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**Sample Papers for Class 12 ****Mathematics**

**Mathematics**

## CBSE Class 12 Mathematics Term 1 Sample Paper Set A

**SECTION-A**

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

**1. sin [π/r – sin-4(-1/2)] is equal to : **

(A) 1/2

(B) 1/3

(C) –1

(D) 1

**Answer**

D

**2. The value of k (k < 0) for which the function f defined as **

**is continuous at x = 0 is : **

(A) ± 1

(B) –1

(C) ± 1/2

(D) 1/2

**Answer**

B

**3. If A = [a _{ij}] is a square matrix of order 2 such that a_{ij} = **

**Answer**

D

**4. Value of k, for which A = **

**is a singular matrix is: **

(A) 4

(B) –4

(C) ±4

(D) 0

**Answer**

C

**5. Find the intervals in which the function f given by f(x) = x2 – 4x + 6 is strictly increasing: **

(A) (– ∞, 2) ∪ (2, ∞)

(B) (2, ∞)

(C) (− ∞, 2)

(D) (– ∞, 2] ∪ (2, ∞)

**Answer**

B

**6. Given that A is a square matrix of order 3 and | A | = – 4, then | adj A | is equal to: **

(A) –4

(B) 4

(C) –16

(D) 16

**Answer**

D

**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 an equivalence relation in A? **

(A) (1, 1)

(B) (1, 2)

(C) (2, 2)

(D) (3, 3)

**Answer**

B

**8. If **

**,then value of a + b – c + 2d is: **(A) 8

(B) 10

(C) 4

(D) –8

**Answer**

A

**9. The point at which the normal to the curve y = x + 1/x , x >0 is perpendicular to the line 3x – 4y – 7 = 0 is: **

(A) (2, 5/2)

(B) (±2, 5/2)

(C) (–1/2, 5/2)

(D) (1/2, 5/2)

**Answer**

A

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

(A) x/√1- x^{2}

(B) 1/√1-x^{2}

(C) 1/√1+x^{2}

(D) x/√1+x^{2}

**Answer**

D

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

(A) {1, 5, 9}

(B) {0, 1, 2, 5}

(C) Φ

(D) A

**Answer**

A

**12. If ex + ey = ex+y, then dy/dx is: **

(A) e^{y – x}

(B) e^{x + y}

(C) –e^{y – x}

(D) 2e^{x – y}

**Answer**

C

**13. Given that matrices A and B are of order 3 × n and m × 5 respectively, then the order of matrix C = 5A + 3B is: **

(A) 3 × 5 and m = n

(B) 3 × 5

(C) 3 × 3

(D) 5 × 5

**Answer**

B

**14. If y = 5 cos x – 3 sin x, then d2y/dx2 is equal to: **

(A) –y

(B) y

(C) 25y

(D) 9y

**Answer**

A

**15. For matrix A =**

**,(adj A)’ is equal to:**

**Answer**

C

**16. The points on the curve x2/9 + y2/16 = 1 at which the tangents are parallel to y-axis are: **

(A) (0, ±4)

(B) (±4, 0)

(C) (±3, 0)

(D) (0, ±3)

**Answer**

C

**17. Given that A = [a _{ij}] is a square matrix of order 3 × 3 and |A| = −7, then the value of ^{3}∑_{i=1} a_{12} A_{12} , where A_{ij} denotes the cofactor of element aij is: **

(A) 7

(B) –7

(C) 0

(D) 49

**Answer**

B

**18. If y = log(cos ex), then dy/dx is: **

(A) cos e^{x−1}

(B) e^{–x} cos e^{x}

(C) e^{x} sin e^{x}

(D) –e^{x} tan e^{x}

**Answer**

D

**19. Based on the given shaded region as the feasible region in the graph, at which point(s) is the objective function Z = 3x + 9y maximum? **

(A) Point B

(B) Point C

(C) Point D

(D) every point on the line segment CD

**Answer**

D

**20. The least value of the function f(x) = 2cos x + x in the closed interval [0,π/2] is: **

(A) 2

(B) π/6 + √3

(C) π/2

(D) The least value does not exist

**Answer**

C

**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) = x ^{3} is: **

(A) One-one but not onto

(B) Not one-one but onto

(C) Neither one-one nor onto

(D) One-one and onto

**Answer**

D

**22. If x = a sec q, y = b tan q, then d ^{2}y/dx^{2} at θ = π/6 is: **

(A) -3√3b/a

^{2}

(B) -2√3b/a

(C) -3√3b/a

(D) -b/3√3 a

^{2}

**Answer**

A

**23. In the given graph, the feasible region for a LPP is shaded. The objective function Z = 2x – 3y, will be minimum at: **

(A) (4, 10)

(B) (6, 8)

(C) (0, 8)

(D) (6, 5)

**Answer**

C

**24. The derivative of sin–1 (2x√1-x2) w.r.t sin–1x, 1/√ < x < 1, is: **

(A) 2

(B) π/2 – 2

(C) π/2

(D) –2

**Answer**

A

**25. If A = **

(A) A–1 = B

(B) A–1 = 6B

(C) B–1 = B

(D) B–1 = 1/6 A

**Answer**

D

**26. The real function f(x) = 2×3 – 3×2 – 36x + 7 is: **

(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, ∞)

**Answer**

B

**27. Simplest form of tan-1 (√1+cose x +√1-cos x/√1+cose x – √1-cos x) , π < x < 3π /2 is: **

(A) π/4 – x/2

(B) 3π/2 – x/2

(C) -x/2

(D) π – x/2

**Answer**

A

**28. Given that A is a non-singular matrix of order 3 such that A2 = 2A, then value of |2A| is: **

(A) 6

(B) 8

(C) 64

(D) 16

**Answer**

C

**29. The value of b for which the function f(x) = x + cosx + b is strictly decreasing over R is: **

(A) b < 1

(B) No value of b exists

(C) b ≤ 1

(D) b ≥ 1

**Answer**

B

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

(A) (2, 4) ∈ R

(B) (3, 8) ∈ R

(C) (6, 8) ∈ R

(D) (8, 7) ∈ R

**Answer**

C

**31. The point(s), at which the function f given by f(x) = **

**is continuous, is/are:**

(A) x ∈ R

(B) x = 0

(C) x ∈ R –{0}

(D) x = –1 and 1

**Answer**

A

**32. If A = **

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

(A) −6, −12, −18

(B) −6, −4, −9

(C) −6, 4, 9

(D) −6, 12, 18

**Answer**

B

**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

(A) a unique point

(B) no point

(C) infinitely many points

(D) two points only

**Answer**

D

**34. The area of a trapezium is defined by function f and given by f(x) = (10 + x) √100 – x ^{2} , then the area when it is maximised is: **

(A) 75 cm

^{2}

(B) 7 √3 cm

^{2}

(C) 75 √3 cm

^{2}

(D) 5 cm

^{2}

**Answer**

C

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

(A) A

(B) I + A

(C) I − A

(D) I

**Answer**

D

**36. If tan–1 x = y, then: **

(A) −1< y <1

(B) -π/2 ≤ y ≤ π/2

(C) -π/2 < y < π/2

(D) y ∈ {-π/2 , π/2}

**Answer**

C

**37. Let A = {1, 2, 3}, B = {4, 5, 6, 7} and let f = {(1, 4), (2, 5), (3, 6)} be a function from A to B. Based on the given information, f is best defined as: **

(A) Surjective function

(B) Injective function

(C) Bijective function

(D) function

**Answer**

B

**38. For A = **

** ,then 14A–1 is given by:**

**Answer**

B

**39. The point(s) on the curve y = x3 – 11x + 5 at which the tangent is y = x – 11 is/are: **

(A) (–2, 19)

(B) (2, – 9)

(C) (±2, 19)

(D) (–2, 19) and (2, – 9)

**Answer**

B

**40. Given that A = **

**and A ^{2} = 3I, then:**

(A) 1 + α

^{2}+ βϒ = 0

(B) 1 – α

^{2}– βϒ = 0

(C) 3 – α

^{2}– βϒ = 0

(D) 3 + α

^{2}+ βϒ = 0

**Answer**

C

**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 of the 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 the maximum Z occurs at both the points (30, 30) and (0, 40) is: **

(A) b − 3a = 0

(B) a = 3b

(C) a + 2b = 0

(D) 2a − b = 0

**Answer**

A

**42. For which value of m is the line y = mx + 1 a tangent to the curve y ^{2 }= 4x? **

(A) 1/2

(B) 1

(C) 2

(D) 3

**Answer**

B

**43. The maximum value of [x(x-1)+1]1/3 , 0 ≤ x ≤ 1 is: **

(A) 0

(B) 1/2

(C) 1

(D) 3√1/3

**Answer**

C

**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 **

(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

**Answer**

B

**45. Let A = **

**, where 0 ≤ a ≤ 2p, then: **

(A) |A|= 0

(B) |A| ∈ (2, ∞)

(C) |A| ∈ (2, 4)

(D) |A| ∈ [2, 4]

**Answer**

D

**CASE-STUDY**

The fuel cost per hour for running a train is proportional to the square of the speed it generates in km per hour. If the fuel costs `48 per hour at speed 16 km per hour and the fixed charges to run the train amount to `1200 per hour.

Assume the speed 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: **

(A) 16/3

(B) 1/3

(C) 3

(D) 3/16

**Answer**

D

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

(A) 15/16 v + 600000/v

(B) 75/4 v + 600000/v

(C) 5/16 v2 + 150000/v

(D) 3/16 v + 6000/v

**Answer**

B

**48. The most economical speed to run the train is: **

(A) 18km/h

(B) 5km/h

(C) 80km/h

(D) 40km/h

**Answer**

C

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

(A) ₹3750

(B) ₹750

(C) ₹7500

(D) ₹75000

**Answer**

C

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

(A) ₹3750

(B) ₹75000

(C) ₹7500

(D) ₹15000

**Answer**

D