WWW.SAKSHI.COM/VIDYA/BHAVITHA çÜμÆý‡®Ä¶æ* Ð]lÆý‡®™ól ѧýlÅ {糆 VýS$Æý‡$-Ðé-Æý‡… Ýë„ìS-™ø E_-™èl… 24-&2&2011 ☞ Connectives: 1. and 2. or 3. if... then 4. if and only if ☞ Compound Statements: 1.Disjunction 2. Conjunction 3. Conditional 4. Biconditional ☞ Function: f:A→B, 1. if for every a∈A there is b ∈ B such that (a/b)f ☞ One-one function (injection): f(x 1 ) = f(x 2 ) ⇒ x 1 = x 2 ☞ Arithmetic progression(A.P) difference (d) is equal ☞ General form of A.P a, a + d, a + 2d, .... ☞ Modulus of 'x', |x| |x| = x if x > 0 or - x if x < 0 or 0 if x = 0 ☞ |x| = a solution: x=a or x= - a ☞ Quadratic equation ax 2 + bx+c=0 ☞ Discriminent Δ = b 2 –4ac ☞ Δ > 0 Roots are real, unequal ☞ Sexagesimal system Degree ☞ Centesimal system grade ☞ Circular measure Radian ☞ Convex Set: X is convex if the line segment joining any two points P, Q in x is contained in x ☞ Linear programming problem L.P.P consists of Minimising/maximising a function f = ax+by, a, b∈R subject to certain constraints ☞ Circum center Concurrence point of perpendicular bisector of the sides of the Triangle. ☞ In center concurrence point of angle bisector of the Triangle. ☞ Equation of X-axis y = K ☞ Equation of Y-axis x = K ☞ Slope of X-axis 0 10 th MATHEMATICS BITBANK SPECIAL ● Practice Bits ● Important Questions ● Question Trends ● Preparation Tips ● Quick Review 10 th MATHEMATICS BITBANK SPECIAL QUICK RE'VIEW'
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
&&&&&&&&&……………………………………
W W W . S A K S H I . C O M / V I D Y A / B H A V I T H A
4 Marks1. Let f : R → R be defined by f (x) = 2x + 3.
find f–1(4),
2. Let f,g,h be functions , f(x) = x+2, g (x) =
3x-1 and h(x)=2x show that
ho(gof)=(hog)of ?
3. If a function f : R → R is defined by f(x) =
3x-5, then find a formula that defines the
inverse function f–1?
4. Let f be given by f(x) = x+2 and f has the
domain {x : 2 ≤ x ≤ 5} find f-1and its domain
and Range?
2 Marks1. Let f : R -{2} → R be defined by
show that ?
2. Define one-one function show that f(x) = 3x
– 2; x ∈ N is one -to-one.?
3. If f(x) = x2 + 2x + 3, x ∈ R find the volue of
when h ≠ 0.?
4. f : R → R be defined by f(x) = 6x + 5, find
f–1 (x).?
5. f(x) = x + 2, g(x) = x2 – 3 find
1) (gof) (-2) 2) (fog) (-2).?
1 Mark1. Define on-to function?
2. Let f : A → B and let f have an inverse func-
tion f–1 : B → A. state the properties of f for
which its inverse exists.
3. Define equal functions?
4. Let f = {(1,2), (2,3), (3,4)} and g = {(2, 5),
(3, 6), (4, 7)} find gof?
5. Define a bijection?
6. Let f : R-{1} → R be defined by f(x) = 1 +
2x, g(x) = 3 – 2x, find (fog) (3)?
( ) ( )f x h f x
h
+ −
2x 1f x.
x 2
+⎛ ⎞ =⎜ ⎟−⎝ ⎠
2x 1f (x)
x 2
+=−
{ }{ }1 1f (x) : 2 x 3 , f (x) : x 5 .− −≤ ≤ ≤
FUNCTIONS: Important Questions
5 Marks1. Using graph of y = x2, solve x2 – 4x+3 = 0
2. Draw the graph of y = x2 + 5x + 6 and find
the solution of x2 + 5x + 6 = 0?
4 Marks1. If ax2 + bx + c is exactly divisible by (x-1),
(x-2) and leaves remainder 6 when divided
by (x+1). find a,b and c?
2. Resolve in to factors of the polynomial
3x4 – 10x3 + 5x2 + 10x – 8?
3. Find the independent term of ‘x’ in the
expansion of?
4. Find a quadratic function is in ‘x’ such that
when it is divided by (x-1),(x-2) and (x-3)
leaves remainders 1,2 and 4 respectively.
2 Marks1. Find the value of ‘m’ in order that x4 –2x3 +
3x2–mx+5 may be exactly divisible by(x-3)?
2. Find the roots of x2+x (c-b)+(c-a) (a-b) = 0.
3. Find the middle term of the expansion of
?
4. Solve the inequation x2 – 6x + 8 > 0?
5. The difference of two numbers is 5 and their
product is 84 find them?
6. Find the 5th term in the expansion
1 Mark1. Define mathematical induction?
2. Comment up on the roots of a quadratic
equation 3x2 – 7x + 2 = 0 ?
3. Find the quadratic equation having roots
?
4. Find the value of K so that x3 – 3x2 + 4x +
K is exactly divisible by x-2?
5. Find the sum and product of the roots of
the equation ?
6. Define Remainder theorem?
7. The product of two consecutive numbers
is 72. Find the number?
8. Write factor theorem?
9. Expand ?
10. Write General term of expression (x+y)n?
( )a a b c+ −∑
23x 9x 6 3 0+ + =
1 2 and 1 2+ −
81
2x3y
⎛ ⎞+⎜ ⎟
⎝ ⎠
71
3x2x
⎛ ⎞−⎜ ⎟⎝ ⎠
82
2
56x
x
⎛ ⎞−⎜ ⎟⎝ ⎠
POLYNOMIALS: Important Questions
(1,0) x
y
(–1,0)(–3,0) (3,0)
Ýë„ìS ¿¶æÑ™èl VýS$Æý‡$ÐéÆý‡… 24 íœ{ºÐ]lÇ, 2011
4
The point on Y axis is... MATHEMATICS BIT BANK LINEAR PROGRAMMING
50. Middle term in the expansion of
is ________ ( March 2008)
51. If (a + b, 1) = (5, a – b) then 2a + 3b =
________ ( March 2006)
52. (x+1) is a factor to ax4 + bx3 + cx2 + dx + e
then the condition is ________
53. If |x| ≤ a then the solution set is ________
54. The middle term of expansion is
________ (March 2010)
55. Sum of the number and its reciprocal is 17/4
then the number is ________
56. Expand
LINEAR PROGRAMMING
1. Any line belonging to the system of parallel
lines of objective function is called _______
2. If none of the feasible solutions maximise
or minimise the objective function, then the
problem has ________
3. A line divides the plane in to ________ sets.
4. If x>0, y<0 then the point (x,y) lies in
________ quadrant (June 2008)
5. If then the value of P at the
point (4,9) is ________
6. The parallel lines that are determined by the
objective function are called ________
7. If the isoprofit line moves away from the
origin then the value of the objective func-
tion ‘f’ is ________
8. Polygon represented by the inequalities
x ≥ 1, y ≥ 1, x ≤ 3, y ≤ 3 is ________
9. The solution set of constraints of linear pro-
gramming is called ________
10. A line segment joining the points P and Q,
where P, Q ∈ x such that then X is
called ________ set
11. If the values of the expression f = ax+by is
attained maximum or minimum at one of
the vertices that is called _______
12. The expression ax+by which is to mini-
mized or maximized is called ________
13. The isoprofit line coincides with the sides of
polygon then it has ________ solution.
14. Any point (x,y) in the feasible region gives
a solution to LPP is called ________
15. If the point (-3,2) lies on 3x-5y+k<0 then
the maximum value of K is ________
16. f = A X + BY is called ________
17. The shaded region represents the inequation
is ________
18. The solution set of x ≥ y and x ≤ y is
________
19. The point on Y axis is ________
20. The value of x+y should not be less than8’ can be written as ________
21. The slope of Y-axis is ________22. Isoprofit lines are ________23. The profit of a chair is Rs 10 and table is
Rs 25. A man purchased x chairs and ytables. Then the total profit is ________
24. The c>0 then ax+by+c <0 represents theregion ________
25. If a<0 then the point (4,-a) lies in________ quadrant.
26. The knowledge of Linear Programming
helps to solve the problem in ________
27. The maximum (or minimum) value of foccurs on atleast one of the vertices of thefeasible region. This is the statement ofthe ________ theorem of LinearProgramming.
28. If Q1 and Q2 are first and second quad-rants then Q1 ∩ Q2 ________
29. If f = ax+by is objective function, thenthe line ax+by = c is called ________ line
30. If x = 0 then (x,y) is a point on ________axis.
31. The value of f = 2x+3y at (1,2) is________
32. Intersection of x ≥ 0, y ≥ 0 is ________33. The value of an objective function
at (0,9) is ________
REAL NUMBERS
1. If 2x+3 = 8x+3 then x = ______ (March 2009)
2. (16)1.25 =________ (March 2009)
3. ________ (March 2009)
4. (16)0.5 = ________ (March 2008)
5. (64)x = then x = _______ (March 08)
6. The limiting position of secant of a circle is
________ (March 2008)
7. If then a = ______ (March 2008)
8. If ax = b; by = c, cz = a then the value of xyz
= ________ (March 2008), (March 2009)
9. (March 2009)
10. If (x2/3)p=x2then the value of p is ____
11. (June 2009, 10)
12. a≠0 and if p + q + r = 0 then a3p + 3q + 3r =
________ (June 2007)
13. If (March'10)
14. If x = –3 then |x2 – 20| = ______ (March'10)
15. (June 2010)
16. If then 3x = ________
17. If then x = ________
18. If then a = ________
19. The rationalising factor of is _____
20.
21.
22.
23.
24.
25. If 35x + 2 = (27)4 then x = ________
1 1 1 1 1 14 4 2 4 4 2x y x x y y ________
⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟+ − + =⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
3
2x 2
x 2x 2Lt ________
2x 3x 5→−
− + =+ +
a b b a
1 1________
1 x 1 x− −+ =+ +
2
x 0
x 5xLt _______
x→
+ =
3 3 3 3
4n
1 2 3 nLt ______
n→∞
+ + + − − − + =
1 13 3a b+
(a a ) aa (a a )=
32x 0.027=
xy
1(64)
(256)=
3 i
i 04 ________
==∑
1 1x 4 then x _______
x x+ = − =
x
2x 3Lt _________
3x 5→∞
+ =+
3 2
n 1(n 1) _______
=+ =∑
2a 3x x=
2 2
x
1Lt
x→∞=
1 2f x y
3 3= +
PQ X⊆
1 2P x y
4 3= +
2a (b c) ______________− =∑
4x y
y x
⎛ ⎞+⎜ ⎟
⎝ ⎠
8x y
y x
⎛ ⎞+⎜ ⎟
⎝ ⎠
KEY
1. -C 2. 9/2 3. (0,c) 4. I & III quadrants 5. 8 6.
42. A.P 43. B = 15° 44. 1 45. 44/7 cm 46. 247. Geometric Progression 48. –Cosecθ49. 180° 50. initial side 51. 2mt
c
4
π
2 2
a
a b+
3 / 2
2Sec 1
Sec
θ −θ
c c c
and6 3 2
π π π
2
2
p 1
p 1
−+
c3
2
π
Ýë„ìS ¿¶æÑ™èl VýS$Æý‡$ÐéÆý‡… 24 íœ{ºÐ]lÇ, 2011
9
Number of rows in a Row matrix...MATHEMATICS BIT BANK MATRICES
21. The most common and widely used measure
is ________
22. Father of statistics is ________23. Given data, frequency of modal class
f = 36, f2 = 24 then Δ2 = ________24. The average which is not affected by the
extraction value is ________25. The median of 7,5,7.5,5.5,6,6.5 is ______26. The mean of 10 observations is 7 and the
mean of 15 observation is 12 then themean of all observations is ________
27. Mid value of the class 1-10 is ________28. In a frequency distribution, the mid value
of a class is 35 and the lower boundary is30 then upper boundary is ________
29. 0-10,10-20,20-30 are ________ type ofclasses.
30. Unlike mean , median is not affected bythe ________ observations
31. A.M = where A is called
________32. In a data having two modes, then it is
called ________33. Sum of 20 observations is 420 then the
mean is ________34. The difference between two consecutive
lower limits of the class is ________35. Circular diagram consists of ________36. The mode of 4,8,9,p,2,6,4,9 is 9 then p =
________37. The Arithmetic mean of sum of the even
natural numbers is ________38. The median of natural numbers from 1 to
9 is ________39. A Histogram Consists of ________40. In a distribution
Δ1 = 6, Δ2 = 4, c = 10 and L=25 thenmode = ________
MATRICES
1. If then |A|=_____ (March09)
2. If then t = ________
3. If then the value of ‘x’
is ________
4. = ________
5. If |A| = 0 then the matrix has ________6. The mathematician who introduced
matrices is ________ (June 2006)7. A,B are two matrices (AB)T = ________8. The condition to multiply two matrices
A,B is ________
9. then order of M =____
10. If has no multiplicative
inverse then x = ________11. If the transpose of a given matrix is equal
to its additive inverse, then the matrix iscalled ________
12. Matrix obtained by interchanging rowsand columns is called ________ (March2009)
13. If the rows and columns of a matrix aresame, then it is called ________ (March09)
14. If then a and b
are ________
15. If then x = ________
16. If then d = ________
17.
then AB = ________
18. If is to be scalar matrix then λ
= ________
19. If A and B are two matrices then (AB)–1 =________
20. If and ad = bc then A is
________ matrix
21. If and AD = A then D is
________ Matrix 22. If A2×3, B3×2 then the order of A×B is
________23. If AB = KI, where K ∈ R, then A–1 =
________24. If A is a matrix then (AT)T =________
25. If then a+b+c+d =
________ (June 2005)26. The order of A is 3 × 2 then the order of
AT is ________
27. is example of ________
28. ________
29. If A is matrix then A.A–1=A–1. A= ______30. Number of rows in a Row matrix
________31. The order of A and B are 3×4 and 5×3
then the order of BA is ________32. If A is 2 × 2 matrix such that A=A–1 then
A2 = ________ (June 2009)
33. A is any 2 × 2 matrix. if then
AB = ________ (June 2009)
1 0B
0 1
⎛ ⎞= ⎜ ⎟
⎝ ⎠
1 3
3 1
2
3 (1 2 3)
4×
×
⎛ ⎞⎜ ⎟ =⎜ ⎟⎜ ⎟⎝ ⎠
4 0 0
0 4 0
0 0 4
⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠
a b 1 2
c d 3 1
⎛ ⎞ ⎛ ⎞=⎜ ⎟ ⎜ ⎟−⎝ ⎠ ⎝ ⎠
1 2A
3 4
−⎛ ⎞= ⎜ ⎟−⎝ ⎠
a bA
c d
⎛ ⎞= ⎜ ⎟
⎝ ⎠
3 0P
0
⎛ ⎞= ⎜ ⎟λ⎝ ⎠
2 3 2 2
1 2 3 1 0A ;B
3 0 1 0 1× ×
⎛ ⎞ ⎛ ⎞= =⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
2 414
d 5
−=
1 3 2 x
0 1 1 1
⎛ ⎞⎛ ⎞ ⎛ ⎞=⎜ ⎟⎜ ⎟ ⎜ ⎟− −⎝ ⎠⎝ ⎠ ⎝ ⎠
a 5 4 6 2 1
8 b 7 2 1 5
−⎛ ⎞ ⎛ ⎞ ⎛ ⎞− =⎜ ⎟ ⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠ ⎝ ⎠
x 3A
3 x
⎛ ⎞= ⎜ ⎟
⎝ ⎠
2 3M (6 10)
0 1
⎛ ⎞× =⎜ ⎟
⎝ ⎠
Tan sec
sec Tan
θ θθ θ
x 3 2 5
1 2 1 0
⎛ ⎞⎛ ⎞ ⎛ ⎞=⎜ ⎟⎜ ⎟ ⎜ ⎟−⎝ ⎠⎝ ⎠ ⎝ ⎠
t
4 34 3
2 32 2 2
−⎛ ⎞−⎛ ⎞= ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
4 3A
2 1
⎛ ⎞= ⎜ ⎟−⎝ ⎠
fxA c
N∑+ ×
KEY
1. 2. 19 3. 4. 7 5. 37.56. Arithmetic mean 7. 20,30 8. 3 9. a 10. f – f1
11. inclusive 12. 13. class inter-
vals 14. Midvalues of the classes 15. 10.5 16.
17. 35 18. Mode = 3Median-2A.M 19. 20
20. Median 21. Arithmetic mean 22. SirRonald A. Fisher 23. 12 24. Median 25. 6.2526. 10 27. 5.5 28. 40 29. Exclusive 30.Extreme 31. Assumed mean 32. Bi modal 33.21 34. Class interval 35. Sectors 36. 9 37.(n+1) 38. 5 39. Rectangles 40. 31
7
12
NF
2L cf
−+ ×
i i1
A f cN
+ Σ μ ×(n 1)
2
+
4 Marks1. Calculate the A,M for the following data
by deviation method?
2. Find the median for the following data ?
2 Marks1. The mean of 20 observation is 135. By an
error, one observation is registered as-25
instead of 25 . Find the correct mean?
2. Write four merits of the Arithmetic mean ?
3. The mean and median of Uni-modal
grouped data are 72.5 and 73.9 respective-
ly. Find the mode of the data?
4. Observations of some data are
where x>0. If the median
of the data is 8. Find the value of ‘x’?
5. The observations of an ungrouped data are
x1, x2 and 2x1 and x1 < x2 < 2x. If the mean
and median of the data are each equal to 6.
Find the observations of the data?
1 Mark1. The mean of 9,11,13,P,18,19, is P. Find the
value of ‘P’?
2. Find the mode of the data 12, 11, 15, 12,
11, 15, 12, 9, 12?
3. Write two properties of mode?
4. A.M= x, Median= y find mode of the data?
5. Find the median of the observations 1.8,
4.0, 2.7, 1.2, 4.5, 2.3 and 3.7?
6. The observation of an ungrouped data in
the assending order is 12, 15, x, 19, 25. If
the median of the data is 18 find the value
of ‘x’?
x x x x,x, , and
5 4 2 3
STATISTICS: Important Questions
Marks 0-10 10-20 20-30 30-40 40-50 50-60No.of
5 7 15 8 3 2students
Class 60-64 65-69 70-74 75-79 80-84 85-89
Frequency 13 28 35 12 9 3
4 Marks
1. If ?
find 1) A–1 2) B–1 3) (AB)–1 4) B–1A–1 ?2. Solve the following linear system of equa-
tions using cramers method 4x–y=16 and
?
3. Solve the following equations by using
Matrix inversion method and y
= 13 – 6x?
4. If show that
A2 – (a+d) A= (bc-ad) I.?
5. If
Show that A(B+C) = AB+AC?
2 Marks
1. If find the order ofM
and determine the Matrix ‘M’ ?
2. If find ‘m’ if
AB=BA.?
3. If find the
Matrix B+A–1?
4. If findx,y?
5. If
find A2 + BC?
1 Mark
1. If find the value of A+AT?
2. If find 3A-2B?
3. If find A+A–1 = 4I?
4. =0 find ‘d’?
5. If find A–1?
6. Define Non-singular Matrix
7. If and then Find
AB?
0 0B
0 1
⎛ ⎞= ⎜ ⎟
⎝ ⎠
1 0A
0 0
⎛ ⎞= ⎜ ⎟
⎝ ⎠
2 3A
1 5
−⎛ ⎞= ⎜ ⎟
⎝ ⎠
d 2 5
4 2
−−
1 2A
1 3
⎛ ⎞= ⎜ ⎟
⎝ ⎠
2 4 4 3A , B
6 5 5 7
−⎛ ⎞ ⎛ ⎞= =⎜ ⎟ ⎜ ⎟−⎝ ⎠ ⎝ ⎠
1 3A
5 6
⎛ ⎞= ⎜ ⎟
⎝ ⎠
1 4 3 2 1 0A ;B ;C
2 1 4 0 0 2
−⎛ ⎞ ⎛ ⎞ ⎛ ⎞= = =⎜ ⎟ ⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠ ⎝ ⎠
3x 2y 6 5 6
2 2x 3y 2 1
+⎛ ⎞ ⎛ ⎞=⎜ ⎟ ⎜ ⎟− −⎝ ⎠ ⎝ ⎠
1 2 2 0A ;B
1 3 5 3
⎛ ⎞ ⎛ ⎞= =⎜ ⎟ ⎜ ⎟−⎝ ⎠ ⎝ ⎠
2 m1 4
A ;B 10 1 0
2
⎛ ⎞⎛ ⎞ ⎜ ⎟= = −⎜ ⎟ ⎜ ⎟− ⎜ ⎟⎝ ⎠ ⎝ ⎠
( )1 2M 2 3
0 5
⎛ ⎞× =⎜ ⎟
⎝ ⎠
2 4 2 5 1 2A , B , C .
3 6 6 1 3 0
−⎛ ⎞ ⎛ ⎞ ⎛ ⎞= = =⎜ ⎟ ⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠ ⎝ ⎠
a b 1 0A and I
c d 0 1
⎛ ⎞ ⎛ ⎞= =⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
7 3yx
2
−=
3x 7y
2
− =
2 1 2 0A ,B
3 1 5 3
−⎛ ⎞ ⎛ ⎞= =⎜ ⎟ ⎜ ⎟− −⎝ ⎠ ⎝ ⎠
MATRICES: Important Questions
Ýë„ìS ¿¶æÑ™èl VýS$Æý‡$ÐéÆý‡… 24 íœ{ºÐ]lÇ, 2011
10
The unit that gains results from C.P.U.. MATHEMATICS BIT BANK COMPUTING
34. The Inverse of an identity matrix is________ (March 2009)
35. If then A–1= ____ (March08)
36. If then x =
________ (March 2008)
37. In a Matrix the element in
2nd row and 3rd column is_____(June 07)
38. , then AB =
________ (June 2007)
39. While solving the equations 3x+4y = 8and x – 6y = 10 by Cramer’s method thenthe matrix B1 =________
40. The determinant of a singular matrix is________
41. If and A+B = A then B is
________ matrix
42. If and P+R=I then R=______
43. If and A–B+X=0
then the Matrix X is ________
44. In a Matrix the number of rows are notequal to number of columns then thematrix is ________
45. A square matrix in which each of theprincipal diagonal elements are equal toone and all other elements are zero iscalled a ________ matrix
46. If the transpose of a given matrix is equalto its additive inverse that matrix is called_______
COMPUTING
1. Small Transistors are used in _______generation of computers. (March 06,June 09)
2. All parts of computer are controlled by________ (2006, 2007, 2009)
3. Input, Output, CPU are ________ of thecomputer. (June 2006)
4. An example for output is ________ (June2006)
5. Vacuum tubes are used in ________ gen-eration of computers
(March 2007)6. The language known to the computers is
called ________ (June 2009)7. ________ is used to make a diagrammat-
ic representation of an algorithm (March2008)
8. The father of computer is ________(March 2008)
9. To express the algorithm in a languageunderstandable by a computer is called________
10. The number of major parts in a computeris ________ (June 2009)
11. C.P.U means ________12. large amount of information is stored in
________ unit of computers.13. The method of solving a problem is
called ________14. ________ are used in fourth generation
of computers.15. All the mathematical operations are car-
ried out in ________ units.16. The input unit, C.P.U and output unit all
together is called ________17. The unit that gains results from C.P.U is
________18. Example for computer language is
________19. The present day computers are made as
________ generation computers.20. In the preparation of flow charts, we use
Rhombus shaped box for ________21. A computer is an ________ device.22. Pictorial representation of algorithm is
called ________23. Printer is example for ________ unit24. COBOL means ________25. The computers built in between 1950-
1960 are called as ________ generationof computers.
26. ________ is example for Input unit27. An algorithm means ________28. The Rhombus shaped box is used in a
flow chart for ________29. Each computer consists of three essential
units, namely Input unit, output unit andthe ________ unit.
30. BASIC is ________ language.31. Father of modern computers is
________
32. ________ are used in third generation ofcomputers.
33. A.L.U means ________
1 2 2 4A ,B
3 4 3 5
⎛ ⎞ ⎛ ⎞= =⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
4 5P
7 6
−⎛ ⎞= ⎜ ⎟−⎝ ⎠
5 7A
0 8
⎛ ⎞= ⎜ ⎟
⎝ ⎠
1 22 1
xA , B (5 2)
y ××
⎛ ⎞= =⎜ ⎟
⎝ ⎠
1 8 4
2 3 0
5 7 4
⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟−⎝ ⎠
x y x y 2 0
2x 3y 2x 3y 5 1
+ −⎛ ⎞ ⎛ ⎞=⎜ ⎟ ⎜ ⎟+ − −⎝ ⎠ ⎝ ⎠
1 4A
0 1
⎛ ⎞= ⎜ ⎟−⎝ ⎠
1. Negation2. And3. Or4. Implie5. If and only if6. For all7. For some8. Belongs9. Not belongs10. Subset11. Superset12. Union13. Intersection14. Powerset15. Null set16. Complement of A17. Cartesian product of
A, B is18. Identity function19. Discriminant20. Transpose of A21. Inverse of A22. Fistle function A to B23. Composite function of f
and g24. Sum of first 'n' natural
numbers25. nth term26. Sum of 'n' terms27. Arithmetic mean28. Sum of frequencies
KEY
1. 10 2. 5 3. 4 4. –1 5. has no multiplicative
inverse 6. Author Cayley 7. BT.AT 8. No.of
Columns in A = Rows in B 9.(1×2) 10. ±3 11.
Skew symmetric 12. Transpose of matrix 13.
Square matrix 14. 6,7 15. –1 16. 1 17. is not
defined 18. 3 19. B–1.A–1 20. Singular matrix
21. Identity matrix 22. 2 × 2 23.
24. A 25. 5 26. 2 × 3 27. 3 × 3 scalar matrix
28. 29. I 30. l 31. 5 × 4 32. I
33. A 34. also identity matrix 35.
(or) A 36. 1 37. 0 38. 39.
40. zero 41. null 42. 43.
44. Rectangle matrix 45. Identity matrix
46. Skew symmetric matrix
1 2
0 1
⎛ ⎞⎜ ⎟⎝ ⎠
3 5
7 7
−⎛ ⎞⎜ ⎟−⎝ ⎠
8 4
10 6
⎛ ⎞⎜ ⎟−⎝ ⎠
5x 2x
5y 2y
⎛ ⎞⎜ ⎟⎝ ⎠
1 4
0 1
⎛ ⎞⎜ ⎟−⎝ ⎠
2 4 6
3 6 9
4 8 12
⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠
1B.
K
KEY
1. Second 2. C.P.U 3. Hardware 4. print-er 5. First 6. Higher language (or) soft-ware programming language 7. Flowchart 8. Charles Babbage 9.Programming language 10. 3 11. CentralProcessing Unit 12. Memory 13.Programme 14. Very large scale integrat-ed circuites 15. Arithmetic and logicalunit 16. Hardware 17. Out put 18.COBOL (or) PASCAL 19. IVth genera-tion 20. Decision box 21. Eelectronic 22. Flowchart 23.Output 24. Common business orientedlanguage 25. Ist generation 26. Key board27. Plan of obtaining a solution to a prob-lem 28. Decision making 29. Central Processing Unit (C.P.U.) 30.Computer 31. Von Newmann 32. Verysmall electronic circuits33. Arithmetic and Logic unit
4 Marks1. Give the principal amount and the rate
of interest write an algorithm to obtain atable of compound interest at the end ofeach year for 1 to 5 years and draw aflow chart?
2. Gopal purchased a radio set for 500 andsold it for 600. Execute a flow chart todetermine loss or gain percentages?
3. Draw the flow chart to find the value ofproduct of the first ‘n’ natural numbers?
2 Marks1. What are the different boxes used in a
flow chart? Write for what functionsthey are used?
2. Write the characteristics of a computer?3. Draw a structure diagram of computer?4. What is Flow chart and define
Algorithm?5. What are the essential components of a
computer?6. What is meant by step-wise refinement
in computer?7. What are the types of operations that a
computer performs?
1 Mark1. What is meant by Computer Hardware?2. Expand C.P.U.?3. Write 4 computer languages?4. What are the essential components of
C.P.U.?5. What are the shapes of terminal and
decision boxes in the flow chart?6. Define a computer?7. What is the difference between Hard
ware and software?
COMPUTING: Important Questions
∼∼∧∧∨∨⇒⇒⇔⇔∀∀∃∃∈∈∉∉⊂⊂⊃⊃∪∪∩∩μμφφ
A1 / Ac
A × BI (A)
ΔΔ or DA T
A–1
f:A→→B
gofΣΣ n
tnsnx
ΣΣf or N
Important symbols
Ýë„ìS ¿¶æÑ™èl VýS$Æý‡$ÐéÆý‡… 24 íœ{ºÐ]lÇ, 2011
11
IMPORTANT POINTS PAPER- 1 AND 2
STATEMENTS and SETS
1. Function: f:A→B, 1. if for every a∈A there is b ∈ B such that(a/b)f
5. Constant function: f:A→B, The range of f(A) = singleton set6. Identity function: I:A→A, I(x)=x ∀ x ∈ A7. Inverse function: If fi is Bijection then f-1is also a function8. Equal function: Two functions domains are equal.9. In to function: f: A→Β, f(Α) ⊂ Β 10. Real function: f: A→Β, A,B are subsets of R.11. Composite function (gof): A,B,C are three sets, f:A→Β, g:B→C then gof:
A→C.
FUNCTIONS
Item Explanation1. Arithmetic progression(A.P) difference (d) is equal
2. General form of A.P a, a + d, a + 2d, ....
3. 'n'th term in A.P (tn) a+ (n–1)d
4. Sum of n terms in A.P. (sn)
5. Geometric progression (G.P) Ratio (r) is equal
6. General form of G.P a, ar, ar2, .....
7. nth term in G.P (tn) a⋅rn – 1
8. Sum of n terms in G.P. (sn)
9. Harmonic progression (H.P) Reciprocal of the terms form an A.P.
10. Arithmetic mean of a, b
11. Geometric mean of a , b
12. Harmonic mean of a, b
13. Σ n = 1 + 2 + 3 + . . . . + n
14. Σ n2 = 12 + 22 + 32 + . . . . + n2
15. Σ n3 = 13 + 23 + 33 + . . . . + n3
2 2n (n 1)
4
+
n(n 1)(2n 1)
6
+ +
n(n 1)
2
+
2ab
a b+
ab
a b
2
+
( ) ( )n na r 1 a 1 rif r 0 or if r 0
r 1 1 r
− −> <
− −
( ) ( )n n2a n 1 d or a l
2 2+ − +⎡ ⎤⎣ ⎦
PROGRESSIONS
Statement Explanation1. Modulus of 'x', |x| |x| = x if x > 0 or - x
if x < 0 or 0 if x = 0
2. |x| = a solution: x=a or x= - a
3. |x| ≤ a solution: – a ≤ x ≤ a
4. |x| ≥ a solution: x≥a or x≤– a
5. n⋅xn – 1
6.
m nma
n−
m m
n nx a
x aLt
x a→
−−
n n
x a
x aLt
x a→
−−
REAL NUMBERS
Item Explanation1. Convex Set: X is convex if the line segment joining any
two points P, Q in x is contained in x
2. Linear programming problem L.P.P consists of Minimising/maximising a
function f = ax+by, a, b∈R subject to certain
constraints.
3. Objective function: f = ax + by, a, b∈R which is to be minimised
or maximised
4. Feasible Region: Solution set of constraints of LPP is convex
set is called
5. Feasible solutions: Anypoint (x, y) in the feasible region gives a
1.Direct method 2.Indirect Method 3. disproof by counter
example.
1. Series combination 2. Parallel combination.
Ýë„ìS ¿¶æÑ™èl VýS$Æý‡$ÐéÆý‡… 24 íœ{ºÐ]lÇ, 2011
12
IMPORTANT POINTS Paper 1, 2
GEOMETRY
STATISTICS
ANALYTICAL GEOMETRY
Preparation Tips and Blue Print
Paper - IChapter 5 Marks 4 Marks 2 Marks 1 Mark 1/2 MarkStatements & Sets & 1 2 1 5Functions (Mappings) & 2 1 1 5Polynomials 1 1 1 1 6Real Numbers & 1 2 1 3Linear Programming 1 1 1 1 6Progressions & 2 1 1 5
Paper - IIChapter 5 Marks 4 Marks 2 Marks 1 Mark 1/2 MarkGeometry 1 1 1 1 5Analytical Geometry & 2 2 1 5Trignometry 1 1 1 1 5Statistics & 1 1 1 5Matrices & 2 1 1 5Computing & 1 2 1 5
Item Formula
1. The slope of (x1, y1)and (x1, y2)
2. Distance between (x1, y1) and (x2, y2)
3. Equation of X-axis y = K
4. Equation of Y-axis x = K
5. Slope of X-axis 0
6. Slope of y-axis Not defined
7. General form of a straight line ax + by +c = 0
8. Slope of ax + by +c = 0
9. Mid point of (x1, y1) and (x2, y2)
10. Given points is divided by internally
in the ratio m : n is
11. Given points is divided by externally
in the ratio m : n is
12. Gradien form of equation y = mx
13. Slope - intercept form y = mx + c
14. Point - slope form y – y1 = m(x – x1)
15. Two intercepts form
16. Two - points form (y – y1) (x2 – x1) = (x – x1) (y2 – y1)
17. Centroid of the Triangle
18. Area of the Triangle
19. Area of the Triangle = 0 given points are collinear.
20. Product of the slopes = –1 Lines are perpendicular
21. Slopes are equal Lines are parallel.
( ) ( ) ( )1 2 3 2 3 1 3 1 21
x y y x y y x y y2
− + − + −
1 2 3 1 2 3x x x y y y,
3 3
+ + + +⎛ ⎞⎜ ⎟⎝ ⎠
x y1
a b+ =
1 1 2 1mx nx my ny,
m n m n
− +⎛ ⎞⎜ ⎟− −⎝ ⎠
2 1 2 1mx nx my ny,
m n m n
+ +⎛ ⎞⎜ ⎟+ +⎝ ⎠
1 2 1 2x x y y,
2 2
+ +⎛ ⎞⎜ ⎟⎝ ⎠
a
b
−
( ) ( )2 22 1 2 1x x y y− + −
2 1
2 1
y y
x x
−−
� 'Mathematics is a hard nut to crack' it isgeneral perception of students. Actuallyregular practice and good command onbasics will make mathematics easy andscoring subject.
� Out of 12 chapters, if you prepare 8chapters thoroughly you can get 90marks easily.
� Before solving the problems you shouldbe well aware of the definitions andlaws of that particular chapter.
� Please do not by-heart steps in theproblem. Understand the problem andmake steps accordingly to arrivesolution.
� Avoid tension and be cool in exam hall.� First of all read all the questions in the
given question paper, then only answerthe question first which you feel wellprepared.
� Avoid illegible writing and striking. Useonly the allotted place for rough work.
� Before answering the question, checktwice whether the number of thequestion written correctly or not in theleft side of the margin.
� In every section answer according tochoice. If you have time after answeringall you may try for some otherproblems.
� Highlight the answers and laws bymaking boxes if necessary.
� You got more scope to get good score inPart B only. So prepare accordingly.
� The allotted time for Part B is only 30minutes. Utilize the allotted time fullyto this part only. Start writing knownanswers first, then only go for otherbits.
� Students must aware of basic factors inexamination hall like quoting correctnumber of question concerned, andindentify the key factor of the problemetc.
PREPARATION TIPS
Statement Explanation1. Circum center Concurrence point of perpendicular bisector of the
sides of the Triangle.
2. In center concurrence point of angle bisector of the Triangle.
3. Centroid Concurrence point of the medians
4. Ortho center Concurrence point of the heights
5. Basic proportionality theorem In ΔABC, DE //BC then
6. Vertical angle bisector theorem In ΔABC, the bisector of A intersects BC in D then