cbse sample question paper 2021-22 class : xii

C.B.S.E SAMPLE PAPER‐[XII] 1

SECTION–AIn this section, attempt any 16 questions out of Questions 1 – 20.Each Question is of 1 mark weightage.

Marks

1. :toequalis21

–sin–3

sin 1–⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛π

1

(a)21

(b) 31

(c) – 1 (d) 12. The value of k (k < 0) for which the function f defined as

⎪⎪⎩

⎪⎪⎨

⎧

=

≠=

0,21

0,sin

cos–1

)(x

xxx

kx

xf is continuous at x = 0 is 1

(a) ± 1 (b) – 1

(c)21

± (d)21

CBSE SAMPLE QUESTION PAPER 2021-22CLASS : XII

MATHEMATICS (CODE-041)

[With Marking Scheme]

TERM - I

Time : 90 minutes Maximum Marks : 40

General Instructions :

1. This question paper contains three sections – A, B and C. Each part is compulsory.

2. Section-A has 20 multiple choice questions. Attempt any 16 out of 20.

3. Section-B has 20 multiple choice questions. Attempt any 16 out of 20.

4. Section-C has 10 multiple choice questions. Attempt any 8 out of 10.

5. There is no negative marking.

6. All questions carry equal marks.

C.B.S.E SAMPLE PAPER‐[XII]2

Marks

3. If A = [aij] is a square matrix of order 2 such that aij = :isAthen,when,0

when,1 2

⎩⎨⎧

=

≠

ji

ji1

(a) ⎥⎦

⎤⎢⎣

⎡0101

(b) ⎥⎦

⎤⎢⎣

⎡0011

(c) ⎥⎦

⎤⎢⎣

⎡0111

(d) ⎥⎦

⎤⎢⎣

⎡1001

4. The value of k, for which A = ⎥⎦

⎤⎢⎣

⎡k

k248

is a singular matrix is : 1

(a) 4 (b) – 4(c) ± 4 (d) 0

5. Find the intervals in which the function f given by f (x) = x2 – 4x + 6 is strictlyincreasing. 1(a) (– ∞, 2) ∪ (2, ∞) (b) (2, ∞)(c) (– ∞, 2) (d) (– ∞, 2] ∪ (2, ∞)

6. Given that A is a square matrix of order 3 and | A | = – 4, then | adj. A | is 1equal to(a) – 4 (b) 4(c) – 16 (d) 16

7. A relation R in set A = {1, 2, 3} is defined as R = {(1, 1), (1, 2), (2, 2), (3, 3)}.Which of the following ordered pair in R shall be removed to make it anequivalence relation in A ? 1(a) (1, 1) (b) (1, 2)(c) (2, 2) (d) (3, 3)

8. If ⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡+

+2411

3–434–52–2

dcdcbaba

, then value of a + b – c + 2d is 1

(a) 8 (b) 10(c) 4 (d) – 8

9. The point at which the normal to the curve y = 0,1

>+ xx

x is perpendicular to

the line 3x – 4y – 7 = 0 is 1

(a) ⎟⎠⎞

⎜⎝⎛

25

,2 (b) ⎟⎠⎞

⎜⎝⎛±

25

,2

(c) ⎟⎠⎞

⎜⎝⎛

25

,21

– (d) ⎟⎠⎞

⎜⎝⎛

25

,21


Marks

10. sin (tan–1 x), where |x| < 1, is equal to : 1

(a)2–1 x

x(b)

2–1

1

x

(c)21

1

x+(d)

21 x

x

+

11. Let the relation R in the set A = { x ∈ Z : 0 ≤ x ≤ 12}, given by R = {(a, b) : |a – b|is a multiple of 4}. Then [1], the equivalence class containing 1, is 1

(a) {1, 5, 9} (b) {0, 1, 2, 5}

(c) φ (d) A

12. If ex + ey = ex + y , then dxdy

is : 1

(a) e y – x (b) e x + y

(c) – e y – x (d) 2e x – y

13. Given that matrices A and B are of order 3 × n and m × 5 respectively, thenthe order of matrix C = 5A + 3B is 1

(a) 3 × 5 and m = n (b) 3 × 5

(c) 3 × 3 (d) 5 × 5

14. If y = 5 cos x – 3 sin x, then 2

2

dx

yd is equal to : 1

(a) – y (b) y

(c) 25y (d) 9y

15. For matrix A = ⎥⎦

⎤⎢⎣

⎡711–52

, (adj. A)′ is equal to : 1

(a) ⎥⎦

⎤⎢⎣

⎡7–115–2–

(b) ⎥⎦

⎤⎢⎣

⎡21157

(c) ⎥⎦

⎤⎢⎣

⎡25–

117(d) ⎥

⎦

⎤⎢⎣

⎡2115–7

16. The points on the curve 1169

22

=+yx at which the tangents are parallel to

y-axis are : 1(a) (0, ± 4) (b) (± 4, 0)

(c) (± 3, 0) (d) (0, ± 3)


Marks

17. Given that A = [aij] is a square matrix of order 3 × 3 and |A| = – 7, then

the value of ∑=

3

122 A

iiia , where Aij denotes the cofactor of element aij is : 1

(a) 7 (b) – 7(c) 0 (d) 49

18. If y = log (cos e x), then dxdy

is : 1

(a) cos e x – 1 (b) e – x cos ex

(c) ex sin ex (b) – e x tan ex

19. Based on the given shaded region as thefeasible region in the graph, at whichpoint(s) is the objective functionZ = 3x + 9y maximum ? 1(a) Point B(b) Point C(c) Point D(d) every point on the line segment CD

20. The least value of the function f (x) = 2 cos x + x in the closed interval ⎥⎦⎤

⎢⎣⎡

2,0π is : 1

(a) 2 (b) 36+

π

(c)2π

(d) The least value does not exist

SECTION–B

In this section, attempt any 16 questions out of the Questions 21 - 40.Each Question is of 1 mark weightage.

21. The function f : R → R defined as f (x) = x3 is : 1(a) one-one but not onto (b) not one-one but onto(c) neither one-one nor onto (d) one-one and onto

22. If x = a sec θ, y = b tan θ, then 2

2

dx

yd at θ =

6π

is :1

(a) 233

–a

b(b)

ab32

–

(c)a

b33– (d) 233

–a

b

5 20 35 50x y + 3 = 60

X

x y + = 10(10,0)

Y′

OX′(0,10) 5

15

25 D(0,20)C(15,15)

(60,0)B(5,5)

x y

= Y


Marks

23. In the given graph, the feasible region for a LPPis shaded. The objective function Z = 2x – 3y, willbe minimum at : 1

(a) (4, 10)

(b) (6, 8)

(c) (0, 8)

(d) (6, 5)

24. The derivative of sin–1 )–12( 2xx w.r.t sin–1 x, ,12

1<< x is : 1

(a) 2 (b) 2–2π

(c)2π

(d) – 2

25. If A = ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

21043201–1

and B = ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

51–24–24–4–22

, then : 1

(a) A–1 = B (b) A–1 = 6B

(c) B–1 = B (d) B–1 = A61

26. The real function f (x) = 2x3 – 3x2 – 36x + 7 is : 1

(a) Strictly increasing in (– ∞, – 2) and strictly decreasing in (– 2, ∞)

(b) Strictly decreasing in (– 2, 3)

(c) Strictly decreasing in (– ∞, 3) and strictly increasing in (3, ∞)

(d) Strictly decreasing in (– ∞, – 2) ∪ (3, ∞)

27. The simplest form of tan–1 2π3π,

cos–1–cos1

cos–1cos1<<

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

+

++x

xx

xx is : 1

(a)2

–4π x

(b)2

–2π3 x

(c)2

–x

(d)2

–π x

O 1 2 3 4 5 6 7

1

2

3

4

5

6

7

8

9

10

11

X

Y

A(0, 8)

B(4, 10)

C(6, 8)

D(6, 5)

E(5, 0)


Marks

28. Given that A is a non-singular matrix of order 3 such that A2 = 2A, then valueof |2A| is : 1(a) 4 (b) 8(c) 64 (d) 16

29. The value of b for which the function f (x) = x + cos x + b is strictly decreasingover R is : 1(a) b < 1 (b) No value of b exists(c) b ≤ 1 (d) b ≥ 1

30. Let R be the relation in the set N given by R = {(a, b) : a = b – 2, b > 6}, then : 1

(a) (2, 4) ∈ R (b) (3, 8) ∈ R

(c) (6, 8) ∈ R (d) (8, 7) ∈ R

31. The point(s), at which the function f given by ⎪⎩

⎪⎨

⎧

≥

<=

0,1–

0,||)(

x

xx

xxf is continuous,

is/are : 1(a) x ∈ R (b) x = 0(c) x ∈ R – {0} (d) x = – 1 and 1

32. If A = ⎥⎦

⎤⎢⎣

⎡4–3

20 and kA = ⎥

⎦

⎤⎢⎣

⎡24230

ba

, then the values of k, a and b respectively are : 1

(a) – 6, – 12, – 18 (b) – 6, – 4, – 9(c) – 6, 4, 9 (d) – 6, 12, 18

33. A linear programming problem is as follows :

Minimize Z = 30x + 50y subject to the constraints,

3x + 5y ≥ 15

2x + 3y ≤ 18

x ≥ 0, y ≥ 0

In the feasible region, the minimum value of Z occurs at 1

(a) a unique point (b) no point

(c) infinitely many points (d) two points only

34. The area of a trapezium is defined by function f and is given by

f (x) = (10 + x) 2–100 x , then the area when it is maximised is : 1

(a) 75 cm2 (b) 2cm37

(c) 2cm375 (d) 5 cm2


Marks

35. If A is square matrix such that A2 = A, then (I + A)3 – 7 A is equal to : 1

(a) A (b) I + A (c) I – A (d) I36. If tan–1 x = y, then : 1

(a) – 1 < y < 1 (b) 22–

ππ≤≤ y

(c)22

–ππ

<< y (d)⎭⎬⎫

⎩⎨⎧∈

2,

2–

ππy

37. Let A = {1, 2, 3}, B = {4, 5, 6, 7} and let f = {(1, 4), (2, 5), (3, 6)} be a functionfrom A to B. Based on the given information, f is best defined as : 1(a) Surjective function (b) Injective function(c) Bijective function (d) function

38. If A = ⎥⎦

⎤⎢⎣

⎡21–13

, then 14A–1 is given by : 1

(a) ⎥⎦

⎤⎢⎣

⎡311–2

14 (b) ⎥⎦

⎤⎢⎣

⎡622–4

(c) ⎥⎦

⎤⎢⎣

⎡3–11–2

2 (d) ⎥⎦

⎤⎢⎣

⎡2–11–3–

2

39. The point(s) on the curve y = x3 – 11x + 5 at which the tangent is y = x – 11is / are : 1(a) (– 2, 19) (b) (2, – 9)(c) (± 2, 19) (d) (– 2, 19) and (2, – 9)

40. Given that A = ,I3Aandγ

βα 2 =⎥⎦

⎤⎢⎣

⎡α–

then : 1

(a) 1 + α2 + βγ = 0 (b) 1 – α2 – βγ = 0(c) 3 – α2 – βγ = 0 (d) 3 + α2 + βγ = 0

SECTION – C

In this section, attempt any 8 questions. Each question is of 1-mark weightage.Questions 46-50 are based on a Case-Study

41. For an objective function Z = ax + by, where a, b > 0 ; the corner points ofthe feasible region determined by a set of constraints (linear inequalities) are(0, 20), (10, 10), (30, 30) and (0, 40). The condition on a and b such that themaximum Z occurs at both the points (30, 30) and (0, 40) is : 1(a) b – 3a = 0 (b) a = 3b(c) a + 2b = 0 (d) 2a – b = 0


Marks

42. For which value of m is the line y = mx + 1 a tangent to the curve y2 = 4x ? 1

(a)21

(b) 1

(c) 2 (d) 3

43. The maximum value of 10,]1)1–([ 31

≤≤+ xxx is : 1

(a) 0 (b)21

(c) 1 (d) 331

44. In a linear programming problem, the constraints on the decision variables

x and y are x – 3y ≥ 0, y ≥ 0, 0 ≤ x ≤ 3. The feasible region 1

(a) is not in the first quadrant (b) is bounded in the first quadrant

(c) is unbounded in the first quadrant (d) does not exist

45. Let A = ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

1ααα

α

sin– 1–sin1sin–

1sin1

, where 0 ≤ α ≤ 2π, then : 1

(a) |A| = 0 (b) |A| ∈ (2, ∞)(c) |A| ∈ (2, 4) (d) |A| ∈ [2, 4]

CASE STUDY

The fuel cost per hour for running a train isproportional to the square of the speed it generatesin km per hour. If the fuel costs ` 48 per hour atspeed 16 km per hour and the fixed charges to runthe train amount to ` 1200 per hour. Assume thespeed of the train as v km/h.Based on the given information, answer the following questions.46. Given that the fuel cost per hour is k times the square of the speed the train

generates in km/h, the value of k is : 1

(a) 316

(b) 31

(c) 3 (d) 163


Marks

47. If the train has travelled a distance of 500 km, then the total cost of runningthe train is given by the function : 1

(a) vv

6000001615

+ (b) vv

6000004

375+

(c) vv

150000165 2 + (d) v

v6000

163

+

48. The most economical speed to run the train is : 1(a) 18 km/h (b) 5 km/h(c) 80 km/h (d) 40 km/h

49. The fuel cost for the train to travel 500 km at the most economical speed is : 1

(a) ` 3750 (b) ` 750

(c) ̀ 7500 (d) ` 75000

50. The total cost of the train to travel 500 km at the most economical speed is : 1

(a) ` 3750 (b) ` 75000

(c) ` 7500 (d) ` 15000

Marking Scheme

Q. No. Correct Hints / SolutionsOption

1. (d) 12π

sin6π

3π

sin =⎟⎠⎞

⎜⎝⎛=⎥

⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛−−

2. (b)21

sincos1

Lim0

=⎟⎟⎠

⎞⎜⎜⎝

⎛ −→ xx

kxx

⇒ 21

sin2

sin2Lim

2

0=

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

→ xx

kx

x

⇒21

sin2

2sin

22Lim

2

2

0=⎟⎟

⎠

⎞⎜⎜⎝

⎛

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

⎟⎠⎞

⎜⎝⎛

→ xx

kx

kxk

x

⇒ k2 = 1 ⇒ k = ± 1 but k < 0 ⇒ k = – 1

3. (d) A2 = ⎥⎦

⎤⎢⎣

⎡1001


4. (c) As A is singular matrix

⇒ |A| = 0

⇒ 2k2 – 32 = 0 ⇒ k = ± 4

5. (b) f (x) = x2 – 4x + 6

f ′(x) = 2x – 4

Let f ′(x) = 0 ⇒ x = 2 2– ∞ ∞

(–) (+)

As f ′(x) > 0 for all x ∈ (2, ∞)

⇒ f (x) is strictly increasing in (2, ∞)

6. (d) As |adj. A| = |A| n – 1, where n is order of matrix A

= ( – 4)2 = 16

7. (b) (1, 2)

8. (a)

4321

2434115

3242

====

⇒

⎪⎪⎭

⎪⎪⎬

⎫

=+=−

−=−=+

dcba

dcdcbaba

∴ a + b – c + 2d = 8

9. (a) f (x) = 0,11

1)(0,1

2

2

2>

−=−=⇒>+ ′ x

x

x

xxfx

xx

As normal to f (x) is perpendicular to the given line

⇒ 143

1 2

2

−=×⎟⎟⎠

⎞⎜⎜⎝

⎛

− x

x[Q m1 . m2 = – 1]

⇒ x2 = 4 ⇒ x = ± 2But x > 0, ∴ x = 2

∴ Point = ⎟⎠⎞

⎜⎝⎛

25

,2

10. (d) sin (tan – 1 x) = 22

1

11sinsin

x

x

x

x

+=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

+

−

11. (a) {1, 5, 9}

12. (c) ex + ey = e x + y

⇒ e – y + e – x = 1

Differentiating w.r.t. x :

xyxy edxdy

edxdy

e −−− −=⇒=−− 0


13. (b) 3 × 514. (a) y = 5 cos x – 3 sin x

⇒ xxdxdy

cos3sin5 −−=

⇒ yxxdx

yd−=+−= sin3cos5

2

2

15. (c) adj. A = ⎥⎦

⎤⎢⎣

⎡−

=⇒⎥⎦

⎤⎢⎣

⎡ −′

25117

)Aadj.(21157

16. (c) 0162

92

1169

22

=+⇒=+dxdyyxyx

⇒ Slope of normal = xy

dydx

169

=−

As curve’s tangent is parallel to y-axis

⇒ 0169

=xy

⇒ y = 0 and x = ± 3

∴ Points are (± 3, 0)

17. (b) |A| = – 7

∴ 7|A|AAAA 32322222121222

3

1

−==++=∑=

aaaa iii

18. (d) y = log (cos ex)

Differentiating w.r.t. x :

xxx

eeedx

dy.)sin(.

)(cos1

−= [By Chain rule]

⇒ xx eedxdy

tan−=

19. (d) Z is maximum 180 at points C(15, 15) and D(0, 20)

⇒ Z is maximum at every point on the line segment CD.

20. (c) f (x) = 2 cos x + x, x ∈ ⎥⎦⎤

⎢⎣⎡ π

2,0

f ′ (x) = – 2 sin x + 1

Let f ′ (x) = 0 ⇒ ⎥⎦⎤

⎢⎣⎡ π

∈π

=2

,06

x

f (0) = 2

362

+π

=⎟⎠⎞

⎜⎝⎛ π

f

22π

=⎟⎠⎞

⎜⎝⎛ π

f ⇒ Least value of f (x) is 2

at2

π=

πx .


Section – B

21. (d) Let f (x1) = f (x2) for all x1 x2 ∈ R Let f (x) = x3 = y for all y ∈ R

⇒ x13 = x2

3 ⇒ x = y1/3

⇒ x1 = x2 Every image y ∈ R has a unique

⇒ f is one-one. pre-image in R

⇒ f is onto.

∴ f is one-one and onto

22. (a) x = a sec θ ⇒ θθ=θ

sectanaddx

y = b tan θ ⇒ θ=θ

2secbddy

∴ θ= cosecab

dxdy

⇒ θ−

=θ

θθ−

= 322

2

cot.cot.coseca

bdxd

ab

dxyd

∴ 2

6

2

2 33

a

b

dx

yd −=

⎥⎥⎦

⎤

π=θ

23. (c) Z is minimum – 24 at (0, 8)

24. (a) Let u = )12(sin 21 xx −− and v = 12

1,sin 1 <<− xx

⇒ sin v = x ...(1)

Using (1), we get : sin – 1 (2 sin v cos v) ⇒ u = 2v

Differentiating w.r.t. v, we get 2=dvdu

25. (d) AB = 6 I ⇒ B – 1 = A61

26. (b) f ′(x) = 6 (x2 – x – 6) = 6 (x – 3) (x + 2) 3– ∞ ∞(–) (+)– 2As f ′(x) < 0 for all x ∈ ( – 2, 3)

⇒ f (x) is strictly decreasing in ( – 2, 3)

27. (a)⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−−+

−++−

xx

xx

cos1cos1

cos1cos1tan 1

= 2

3,

2sin2

2cos2

2sin2

2cos2

tan 1 π<<π

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

−−

+−− x

xx

xx


= 24

2sin

2cos

2sin

2cos

tan 1 xxx

xx

−π

=⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

+

−−

28. (c) A2 = 2A⇒ |A2| = |2A|

⇒ |A|2 = 23 |A| [Q |kA| = kn|A| for a matrix of order n]

⇒ Either |A| = 0 or |A| = 8

But A is non-singular matrix

∴ |Α| = 82 = 64

29. (b) f ′(x) = 1 – sin x ⇒ f ′(x) > 0 for all x ∈ R

⇒ No value of b exists.

30. (c) a = b – 2 and b > 6

⇒ (6, 8) ∈ 6

31. (a) f (x) = ⎪⎩

⎪⎨⎧

≥−

<−=−

0,1

0,1

x

xx

x

⇒ f (x) = – 1 for all x ∈ R

⇒ f (x) is continuous for all x ∈ R as it is a constant function.

32. (b) kA = ⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡− 242

3043

20b

akk

k

⇒ k = – 6, a = – 4 and b = – 9

33. (d) Corner points of feasible region Z = 30x + 50y

(5, 0) 150(9, 0) 270(0, 3) 150(0, 6) 300

Minimum value of Z occurs at two points

34. (c) f ′(x) = 2

2

100

100102

x

xx

−

+−−

f ′(x) = 0 ⇒ x = – 10 or 5. But x > 0 ⇒ x = 5

f ′′(x) = 075

30)5(

)100(

100030022/3

3

<−

=⇒−

−− ′fx

xx

⇒ Maximum area of trapezium is 375 cm2 when x = 5

35. (d) (I – A)3 – 7A = I + A + 3A + 3A – 7A = I


36. (c)22π

<<π

− y

37. (b) As every pre-image x ∈ A has a unique image y ∈ B

⇒ f is injective function.

38. (b) |A| = 7, (adj. A) = ⎥⎦

⎤⎢⎣

⎡ −3122

∴ 14A– 1 = ⎥⎦

⎤⎢⎣

⎡ −6224

39. (b) y = x3 – 11x + 5 ⇒ 113 2 −= xdxdy

Slope of line y = x – 11 is 1 ⇒ 3x2 – 11 = 1 ⇒ x = ± 2

∴ Point is (2, – 9) as ( – 2, 19) does not satisfy the given line.

40. (c) A2 = 3I

⇒ ⎥⎦

⎤⎢⎣

⎡=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

α+βγβγ+α

3003

00

2

2

⇒ 3 – α2 – βγ = 0

Section – C

41. (a) As Z is maximum at (30, 30) and (0, 40)

⇒ 30a + 30b = 40b ⇒ b – 3a = 0

42. (b) y = mx + 1 ...(1) and y2 = 4x ...(2)

Substituting (1) in (2), we have (mx + 1)2 = 4x

⇒ m2x2 + (2m – 4) x + 1 = 0 ...(3)

As line is tangent to the curve

⇒ Line touches the curve at only one point

⇒ (2m – 4)2 – 4m2 = 0 ⇒ m = 1

43. (c) Let f (x) = [x (x – 1 ) + 1] 1/3, 0 ≤ x ≤ 1

f ′(x) = 3/22 )1(3

12

+−−

xx

x and let f ′(x) = 0 ⇒ x = ]1,0[

21

∈

f (0) = 1, 1)1(and43

21 3

1

=⎟⎠⎞

⎜⎝⎛=⎟

⎠⎞

⎜⎝⎛ ff

∴ Maximum value of f (x) is 1

44. (b) Feasible region is bounded in the first quadrant.

45. (d) |A| = 2 + 2 sin2 θ

As – 1 ≤ sin θ ≤ 1, for all 0 ≤ θ ≤ 2π

⇒ 2 ≤ 2 + 2 sin2 θ ≤ 4 ⇒ |A| ∈ [2, 4]


46. (d) Fuel cost = k (speed)2

⇒ 48 = k .162 ⇒ k = 163

47. (b) Total cost of running train (let C) = ttv 1200163 2 +

Distance covered = 500 km ⇒ Time = hrs500

v

Total cost of running the train for 500 km = ⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛

vvv

5001200

500163 2

⇒ C = 2

6000004

375v

v +

48. (c) 2

6000004

375Cvdv

d−=

Let km/h800C

=⇒= vdvd

49. (c) Fuel cost for running 500 km = 804

3754

375×=v = ` 7500

50. (d) Total cost for running 500 km = v

v600000

4375

+

= 80

6000004

80375+

× = ` 15000


SAMPLE OMR SHEET

MODEL TEST PAPER – 1Time : 90 minutes Maximum Marks : 40

General Instructions :1. This paper contains three sections - A, B and C. Each part is compulsory.

2. Section A consists of 20 multiple choice questions. Attempt any 16 out of 20.

3. Section B consists of 20 multiple choice questions. Attempt any 16 out of 20.

4. Section C consists of 10 multiple choice questions. Attempt any 8 out of 10.



SECTION – A

In this section, attempt any 16 out of questions 1-20. Each question is of 1 markweightage.

1. If cij is the cofactor of aij of the determinant

7–51

406

53–2

, then the value of a32.c32 is

(a) 110 (b) 22

(c) – 110 (d) – 22

2. If )(cos1)( 22 xxf += , then the value of ⎟⎟⎠

⎞⎜⎜⎝

⎛ π′

2f is

(a)6π

(b)6

–π

(c)6

1(d) 6

π

3. Which of the following is the principal value branch of cosec–1 x ?

(a) ⎟⎠⎞

⎜⎝⎛ ππ

2,

2– (b) ( ) ⎥⎦

⎤⎢⎣⎡π

π2

–,0

(c) ⎥⎦⎤

⎢⎣⎡ ππ

2,

2– (d) }0{–

2,

2– ⎥⎦

⎤⎢⎣⎡ ππ

4. The function f : R → R given by f (x) = x3 – 1 is

(a) a one-one function (b) an onto function

(c) a bijection (d) neither one-one nor onto

MODEL TEST PAPERS‐[XII]18

5. If matrix ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

760

543

1–01

A and its inverse is denoted by

333231

232221

1312111–A

aaa

aaa

aaa

= , then the

value of a23 is

(a) 2021

(b) 51

(c)52

– (d) 52

6. If A is a non-singular matrix of order 3 and |adj A| = |A|k, then the value of k is

(a) 2 (b) 4

(c) 6 (d) 8

7. If ⎪⎩

⎪⎨

⎧

=

≠=

1,0

1,1–1–][

)(x

xxx

xf , then f(x) is

(a) continuous as well as differentiable at x = 1

(b) differentiable but not continuous at x = 1

(c) continuous but not differentiable at x = 1

(d) neither continuous nor differentiable at x = 1

8. Let A = {1, 2, 3} and R = {(1, 2), (2, 3)} be a relation in A. How many minimum number ofordered pairs may be added to the R, so that R becomes an equivalence relation ?

(a) 7 (b) 5

(c) 1 (d) 4

9. If 011 =+++ xyyx , then dxdy

is equal to

(a) xx 1+

(b) 11+x

(c) 2)1(

1–

x+(d) x

x+1

10. If ⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡=

242

30Aand

4–3

20A

b

ak , then the values of k, a, b respectively are

(a) – 6, –12, –18 (b) – 6, 4, 9(c) – 6, – 4, – 9 (d) – 6, 12, 18

11. The equation of all lines having slope 2 which are tangent to the curve 3,3–

1≠= x

xy ,

is(a) y = 2 (b) y = 2x

(c) y = 2x + 3 (d) none of these

MODEL TEST PAPERS‐[XII] 19

12. The difference between the greatest and least values of the function f(x) = sin 2 x – x, on

⎥⎦⎤

⎢⎣⎡ ππ

2,

2– is

(a)2π

(b) π

(c)2

3π(d)

4π

13. If 0

2–

1

230

150

231

]11[ =⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ x

x , then x is equal to

(a) 21

– (b)21

(c) 1 (d) – 1

14. The interval in which the function f (x) = cot–1 x + x increases is

(a) (1, ∞) (b) (– 1, ∞)

(c) (0, ∞) (d) (– ∞, ∞)

15. If AX = B such that B = ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

4/3–4/3–2

4/54/34–

2/1–2/1–3

Aand

0

52

91– , then X is equal to :

(a) ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

4/3–

4/3

3

(b)⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

7

2/1–

3

(c) ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

3

21

4–

(d)⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

21–

3

1

16. If f (x) = x2 + 4x – 5 and ⎥⎦

⎤⎢⎣

⎡=

3–4

21A , then f (A) is equal to

(a) ⎥⎦

⎤⎢⎣

⎡88

4–0(b) ⎥

⎦

⎤⎢⎣

⎡02

12

(c) ⎥⎦

⎤⎢⎣

⎡01

11(d) ⎥

⎦

⎤⎢⎣

⎡08

48

17. If y = (tan x)sin x, then dxdy

is equal to

(a) (tan x)sin x [sec x + cos x] (b) (tan x)sin x [sec x + log tan x]

(c) (tan x)sin x sec x (d) (tan x)sin x [sec x + cos x (log tan x)]


18. The principal value of sec–1 (2) is

(a) 6π

(b) 3π

(c) 32π

(d) π

19. On the interval [0, 1], the function x25 (1 – x)75 takes its maximum value at the point

(a) 0 (b)41

(c)21

(d) 31

20. The region represented by the inequalities 2x + y ≤ 10, x ≥ 0, y ≥ 0 and x ≥ 6, y ≥ 2 is

(a) unbounded (b) a polygon

(c) exterior of a triangle (d) empty set

SECTION B


21. The number of equivalence relations in the set {1, 2, 3} containing (1, 2) and (2, 1) is

(a) 2 (b) 3

(c) 1 (d) 4

22. If |x| ≤ 1, then 2 tan–1 x + sin –1 ⎟⎟⎠

⎞⎜⎜⎝

⎛

+ 212

x

x is equal to

(a) 4 tan–1x (b)2π

(c) 0 (d) π

23. If

⎪⎪⎩

⎪⎪⎨

⎧

=+

≠+=

0,21

0,2

5sin

)(2

xk

xxx

x

xf is continuous at x = 0, then the value of k is

(a) 1 (b) – 2(c) 2 (d) 1/2

24. If ⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡10

01

3–5

23–A

23

12, then matrix A is

(a) ⎥⎦

⎤⎢⎣

⎡01

11(b) ⎥

⎦

⎤⎢⎣

⎡10

11

(c) ⎥⎦

⎤⎢⎣

⎡11

01(d) ⎥

⎦

⎤⎢⎣

⎡11

10


25. If y = |sin x||x|, then the value of dxdy

at 6

–π

=x is

(a) [ ]π

π

3–2log66

2 6–

(b) [ ]π+

π

32log66

26

(c) [ ]π+

π

32log66

2 6–

(d) none of these

26. Let A = {1, 2, 3} and B = {a, b, c}. Then the number of bijective functions from A to Bare

(a) 2 (b) 8

(c) 6 (d) 4

27. Let f : R → R be function defined by f (x) = sin (2x – 3). Then f is

(a) injective (b) surjective

(c) bijective (d) neither injective nor surjective

28. The value of ⎟⎟⎠

⎞⎜⎜⎝

⎛

35

cos21 1– is

(a) 2

53 +(b)

25–3

(c) 65

(d) none of these

29. The roots of the equation 0

90

75

160

=

x

x

x

are

(a) 0, 12, 12 (b) 0, 12, –12(c) 0, 12, 16 (d) 0, 9, 16

30. If ⎟⎟⎠

⎞⎜⎜⎝

⎛θθ+

=2cos–12cos1

y , then θddy

at θ = 43π

is equal to

(a) – 2 (b) 2

(c) ±2 (d) none of these

31. The value of the derivative of |x – 1| + |x – 3| at x = 2 is

(a) – 2 (b) 0

(c) 2 (d) not defined


32. If ⎪⎩

⎪⎨⎧

=

≠=0,0

0,1

cos)(x

xx

xxfk

is continuous at x = 0, then

(a) k < 0 (b) k > 0

(c) k = 0 (d) k ≥ 0

33. If the function ⎪⎩

⎪⎨

⎧

=

≠++

=2,

2)2–(

2016–)( 2

23

xk

xx

xxxxf is continuous for all x, then k is equal to

(a) 3 (b) 5

(c) 7 (d) 9

34. If A is symmetric as well as skew-symmetric matrix, then A is a

(a) diagonal matrix (b) null matrix

(c) triangular matrix (d) none of these

35. If ⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡=

1–

1B,

1–2

1–1A

y

x and (A + B)2 = A2 + B2, then x + y is equal to

(a) 2 (b) 3

(c) 4 (d) 5

36. If )(tan 2

.log xxex + , then dxdy

is equal to

(a) ⎥⎦⎤

⎢⎣⎡ +++ xxxx

e xx log)(sec1 2)(tan 2

(b) ⎥⎦⎤

⎢⎣⎡ ++ xxxx

e xx log)–(sec1 2)(tan 2

(c) ⎥⎦⎤

⎢⎣⎡ +++ xxxx

e xx log)2(sec1 2)(tan 2

(d) ⎥⎦⎤

⎢⎣⎡ ++ xxxx

e xx log)2–(sec1 2)(tan 2

37. Region represented by x ≥ 0, y ≥ 0 is

(a) first quadrant (b) second quadrant

(c) third quadrant (d) fourth quadrant

38. If ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

ba 2

2–12

221

A is a matrix satisfying AA′ = 9I3, then the values of a and b are

(a) a = 1, b = 2 (b) a = – 2, b = – 1(c) a = – 1, b = 2 (d) a = – 2, b = 1


39. Let ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

001–

01–0

1–00

A . Which of the following statement is correct :

(a) A2 = I (b) A–1 = – A(c) A–1 does not exist (d) A is a zero matrix

40. The solution region satisfied by the inequalities

x + y ≤ 5, 5x + y ≥ 5, x + 6y ≥ 6, x ≤ 4, y ≤ 4, x ≥ 0, y ≥ 0 is bounded by

(a) 4 straight lines (b) 5 straight lines

(c) 6 straight lines (d) unbounded

SECTION C

In this section, attempt any 8 questions. Each question is of 1 mark weightage.Q. 46 to 50 are based on a case study.

41. The angle of intersection of the curve y2 = x and x2 = y is

(a) ⎟⎠⎞

⎜⎝⎛

23

tan 1– (b) ⎟⎠⎞

⎜⎝⎛

43

tan 1–

(c) ⎟⎠⎞

⎜⎝⎛

21

tan 1– (d) ⎟⎠⎞

⎜⎝⎛

51

tan 1–

42. If at x = 1, the function x4 – 62x2 + ax + 9 attains its maximum value on the interval [0, 2],then a is equal to

(a) 110 (b) 120

(c) 55 (d) 105

43. The minor of the element a11 in the matrix ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

541

871

962

is

(a) 0 (b) 3

(c) 5 (d) 7

44. If a point (h, k) satisfies an inequation ax + by ≥ 4, then the half plane represented by theinequation is

(a) half plane containing the point (h, k) but excluding the points on ax + by = 4

(b) half plane containing the point (h, k) including the points on ax + by = 4

(c) whole xy-plane (d) none of these45. The maximum value of Z = x + 3y subject to constraints 2x + y ≤ 20, x + 2y ≤ 20, x ≥ 0,

y ≥ 0 is(a) 10 (b) 60(c) 30 (d) 20


Case Study

A mini pocket sized music player comes with 5000 hand-picked high quality songs inside.Company’s financial officer is trying to create a simple profit function to identify how smalldistributors can be provided a higher discount during festival time without eating into theprofit margin too much.

The officer has come up with the plan of production of x music players per

day at a total cost of ̀ ⎟⎟⎠

⎞⎜⎜⎝

⎛++ 2535

4

2

xx

and sell these to the distributors

at a price of ` )100(21

x− per piece.

Based on the above information, answer the following :

46. What is the profit function P(x) ?

(a) ⎟⎟⎠

⎞⎜⎜⎝

⎛++−− 2535

4)100(

21 2

xx

xx (b) )100(21

25354

2

xxxx

−−⎟⎟⎠

⎞⎜⎜⎝

⎛++

(c) ⎟⎟⎠

⎞⎜⎜⎝

⎛++ 2535

4

2

xx

(d) )100(21

xx −

47. What is the profit made when 8 pocket units are produced?

(a) ` 32 (b) ` 35

(c) ` 47 (d) ` 54

48. What is the number of units produced to give the maximum profit?

(a) 10 (b) 12

(c) 15 (d) 18

49. What is the maximum profit ?

(a) ` 38 (b) ` 45

(c) ` 50 (d) ` 60

50. What is the total revenue when 6 units are sold ?

(a) ` 8 (b) ` 274 (c) ` 282 (d) none of these

ANSWERS

1. (a) 2. (b) 3. (d) 4. (c) 5. (c) 6. (a)7. (d) 8. (a) 9. (c) 10. (c) 11. (d) 12. (b)

13. (b) 14. (d) 15. (d) 16. (d) 17. (d) 18. (b)19. (b) 20. (d) 21. (a) 22. (a) 23. (c) 24. (a)25. (a) 26. (c) 27. (d) 28. (b) 29. (b) 30. (a)31. (b) 32. (b) 33. (c) 34. (b) 35. (d) 36. (c)37. (a) 38. (b) 39. (a) 40. (b) 41. (b) 42. (b)43. (b) 44. (b) 45. (c) 46. (a) 47. (c) 48. (a)49. (c) 50. (c)









SECTION – A


1. If y = log tan x , then dxdy

is

(a)x2

1(b)

xx

x

tan

sec2

(c) x2sec2 (d)xx

x

tan2

sec2

2. Let function f : R → R is defined as f (x) = 2x + sin x for x ∈ R. Then f is

(a) one-one and onto

(b) one-one but not onto

(c) onto but not one-one

(d) neither one-one nor onto

3. The principal value of ⎟⎟⎠

⎞⎜⎜⎝

⎛

23

–sin 1– is :

(a) 3–

π (b) 6

π

(c) 32

–π

(d) 32π


4. If y = 3– x, then dxdy

is equal to :

(a) 13– +x

x(b) 3x log 3

(c) 3– x log 3 (d)x331

log

5. If ⎥⎦

⎤⎢⎣

⎡ +=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=⎥

⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡=

1

15Dand

1

2

1

C,13

1–02B,

22–1

014A

x

xsuch that (2A – 3B) C = D,

then x is equal to

(a) 3 (b) – 4

(c) – 6 (d) 6

6. If area of triangle is 4 sq units with vertices (– 2, 0), (0, 4) and (0, k), then k is equal to

(a) 0, – 8 (b) 8

(c) – 8 (d) 0, 8

7. The relation R on the set Z defined by R = {(a, b) : (a – b) is divisible by 5} divides the set Zinto how many disjoint equivalence classes ?

(a) 5 (b) 2

(c) 3 (d) 4

8. If a function f (x) is defined as ⎪⎩

⎪⎨

⎧

=

≠=

0,0

0,)( 2

x

xx

x

xf then :

(a) f(x) is continuous at x = 0 but not differentiable at x = 0

(b) f(x) is continuous as well as differentiable at x = 0

(c) f(x) is discontinuous at x = 0

(d) none of these.

9. If )tan1(log

21 2 x

ey+

= , then dxdy

is equal to

(a) x2sec21

(b) sec2 x

(c) sec x tan x (d))tan1(log

21 2 x

e+

10. If y = 5x⋅ x5, then dxdy

is

(a) 5x (x5 log 5 – 5x4) (b) x5 log 5 – 5x4

(c) x5 log 5 + 5x4 (d) 5x (x5 log 5 + 5x4)


11. If y = exx , then

dxdy

is

(a) y (1 + loge x) (b) yxx (1 + loge x)

(c) yex (1 + loge x) (d) none of these

12. The graph of inequations x ≤ y and y ≤ x + 3 is located in

(a) II quadrant (b) I, II quadrants

(c) I, II, III quadrants (d) II, III, IV quadrants

13. If ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

100

010

001

A , then A2 + 2A = _________ .

(a) A (b) 3A

(c) 2A (d) 5A

14. For what values of x and y are the following matrices equal

⎥⎥⎦

⎤

⎢⎢⎣

⎡ ++=⎥

⎦

⎤⎢⎣

⎡ +=

6–0

23B,

5–0

312A

2

2yx

yy

yx

(a) x = 2, y = 3 (b) x = 3, y = 4

(c) x = 2, y = 2 (d) x = 3, y = 3

15. For a matrix A, AI = A and AA′ = I is true when

(a) A is a square matrix (b) A is a non-singular matrix.

(c) A is a symmetric matrix (d) A is any matrix.

16. The co-factors of elements a12 and a32 of the matrix ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

θ

θθ

θ

1sin–1–

sin1sin–

1sin1

are

respectively(a) 0 and – 2 sin θ (b) 2 and 2 sin θ

(c) 2 and – 2 sin θ (d) – 2 sin θ and 0

17. If the matrix ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ +λ

1053

842

231

is singular, then λ is equal to

(a) – 2 (b) 4

(c) 2 (d) – 4

18. The derivative of ex3 with respect to log x is

(a) ex3 (b) 3x2 . 2ex3

(c) 3x3 ex3 (d) 3x3 ex3 + 3x2


19. A wire 34 cm long is to be bent in the form of a quadrilateral of which each angle is 90°.What is the maximum area which can be enclosed inside the quadrilateral ?

(a) 68 cm2 (b) 70 cm2

(c) 72.25 cm2 (d) 71. 25 cm2

20. If 0cos22

sincos 1–1– =⎟⎠⎞

⎜⎝⎛ + x , then x is equal to

(a) 51

(b) 52

(c) 0 (d) 1

SECTION B


21. Which of the following functions is decreasing on ⎟⎠⎞

⎜⎝⎛ π

2,0 ?

(a) sin 2x (b) tan x

(c) cos x (d) cos 3x

22. f : X → Y is onto, if and only if

(a) range of f = Y (b) range of f ≠ Y

(c) range of f < Y (d) range of f ≥ Y

23. The value of tan–1 (1) + tan–1(0) + tan–1(2) + tan–1 (3) is

(a) π (b)45π

(c) 2π

(d) none of these

24. If 3tan221

sin5sec3

1tan2)2(sec

51

sin 1–1–1–1–1–1– +⎟⎠⎞

⎜⎝⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛++⎟

⎠⎞

⎜⎝⎛ = kπ, then k

is equal to

(a) 1 (b) 2

(c) 4 (d) 5

25. Let ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡α=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

32–1

05–

224

B01and

111

3–12

11–1

A . If B is the inverse of matrix A, then α is

equal to

(a) 5 (b) – 1

(c) 2 (d) – 2


26. The number of points at which the function ][–

1)(

xxxf = , where [.] denotes the greatest

integer function, is not continuous is

(a) 1 (b) 2

(c) 3 (d) infinite

27. The domain of xx

y–||

1= is

(a) [0, ∞) (b) (– ∞, 0)

(c) (– ∞, 0] (d) [1, ∞)

28. Angle formed by the positive y-axis and the tangent to y = x2 + 4x – 17 ⎟⎠⎞

⎜⎝⎛

43–

,25

at is

(a) tan–1 9 (b) cot–1 9

(c) ⎟⎠⎞

⎜⎝⎛ +

π9tan

21–

(d)2π

29. The point at which the maximum value of Z = 3x + 2y subject to the constraints x + y ≤ 2,x ≥ 0, y ≥ 0 is obtained, is

(a) (0, 0) (b) (1.5, 1.5)

(c) (2, 0) (d) (0, 2)

30. Inequation y – x ≤ 0 represents

(a) the half plane that contains the positive x-axis

(b) closed half plane above the line y = x, which contains positive y-axis

(c) half plane that contains the negative x-axis

(d) none of these

31. Let R be a relation on the set N be defined by {(x, y) : x, y ∈ N, 2x + y = 41}. Then, R is

(a) reflexive (b) symmetric

(c) transitive (d) none of these

32. If A is a 3 × 3 matrix whose elements aij are given by 2

)( 2ji +, then matrix A is

(a)⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

18258

549

82/72

(b)⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

251825

52/52/9

2/252/92

(c)⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

182/258

2/2582/9

82/92

(b) none of these


33. If sin y + e– x cos y = e, then dxdy

at (1, π) is equal to

(a) sin y (b) – x cos y

(c) e (d) sin y – x cos y

34. If ⎥⎦

⎤⎢⎣

⎡=

65–4–

321A , then | A | is

(a) 2 (b) 0

(c) – 2 (d) does not exist

35. If ⎥⎦

⎤⎢⎣

⎡δγ

βα=A , then adj. A is equal to

(a) ⎥⎦

⎤⎢⎣

⎡αβ

γδ

–

–(b) ⎥

⎦

⎤⎢⎣

⎡αγ

βδ

–

–

(c) ⎥⎦

⎤⎢⎣

⎡αγ

βδ

–

–(d) ⎥

⎦

⎤⎢⎣

⎡αγ

βδ ––

36. If y = (4x – 5) is a tangent to the curve y2 = px3 + q at (2, 3), then the values of p and q are

(a) p = – 2, q = – 7 (b) p = – 2, q = 7

(c) p = 2, q = – 7 (d) p = 2, q = 7

37. The relationship between a and b so that the function defined by ⎩⎨⎧

>+

≤+=

3if,3

3if,1)(

xbx

xaxxf is

continuous at x = 3, is

(a) 32

+= ba (b)23

– =ba

(c) 32

=+ ba (d) a + b = 2

38. If ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

01–1

312

102

A , then A2 – 5A + 6I is equal to

(a) ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

410–3–

41–1–

5–1–1

(b)⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

445–

10–1–1–

3–1–1

(c) O (d) I

39. If ⎥⎦

⎤⎢⎣

⎡α

α=

2

2A and |A3| = 125, then the value of α is

(a) ± 1 (b) ± 2

(c) ± 3 (d) ± 5


40. If )4–()1–(

–xxbax

y = has a turning point at P(2, – 1), then the values of a and b are

(a) a = 1, b = 0 (b) a = 2, b = 1

(c) a = 0, b = 1 (d) a = 1, b = 2

SECTION C

In this section, attempt any 8 questions. Each question is of 1 mark weightage. Q. 46to 50 are based on a case study

41. If xx = yy, then dxdy

is equal to

(a) xy

– (b) yx

–

(c) ⎟⎟⎠

⎞⎜⎜⎝

⎛+

yx

log1 (d)yx

log1log1

++

42. The second derivative of a sin3 t with respect to a cos3 t at t = 4π

is

(a) a324

(b) 2

(c) a121

(d) none of these

43. The corner points of the feasible region determined by the system of linear constraints are(0, 3), (1, 1) and (3, 0). Let Z = px + qy, where p, q > 0. Condition on p and q so that theminimum of Z occurs at (3, 0) and (1, 1) is

(a) p = 2 q (b)2q

p =

(c) p = 3q (d) p = q

44. If ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=⎥

⎦

⎤⎢⎣

⎡=

11

01

10

B,02–2

21–0A and C = AB, then the value of C–1 is

(a) ⎥⎦

⎤⎢⎣

⎡5/43/1

3/1–3/2(b) ⎥

⎦

⎤⎢⎣

⎡6/13/1

3/1–3/1

(c) ⎥⎦

⎤⎢⎣

⎡03/2

3/1–2(d) none of these

45. The objective function Z = – 60x + 30y, subject to constraints 2x – y ≥ – 5; 3x + y ≥ 32x – 3y ≤ 12 ; x, y ≥ 0 has

(a) no maximum value (b) no minimum value

(c) minimum value as well as maximum value (d) none of these


Case Study

An engineer designs a metal box with a square base and vertical sidesto contain 1024 cm3 of water.

Based on the above information answer the following :

46. If x cm is the length of the side of square base and y cm isthe height of the box, then the relation between thevariables is

(a) xy2 = 512 (b) x2y = 1024

(c) xy = 1024 (d) xy = 512

47. The total surface area S of the metal box expressed as a function of x and y is

(a) S = x2 + 2xy (b) S = x2 + 4xy

(c) S = 2x2 + 4xy (d) none of these.

48. If the material for the top and bottom costs ` 5 per cm2 and the material for the sidescosts ` 2.50 per cm2, then the expression for total cost C of the metal box expressed asa function of x is

(a)2

2 10245C

xx += (b) 2

2 102410C

xx +=

(c) x

x10240

5C 2 += (d) x

x10240

10C 2 +=

49. The value of x for which the cost of the metal box is least is

(a) x = 6 cm (b) x = 8 cm

(c) x = 10 cm (d) none of these.

50. The least cost of the box with fixed volume of water 1024 cm3 is

(a) ` 2400 (b) ` 1920(c) ` 2880 (d) none of these.

ANSWERS

1. (d) 2. (a) 3. (a) 4. (d) 5. (c) 6. (d)

7. (a) 8. (c) 9. (c) 10. (d) 11. (b) 12. (c)

13. (b) 14. (c) 15. (a) 16. (a) 17. (b) 18. (c)

19. (c) 20. (b) 21. (c) 22. (a) 23. (a) 24. (b)

25. (a) 26. (d) 27. (b) 28. (b) 29. (c) 30. (a)

31. (d) 32. (c) 33. (c) 34. (d) 35. (b) 36. (c)

37. (a) 38. (b) 39. (c) 40. (a) 41. (d) 42. (a)

43. (b) 44. (b) 45. (b) 46. (a) 47. (c) 48. (a)

49. (c) 50. (c)

yy y y

x

x

x

x









SECTION – A


1. If A = [aij]2 × 2, where aij = 2

)2( 2ji +, then A is equal to

(a) ⎥⎦

⎤⎢⎣

⎡188

259(b)

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

1882

2529

(c) ⎥⎦

⎤⎢⎣

⎡94

259(d)

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

942

1529

2. The value of °°

°°

80cos80sin

10cos–10sin is

(a) –1 (b) 1

(c) 2 (d) 0

3. A function f from the set of natural numbers to integers defined by

⎪⎩

⎪⎨

⎧

=eveniswhen,

2–

oddiswhen,2

1–

)(n

n

nn

nf is

(a) neither one-one nor onto (b) one-one but not onto

(c) onto but not one-one (d) one-one and onto


4. If ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

+

+=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

+

+

++

021–42

223–2

2–360

233–

01–64

7–243

b

cx

y

czbb

ax

yzx, then

(a) a = – 2, b = – 7, c = – 1, x = – 3, y = – 5, z = 2

(b) a = 2, b = 7, c = 1, x = 3, y = 5, z = – 2

(c) a = 1, b = 3, c = 4, x = 2, y = 8, z = 9

(d) a = – 1, b = 3, c = – 2, x = –7, y = 4, z = 5

5. If ⎥⎦

⎤⎢⎣

⎡=

xx

xx

cossin

sin–cosA , then AA′ is

(a) zero matrix (b) I2

(c) ⎥⎦

⎤⎢⎣

⎡11

11(d) none of these

6. For any 2 × 2 matrix A, if A (adj. A) = ⎥⎦

⎤⎢⎣

⎡100

010, then | A | is equal to

(a) 0 (b) 10

(c) 20 (d) 100

7. The derivative of ⎟⎟⎠

⎞⎜⎜⎝

⎛

+ 2

21–

1

–1cos

x

x w.r.t. ⎟

⎟⎠

⎞⎜⎜⎝

⎛3

21–

–3

3–1cot

xx

x is

(a)23

(b) 1

(c)21

(d) 32

8. Let A = {1, 2, 3}. Then, the number of relations containing (1, 2) and (1, 3), which arereflexive and symmetric but not transitive, is

(a) 1 (b) 2

(c) 3 (d) 4

9. The coefficient of x in

01

2)1(log1

–sin1

)(22 xx

x

yxxx

xf

+

+

+

= , – 1 < x ≤ 1 is

(a) 1 (b) – 2

(c) – 1 (d) 0


10. The function x

xftan11

)(+

=

(a) is a continuous, real-valued function for all x ∈ (– ∞, ∞)

(b) is discontinuous only at 43π

=x

(c) has infinitely many discontinuities on (– ∞, ∞)

(d) none of these

11. If matrix ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

110

1–21

42–3

A and A–1 = k1

(adj. A), then k is equal to

(a) 7 (b) – 7

(c) 15 (d) 11

12. If y = tan–1 ⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

+ 23

1

–

x

xx , then y′ (1) is equal to

(a) 0 (b)21

(c) – 1 (d) – 41

13. If y = t10 + 1 and x = t8 + 1, then 2

2

dx

yd is equal to

(a) t25

(b) 20 t8

(c) 6165t

(d) none of these

14.⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛+

4/3

22–

logxx

edxd x is equal to

(a) 1 (b) 4–

12

2

x

x +

(c) 4–

1–2

2

x

x(d)

4–

1–. 2

2

x

xex

15. The function f (x) = 2x3 – 3x2 – 12x + 4, has

(a) two points of local maximum (b) two points of local minimum

(c) one maxima and one minima (d) no maxima or minima


16. The slope of tangent to the curve x = t2 + 3t – 8, y = 2t2 – 2t – 5 at the point (2, – 1) is

(a) 722

(b) 76

(c) 76–

(d) – 6

17. The maximum value of the objective function Z = 4x + 6y,

subject to constraints 3x + 2y ≤ 12, x + y ≥ 4, x, y ≥ 0, is

(a) 16 at (4, 0) (b) 24 at (0, 4)

(c) 24 at (6, 0) (d) 36 at (0, 6)

18.⎭⎬⎫

⎩⎨⎧

⎟⎠⎞

⎜⎝⎛

53

–cos2sin 1– is equal to

(a) 256

(b) 2524

(c) 54

(d) 2524

–

19. If ⎪⎩

⎪⎨

⎧

≤≤+

<≤+

=10for,2–32

01–for,–1–1

)(2 xxx

xx

kxkxxf is continuous at x = 0, then k is equal to

(a) – 4 (b) – 3(c) – 2 (d) – 1

20. The mapping f : N → N given by f(n) = 1 + n2, n ∈ N, where N is the set of natural numbers,is

(a) one-one and onto (b) onto but not one-one

(c) one-one but not onto (d) neither one-one nor onto

SECTION B

In this section, attempt any 16 out of questions 21-40 . Each question is of 1 markweightage.

21. The corner points of the feasible region for an LPP are (0, 2), (3, 0), (6, 0), (6, 8) and (0, 5).If Z = 4x + 6y is the objective function, then the minimum value of Z occurs at

(a) (0, 2) only (b) (3, 0) only

(c) the mid point of the line segment joining the points (0, 2) and (3, 0).

(d) any point on the line segment joining the points (0, 2) and (3, 0).

22. If A is a square matrix such that (A – 2I) (A + I) = 0, then A–1 = ________.

(a) 2

I–A(b)

2IA +

(c) 2 (A – I) (d) 2A + I


23. The slope of the normal at the point (at2, 2at) of the parabola y2 = 4ax is

(a) t1

(b) t

(c) – t (d) – t1

24. If Aij denotes the co-factor of the element aij of the determinant 7–51

406

53–2

, then the

value of a11A31 + a13A32 + a13A33 is(a) 0 (b) 5

(c) 10 (d) – 5

25. The minimum value of Z = 7x + y, subject to 5x + y ≥ 5, x + y ≥ 3, x ≥ 0, y ≥ 0 occurs at

(a) (3, 0) (b) ⎟⎠⎞

⎜⎝⎛

25

,21

(c) (7, 0) (d) (0, 5)

26. ⎥⎦⎤

⎢⎣⎡ ⎟

⎠⎞⎜

⎝⎛ 21– –1––1sin xxxx

dxd

is equal to

(a) 2–1

1–

)–1(2

1

xxx(b)

2)–1()–1(

1

xxx

(c) )–1(2

1–

–1

12 xxx

(b) none of these

27. The number of points at which the function ||log

1)(

xxf = is discontinuous is

(a) 1 (b) 2

(c) 3 (d) 4

28. If f (x) = cos x, then

(a) f (x) is strictly decreasing in (0, π)

(b) f (x) is strictly increasing in (0, 2π)

(c) f (x) is neither increasing nor decreasing in (π, 2π)

(d) All the above are correct

29. If ⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡=

27

14Band

47–

1–2A , then which of the following statement is true ?

(a) AA′ = I (b) BB′ = I

(c) AB ≠ BA (d) (AB)′ = I


30. If y = cot –1 (x2), then the value of dxdy

is

(a) 412

x

x

+ (b)x

x

41

2

+

(c) – 412

x

x

+(d)

21

2–

x

x

+

31. If A is a square matrix of order m, then the matrix B of same order is called the inverse ofthe matrix A, if

(a) AB = A (b) BA = A

(c) AB = BA = I (d) AB = –BA

32. If the area of the triangle formed by the points (1, 2), (k, 5) and (7, 11) is zero, then thevalue of k is

(a) 0 (b) 3

(c) 5 (d) 733. The maximum value of [x (x – 1) + 1]1/3, 0 ≤ x ≤ 1 is

(a) 3/1

31

⎟⎠⎞

⎜⎝⎛

(b)21

(c) 1 (d) 0

34. If y = x – x2, then the derivative of y2 with respect to x2 is

(a) 1 – 2x (b) 2 – 4x

(c) 3x – 2x2 (d) 1 – 3x + 2x2

35. If A and B are two matrices such that A + B and AB are both defined, then

(a) A and B are two matrices not necessarily of same order.

(b) A and B are square matrices of same order.

(c) number of columns of A = number of rows of B.

(d) none of these.

36. If the sum of two numbers is 3, then the maximum value of the product of first number andthe square of second number is

(a) 4 (b) 3

(c) 2 (d) 1

37. The slope of the tangent to the curve x = 3t2 + 1, y= t3 –1 at x = 1 is :

(a) 21

(b) 0

(c) – 2 (d) ∞

38. The x-coordinate of the point on the curve f (x) = )6–7( xx , where the tangent is parallelto x-axis is

(a) 31

– (b) 72

(c) 76

(d)21


39. The smallest value of the polynomial x3 – 18x2 + 96x in the interval [0, 9] is :

(a) 126 (b) 0

(c) 135 (d) 160

40. If x

xxf1

sin)( 2= , where x ≠ 0, then the value of the function f at x = 0, so that the function

is continuous at x = 0, is(a) 0 (b) – 1

(c) 1 (d) none of these

SECTION C

In this section, attempt any 8 questions. Each question is of 1 mark weightage.Q. 46 to 50 are based on a case study

41. Which of the following cannot be considered as the objective function of a linearprogramming problem ?

(a) Maximize Z = 3x + 2y (b) Minimize Z = 6x + 7y + 9z

(c) Maximize Z = 2x (d) Minimize Z = x2 + 2xy + y2

42. The maximum value of x

xlogin the interval (2,∞) is

(a) 1 (b) e

(c) e2

(d) e1

43. If ⎥⎦

⎤⎢⎣

⎡1–1

4–3is the sum of a symmetric matrix B and a skew-symmetric matrix C, then C is

(a) ⎥⎦

⎤⎢⎣

⎡02/5

2/5–1(b) ⎥

⎦

⎤⎢⎣

⎡12/5

2/5–1

(c) ⎥⎦

⎤⎢⎣

⎡02/5

2/5–0(d) ⎥

⎦

⎤⎢⎣

⎡12/5

2/3–1

44. The points at which the tangent passes through the origin for the curve y = 4x3 – 2x5 are

(a) (0, 0), (2, 1) and (– 1, – 2) (b) (0, 0), (2, 1) and (– 2, – 1)

(c) (2, 0), (2, 1) and (– 3, 1) (d) (0, 0), (1, 2) and (– 1, – 2)

45. The corner points of the feasible region determined by the system of linear constraints are(0, 10), (5, 5) (15, 15), (0, 20). Let Z = px + qy, where p, q > 0. The condition on p and q so thatthe maximum of Z occurs at both the points (15, 15) and (0, 20) is

(a) p = q (b) p = 2q

(c) q = 2p (d) q = 3p


Case Study

In an annual sports days of a school, 50 boys and25 girls participated in the various events. 5 boysand 3 girls were selected to represent the schoolin the upcoming state level competition. The setsof boys selected is represented by B = {b1, b2, b3,b4, b5} and the sets of girls selected is representedby G = {g1, g2, g3}.

Based on the above information, answerthe following questions :

46. How many relations are possible from B to G ?

(a) 25 (b) 215

(c) 28 (d) 23

47. The number of possible reflexive relations defined on set G are :

(a) 23 (b) 29

(c) 26 (d) 212

48. The number of symmetric relations defined on set B are :

(a) 26 (b) 220

(c) 215 (d) 25

49. Let R : B → G be defined by R = {(b1, g1), (b2, g2), (b3, g3), (b4, g2),(b5, g1)}, then Ris__________

(a) one-one and onto (b) onto

(c) one-one (d) neither one-one nor onto

50. The number of onto functions possible from set B to itself are

(a) 120 (b) 24

(c) 720 (d) 6

ANSWERS

1. (b) 2. (b) 3. (d) 4. (a) 5. (b) 6. (b)

7. (d) 8. (a) 9. (b) 10. (c) 11. (c) 12. (d)

13. (c) 14. (c) 15. (c) 16. (b) 17. (d) 18. (d)

19. (c) 20. (c) 21. (d) 22. (a) 23. (c) 24. (a)

25. (d) 26. (c) 27. (c) 28. (a) 29. (d) 30. (c)

31. (c) 32. (b) 33. (c) 34. (d) 35. (b) 36. (a)

37. (b) 38. (b) 39. (b) 40. (a) 41. (d) 42. (d)

43. (c) 44. (d) 45. (d) 46. (b) 47. (c) 48. (c)

49. (b) 50. (a)

cbse sample question paper 2021-22 class : xii

Documents