KV Pre Boardmathspaper

Pratima Nayak,KV,Fort William

BLUE PRINT -2nd pre board- 2013

XII mathematics

SL. NO TOPIC

VSA

(1

Mark)

SA

(4 Mark)

LA

(6 Mark) TOTAL

(a)

(b)

Relations & Functions

Inverse Trigonometric Functions

1(1)

1(1)

4(1)

4(1)

-

- 10(4)

(a)

(b)

Matrices

Determinants

2(2)

1(1)

-

4(1)

6(1)

- 13(5)

(a)

(b)

(c)

(d)

Continuity & differentiability

Applications of Derivatives

Integrals

Applications of Integrals

Differential Equations

2(2)

8(2)

4(1)

4(1)

4(2)

-

6(1)

6(1)

6(1)

44(11)

(a)

(b)

Vectors

Three Dimensional Geometry

2(2)

1(1)

4(1)

4(1)

-

6(1) 17(6)

Linear Programming

- - 6(1) 6(1)

Probability

- 4(1) 6(1) 10(2)

TOTAL

10(10) 48(12) 42(7) 100(29)

CBSE –Annexure –F 2013-14


MATHEMATICS (041)

CLASS XII

Time allowed : 3hours Max Marks: 100

General Instructions

1. All questions are compulsory.

2. The question paper consist of 29 questions divided into three sections A, B and C.

Section A comprises of 10 questions of one mark each, section B comprises of 12

questions of four marks each and section C comprises of 07 questions of six marks

each.

3. All questions in Section A are to be answered in one word, one sentence or as per

the exact requirement of the question.

4. There is no overall choice. However, internal choice has been provided in 04

questions of four marks each and 02 questions of six marks each. You have to

attempt only one of the alternatives in all such questions.

5. Use of calculators in not permitted. You may ask for logarithmic tables, if required.

Section A

Q1. Let NNA and let * be the binary operation on A is defined by

a * b =

Find 3 * 2 .

Q2. Find the principal value of .3

5coscos 1

Q3. For what value of k, the matrix

is skew symmetric?

Q4. A is a non- singular square matrix of order 3 and 4A .Find .adjA

Q5. Find x and y, if

+

=


Q6. If and are two unit vectors inclined to x–axis at angles 300 and 1200

respectively,

write the value of |

Q7. Evaluate: dx

x

)x(logsec2

.

Q8. Find the projection of - + on -2 +

Q9. Evaluate

.

Q.10 Find the value of p for which the following two lines are perpendicular to each

other.

1

11

7

1 and

1

7

2

5

1

3

z

p

yxzyx

Section B

Q11. Consider f : R + given by f(x) = x2 + 4.Show that f is invertible with the

inverse f-1 of given by , where R + is the set of all non negative real

numbers.

Q12. Prove that

2

1tan 1

x

x+

2

1tan 1

x

x=

OR

+

=

Q13. Using the properties of determinants, prove that –

))()()((

31

31

31

zyxxzzyyx

zz

yy

xx

Q14. Show that the function f (x) = |x + 2| is continuous at every x R but fails to be

differentiable at x = –2.

Q15. If , find

OR


If x sin ( a + y ) + sin a cos ( a + y) = 0 prove that

=

Q16. Using differentials, find the approximate value of

Q17. . Evaluate: )1( nxx

dx

OR

Evaluate:

dxex

x x

3)3(

1

Q18. Using vectors, fine the area of a triangle ABC whose vertices are

A (1, 1, 2), B (2, 3, 5) and C(1, 5, 5)

OR

If a,b

and c

are unit vectors such that a is perpendicular to the plane of b

, c

and

the angle between b

, c

is 3

then find cba

Q19. Solve the differential equation

x

yxy

dx

dyx tan

Q20. Solve the following differential equation

OR

Form the differential equation of the family of circles of radii 3.

Q21. Find the shortest distance between the lines

)k2j5-i(3k-ji2r

and )ˆˆˆ2(ˆˆ

kjijir


Q22. A target is displayed as ‘’ Be truthful ‘’ the probability of A’s hitting a target is

4/5 and that of B’s hitting is 2/3.They both fire the target .Find the probability

at least one of them will hit the target

Only one of them will hit the target .

Which value is emphasized in the question?

OR

Assume that each child born is equally likely to be a boy or a girl. If a family has

two children, what is the conditional probability that both are girls given that

the youngest is a girl ?

At least one is a girl ?

Pre-natal sex determination is a crime. What will you do if you come to know that

some of our known is indulging in pre-natal sex determination?

SECTION -C

Q23.

Two schools A and B want to award prizes their students for the values of honesty

(X) , punctuality ( Y ) and obedience( Z ) .The sum of all the awardees is 12.Three

times of the sum of awardees for obedience and punctuality added to two times of

the number of awardees for honesty is 33.The sum of the number of awardees for

honesty and obedience is twice the number of awardees for punctuality, using

matrix method, find the number of awardees for each category. Apart from these

values suggest one more other value which could be considered for award?

Q24 A window in the form of a rectangle is surmounted by a semi circular

opening. The total perimeter of the window is 30 m. find the dimensions of the

rectangle part of the window to admit maximum light through the whole opening.

OR

Show that the volume of greatest cylinder that can be inscribed in a cone of

height h and semi vertical angle α is, 23 tan27

4h .


Q25. Using properties of definite integrals, evaluate:

Q26. Find the equation of plane passing through the line of intersection of the

planes

x + 2y + 3z = 4 and 2x + y – z + 5 =0 and perpendicular to the plane

5x + 3y - 6z + 8 = 0.

Q27. Find the area bounded by the curves

OR

Find the area of the region bounded by the two parabolas y = x2 and y2 = x.

Q28. In a group of 400 people,160 are smokers and non vegetarians,100 are

smokers and vegetarians and remaining are non smokers and vegetarian. The

probability of getting a special chest pain disease are 35% , 20% and 10%

respectively. A person is chosen from the group at random and found to be suffering

from the disease. What is the probability is that the selected person is smoker and

no vegetarian? What value is reflected in the question?

Q29. A dietician wishes to mix two types of foods in such a way that vitamin

contents of the mixture contain at least 8 units of vitamin A and 10 units of vitamin C.

Food ‘I’ contains 2 units/kg of vitamin A and 1 unit/kg of vitamin C. Food ‘II’ contains

1 unit/kg of vitamin A and 2 units/kg of vitamin C. It costs Rs 50 per kg to purchase

Food ‘I’ and Rs 70 per kg to purchase Food ‘II’.

Formulate this problem as a linear programming problem to minimize the cost of

such a mixture. In what way a balanced and healthy diet is helpful in performing your

day-to-day activities?


Answer ans Marking Scheme

Q1. 4 / 3 Q2. . Q3. K =

.Q4 16 Q5. . x = 3, y = 3

Q6. 2 Q7. Tan ( log x) Q8. 7 Q9. Cx

2

)(sin 21

Q10. P = - 4

Q11. For one- one

For onto

1 12 Use the formula for tan-1x + tan-1y 1

Correct solution 3

OR

1

1

tan-1x + tan-1y and result 2

13. Applying R1 R1 – R2 , R2 R2 – R3 1

For taking common 1

For expansion along C1 1

For getting required result 1

Q14. L.H.L , R.H.L , L.H.L =, R.H.L , ½ + ½ +½

Continuous ½

L.H.D, RHD LHD ≠ RHD ½ + ½ +½

Not differentiable ½

Q15 u = xy v = yx

=0 1

Putting the values and simplifying 1+1

Answer

1

OR

x = -[sin a cos ( a + y) ]/ sin ( a + y ) 1

dx/dy = sin a/ sin2(a+y) 2

=

1

Q16. Given


1

Ans. 1

=54+68(0.01)=54.68. 2

Ans.17

I = )1( nxx

dx

Substitute xn = t dx = dtxn n 1

1

1

I = )1(

1

tt

dt

n 1

= cx

x

n n

n

1

log1

2

OR

I= dxex

x x

3)3(

1= dxe

x

x x

3)3(

23

1

= dxexx

x

32 )3(

2

)3(

1

1

= dxxfxfe x )()(

1

= ex f(x) + c = ex

2)3(

1

x+c 1

Q18. =1 +2 +3

=0 +4 +3

Area of triangle=

1

=-6 -3 +4 ans. 1+1

OR

1 cba

0. ba

; 0. ca

, 3cos. cbcb

Or

3cos =

21 . ½ + ½ +½ +1/2

)).((2

cbacbacba


= 1 + 1 + 1 + 2 2

1 . = 4 1+1

Q19.

dx

dy= )tan(

x

y

x

y

Put y = vx dx

dy=v + x

dx

dv

v + xdx

dv= v+tan v 1

x

dxvdvcot

&

=cx 1+1

Q20.

dy 1+1

Or,

1+1

OR

( x - a )2 + ( y - b )2 = 9 1

Formation of equation 3

Q21.

3 59bb

ˆ7ˆˆ3

2 5- 3

1 1- 2

k j i

1 ,ˆˆ

21

212

kji

bbkiaa

shortest distance =

21

2221 )).((

bb

aabb

=

59

10 1

Ans 22. P(A) =4/5 P(B) = 2/3 1

(i) P(at least one) =1 – P(0)= 1- P( . .) 1

(ii) P (only one) = P(A + . B) 1

(iii) Truthfulness 1

OR

Ans22.

(i). A: Both are girls ={GG}, B: Youngest is the girl GGBG, 1


2

1

42

41

)(

)()(

Bp

BAp

B

AP

1

(ii). A: Both are girls B: At least one of them is girl GBGGBG ,,

1 3

1

43

41

)(

)()(

Bp

BAp

B

AP

(iii) Every valuable answer given by the student 1

Q23.

x + y + z = 12, 3(y + z ) + 2x = 33 , x + z -2y =0 1

Matrix multiplication form 1

|A| =3

Cofactors

2

x = 3,y = 4 z = 5 1½

One appropriate value ½

Q24.Figure 1

30= 2x+2y+2y+ 1

A=2x

1

= 1

=-( 1

Length =

m ,breath =

1

OR

Can be marked in similar way.

Q25. Use of property

, I =

1

2I =

1½

Use of property


2I =

1½

tanx = t sec2xdx =dt & Correct result I =

2

Ans26

The required plane is (x+2y + 3z ) + k (2x + y – z +5 )= 0 1

Or (1+2k)x +(2+k)y +(3-k)z-4+5k=0 1

5(1+2k) +3 (2+k) -6 (3-k)=0, i.e k= 7/19, 3

The equation of the plane is : 33x+45y +50z = 41 1 Q27.

Correct figure 1½

Point of intersection

Required area = 2( Area of shaded portion)

1+2

Finding integral and getting answer

sq.unit 2½

OR

Figure 1

Intersection points ( 0,0) and ( 1,1). 1

Area of the shaded region

=

dx =

dx= [ 2/3 x3/2 – x3 /3 ]

3

= 1/3 sq. unit. 1

Ans28.

P(E1) = 2/5 P ( E2) = ¼ p(E3) = 7/20 1

P(A/E1) = 35% P(A/E2) = 20% P(A/E3) = 10% 1

Formula for P( E1/A) and expression 1+1 Correct answer 1 Ans 29

Let the mixture contain x kg of Food ‘I’ and ‘y’ kg of Food ‘II’

00

102

82

7050

yx

yx

yx

yxZMin

½ +1/2 +1/2+1/2 Drawing the graph feasible region has no point in common. 2 x=2 & y=4


MinZ=380 2 Marking may be done for all alternative correct answer.

KV Pre Boardmathspaper

Education

x y y z z x x y z z3q14

planes x

x differentiable

log x dx x q7

function f x

values of honesty x

punctuality y

parabolas y