles)

~ L AGF =65° Since AB II CD,

LGHD = LAGH = L AGF ~ LGHD =65° ~ LGHO + LOHD = 65°

E

F Draw a line XY through '0' parallel toAB and CD. Since XY II AB, LXOG = L BGO

~ LXOG = 45° ( Alternate angles)

Explanatory Answers

4.

5.

and XY II CD => LXOH = LOHD ~ LXOH = 25°

But po = LXOG + LXOH p = 45° + 25° = 70° p = 70° & q = 25°

(A) Given a II band c II d and L1 = 75° Since a II b, L1 = L2

a b

Also c II d, ~ L2 +L3 = 180° . . L3 = 105°

(B) 6. (C)

7. (A) Given AD II BC => x = y (Corresponding angles)

8. (A)

Also AB II CD => x + 2x = 180° ~ x = 60• and y = 60•

Clearly p = 360°- 270° = 90° (Angles at a point) Through C, draw a line l parallel toAB and DE.

:. 42° + x = 180° and q + y = 180°

=> X = 180°-42° = 138° :. y = 270°-138° = 132° :. q = 180° - 132° = 48°

I 6. Triangles I ~ Multiple Choice Questions 1. (C) AB is the hypotenuse

~ C has the right angle.

II


2. (B) Since L C = 120° >goo, the triangle formed is obtuse.

3. (D) The sum of any two sides is greater than the third side. Since AB + BC < AC, no triangle is formed.

4. (A) Since sum of the angles in a triangle = 180° 65° + 85° +X= 180° ~ X= 30° :. Third angle = 30°

5. (C) Let the least angle be X 0•

The greatest angle= X 0 + 60°

6. (A)

10. (B)

11. (A)

12. (B)

13. (D )

18. (B)

II

x + x+ 60o Third angle = -----::-2-- = x + 30°

We have, X + X + 30° + X + 60° = 180° ~ 3x + go• = 180° ~ x = 30°

:. The angles are 30°, 60° and goo. Since one of the angles is go•, the triangle formed is a right angled triangle.

7. (C) 8. (D) g, (A)

We have, by Pythagoras' theorem, x2 + x2 = 100

~ x = J50 =·J25 x2 = 5../2 Since PQ = PR, L Q = L R. Given that L Q = 2 L P, we have L P + L Q + L R = 180°

LQ ~ 2 + LQ + LQ = 180°

5 ~ 2 L. Q = 180° ~ L Q =72°

Length of the third side should be 8 em, because if we take third side as 3 em, then the sum of two sides 3 em + 3 em = 6 em is less than third side.

14. (A) 15. (A) 16. (C) 17. (D)

Let the measures of the angles be lx, 2x and 7x. We have, 1x + 2x + 7x = 180° ~ X= 18° The angles are 18°, 36° and 126°. :. The triangle is obtuse angled.

1g. (C) 20. (A) 21. (C) 22. (A) 23. (D) An isosceles triangle can be obtuse

angled but it may not be true always: e.g., a triangle with angle measures g1 •, 45•, 45•, is both isosceles and obtuse but the triangle cannot be formed as the sum of the angles is greater than 180•.

24. (D) According to the properties of a triangle, all the given statement are true.

25. (C)

26. (B)

A

~t c B E

E 00 .... M

~ ....

12m

To find the distance between the tops of chimneys, we have to find AC. By Pythagoras' theorem, AC2 = AB2 + BC2

(Since ~ABC is a right triangle.) ~ AC2 =52 + 122 = 16g

~ AC = .J16g = 13 m

c 15m B

Since ~ABC is right angled, AC2 = AB2 + BC2

~ AC2 = 82 + 152 = 28g

~ AC = .J28g = 17 m . . Actual length of tree

=AB +AC= 25m 27. (C) 28. (D) 2g. (D)

Explanatory Answers


30. (B) The centroid divides the median in the ratio 2 : 1.

31. (C)

32. (D)

35. (B )

36. (D )

37. (C) 40. (A )

AG: GD = 2: 1 So, AG = 2 X GD = 2 x 2 = 4 em

AD = AG+GD = 4+2 = 6cm Since G divides BE in the ratio 2 : 1, BE : GE = 3 : 1.

33. (A) 34. (C) Since XP is the median, P is the midpoint ofYZ. So, YP = PZ. L x + L y forms the exterior angle of 6. ABC , which is equal to the sum of interior opposite angles L l and L2 .

38. (C) 3g. (C) 60° & y are vertically opposite angles which are equal ~ y = 60° In the triangle, X +60°+40° = 180°

(Angle sum property) ~ X = 180° -100° = 80°

~ Previous Contest Questions 1. (B) Given that the angles of the triangle

are (x + 10°), (x + 40°) and (2x- 30°). Sum of the angles of a triangle = 180° ~ x + 10° + x +40° +2x- 30° = 180° ~ 4x = 160° ~ x =40°

2. (C) Given AB = BD ~ LBAD = LBDA = 35°

Lb = L BDA + LBAD ~ Lb = 35° + 35° = 70°

Also given AC = CE ~ LCAE = LCEA = 46° Using exterior angle property, ~ Lc = LCAE + LCEA

= 46° + 46° = g2o In tlABC, La + Lb + L c = 180° ~ L a= 180° -Lb- Lc ~ L a = 180° - 70° - g2o = 18° :. L a = 18°, Lb = 70° and L c = g2o

Explanatory Answers

3. (A) Given, AB = AC

4. g.

(D)

(C)

~ L ABC=LACB Also LOBC = LOCA ~ OB and OC are angular bisectors

LA :. LBOC =goo+-2

40° =goo+-= goo+ 20° = 110° 2

5. (B ) 6. (C) 7. (A ) 8. (B )

Given BE II CD, xo = 45°. (Corresponding angles) In M CD, X0 + y 0 + 108° = 180° (Angle sum property.) ==> 45° + y + 108° = 180° ==> y = 180° -153° = 27°

:. X = 45°, y = 27°

10. (C) In tlABC, LACB = L180° -

(35° + 3g0) = 180° -74° = 106°

AE and BD intersect at C ==> LDCE = LACB = 106° (Vertically opposite angles) ::> X = 180°- (106° + 48°) = 26°

11. (A) XW II yz ==> LYWX = L'ZYW = 28° ::> X = 180° -(40° + 28°)

= 180° - 68° = 112°

I 7. Congruence of Triangles I ~ Multiple Choice Questions 1. (C) According to the question, the

triangles formed are right angled triangles, as shown in the figure.

A

ill B D c

So, MDB :: MDC (R.H.S. property)

II


2. (D) Line segments of equal length are congruent.

3. (B) In the given figure,

4. (B)

5. (A) 9. (A)

10. (A)

11. (B)

12. (C) 16. (C)

17. (C)

18. (A)

19. (A)

II

AD = BC, AC = AC, and LDAC= LBCA

( ·:AD I I BC alternate angles)

By S.A.S. theorem ~C = ~CDA So, AB =DC. (Corresponding parts of congruent triangles.)

Q R y

In ~ PQR and ~ XYZ , LP =LX= 50°, PQ=XY and PR=XZ.

z

:. ~ PQR = ~ XYZ (SA.S. property) 6. (C) 7. (B) 8. (C)

Corresponding parts of congruent triangles are equal. So, PQ = AB = x em, QR = BC = y em and RP= CA= z em. Corresponding sides in ~D and ~CBD .

Since ~D and ~CBD are congruent, LADB = LCDB (Corr-esponding parts of congruent triangles). ~ BD bisects L ADC

13. (B) 14. (C) 15. (B) BC is the hypotenuse of the given triangles. Whose measure must be known. A.A.A. criterion does not exist. Ifthe three angles of a triangle are congruent to corresponding angles of the other, it is an enlarged copy of the triangle.

A R DD B c Q p

CB corresponds to PQ

20. (B) AC corresponds to RP [Note that CA corresponds to PR.]

21. (A ) ~C=~

=> LA = L.x = 180°- (60° + 40°) = 80° Similarly, Ly = LB = 60°.

22. (A) The sides corresponding to AC and DE respectively are DF and AB.

23. (A)

24. (C)

25. (A)

26. (A)

30. (B)

31. (C)

32. (C)

33. (A)

34. (C)

:. AC = 10 em, and DE = 3 em. L F corresponds to L A in ~C . Hence, LF = 180° -110° = 70° Corresponding parts of congruent triangles are equal. Corresponding angles of congruent figures are equal (proportional). :. LOBA = LOCA

27. (C) 28. (A) 29. (A)

~C = ~PQR by S.S.S. condition as three sides of ~C are correspondingly equal to three sides of ~QR . ~C =~B. So, option (C) is the required answer. In MBD and ~CDB, BD is the common side. So, ~D = ~CDB by S.S.S. condition. ~z =: ~MN by R.H.S. condition as YZ = MN, LY = LM andXZ=LN. All statements are correct except that in (C). Two triangles can be congruent according to any of the 4 properties of congrudence.

[i' Previous Contest Questions

1. (B) In ~PQS & ~RS , we observe that, PQ = PR, PS = PS, LQPS = LRPS So, ~ PQS =: ~ PRS (By SAS property.)

2. (D) Right angle, hypotenuse and side of the triangles are congruent.

3. (C) 4. (A ) 5. (C)

6. (A) Order of le tters of congruent triangles must be maintained.

7. (B) By definition of congruence of triangles.

Explanatory Answers


8. (A) The measurements in option (A) are the measurements of two triangles under S .A.S condition.

9. (B) 10. (A)

I 8. Comparing Quantities I ~ Multiple Choice Questions

1. (C) Office hours = 10:30 a.m. to 5:30 p.m. = 7 hrs = 420 minutes Lunch time = 30 minutes :. Required ratio= 420 : 30 = 14: 1

2 (D) 3. (B) 4. (B) 5. (C) Let the original cost of each book

be ~ x. According to the problem, 20 x x = 22x- 5.5 + 0.70

~ X = ~ 2.40 : . The boy had 20 x ~ 2.40

= ~ 48 to buy books 6. (A) Let the number be x. Then,

x -4 = 80% ofx

X ~ 5 =4 ~ x=20

7. (C) Let the number be x .

8. (B)

9. (C)

1 2 2 % ofx = 0.2

200 ~ X= 0.2 X - 5- = 8

~ 120% of8 = 120 x 8 = 9.6 100

10% increase per year ~ Increase for 2 years

110 110 = 400 X 100 X 100 = 484

Let total votes polled be x. Winning candidate got 70% votes.

70x 30x So, 100 - 100 = 15000

100 ~ X = 15QQQ X 40 = 37500

: . Votes polled for the winning candidate

Explanatory Answers

70x 70 =- = - x37500 = 26250 100 100

10. (B) Let the number of games won in a row bex.

11. (A) 15. (A)

16. (A)

(30% of 60) + x 50 So 60 +x = 100 ~ 36 + 2x = 60 + X ~ X = 24 12. (D) 13. (C) 14. (C) Remaining amount after successive deduction of 5% and then 10% of original sum

(100- 5) (100 -10) -~-x-~~-

100 100 x original sum = 171

100 100 · Orimnal sum = 171 x-x-.. o· 95 90

= ~ 200 Gain%

No. of articles on C.P-No. of articles on S.P x 100% No. of articles on S.P

25 - 20 =---x100 % = 25 %

20 17. (B) The val ue of the machine after

depreciation of 5% per year for 2 years = (where x1 = loss and

18. (C)

19. (A)

24. (A)

x2 =gain) 95 95

~ 100000 X 100 X 100 = ~ 90250

S.P1 S.P2 We have 100 + x - 100 + x

1 2

600 SP2 ~ 100 -20 100+ 25

125 S.P2 = ~ 600 X 80 = ~ 937.50

20. (C) 21. (B) 22. (B) 23. (B)

C.PSncha = ~ 480 C.PNeba = S.PSncha

=~x~480 100

= ~ 510 C.P Devi S .P Neha

(100 + 10) 510 = 1QQ X = ~ 561

: . Devi had to pay ~ 561.

II


25. (B) S.I = ( 810; R = 9%; T = 6 years, P = ? We have

26. (C)

28. (C)

29. (D )

30. (A )

33. (D)

100x l 100x810 ~P = TR = 6x9

= ~ 1500 27. (A) Total S.I = ~ 275 Let x be the sum borrowed at 7% rate. So,

(2500-x)x5x2 + xx7x2 = 275 100 100

~ x = ~ 625

Re . . 1 (1 1) 1 mammg part = -3

+6 ='2

Average rate % per annum (R)

=(~ x 3j+(i x 6 }(%x8 )=6%

S.I = ~ 600 T = 2 years, P = ?

100 xl 100 x600 ~ P = TR= 2 x 6 = ~ 5000

31. (C) 32. (B) We have

PTR A= p + I = p + 100

PTR A-P = 100

A 1 - P1 _ P1T1R 1 So,-----A2 - P2 P2 T2R2

95-85 85x3xR => = --:-~--~ A2 -102 102x5xR ~ A2 - 102 = 20 ~ A2 = ~ 122

[i' Previous Contest Questions

1. (D ) Let 2A = 3B = 4C = k

Ill

k k k ~ A:B:C =2:3-:4

=(~x12 }(~x12 }(~x12) [L.C.M of2, 3, 4 is 12.] Hence, A : B : C = 6 : 4 : 3

2. (A) 3. (C) 4. (B) 5. (C) Let the C.P of the heater be ~ x.

6. (B)

Then, gain = ( ( ~ J

~ S.P= ~(x+~)=~ 7: 7

~ - x=322 6

~ x=(322x%J=276

: . C.P = ~ 276 and S.P = ( 322 ~ Gain = (S.P) - (C.P)

= ~ (322 - 276) = ~ 46

~ Gain% = ( ~~ x100 J%

( 46 J 50 2 = - x100 % =- % =16 - % 276 3 3

2 Hence gain% = 16-% , 3 Let the sum be ~ x. Then amount

= ~ 2x : . S.I = ~ (2x- x) = ~ x

4 25 Time = 8 12 years = 3 years

Thus, P = ~ x, S.I = ~ x 25 and T = - years 3

. R _100 xS.I . . ate - PxT

= (10~xx x :5 )% p.a = 12% p.a.

Again, Sum = ~ x, amount = ~ 3x and

rate = 12% p.a. Then, S.I = ~ (3x -x) = ~ 2x : . p = ~ X, S.I = ~ 2x and R = 12% p.a.

100xS.I Time= P xR

50 (

100 x2x J = years = - years xx12 3

= 16 years 8 months

Explanatory Answers


7. (D) 10. (D)

8. (B ) 9. (A ) Let one part be ~ x. Then the other part is ~ (8000- x).

T1 = 5 years; R, = 12 % :. S.I (on first part)

xx5x 12 3x 100 = ~5

T2 = 2 years; R2 = 18 % S.I (on second part)

(8000 - xx2x18) 100

=(8000 - x)~ 25

3x 9 - = - (8000 - x) 5 25

~ 5x = 24000 - 3x 24000

~ X = - 8- = ~ 3000

The other part = ~ (8000- 3000) = ~ 5000

11. (C) S.I = ~ 31.50 1 5 T = 1- years =- years 4 4 1 21 R=5 - % = - % 4 4

5 21 => 31.50 = P x - x -4 400

~ p = ~ 31.50 x 4x400 5x 21

~ P = ~ 480

I 9. Rational Numbers I ~ Multiple Choice Questions

3 1. (A) Let us consider a fraction 5 and a

- 4 rational number = - . 7

3 By definition, 5 is a rational number.

- 4 So, p is true. By definition, 7 is not a fraction, since, - 4 is not a natural number. So, q is false .

Explanatory Answers

2.

3.

4 .

5.

(D)

(B)

(D )

(C)

J2 3 is not a rational number.

Since, every integer having a denominator 1 can be expressed in p D . q orm, p IS true.

Since, a rational number with denominator other than 1 is not an

3 integer (e.g., 5 ), q is false.

Since, denominator is 0, it is not a rational number.

6. (D ) 7. (B) 8. (A ) 9. (C) 10. (B ) L.C.M. of 5, 4 and 20 is 20.

4 4 16 3 3x 5 15 - X- = - · - =-- = -- 5 4 -20 ' 4 4 x 5 20

4 -11 3 -2 < - < - < - or - 5 20 4

3 -11 4 - > - > - > -2 is the required 4 20 - 5 descending order. Rewrite equivalent fractions of the given fractions and then compare then. Arrange them in descending order.

11. (D) The ascending order of 412 5 1254 - - - and - is - < - < - < -. 7 '3' 5 9 3 5 9 7

2 The two middle numbers are 5

5 and g·

2 5 - + - 3

:. Average = ~ = ±.._ 2 90

12. (A) The given numbers can be arranged in ascending order as 1 3 7 9 - < - < - < - . 5 5 5 5

9 The greatest number = 5 ;

1 The least number = 5

9 X 1 We have, 5x 100 = 5

100 1 x = - = 11-% 9 9

Ill


13. (C)

14. (D)

15. (B)

16. (B) 19. (D)

20. (A)

21. (B) 25. (D)

26. (B)

Ill

The ascending order of given 1 5 11

numbers is 9' 9' 9· :. Required difference

11 1 10 =---=-

9 9 9 - 24 15 is the equivalent rational

- 8 number of 5 ·

1 Since, 0 is not rational, the quotient of two integers is not always rationaL

17. (C) 18. (C) All the given statements are correct.

4 Consider 5.

4 4 (A) Since 5 + 0 = 5, is the additive

identity of rational numbers. 4 4

(B) Since -x1 =- 1 is the 5 5' multiplicative identity of rational numbers.

(C) Since 0 + 0 = 0, 0 is the additive inverse ofO.

Let x be the required number.

So, x+( ~7 )= -3 7 -15+7 -8

~ x=-3+- =--=-5 5 5

22. (D ) 23. (C) 24. (B)

Let 'x' be the other number.Let 'x' -4 - 9

be the other number xx3 = 16

-9 16 -9 -3 27

~ X = -4 = 16 X 4 = 64 3

Let the required rational number -8 be 'x'. Then xx 39

= 26

::> X = ~ = 26 X 39 = -507 -8 -8 4 39

27. (A) Total length of the rope = 30 m.

3~m Length of each piece = 4 . N um her of pieces

= ~ = 30 = 30 X_±_ = 8 3~ 15 15

4 4

28. (B) Given P = ( -2%) and q = ( -1~).

p+q = ( - 2!)+(- 1!): -11 - i

5 3) 5 3

= -(~:)= -3 1~

29. (A) 1 (-8) (-5) 14+ 3- 9

-65-96 -(-31) - -- - -36 36

30. (C) -6~x~ = -23° x~ = ( -2~)

31. (A )

32. (A) 34. (D)

35. (C)

'!._-(- 11 )+x = 3~ 8 4 24

=>X=~: - ~9 = ( ~ )= ( -31 j 33. (B) Total time taken by Rohit, Peter and Santosh to walk around a

(1 2 5 ) circular park = - +-+- h 3 5 12

69 60 . = 60 x mmutes = 69 minutes

~=0.75· ! - 0.5· 69 = 0.78 4 , 2 , 88

9 13 1 - = 0.82· - = 1.18· - = 0.25 11 , 11 , 4

Of the given rational numbers as 0.78 lies between 0.75 and 0.82,

69 3 9 88 lies between 4 and 11 .

Explanatory Answers


~ Previous Contest Questions 1. (B) Let the number to be added be x.

2. (B)

3. (C)

5. (C)

6. (B)

Then, -; + x = ~ ~ x = ~ - ( -; )

=~+~ [since-(;)=~-] (32+63) 95 --- = -

72 72 95

Hence, the required number is 72 .

Let the number to be subtracted be 'x'. Then, -2 5 -2 5 --x=- ~ -=-+x 3 6 3 6

- 2 5 - 2 - 5 ~ x= - - - = - + -

3 6 3 6 (- 4)+(- 5) - 9 - 3

6 6 2 - 3

Hence, the required number is 2 . 4. (D)

. -33 -11 Reqmred number =-+-8 2

- 33 2 - 33 - 2 = - X - = - X -

8 -11 8 11 (-33) X (-2) 33 X 2 3

8xll 8xll 4

3 Hence, the required number is 4 .

7. (A)

I 10. Practical Geometry I ~ Multiple Choice Questions 1. (A) S.S.S. criterion can be used

indirectly to construct a triangle given the lengths of its three sides.

2. (C) ~PQR is isosceles since PQ = QR. :. LQPR = LQRP = 60° LRQT is the exterior angle of ~PQR which is equal to the sum of interior opposite angles LPandLR. Hence, LRQT = 60• + 60• = 120•.

Explanatory Answers

3. (B) 4. (B ) 5. (C) 6. (D)

7. (C) The difference of any two sides of a triangle must be less than the third side. This property of triangles is not satisfied by the given measurements as 10 - 7 = 3 > 2 and 10 - 2 = 8 > 7, though 7-2 = 5 < 10 is true.

8. (A) Clearly, l ll m is true. 9. (B)

10. (C)

11. (D )

12. (B )

13. (D)

16. (C)

A 90° angle is formed at the intersection of l and n. So l .l n. 'n' cuts l and m at distinct points and also l and m are parallel. So, n is called the transversal. A line parallel to a given line can be drawn using a ruler and a compass. Through a given point, an infinite number oflines can be drawn. But only one of them will be parallel to the given line.

14. (C) 15. (D)

The triangle cannot be constructed as it does not satisfy the angle sum property.

17. (A) The difference of two sides of a triangle is less than its third side.

18. (A) Bisecting a 60° angle results in a 30° angle.

19. (C) The given steps of construction are to construct an angle of 120°.

D

20. (C) Following steps 1 and 2, an angle of 45° is constructed. So step 3 is not required.

21. (B ) 22. (A ) 23. (D) 24. (A ) 25. (C)

26. (D )

27. (A) p

Ill


As can be seen from the given figure, one and only one perpendicular line can be drawn to a given line from a point not on it.

z

28. (C) ~ x~v Given XY > YZ > ZX => L Z > LX > LY ~ The smallest angle is LY .

2g. (A) 30. (A) 31. (B) 32. (C) 33. (B) From the given measurements, BC

is the hypotenuse. The angle opposite to BC is LA which is a right angle.

34. (A )

35. (C) In ~PQR since all the angles are acute, it is acute angled. Also since all the angles are equal, it is equilateral.

36. (B) Since the measures of all the three sides are given, the triangle can be constructed using the S.S.S. criterion.

IJW Previous Contest Questions 1. (A) Corresponding angles of parallel

lines are equal. 2. (C) 3. (C) 4. (D ) 5. (A) 6. (B )

7. (D) Since 6LMN is equilateral the measurement of one side is used for the other two sides of the triangle. Hence ~ LMN can be constructed by S.S.S. criterion.

8. (D) By Pythagoras' theorem, one of the

g_ (A )

Ill

perpendicular sides is 3 em. and by angle sum property, L D+ LE = 180° -LF

= 180°-goo = goo

T

.il, Scm

Clearly, from the figure two angles and the included side are given. So, A.S .A. criterion can be used to construct ~RST.

I 11. Perimeter and Area I IJW Multiple Choice Questions

1. (C)

2. (A )

3. (D)

4.. (B )

5. (A )

6. (D ) 10. (D)

44 27tr = 44 => r = ~ = 7 m

2x-7

Area of a circle = 1tr2

22 2 = - X 7 X 7 = 154 m 7

1t d2

= 2464 4 ~ d = .J3136 =56 m

@ r = 35 m; R = 35 + 7 = 42 m Area of circular path = 7t(R + r)(R- r)

= 22 (42 + 35)(42 - 35) = 16g4 m2

7 27tr-r = 37

37 37 r =-- = =7m 27t-1 2 x 22 _ 1

7 Circumference

22 = 27tr = 2 x- x 7 = 44 m 7

Circumference = 30 em

C2 30 X 30 2 Area = - = --- = 71.6 em 47t 4 X 22

7 A 71.6 Number of plants = - = --4 4

= 17.g :::18 7. (B ) 8. (A) g_ (C) The perimeter of the wall hanging is given by the sum of circumferences of the 4 semicircles - 4x diameter. Clearly, the diameter of each semicircle is 14 em. The requi red perimeter = 2 x circumference of circle of radius 7 em.

22 = 2 X 2 X - X 7 = 88 em 7

Explanatory Answers


11. (D) Lengthofwire =2nx42 = 84n em Let x be the side of the square. We have, 4x = 84n ~ x = 21n Area of the circle : Area of the square = n(42t: (21n)2

22 = 4 : 1t = 4 : 7 = 14 : 11

12. (C) :. Area= 400 x 250 m2

= 100000 m2

13. (C)

14. (A)

15. (D)

16. (A) 19. (A)

20. (B) 24. (B)

Cost of the land per square metre = ~ 1000 :. Cost of total land 100000 x ~ 1000 = ~ 10 crores Here we equate the areas, i.e., AB x DL = BC x DM ~ 18 X DL = 12 X 10

12x10 20 2 ~DL=~=s=63 cm

We have, area = 156 cm2

~ b X h = 156 AB X DL= 156

156 156 DL = AB = J:3 = 12 em In right 11 ADL, AD2 = DL2 + LA2

AL2 = AD 2 - DU = 132 - 122 = 25 . . AL = ..J25 = 5 em Distance between two opposite

corners = -/ [2 + b2

40 em

= -/402 + 302 =50 em 17. (B) 18. (C) 2.4 dam = 2.4 x 10 = 24 m (Since 10m= 1 dam.) We have, area = 576 m2

~ 24 x altitude = 576 576

:. altitude = 24 = 24 m 21. (D) 22. (B) 23. (A) Area of rectangle = l x b = 30 x 20

= 600 cm2

1 Area of 11DGF = 2xbxh

= 75 cm2

Explanatory Answers

Similarly area of 11 AGE = 75 cm2

Area of unshaded region = 600 - (75 + 75) cm2

= 450 cm2

25. (A) Radius of outer circle = 14 em

26. (B)

27. (B)

3.5cm ~ Circumference = 88 em Radius of inner circle= 14- 3.5 = 10.5 em ~ Circumference = 66 em . . Difference of circumferences

= 88 - 66 em = 22 em

D C

.0. In 11ABC, base = AC = 34 m, height = BM = 12 m

1 Area of 11ABC = 2 x 34 x 12

=204m2

Similarly the area of 11 ADC = 204 m2

Area of parallelogram = (204 + 204) m2 =408m2

Consider PR = 25 em as the base of two triangles, 11 PQR and 11 PSR.

1 Area of 11 PQR = 2 x 25 x 15

375 - 2

Similarly area of 1

11 PSR = 2 X 25 X 15

375 = 2 cm2

: . Area of shaded region =Area of rectangle - (area of 11 PQR + area of 11PSR)

= (25x15)-(3~5 + 3~5) = 750 - 375 cm2 = 375 cm2

II


28. (B) 29. (A) 30. (A) 31. (A) 32. (C) The length of the hour hand= 4.5 em

(= r). The distance it covers in 12 hours = The circumference of the circle with radius 4.5 em

22 = 2x7 x4.5 em = 28.28 em

33. (A) Area of a parallelogram

= bxhsq. units

= 20x4 = 100 cm 2

34 (C) Clearly, the figure is a rectangle. :. 2(x + 3) = x + 8 ~ 2x+6=x+8 ~ x=2cm :. Area= (x + 8) (6x + 9)

= (2 + 8) (6 X 2 + 9) = 10 x 21 = 210 cm2

l@f Previous Contest Questions 1. (C) Given length of the diagonal of a

square = 12../2 em If'a' is the side of the square, then the length of diagonal is J2a .

:. J2a = 12../2 ~a = 12 :. Perimeter= 4a = 4 x 12 = 48 em

2. (A) Given perimeter of a semicircle = 144 em

3. (B)

Ill

~ r(7t+2)=144

36 ~ rx - =144 ~ r=28cm

7 Area of semicircle

1 2 1 22 = - nr = - X - X 28 X 28 2 2 7

= 1232 cm2

70 Radius of a wheel = 2 = 35 em

In one revolution, the wheel covers a distance equal to its circum-ference

22 :. 2nr = 2x - x35 = 220 em 7

In 24 complete revolutions, distance covered = 24x220 = 5280 em

4. (C) 5. (B) 6. (B) 7. (A) 8. (B) 9. (A) Perimeter = 21m= 2100 em

Length = 5 m 60 em = 560 em

Width =~-l 2

= (21200 - 560 )em

= 490 em = 4 m 90 em 10. (C) Perimeter= 56 em

Given length is three times its width. Let the width be x em. Then length = 3x em :. Perimeter= 2(x + 3x) = 8x em

::::> 8x = 56 ::::> X = 7 :. Width = 7 em

11. (A) Length of the floor = 12 m Breadth = 10 m :. Area = 12 x 10 = 120 m2

:. Area of the carpet= 120m2

I 12. Algebraic Expressions I l@f Multiple Choice Questions

1. (A ) x+y=5

2. (B)

3. (A)

4. (C)

y + z =7 + Z +X= 12

2(x + y + z) = 24 ~ X + y + Z = 12 3 formulae with the three unknowns can be formed from the given expression. (3a + 2b) - (-2a - 5b) = 3a + 2b + 2a + 5b = 5a + 7b

w .......

w{~t t=tD+ w ~z----..

Area of the path along length = l x w Area of the path along breadth = l x w The common area of the paths= w2

: . Total area of the path = lw + lw- w2 = (l + b - w) w

Explanatory Answers


5. (B) 6. (B) 7. (D) 8. (D ) 9. (A )

10. (A )

11. (B )

16. (C)

C = x- a ~ C(x - b) = x - a x-b

bC - a ~ X= C - 1

12. (C) 13. (B ) 14. (C) 15. (A )

as - 2a2 + 4a - 5 (- ) -as +2a2-8a+5

+ +

2a3 - 4a2+12a-10 17. (B ) x4 + Sx2y2 + y4

(- ) x4-4x2y2+y4 - +

(C) (a2 + b3 _ c3

) + (2a2 + 3b3 _ 4c3

}

2 3 4 3 4 5) 18.

+ (a2 + bs +ca)

= 13 a2 + 25 ba _2._cs 6 12 20

19. (D) Substitute the given values in the expressions and evaluate. a3 - b3 = 33 - 23 = 9 - 8 = 1 z3 - 3 (z - 10) = 103 - 3 (10 - 10) = 103- 0 = 1000 x2 + 2x + 1 = (-1)2 + 2(1) + 1 =1-2+1=0 5P- 2 = 5 (-2) -2 = -10-2 = -12

20. (A ) x (y- z) - y (z- x)- z (x - y)

21. (B)

22. (A )

25. (B )

= xy - xz - yz + xy- zx + yz = 2xy - 2zx = 2x (y - z)

Substitute n = 4 in n2 + 1 and simplify. n2 + 1 = (4)2 + 1 = 16 + 1 = 17

23. (B ) 24. (C)

2n denotes an even number as it is exactly divisible by 2.

28. (A) 29. (D) 30. (A) 3 2 1 1 31. (A) 4x-5 ax-y+ 3 ax-8x

5 1 =sx- 15ax - y

When a = 3, x = (-2) andy = (-6), the value of the expression is

%<-2)-1~ (3)(-2)-(-6)

=103=5~ 20 20

32. (B) Perimeter = 2 (l +b) = 2 (3p + 2p) = 2 (5p) = lOp em

:. Perimeter when p = 12 em is 10(12) = 120 em.

33. (B ) The required difference= (7x - 2y-3z) + (3x + 5y- 8z) - (x - 3z) = (lOx + 3y- llz) - (x - 3z) = 9x + 3y- 8z

34. (B) Sum of angles in a quadrilateral is 360°. ~ 6p + 30° = 360°

~ p = 60° +5° = 55°

:. (p + 25)0 = 90°, 2p0 = 130° (p + 20)0 = 65° + 20° = 85°

:. The smallest angle is 65°. 35. (B ) Perimeter of the triangle

= [2a + (2a + 2) + (4a - 2)] em ~ 8a em = 24 em or a = 3 em : . The length of the shortest side is

2a = 6 em.


1. (B ) 3y2 +5yz - 2y2

- 2yz - z2

- yz+2z2

26. (B ) 2n + 1 denotes an odd number since 2. (A) 3. (A) 4. (B ) 5. (D) 6. (B) it leaves a remainder 1 when 7. divided by 2.

27. (C) Like terms have the same literal coefficients.

Explanatory Answers

(C) Total number of sweets = 6x Number of sweets each child gets

6x 3x =-=-

10 5

Ill


8.

9.

(D) Length = 2(x + 6) em

1 Width = - x length

2 1

= 2 x 2(x + 6) em = (x + 6) em

: . Perimeter = 2(l +b) = 2[2(x + 6) + (x + 6)) em = 6x + 36 em

(A ) ~(66x+44)+~(33x-33) 11 11 = 33x + 7

9 5 10. (C) -(30+5t)+-(18t-12) 10 6

= (17 + 3: t)

I 13. Exponents and Powers I ~ Multiple Choice Questions

1. (B ) (3° -4°)x52 = {1-1)x 52 =0

2. (C) ~~ = ~: =(~J

3. (D ) p0 = (1000)0 = 1

4. (C) Since a0 = 1, 12 x 3° = 12 and 8 x 5°= 8.

5. (A ) 432 = 2 X 2 X 2 X 2 X 3 X 3 X 3 = 24 X 33

6. (B) Expand 27 x 53 and find the product of the prime factors. 27 x 53 = 2X2X2X2X2X2X2X5X5 X 5 = 128 X 125 = 16000

7. (B) (- 10)3 = (- 1000) is the largest of the given number with a minus sign. So it has the least value.

8. (D) 9. (C) 10. (A) 11. (C) 12. (A)

13. (D) 108 = 22 x33 and

192 = 26 X 31

: . The required sum is 2 + 3 + 6 + 1 = 12

Ill

14. (B ) am+bm=(~J' ~ (-2)3 +ms

=(:J 15. (C) [(~J-(~J]x26

16.

1 6 =3x- x2 = 3 26

2x34 x25 26 x34 (A ) 9x42 = 32 x24

=26-4 X 34-2 =2232 =4 X 9 = 36

17. (B ) 18. (C) 19. (A ) 20. (C)

.=;! ~ 1 1 21. (D ) (4096) 4 = (212 ) 4 = 2-9 = 9 =-

23. (C) 94 x + 32x = 2187

25. (D)

7 ~ 6x=7 ~x=-

6

p•qs x p2r• xq-2rs p-3 q3 r3

3 26. (A) (256)0·75 = (256)4 = 64

= a-2mn-4m = a-2m(n+2)

2 512

28. (B) 29. (B) 30. (D) 31. (A)

Explanatory Answers


~ Previous Contest Questions 1. (A) In standard form, a number is

expressed as a decimal number between 1.0 and 10.0.

2. (D ) 3. (B) 4. (C) 5. (A ) 6. (C)

7. (B) (95)x = (94)x + 92

=> 5x = 4x- 2 => x = -2

8. (A)

I 14. Symmetry I ~ Multiple Choice Questions 1. (A) Recall definition ofline symmetry.

2. <D) ~·G·• ~·K·• ~·B·• 3. (C) P has no line of symmetry. 4. (B) 0 has many lines of symmetry. 5. (C)

6. (A )

A scalene triangle has no line of symmetry.

B /JJ/JJ/J/77111

B ll

A

7. (C) <~l2 VI

8. (D ) 9. (A ) 10. (C)

11. (B)

12. (A)

Explanatory Answers

13. (B) . '

<P The figure formed is a rhombus.

14. (C) A diameter divides the circle into 2 equal parts. So, it can be considered as a line of symmetry.

15. (A )

16. (C)

21. (A)

22. (A )

23. (A)

X

& y M z Since XY = XZ and XM _l YZ, LYXM = LZXM . So, the triangle is symmetrical about XM as it bisects the apex angle.

17. (B ) 18. (A ) 19. (B ) 20. (C)

The given figure has 4 lines of symmetry, as shown by the dotted lines.

The given figure has only 2 lines of symmetry, as shown by the dotted lines.

Ill


24. (C) CD .. ·@+0

t The figure inside the circle is a rectangle which has only 2 lines of symmetry. Therefore the given figure has 2 lines of symmetry, as shown by the dotted lines.

25. (A) ~

+ The given figure has 4 lines of symmetry as shown by the dotted lines.

26. (B)

'f The given figure has only one line of symmetry.

27. (C) The square inscribed in a circle has a rotational symmetry of order 4.

28. (C) The angle between identical parts of the given figure is 90°. Hence, the order of rotational symmetry is 4.

29. (D) 30. (B ) 31. (C) 32. (D) 33. (B) Option (B) is the correct figure.

·(>~ ~

Ill

34. (A) The given figure fits onto itself three times in one complete turn. So its order of rotational symmetry is 3.

35. (D ) 36. (A) 37. (A)

[@f Previous Contest Questions

1. (B ) A reqular polygon of 'n' sides has 'n' lines of symmetry.

2. (C) An equilateral triangle has 3 lines of symmetry.

3. (A) U appears the same in mirror too. 4. (D) p = 4 and q = 2

:. p > q 5. (B) 6. (A) 7. (C) 8. (D )

I 15. Visualizing Solid Shapes I [@f Multiple Choice Questions

1. (B) 8 edges -4 sides of square and 4

2. (B)

3. (D)

4. (B)

5. (C)

6. (A )

7. (C)

10. (B)

11. (B)

sides of triangular faces. The number of faces of a cube is 6. Observe the given figures. In the figure in option (D ), the shape is divided equally.

A X

The five triangles are, APQ, QXR, PZB, ABC and XYZ. Clearly, the two halves of the rectangles when joined edge to edge form a rectangle. This figure has 5 lines unlike all other figures which have four sides. 8. (C) 9. (B)

10 one-rupee coins piled up results in a cylinder. A cube has 8 vertices.

Explanatory Answers


12. (B ) Column - I Column - II

[J <> ~ oW @ . ~ 4 ~

13. (C) A cube of edge 4 em has square faces of side 4 em.

14. (D) The sum of numbers on opposite faces is 7. So, the total on the face opposite to 4 + 3 is 3 + 4 = 7.

15. (B) The number on opposite faces of a die add up to 7. So, the number on the face opposite to 5 is 2.

16. (A) The height and width of the cuboid formed by placing 3 dice remain the same. Only length will be tripled.

17. (C) The cross section of a cube is a square.

18. (C) The 2-D image of a sphere (ball) is a circle.

19. (A) A book is cuboidal whose shadow is a rectangle.

20. (A) The pipe is cylindrical whose shadow is circular (front view).

21. (C) A pyramid has triangular faces. 22. (A) A die is a cube which has all its

faces square shaped. 23. (B) A honey comb has a combination

of hexagonal shapes when seen from the front.

24. (D) The front view of the given object (as seen from the arrow) is a circle.

25. (B) The side view of the given object is a parallelogram as in option (B).

Explanatory Answers

26. (A) 27. (B) 28. (C) 29. (A) 30. (C) 31. (A) From the side indicated by the

arrow, the pipe looks like a ring. 32. (C) Viewing the visible faces of the

arrangement that are painted, we get 18. In other words, 6 common faces of the 4 cubes are hidden. So the number of sides of the cube painted = 4 X 6 - 6 = 24 - 6 = 18.

33. (B) The given views are that of a cuboid .

~ Previous Contest Questions 1. (C) A quadrilateral has 4 sides as in

the given figure. 2. (B) Option (B) gives the correct

difference of a cube & a cuboid. 3. (C)

4. (D)

5. (D )

A cuboid has 8 vertices, which is the maximum number of vertices of the given figures.

The two cubes of given dimensions when placed side by side result in a cuboid as shown in the figure. It has a length 6 em, width 3 em and height 3 em. A net when folded should result in a solid, in this case, a cube. The net in option (D) does not form a cube.

6. (A) The brick of given dimensions is cut along the 10 em side which results in a cuboid.

7. (C) The right angled triangle PQR when rotated about its height results in a cone.

8. (A ) 9. (B) 10. (A )

11. (B) The given solid has a triangular base and 3 triangular faces.

12. (C) The solid in option (C) has 3 rectangular and 2 triangular faces.

Ill


1.

2.

II

Suppose that the cost of one book, in rupees, is C.

Then Shruti has ~C and Fida has 4

.!c 2 Combining their money, together Shruti and Fida have

3 1 3 2 5 - C+- C= - C +- C= - C 4 2 4 4 4

If the book was~ 3 cheaper, then the cost to buy one book would be C - 3. If the cost of one book was C- 3, then the cost to buy two at this price would be 2(C- 3) or 2C- 6. Combined, Shruti and Fida would have enough money to buy exactly two books at this reduced price.

5 Thus 2C- 6 = - C ' 4

5 Solving, 2C- 6 = 4C

3 -C=6~C= 8 4

Therefore, the original price of the book is~ 8. Without changing the overall class mean, we may consider that the class has 100 students. That is, 20 students got 0 questions correct, 5 students got 1 question correct, 40 students got 2 questions correct, and 35 students got 3 questions correct. The combined number of marks achieved by all 100 students in the class is then, (20 X 0) + (5 X 1) + (40 X 2) + (35 X

3) = 0 + 5 + 80 + 105 = 190. Since the 100 students earned a total of 190 marks, then the overall class

190 average was 100 = 1.9.

3.

4.


Draw line segment QR parallel to DC, as in the following diagram. This segment divides square ABCD into two halves. Since triangles ABQ and RQB are congruent, each is half of rectangle ABRQ and therefore one quarter of square ABCD. Draw line segment PS parallel to DA, and draw line segment PR. Triangles PDQ, PSQ, PSR and PCR are congruent. Therefore each is one quarter of rectangle DCRQ and therefore one eight of square ABCD.

A B

D P C Quadrilateral QBCP therefore

1 1 1 1 5 represents - + - + - + - - - of 4 8 8 8 - 8 square ABCD. Its area is therefore 5 8 of the area of the square.

5 Therefore, 8 of the area of the square

1 is equal to 15. Therefore, 8 of the

area of the square is equal to 3. Therefore the square has an area of 24. Since the two large triangles are equilateral, then each of their three angles equals 60•. Therefore, each of 6 small triangles in the star has an angle of 60• between the two equal sides. But each of these 6 small triangles is isosceles so each of the remaining

1 two angles must equal 2 (180• - 60•)

or 60•. Therefore, each of the small triangles is equilateral.

Explanatory Answers


5.

This shows us that the inner hexagon has all sides equal, and also that each angle is 180•- 60• or 120•, so the hexagon is regular. Next, we draw the three diagonals of the hexagon that pass through its centre (this is possible because of the symmetry of the hexagon).

$ Also, because of symmetry, each of the angles of the hexagon is split in half, to get 120• + 2 = 60•. Therefore, each of the 6 new small triangles has two 60• angles, and so must have its third angle equal to 60• as well. Thus, each of the 6 new small triangles is equilateral. So all 12 small triangles are equilateral. Since each has one side length marked by a single slash, then these 12 small triangles are all identical. Since the total area of the star is 36, then the area of each small triangle is 36 + 12 = 3. Since the shaded area is made up of 9 of these small triangles, its area is 9 x 3 = 27.

p u

LPOU = LROS (Vertically opp. angles) => LPOQ + L ROS + LUOT = 180° Sum of angles in a triangle = 180• ~ Sum of angles in D. PQO, D. OUT, D. ORS = 180• + 180• + 180• = 540• :. LP +LQ +LR +L S+ LU + LT = 540° - 180• = 360•

Explanatory Answers

6.

7.

8.


Sincex +xy= 391, thenx(1 + y) = 391.

We note that 391 = 17X23. Since 17 and 23 are both prime, then if391 is written as the product of two positive integers, it must be 1 x 391 or 17 x 23 or 23 x 17 or 391 x 1. Matching x and 1 + y to these possible factors, we obtain (x, y) = (1, 390) or (17, 22) or (23, 16) or (391, 0). Since y is a positive integer, the fourth pair is not possible. Since x > y, the first two pairs are not possible. Therefore, (x, y) = (23, 16) and so X+ y = 39. Suppose that each of the smaller rectangles has a longer side oflength x em and a shorter side of length y em.

X

y

y X

y 1----r--1

X

1...-----'---' X y

Since the perimeter of each of the rectangles is 40 em, then 2x + 2y = 40 orx + y = 20. But the side length of the large square is x + y em. Therefore, the area of the large square is (x + y)2 = 202 = 400 cm2 .

If we multiply the second and third equations together, we obtain

7 5 x(y + 1) y(x + 1) = 9x 18

35 or xy(y + 1)(x + 1) = 162 .

1 From the first equation, xy = 9·

1 35 Therefore, 9 (x + 1)(y + 1) = 162

( 35 ) 35 or (x + 1)(y + 1) = 9 162 = 18 ·

Ill


9.

10.

11.

Since the sale price has been reduced by 20%, then the sale price of ~ 1120

4 is 80% or 5 of the regular price.

1 Therefore, 5 of the regular price is

~ 1120 + 4 = ~ 280. Thus, the regular ptice is ~ 280 x 5 = ~ 1400. If the regular price is reduced by 30%, the new sale price would be 70% of

7 the regular price, or 10 (~ 1400)

= ~ 980.

To make ~1+ 2+ 3 +4+ x an integer, we need 1 + 2 + 3 + 4 + x = 10 + x to be a perfect square. Since xis between 1 and 99, then 10 + x is between 11 and 109. There are 7 perfect squares in this interval: 16, 25, 36, 49, 64, 81 and 100, so there are 7 possible values of x : 6, 15, 26, 39, 54, 71, and 90. Let the three numbers be a, b and c. We construct the first equation to be,

b +c a +--=65. 2

Or, 2a + b + c = 130. Similarly we construct the two other equations to be, a + 2b + c = 138 and a + b + 2c = 152. If we add the three equations we obtain, 4a + 4b + 4c = 420.

4(a + b +c) 420 The average is 3 = - 3-

~ a +b+ c = 35. 3

12. Since 2000 = 24 >03 , the smallest possible positive integer satisfying the required conditions is 25 558 which gives the sum 2 + 5 + 5 + 5 + 8 = 25. A natural answer might be 23 since 44 555 satisfies the given conditions. However, since 25 558 < 44 555 and the question requires the smallest number then the answer must be 25 and not 23.

Ill

13.

14.

15.


Since the flag shown is rectangular, then i ts total area is its height multiplied by its width, or h x 2h = 2h2•

Since the flag is divided into seven stripes of equal height and each stripe has equal width, then the area of each stripe is the same. Since the four shaded stips have total area 1400 cm2, then the area of each strip is 1400 + 4 = 350 cm2•

Since the flag consists of 7 strips, then the total area of the flag is 350 cm2 x 7 = 2450 cm2•

Since the flag ish by 2h, then 2h2

= 2450 cm2 or h2

= 1225 cm2•

Therefore, h = -/1225cm 2 = 35 em (since h > 0). The height of the flag is 35 em. Since a and b are both odd, then ab is odd. Therefore, the largest even integer less than ab is ab - 1. Since every other positive integer less than or equal to ab - 1 is even, then the number of even positive integers less than or equal to ab - 1 (thus,

ab-1 less than ab) is - 2- = 45.

1 25 6 WeknowP + --= - -1+ - - 1 q+! 19- 19-

r 1 1

+ w = 1 + --1. - 3+-6 6

Therefore, comparing the two fractions, p = 1, q = 3 and r = 6.

Explanatory Answers

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