Section A

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

1. The point at which the tangent to the curve y =√4x-3 – 1 has its slope 2/3 , is 
(a) (2, 3)
(b) (3, 2)
(c) (3, 1)
(d) (1, 3)

2. The value of     

3. If for any 2 x 2 square matrix A, A adj(A) =     

(a) 64
(b) 1
(c) 5
(d) 32

4. If   

then the value of xyz is
(a) 10
(b) 12
(c) 15
(d) 0

5. If x = sinθ, y = tanθ, then dy/dx at θ = π/3 is equal to 
(a) 1
(b) 8
(c) 3
(d) 4

6. If A is a square matrix such that A2 = I , then A +A-1 is equal to 
(a) A+ I
(b) A
(c) 0
(d) 2A

7. The equation of tangent to the curve y = 2x² + 3 sin and (0, 0) is 
(a) y = 3x
(b) y = – 3x
(c) x = 3y
(d) x = – 3y

8. If [x 1] 

= 0 , then x² is equal to
(a) 2
(b) 4
(c) 8
(d) 1

9. The value of cos(1/2 sin^-1 √3/2) is 
(a) 1/2
(b) √3/2
(c) 1/√2
(d) None of these

10. The principal value of sin-1(-√3/2) is 
(a) -2π/3
(b) -π/3
(c) 4π/3
(d) 5π/3

11. Let X be the set of all persons living in a city. Persons x, y in X are said to be related as x < y, if y is atleast 5 yr older than x. Which one of the following is correct? 
(a) The relation is an equivalence relations on X
(b) The relation is transitive but neither reflexive nor symmetric
(c) The relations is reflexive but neither transitive nor symmetric
(d) The relation is symmetric but neither transitive nor reflexive

12. Let S denote set of all integers. Define a relation R on S as ‘aRb’ if ab ≥ 0, where a, b ∈ S. 
Then, R is
(a) reflexive but neither symmetric nor transitive relation
(b) reflexive, symmetric but not transitive relation
(c) an equivalence relation
(d) symmetric but neither reflexive nor transitive relation

13. If 

(a) (1, 1)
(b) (1, – 1)
(c) (- 1, 1)
(d) (- 1, – 1)

14. If Δ = 

(a) 1
(b) 2
(c) 3
(d) 4

15. If 

then x is equal to
(a) 6
(b) ± 3
(c) – 3
(d) 3

16. The function f (x) = log x is strictly increasing on   
(a) [- 1, 0]
(b) (0, ∞)
(c) (- ∞, ∞)
(d) None of these

17. The function f (x) = x³ – 3x² + 3x – 10 in the interval (- ∞, ∞) is 
(a) decreasing
(b) increasing
(c) strictly increasing
(d) strictly decreasing

18. The slope of the tangent to the curve x = 3t² + 1 , y = t³ – 1 at x = 1 is 
(a) – 1
(b) 1
(c) 0
(d) 2

19. If x 

(a) 4
(b) 16
(c) 2
(d) 1

20. The point on the curve y = x² – 4x + 5, where tangent to the curve is parallel to the X-axis is   
(a) (0, 5)
(b) (- 1, 0)
(c) (1, 2)
(d) (2, 1)

Section B

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

21. If y =(cot^-1 x²) and (x²+1)² d²y/dx² + 2x (x²+1) dy/dx = k , then K is equal to   
(a) 1
(b) 2
(c) 5
(d) 7

22. If A = 

then A² + 3A – 2I is equal to 

23. The equation of normal at the point (1, 1) on the curve 2y + x² = 3 is 
(a) x + y = 0
(b) x – y = 0
(c) x + y + 1 = 0
(d) x – y = 1

24. If x = a(t – sin t), y = a(1 + cos t), then dy/dx at t = π/2 is equal to 
(a) – 1
(b) 0
(c) 3
(d) 8

25. If y = log(xy), then dy/dx at (1, 2) is equal to 
(a) 1
(b) 2
(c) 3
(d) 4

26. The maximum value of the function f (x) = -(x-1)² + 8 is   
(a) 7
(b) 8
(c) 0
(d) 1

27. The value of sin[π/2 – sin-1(-√3/2)] is 
(a) 1/2
(b) -1/2
(c) 1
(d) – 1

28. The function f : R → R defined by f(x) = x/x²+1 , ∀ x ∈ R is 
(a) one-one
(b) not one-one
(c) bijective
(d) None of these

29. Let f : [0, 1] → [0, ∞) be defined by f(x) = x/1+x , then f is 
(a) one-one but not onto
(b) onto but not one-one
(c) both one-one and onto
(d) neither one-one nor onto

30. If A = 

(a) AB= BA
(b) AB ≠ BA
(c) A² = B 
(d) None of these

31. If A = 

then |3A|is equal to
(a) 3|A|
(b) 9|A|
(c)|A|
(d) 27|A|

32. If at x = 1, then function f(x) = x⁴ – 62x² + ax + 9 attains its maximum value on the interval [0, 2], then the value of a is   
(a) 124
(b) 120
(c) – 120
(d) 128

33. If y = 3 cos x + 3 sin x, then d²y/dx² + y is equal to 
(a) 0
(b) 1
(c) 2
(d) 3

34. If y = e-^3x and d²y/dx² = ky , then K is equal to 
(a) 1/9
(b) 9
(c) 2
(d) 1

35. If y = 2 log sin x, then d²y/dx² is equal to 
(a) – 2cosec² x
(b) 2 cosec ²x
(c) 2 cot² x
(d) sec²x

36. If x = at² and y = at³ , then d²y/dx² at t = 3/4 is equal to 
(a) a
(b) 1/a
(c) 4
(d) -1

37. If y = a cos³ t and x = a sin³ t , then dy/dx at t = π/4 is   
(a) 1
(b) -1
(c) 0
(d) 2

38. The function f(x) = (x-1)ex + 2 on [0, ∞) is 
(a) increasing
(b) decreasing
(c) strictly decreasing
(d) None of these

39. For real numbers x and y, define a relation R, xRy if only if x – y + √2 is an irrational number. The, the relation R is 
(a) reflexive
(b) symmetric
(c) transitive
(d) an equivalence relation

40. The function f (x) = x³ + x² + x + 1 has 
(a) maximum value at x = – 1
(b) minimum value at x = – 1
(c) neither maximum nor minimum value
(d) None of these

Section C

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

41. If x = tan(1/a log y) and (1+x²) d²y/dx² + 2x dy/dx = k dy/dx , then k is equal to 
(a) a
(b) a/2
(c) 2a
(d) 1

42. If A = 

then A⁵⁰ is equal to
(a) 2 ⁴⁹ A
(b) 2A
(c) 49A
(d) 2⁹⁹ A

43. The function f given by f (x) = x² – x + 1 on (- 1, 1) is 
(a) strictly decreasing
(b) strictly increasing
(c) neither strictly increasing nor strictly decreasing
(d) None of these

44. If y = sin-1 x and (1-x²) d²y/dx² = k dy/dx , then k is equal to 
(a) x
(b) x²
(c) 1
(d) 0

45. If λ³ = – 2, then the value of 

(a) – 11
(b) – 12
(c) – 13
(d) 0

CASE STUDY

Suppose a dealer in rural area wishes to propose a number of sewing machines. He has some money to invest and has space for few items for storage.
Let x denotes the number of electronic sewing machines and y denotes the number of manually operated sewing machines purchased by the dealer. For the same, constraint related to investment is given by 3x +2y ≤ 48.

And objective function is Z = 22x +18y.
And other constraints consists the following x + y ≤ 20, x, y ≥ 0.
Based on above information, answer the following questions.   

46. Number of corner points of the feasible region is   
(a) 3
(b) 4
(c) 5
(d) 6

47. Sum of values of Z at all the corner points is 
(a) 1008
(b) 1104
(c) 1100
(d) 1108

48. To get the maximum profit (i.e. maximise Z) how many electronic sewing machines should be purchased by the dealer. 
(a) 12
(b) 8
(c) 10
(d) 5

49. To get the maximum profit (i.e. maximise Z) how many manually operated sewing machines should be purchased by the dealer. 
(a) 10
(b) 5
(c) 8
(d) 12

50. Z| _max – Z| _min is equal to  
(a) 360
(b) 392
(c) 352
(d) None of these