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MATHEMATICS

MULTIPLE CHOICE

QUESTIONS WITH

ANSWERS

M.SALMAN SHERAZI

Contents

UNIT

Title

Page

(^1) Functions and Limits 3

(^2) Differentiation 5

(^3) Integration 10

(^4) Introduction to Analytic Geometry 12

(^5) Linear Inequalities and Linear

Programming

(^6) Conic Sections 15

(^7) Vectors 17

24. If 𝒙 = 𝒂𝒚^ , 𝒕𝒉𝒆𝒏 𝒚 = 𝒍𝒐𝒈𝒂𝒙^ is called logarithmic function if (a) 𝑎 < 0 (b) 𝑎 > 0 (c) 𝑎 = 0 (d) ✔𝑎 > 0 , 𝑎 ≠ 1 25. If 𝒄𝒐𝒔𝒉𝒙 = 𝒆𝒙+𝒆−𝒙 𝟐 ,^ then its domain is set of real numbers and range is (a) Set of all real numbers (b) ✔ Set of +iv real numbers (c) [1,∞) (d) [-1,∞) 26. In logarithmic form 𝒄𝒐𝒔𝒉−𝟏𝒙 can be written as

(a) ✔ln(𝑥 + √𝑥^2 + 1) (b) ln(𝑥 + √𝑥^2 − 1) (c) ln(𝑥 − √𝑥^2 + 1) (d) ln(𝑥 − √𝑥^2 − 1)

27. In logarithmic function 𝒔𝒊𝒏𝒉−𝟏𝒙 is written as

(b) ln(𝑥 + √𝑥^2 + 1) (b) ✔ ln(𝑥 + √𝑥^2 − 1) (c) ln(𝑥 − √𝑥^2 + 1) (d) ln(𝑥 − √𝑥^2 − 1)

28. In logarithmic form, 𝒕𝒂𝒏𝒉−𝟏𝒙 can be written as

(a) ✔^12 ln (𝑥+1𝑥−1) , |𝑥| < 1 (b) 12 ln (1+𝑥1−𝑥) , |𝑥| < 1 (c) ln(^1 𝑥 + √1−𝑥

2 𝑥 ) , 0 ≤ 𝑥 ≤ 1 (d) ln (^1 𝑥 + √1−𝑥

2 |𝑥| ) , 𝑥 ≠ 0

29. In logarithmic form, 𝒄𝒐𝒕𝒉−𝟏^ can be written as

(a) 12 ln (𝑥+1𝑥−1) , |𝑥| < 1 (b) ✔ 12 ln (1+𝑥1−𝑥) , |𝑥| < 1 (c) ln(^1 𝑥 + √1−𝑥

2 𝑥 ) , 0 ≤ 𝑥 ≤ 1 (d) ln (^1 𝑥 + √1−𝑥

2 |𝑥| ) , 𝑥 ≠ 0

30. In logarithmic form, 𝑺𝒆𝒄𝒉−𝟏^ can be written as

(b) 1 2 ln (

𝑥+ 𝑥−1) , |𝑥| < 1^ (b)^

1 2 ln (

1+𝑥 1−𝑥) , |𝑥| < 1^ (c)^ ✔^ ln(

1 𝑥 +^

√1−𝑥^2 𝑥 ) , 0 ≤ 𝑥 ≤ 1 (d) ln ( 1 𝑥 +^

√1−𝑥^2 |𝑥| ) , 𝑥 ≠ 0

31. In logarithmic form, 𝑪𝒐𝒔𝒆𝒄𝒉−𝟏^ can be written as

(c) 1 2 ln (

𝑥+ 𝑥−1) , |𝑥| < 1^ (b)^

1 2 ln (

1+𝑥 1−𝑥) , |𝑥| < 1^ (c)^ ln(

1 𝑥 +^

√1−𝑥^2 𝑥 ) , 0 ≤ 𝑥 ≤ 1 (d) ✔ ln ( 1 𝑥 +^

√1−𝑥^2 |𝑥| ) , 𝑥 ≠ 0

32. 𝑥^2 + 𝑥𝑦 + 𝑦^2 = 2 is an example of (a) Linear function (b) quadratic function (c) explicit function (d) ✔ Implicit function 33. 𝒙 = 𝒂𝒕𝟐, 𝒚 = 𝟐𝒂𝒕 are the parametric equations of (a) Circle (b) ✔ Parabola (c) Ellipse (d) Hyperbola 34. 𝒙 = 𝒂𝑪𝒐𝒔𝜽 , 𝒚 = 𝒂𝑺𝒊𝒏𝜽 are parametric equations of (a) Circle (b) Parabola (c) ✔ Ellipse (d) Hyperbola 35. 𝒙 = 𝒂𝒔𝒆𝒄𝜽 , 𝒚 = 𝒃𝒕𝒂𝒏𝜽 are parametric equations of (b) Circle (b) Parabola (c) Ellipse (d) ✔Hyperbola 36. The function , 𝒇(𝒙) = 𝟑𝒙𝟒^ + 𝟕 − 𝟑𝒙𝟐^ is (a) ✔Even (b) Odd (c) Neither (d) None of these 37. The function , 𝒇(𝒙) = 𝑺𝒊𝒏𝒙 + 𝑪𝒐𝒔𝒙 is (a) Even (b) Odd (c) ✔ Neither (d) None of these 38. If 𝒇(𝒙) = 𝟐𝒙 + 𝟏 , 𝒈(𝒙) = 𝒙𝟐^ − 𝟏 , then (𝒇𝒐𝒈)(𝒙) = (a) ✔2𝑥^2 − 1 (b) 4𝑥^2 + 4𝑥 (c) 4𝑥 + 3 (d) 𝑥^4 − 2𝑥^2 39. If 𝒇(𝒙) = 𝟐𝒙 + 𝟑 , 𝒈(𝒙) = 𝒙𝟐^ − 𝟏 , then (𝒈𝒐𝒇)(𝒙) = (a) 2𝑥^2 − 1 (b) ✔ 4𝑥^2 + 4𝑥 (c) 4𝑥 + 3 (d) 𝑥^4 − 2𝑥^2 40. If 𝒇(𝒙) = 𝟐𝒙 + 𝟑 , 𝒈(𝒙) = 𝒙𝟐^ − 𝟏 , then (𝒇𝒐𝒇)(𝒙) = (b) 2𝑥^2 − 1 (b) 4𝑥^2 + 4𝑥 (c) ✔ 4𝑥 + 3 (d) 𝑥^4 − 2𝑥^2 41. If 𝒇(𝒙) = 𝟐𝒙 + 𝟑 , 𝒈(𝒙) = 𝒙𝟐^ − 𝟏 , then (𝒈𝒐𝒈)(𝒙) = (c) 2𝑥^2 − 1 (b) 4𝑥^2 + 4𝑥 (c) 4𝑥 + 3 (d) ✔ 𝑥^4 − 2𝑥^2 42. The inverse of a function exists only if it is (a) an into function (b) an onto function (c) ✔ (1-1) and into function (d) None of these 43. If 𝒇(𝒙) = 𝟐 + √𝒙 − 𝟏 , then domain of 𝒇−𝟏^ = (a) ]2,∞[ (b) ✔ [2,∞[ (c) [1,∞[ (d) ]1,∞[ 44. If 𝒇(𝒙) = 𝟐 + √𝒙 − 𝟏 , then range of 𝒇−𝟏^ = (b) ]2,∞[ (b) [2,∞[ (c) ✔ [1,∞[ (d) ]1,∞[

45. 𝐥𝐢𝐦𝒙→𝟎𝑺𝒊𝒏𝒙𝒙 = 𝟏 𝒊𝒇 𝒂𝒏𝒅 𝒐𝒏𝒍𝒚 𝒊𝒇

(a) 𝑥 is Obtuse angle (b) 𝑥 is right angle (c) 0 < 𝑥 < 𝜋 2 (d)^ ✔𝑥𝜖(−^

𝜋 2 ,^

𝜋 2 )

46. A function is said to be continuous at 𝒙 = 𝒄 if (a) lim𝑥→𝑐 𝑓(𝑥) exists (b) 𝑓(𝑐)is defined (c) lim𝑥→𝑐 𝑓(𝑥) = 𝑓(𝑐) (d) ✔ All of these 47. 𝒇(𝒙) = 𝒂𝒙 + 𝒃 𝒘𝒊𝒕𝒉 𝒂 ≠ 𝟎 is (a) ✔A linear function (b) A quadratic function (c) A constant function (d) An identity function 48. If 𝒇: 𝑿 → 𝒀 is a function then the subset of 𝒀 containing all the images is called : (a) Domain of 𝑓 (b) ✔ range of 𝑓 (c) Co domain of 𝑓 (d) Subset of 𝑋 49. The graph of 𝟐𝒙 − 𝟏𝟎 = 𝟎 is a line (a) Parallel to 𝑥 − 𝑎𝑥𝑖𝑠 (b) ✔ Parallel to 𝑦 − 𝑎𝑥𝑖𝑠 (c) inclined at angle 𝜃 (d) None of these 50. 𝑪𝒐𝒔𝒆𝒄𝒉𝒙 is equal to

(a) 𝑒

𝑥−𝑒−𝑥 2 (b)^

𝑒𝑥+𝑒−𝑥 2 (c)^

2 𝑒𝑥−𝑒−𝑥^ (d)^ ✔^

2 𝑒𝑥+𝑒−𝑥 51. 𝒆𝟐𝒙+𝒆−𝟐𝒙 𝒆𝟐𝒙−𝒆−𝟐𝒙^ equals to (a) 𝑠𝑖𝑛ℎ2𝑥 (b) 𝑐𝑜𝑠ℎ2𝑥 (c) 𝑡𝑎𝑛ℎ2𝑥 (d) ✔ 𝑐𝑜𝑡ℎ2𝑥

52. The function 𝒇(𝒙) = (^) 𝒙+𝟏𝟏 is discontinuous at 𝒙 =

(a) 1 (b) ✔ 0 (c) -1 (d) all real numbers

53. If 𝒇(𝒙) = 𝒙𝟑^ − 𝟐𝒙𝟐^ + 𝟒𝒙 − 𝟏 , then 𝒇(−𝟏) = (a) 8 (b) ✔ -8 (c) 0 (d) - 54. The quantity which is used as a variable as well as constant is called (a) ✔Parameter (b) Constant (c) Real Number (d) None of these 55. If 𝒇(𝒙) = 𝒙−𝟏 𝒙+𝟒 , 𝒙 ≠ −𝟒^ then range of^ 𝒇^ is (a) ✔𝑅 − {1} (b) 𝑅 − {−4} (c) 𝑅 − {0} (d) all real numbers 56. 𝐥𝐢𝐦𝒙→∞ 𝒆𝒙^ = (a) 1 (b) ∞ (c) ✔ 0 (d) - 57. 𝐥𝐢𝐦𝒙→𝟎𝐬𝐢𝐧(𝒙−𝟑)𝒙−𝟑 =

(a) ✔ 1 (b) ∞ (c) 𝑠𝑖𝑛 3 (d) -

58. 𝐥𝐢𝐦𝒙→𝟎𝐬𝐢𝐧(𝒙−𝒂)𝒙−𝒂 =

(a) ✔ 1 (b) ∞ (c) 𝑠𝑖𝑛𝑎𝑎 (d) -

59. 𝒇(𝒙) = 𝒙𝟑^ + 𝒙 is : (a) Even (b) ✔ Odd (c) Neither even nor odd (d) None 60. 𝐥𝐢𝐦𝒙→𝟎(𝟏 + 𝒙)

𝟏 𝒙 (^) = (a) ✔𝑒 (b) 𝑒−1^ (c) 0 (d) 1

61. If 𝒇: 𝑿 → 𝒀 is a function , then elements of 𝒙 are called (a) Images (b) ✔ Pre-Images (c) Constants (d) Ranges 62. 𝐥𝐢𝐦𝒙→𝟎 ( (^) 𝟏+𝒙𝒙) =

(a) 𝑒 (b) ✔ 𝑒−1^ (c) 𝑒^2 (d) (^) √𝑒

63. If the degree of a polynomial function is 1, then it is (a) Identity function (b) Constant function (c) ✔ Linear function (d) Exponential function 64. 𝑪𝒐𝒔𝒉𝟐𝒙 + 𝑺𝒊𝒏𝒉𝟐𝒙 = (a) 1 (b) ✔ 𝐶𝑜𝑠ℎ2𝑥 (c) 𝑆𝑖𝑛ℎ2𝑥 (d) 0 65. 𝐥𝐢𝐦𝒙→𝟎𝑺𝒊𝒏𝒙𝒙 =

(a) 0 (b) ✔ 1 (c) -1 (d) Undefined

66. The function of the form 𝒙 = 𝒂𝒄𝒐𝒔𝒕 ; 𝒚 = 𝒃𝒔𝒊𝒏𝒕 (a) Odd function (b) Explicit function (c) ✔Parametric function(d) Even function 67. If 𝒇(𝒙) = √𝒙 + 𝟐 then range of 𝒇−𝟏^ is : (a) ✔ [-2,∞) (b) [2,∞) (c) (−∞, +∞) (d) [1,∞)

15. [𝒇(𝒙)𝒈(𝒙)]′ = Remember that [𝒇(𝒙)𝒈(𝒙)]′ = (^) 𝒅𝒙𝒅 [𝒇(𝒙)𝒈(𝒙)]

(a) 𝑓′(𝑥) + 𝑔′(𝑥) (b) 𝑓′(𝑥) − 𝑔′(𝑥) (c) ✔ 𝑓(𝑥)𝑔′(𝑥) + 𝑔(𝑥)𝑓′(𝑥) (d) 𝑓(𝑥)𝑔′(𝑥) − 𝑔(𝑥)𝑓′(𝑥)

16. (^) 𝒅𝒙𝒅 ( (^) 𝒈(𝒙)𝟏 ) =

(a) 1 [𝑔(𝑥)]^2 (b)^

1 𝑔′(𝑥) (c)^

𝑔′(𝑥) [𝑔(𝑥)]^2 (d)^ ✔^

−𝑔′(𝑥) [𝑔(𝑥)]^2

17. If 𝒇(𝒙) = 𝟏 𝒙 , 𝒕𝒉𝒆𝒏 𝒇

′′(𝒂) =

(a) − 2 (𝑎)^3 (b)^ −^

1 𝑎^2 (c)^

1 𝑎^2 (d)^ ✔^

2 𝑎^3

18. (𝒇𝒐𝒈)′(𝒙) = (a) 𝑓′𝑔′ (b) 𝑓′𝑔(𝑥) (c) ✔ 𝑓′(𝑔(𝑥))𝑔′(𝑥) (d) cannot be calculated 19. (^) 𝒅𝒙𝒅 (𝒈(𝒙)) 𝒏 = (a) 𝑛[𝑔(𝑥)]𝑛−1^ (b) 𝑛[(𝑔(𝑥)]𝑛−1𝑔(𝑥) (c) ✔ 𝑛[(𝑔(𝑥)]𝑛−1𝑔′(𝑥) (d) [𝑔(𝑥)]𝑛−1𝑔′(𝑥)

20. 𝒅 𝒅𝒙 𝒔𝒆𝒄

−𝟏𝒙 =

(a) ✔ 1 |𝑥|√𝑥^2 −1 (b)^

− |𝑥|√𝑥^2 −1 (c)^

1 |𝑥|√1+𝑥^2 (d)^

− |𝑥|√1+𝑥^2 21. 𝒅 𝒅𝒙 𝒄𝒐𝒔𝒆𝒄

−𝟏𝒙 =

(a) 1 |𝑥|√𝑥^2 −1 (b)^ ✔^

− |𝑥|√𝑥^2 −1 (c)^

1 |𝑥|√1+𝑥^2 (d)^

− |𝑥|√1+𝑥^2

22. The function 𝒇(𝒙) = 𝒂𝒙, 𝒂 > 0 , 𝑎 ≠ 0 , and 𝒙 is any real number is called (a) ✔Exponential function (b) logarithmic function (c) algebraic function (d) composite function 23. If 𝒂 > 0 , 𝒂 ≠ 𝟏, and 𝒙 = 𝒂𝒚^ then the function defined by 𝒚 = 𝒍𝒐𝒈𝒂𝒙^ (𝒙 > 0) is called a logarithmic function with base (a) 10 (b) 𝑒 (c) ✔ 𝑎 (d) 𝑥 24. 𝒍𝒐𝒈𝒂𝒂^ = (a) ✔ 1 (b) 𝑒 (c) 𝑎^2 (d) not defined

25. 𝒅 𝒅𝒙 𝒍𝒐𝒈𝟏𝟎𝒙^ = (a) 1 𝑥 log 10^ (b)^ ✔^

1 𝑥𝑙𝑜𝑔10 (c)^

𝑙𝑛𝑥 𝑥𝑙𝑛𝑥 (d)^

𝑙𝑛 𝑥𝑙𝑛𝑥

26. (^) 𝒅𝒙𝒅 𝒍𝒏[𝒇(𝒙)] =

(a) 𝑓′(𝑥) (b) 𝑙𝑛𝑓′(𝑥) (c) ✔ 𝑓

′(𝑥) 𝑓(𝑥) (d)^ 𝑓(𝑥). 𝑓

′(𝑥)

27. 𝒚 = 𝒔𝒊𝒏𝒉−𝟏𝒙 if and only if 𝒙 = 𝒔𝒊𝒏𝒉𝒚 is valid when (a) 𝑥 > 0, 𝑦 > 0 (b) 𝑥 < 0, 𝑦 < 0 (c) 𝑥 ∈ 𝑅, 𝑦 > 0 (d) ✔𝑥 ∈ 𝑅, 𝑥 > 0 28. 𝒚 = 𝒄𝒐𝒔𝒉−𝟏𝒙 if and only if 𝒙 = 𝒄𝒐𝒔𝒉𝒚 is valid when (a) ✔ 𝑥 ∈ [1, ∞), 𝑦 ∈ [0, ∞) (b) 𝑥 ∈ [1, ∞), 𝑦 ∈ (0, ∞] (c) 𝑥 < 0, 𝑦 < 0 (d) 𝑥 ∈ 𝑅, 𝑦 ∈ 𝑅 29. 𝒚 = 𝒕𝒂𝒏𝒉−𝟏𝒙 if and only if 𝒙 = 𝒕𝒂𝒏𝒉𝒚 is valid when (a) 𝑥 ∈ 𝑅, 𝑦 ∈ 𝑅 (b) ✔ 𝑥 ∈] − 1,1[, 𝑦 ∈ 𝑅 (c) 𝑥 ∈ 𝑅[−1,1], 𝑦 ∈ 𝑅 (d) 𝑥 > 0, 𝑦 > 0 30. 𝒚 = 𝒄𝒐𝒕𝒉−𝟏𝒙 if and only if 𝒙 = 𝒄𝒐𝒕𝒉𝒚 is valid when (a) 𝑥 ∈ 𝑅, 𝑦 ∈ 𝑅 (b) 𝑥 ∈] − 1,1[, 𝑦 ∈ 𝑅 (c) ✔ 𝑥 ∈ [−1,1], 𝑦 ∈ 𝑅 − {0} (d) 𝑥 > 0, 𝑦 > 0 31. 𝒚 = 𝒔𝒆𝒄𝒉−𝟏𝒙 if and only if 𝒙 = 𝒔𝒆𝒄𝒉𝒚 is valid when (a) 𝑥 ∈ 𝑅, 𝑦 ∈ 𝑅 (b) 𝑥 ∈] − 1,1[, 𝑦 ∈ 𝑅 (c) 𝑥 ∈ [−1,1], 𝑦 ∈ 𝑅 − {0}(d)✔ 𝑥 ∈ (0,1], 𝑦 ∈ [0, ∞) 32. 𝒚 = 𝒄𝒐𝒔𝒆𝒄𝒉−𝟏𝒙 if and only if 𝒙 = 𝒄𝒐𝒔𝒆𝒄𝒉𝒚 is valid when (a) 𝑥 ∈ 𝑅, 𝑦 ∈ 𝑅 (b) 𝑥 ∈] − 1,1[, 𝑦 ∈ 𝑅 (c)✔ 𝑥 ∈ 𝑅 − {0}, 𝑦 ∈ 𝑅 − {0}(d) 𝑥 ∈ (0,1], 𝑦 ∈ [0, ∞) 33. If 𝒚 = 𝒔𝒊𝒏𝒉−𝟏(𝒂𝒙 + 𝒃), then 𝒅𝒚𝒅𝒙 =

(a) cos−1(𝑎𝑥 + 𝑏) (b) 1 √1+(𝑎𝑥+𝑏)^2 (c)^ ✔^

𝑎 √1+(𝑎𝑥+𝑏)^2 (d)^ 𝑎 cosh

−1(𝑎𝑥 + 𝑏)

34. If 𝒄𝒐𝒔𝒉−𝟏(𝒔𝒆𝒄𝒙), 𝒕𝒉𝒆𝒏 𝒅𝒚 𝒅𝒙 = (a) 𝑐𝑜𝑠𝑥 (b) ✔𝑠𝑒𝑐𝑥 (c) – sin(𝑠𝑒𝑐𝑥) (d) – sinh−1(𝑠𝑒𝑐𝑥). 𝑡𝑎𝑛𝑥

35. If 𝒚 = 𝒆−𝒂𝒙^ , 𝒕𝒉𝒆𝒏 𝒚𝟐 = (a) – 𝑎𝑒𝑎𝑥^ (b) – 𝑎^2 𝑒𝑎𝑥^ (c) ✔ 𝑎^2 𝑒−2𝑎𝑥^ (d) – 𝑎^2 𝑒−2𝑎𝑥 36. If 𝒚 = 𝒆−𝒂𝒙, 𝒕𝒉𝒆𝒏𝒚 𝒅𝒚𝒅𝒙 =

(a) ✔– 𝑎𝑒−2𝑎𝑥^ (b) – 𝑎^2 𝑒𝑎𝑥^ (c) 𝑎^2 𝑒−2𝑎𝑥^ (d) – 𝑎^2 𝑒−2𝑎𝑥

37. If 𝒄𝒐𝒔(𝒂𝒙 + 𝒃), 𝒕𝒉𝒆𝒏 𝒚𝟐 = (a) 𝑎^2 sin(𝑎𝑥 + 𝑏) (b) – 𝑎^2 sin(𝑎𝑥 + 𝑏) (c) ✔– 𝑎^2 cos(𝑎𝑥 + 𝑏) (d) 𝑎^2 cos(𝑎𝑥 + 𝑏) 38. 𝒇(𝒙) = 𝒇(𝟎) + 𝒙𝒇′(𝒙) + 𝒙𝟐 𝟐! 𝒇

′′(𝒙) + 𝒙𝟑 𝟑! 𝒇

′′′(𝒙) + ⋯ …. + 𝒙𝒏 𝒏! 𝒇

𝒏(𝒙) … is called_____ series. (a) ✔Machlaurin’s (b) Taylor’s (c) Convergent (d) Divergent

39. 𝟏 − 𝒙 + 𝒙𝟐^ − 𝒙𝟑^ + 𝒙𝟒^ − ⋯ =

(a) ✔ 1 1+𝑥 (b)^

1 1−𝑥 (c)^ −^

1 1+𝑥 (d)^

1 𝑥−

[ 𝑯𝒊𝒏𝒕: 𝑼𝒔𝒆 𝑺∞ = (^) 𝟏−𝒓𝒂 , 𝒘𝒊𝒕𝒉 𝒂 = 𝟏 , 𝒓 = −𝒙]

40. 𝒅𝒚𝒅𝒙 |(𝒙𝟏,𝒚𝟏) represents

(a) Increments of 𝑥 1 and 𝑦 1 at (𝑥 1 , 𝑦 1 ) (b) ✔ slope of tangent at (𝑥 1 , 𝑦 1 ) (c) slope of normal at (𝑥 1 , 𝑦 1 ) (d) slope of horizontal line at (𝑥 1 , 𝑦 1 )

41. 𝒇 is said to be increasing on ]𝒂, 𝒃[ if for 𝒙𝟏, 𝒙𝟐 ∈]𝒂, 𝒃[ (a) ✔𝑓(𝑥 2 ) > 𝑓(𝑥 1 ) whenever 𝑥 2 > 𝑥 1 (b) 𝑓(𝑥 2 ) > 𝑓(𝑥 1 ) whenever 𝑥 2 < 𝑥 1 (c) 𝑓(𝑥 2 ) < 𝑓(𝑥 1 ) whenever 𝑥 2 > 𝑥 1 (d) 𝑓(𝑥 2 ) < 𝑓(𝑥 1 ) whenever 𝑥 2 < 𝑥 1 42. 𝒇 is said to be decreasing on ]𝒂, 𝒃[ if for 𝒙𝟏, 𝒙𝟐 ∈]𝒂, 𝒃[ (b) 𝑓(𝑥 2 ) > 𝑓(𝑥 1 ) whenever 𝑥 2 > 𝑥 1 (b) 𝑓(𝑥 2 ) > 𝑓(𝑥 1 ) whenever 𝑥 2 < 𝑥 1 (c) ✔ 𝑓(𝑥 2 ) < 𝑓(𝑥 1 ) whenever 𝑥 2 > 𝑥 1 (d) 𝑓(𝑥 2 ) < 𝑓(𝑥 1 ) whenever 𝑥 2 < 𝑥 1 43. If a function 𝒇 is increasing within ]𝒂, 𝒃[ ,then slope of tangent to its graph within ]𝒂, 𝒃[ remains (a) ✔Positive (b) Negative (c) Zero (d) Undefined 44. If a function 𝒇 is decreasing within ]𝒂, 𝒃[ ,then slope of tangent to its graph within ]𝒂, 𝒃[ remains (b) Positive (b) ✔ Negative (c) Zero (d) Undefined 45. A point where 1st^ derivative of function is zero , is called (a) ✔Stationary point (b) corner point (c) point of concurrency (d) common point 46. 𝒇(𝒙) = 𝒔𝒊𝒏𝒙 is (a) Linear function (b) ✔ odd function (c) even function (d) identity function 47. The maximum value of the function 𝒇(𝒙) = 𝒙𝟐^ − 𝒙 − 𝟐 is

(a) − 9 2 (b)^ ✔^ −^

9 4 (c) -1^ (d) 0

48. 𝒅 𝒅𝒙 (𝒄𝒐𝒔𝒙) −^

𝒅𝟐 𝒅𝒙𝟐^ (𝒔𝒊𝒏𝒙) = (a) 2𝑠𝑖𝑛𝑥 (b) 2𝑐𝑜𝑠𝑥 (c) ✔ 0 (d) −2𝑠𝑖𝑛𝑥

49. If 𝒇(𝒙) = 𝒙𝟑^ + 𝟐𝒙 + 𝟗 then 𝒇′′(𝒙) = (a) 3𝑥^2 + 2 (b) 3𝑥^2 (c) ✔ 6𝑥 (d) 2𝑥 50. If 𝒇(𝒙) = 𝒔𝒊𝒏𝒙 then 𝒇′(𝒄𝒐𝒔−𝟏𝟑𝒙) =

(a) 𝑐𝑜𝑠𝑥 (b) (^) √1−9𝑥−3 2 (c) (^) √1−9𝑥^32 (d) ✔ 3𝑥

51. (^) 𝒅𝒙𝒅 (𝟏𝟎𝒔𝒊𝒏𝒙) =

(a) 10 𝑐𝑜𝑠𝑥^ (b) ✔ 10 𝑠𝑖𝑛𝑥. 𝑐𝑜𝑠𝑥. 𝑙𝑛10 (c) 10 𝑠𝑖𝑛𝑥. 𝑙𝑛10 (d) 10 𝑐𝑜𝑠𝑥. 𝑙𝑛

52. (^) 𝒅𝒙𝒅 (√𝒙 − (^) √𝒙𝟏)

𝟐

(a) 1 − 1 2𝑥 (b)^ ✔^ 1 −^

1 𝑥^2 (c)^ 1 +^

1 𝑥^2 (d) 0

53. At 𝒙 = 𝟎 , the function 𝒇(𝒙) = 𝟏 − 𝒙𝟑^ has (a) Maximum value (b) minimum value (c) ✔point of inflexion (d) no conclusion 54. If 𝑺𝒊𝒏 √𝒙 , then 𝒅𝒚𝒅𝒙 is equal to

(a) ✔𝑐𝑜𝑠√𝑥 2√𝑥 (b) 𝑐𝑜𝑠√𝑥 √𝑥 (c) 𝑐𝑜𝑠√𝑥 (d) 𝑐𝑜𝑠𝑥√𝑥

(a) (^) √1+𝑥^12 (b) 3𝑥

2 √1+𝑥^2 (c)^

1 √1+𝑥^6 (d)^ ✔^

3𝑥^2 √1+𝑥^6

77. A function 𝒇(𝒙) is such that , at a point 𝒙 = 𝒄 , 𝒇′(𝒙) > 0 at 𝒙 = 𝒄 , then 𝒇 is said to be (a) ✔Increasing (b) decreasing (c) constant (d) 1-1 function 78. A function 𝒇(𝒙) is such that , at a point 𝒙 = 𝒄 , 𝒇′(𝒙) < 0 at 𝒙 = 𝒄 , then 𝒇 is said to be (a) Increasing (b) ✔ decreasing (c) constant (d) 1-1 function (b) A function 𝒇(𝒙) is such that , at a point 𝒙 = 𝒄 , 𝒇′(𝒙) = 𝟎 at 𝒙 = 𝒄 , then 𝒇 is said to be (a) Increasing (b) decreasing (c) ✔ constant (d) 1-1 function 79. A stationary point is called ______ if it is either a maximum point or a minimum point (a) Stationary point (b) ✔ turning point (c) critical point (d) point of inflexion 80. If 𝒇′(𝒄) = 𝟎 or 𝒇′(𝒄) is undefined , then the number 𝒄 is called critical value and the corresponding point is called_______ (a) Stationary point (b) turning point (c) ✔ critical point (d) point of inflexion 81. If 𝒇′(𝒄) does not change before and after 𝒙 = 𝒄 , then this point is called_______ (a) Stationary point (b) turning point (c) critical point (d) ✔ point of inflexion 82. Let 𝒇 be a differentiable function such that 𝒇′(𝒄) = 𝟎 then if 𝒇′(𝒙) changes sign from +iv to - iv i.e., before and after 𝒙 = 𝒄 , then it occurs relative ______ at 𝒙 = 𝒄 (a) ✔Maximum (b) minimum (c) point of inflexion (d) none 83. Let 𝒇 be a differentiable function such that 𝒇′(𝒄) = 𝟎 then if 𝒇′(𝒙) changes sign from -iv to +iv i.e., before and after 𝒙 = 𝒄 , then it occurs relative ______ at 𝒙 = 𝒄 (b) Maximum (b) ✔ minimum (c) point of inflexion (d) none 84. Let 𝒇 be a differentiable function such that 𝒇′(𝒄) = 𝟎 then if 𝒇′(𝒙)^ does not change sign i.e., before and after 𝒙 = 𝒄 , then it occurs ______ at 𝒙 = 𝒄 (c) Maximum (b) minimum (c) ✔point of inflexion (d) none 85. If 𝒇(𝒙) = 𝒆√𝒙−𝟏^ then 𝒇′(𝟎) = (a) 𝑒−1^ (b) 𝑒 (c) ✔∞ (d) 1 2 86. (^) 𝒅𝒙𝒅 (𝒕𝒂𝒏−𝟏𝒙 − 𝒄𝒐𝒕−𝟏𝒙) = (a) 2 √1+𝑥^2 (b)^ ✔^

2 1+𝑥^2 (c) 0^ (d)^

− 1+𝑥^2

87. If 𝒇 ( 𝟏 𝒙) = 𝒕𝒂𝒏𝒙^ , then^ 𝒇

′ (^) (𝟏 𝝅) = (a) 𝜋^2 (b) ✔ – 𝜋^2 (c) 1 (d) − 𝜋^2

88. 𝐥𝐢𝐦𝒉→𝟎𝒇(𝒂+𝒉)−𝒇(𝒂)𝒉 = (a) 0 (b) 𝑓(𝑎) (c) 𝑓(ℎ) (d) ✔ 𝑓′(𝑎) 89. If 𝒇(𝒙) = 𝟏 𝒙 , then a critical point of^ 𝒇^ is (a) ✔ 0 (b) 1 (c) -1 (d) no point

UNIT # 03 Integration

Each question has four possible answer.Tick the correct answer.

1. If 𝒚 = 𝒇(𝒙), then differential of 𝒚 is (a) 𝑑𝑦 = 𝑓′(𝑥)^ (b) ✔ 𝑑𝑦 = 𝑓′(𝑥)𝑑𝑥 (c) 𝑑𝑦 = 𝑓(𝑥)𝑑𝑥 (d) 𝑑𝑦 𝑑𝑥 2. If (^) ∫ 𝒇(𝒙)𝒅𝒙 = 𝝋(𝒙) + 𝒄 ,then 𝒇(𝒙) is called (a) Integral (b) differential (c) derivative (d) ✔ integrand 3. If 𝒏 ≠ 𝟏 , then (^) ∫(𝒂𝒙 + 𝒃)𝒏𝒅𝒙 = (a) 𝑛(𝑎𝑥+𝑏)

𝑛− 𝑎 + 𝑐^ (b)^

𝑛(𝑎𝑥+𝑏)𝑛+ 𝑛 + 𝑐^ (c)^

(𝑎𝑥+𝑏)𝑛− 𝑛+1 + 𝑐^ (d)^ ✔^

(𝑎𝑥+𝑏)𝑛+ 𝑎(𝑛+1) + 𝑐

4. ∫ 𝐬𝐢𝐧(𝒂𝒙 + 𝒃) 𝒅𝒙 =

Note:- Every stationary point is also called critical point but then converse may or may not be true.

(a) ✔ − 𝑎 cos(𝑎𝑥 + 𝑏) + 𝑐^ (b)^

1 𝑎 cos(𝑎𝑥 + 𝑏) + 𝑐^ (c)^ 𝑎 cos(𝑎𝑥 + 𝑏) + 𝑐^ (d)−𝑎 cos(𝑎𝑥 + 𝑏) + 𝑐

5. (^) ∫ 𝒆−𝝀𝒙𝒅𝒙 =

(a) 𝜆𝑒−𝜆𝑥^ + 𝑐 (b) – 𝜆𝑒−𝜆𝑥^ + 𝑐 (c) 𝑒−𝜆𝑥 𝜆 + 𝑐^ (d)^ ✔^

𝑒−𝜆𝑥 −𝜆 + 𝑐

6. ∫ 𝒂𝝀𝒙𝒅𝒙 =

(a) 𝑎

𝜆𝑥 𝜆 (b)^

𝑎𝜆𝑥 𝑙𝑛𝑎 (c)^ ✔^

𝑎𝜆𝑥 𝑎𝑙𝑛𝑎 (d)^ 𝑎

𝜆𝑥𝜆. 𝑙𝑛𝑎

7. ∫[𝒇(𝒙)]𝒏𝒇′(𝒙)𝒅𝒙 =

(a) 𝑓

𝑛(𝑥) 𝑛 + 𝑐^ (b)^ 𝑓(𝑥) + 𝑐^ (c)^ ✔^

𝑓𝑛+1(𝑥) 𝑛+1 + 𝑐^ (d)^ 𝑛𝑓

𝑛+1(𝑥) + 𝑐

8. ∫ 𝒇′(𝒙) 𝒇(𝒙) 𝒅𝒙 = (a) 𝑓(𝑥) + 𝑐 (b) 𝑓′(𝑥) + 𝑐 (c) ✔ln|𝑥| + 𝑐 (nd) ln|𝑓′(𝑥)| + 𝑐 9. (^) ∫ 𝒅𝒙 √𝒙+𝒂+√𝒙^ can be evaluated if

(a) ✔𝑥 > 0, 𝑎 > 0 (b) 𝑥 < 0, 𝑎 > 0 (c) 𝑥 < 0, 𝑎 < 0 (d) 𝑥 > 0, 𝑎 < 0

10. (^) ∫ 𝒙 √𝒙𝟐+𝟑 𝒅𝒙 =

(a) ✔√𝑥^2 + 3 + 𝑐 (b) −√𝑥^2 + 3 + 𝑐 (c) √𝑥^2 + 2 + 𝑐^ (d)^ −^

1 2 √𝑥

(^2) + 3 + 𝑐

11. (^) ∫ 𝒆𝒙 𝟐 . 𝒙𝒅𝒙 =

(a) 𝑎

𝑥^2 𝑙𝑛𝑎 + 𝑐^ (b)^ ✔^

𝑎𝑥^2 2𝑙𝑛𝑎 + 𝑐^ (c)^ 𝑎

𝑥^2 𝑙𝑛𝑎 + 𝑐 (d) 𝑎𝑥 2 2 + 𝑐

12. ∫ 𝒆𝒂𝒙[𝒂𝒇(𝒙) + 𝒇′(𝒙)]𝒅𝒙 = (a) ✔𝑒𝑎𝑥𝑓(𝑥) + 𝑐 (b) 𝑒𝑎𝑥𝑓′(𝑥) + 𝑐 (c) 𝑎𝑒𝑎𝑥𝑓(𝑥) + 𝑐 (d) 𝑎𝑒𝑎𝑥𝑓′(𝑥) + 𝑐 13. ∫ 𝒆𝒙[𝒔𝒊𝒏𝒙 + 𝒄𝒐𝒔]𝒅𝒙 = (a) ✔𝑒𝑥𝑠𝑖𝑛𝑥 + 𝑐 (b) 𝑒𝑥𝑐𝑜𝑠 + 𝑐 (c) – 𝑒𝑥𝑠𝑖𝑛𝑥 + 𝑐 (d) – 𝑒𝑥𝑐𝑜𝑠𝑥 + 𝑐 14. To determine the area under the curve by the use of integration , the idea was given by (a) Newton (b) ✔ Archimedes (c) Leibnitz (d) Taylor 15. The order of the differential equation : 𝒙 𝒅𝟐𝒚 𝒅𝒙𝟐^ +^

𝒅𝒚 𝒅𝒙 − 𝟐 = 𝟎 (a) 0 (b) 1 (c) ✔ 2 (d) more than 2

16. The equation 𝒚 = 𝒙𝟐^ − 𝟐𝒙 + 𝒄 represents ( 𝒄 being a parameter ) (a) One parabola (b) family of parabolas (c) family of line (d) two parabolas 17. (^) ∫ 𝒆𝒔𝒊𝒏𝒙. 𝒄𝒐𝒔𝒙𝒅𝒙 =

(a) ✔𝑒𝑠𝑖𝑛𝑥^ + 𝑐 (b) 𝑒𝑐𝑜𝑠𝑥^ + 𝑐 (c) 𝑒𝑠𝑖𝑛𝑥 𝑐𝑜𝑠𝑥 (d)^

𝑒𝑐𝑜𝑠𝑥 𝑠𝑖𝑛𝑥

18. (^) ∫(𝟐𝒙 + 𝟑)

𝟏 𝟐𝒅𝒙 =

(a) 1 3 (2𝑥 + 3)

3 (^2) (b) 1 3 (2𝑥 + 3)

− (^12) (c) 1 3 (2𝑥 + 3)^ (d) None

19. ∫ 𝒙𝒏𝒅𝒙 = 𝒙+𝟏 𝒏+𝟏 + 𝒄^ is true for all values of^ 𝒏^ except (a) 𝑛 = 0 (b) 𝑛 = 1 (c) ✔𝑛 ≠ −1 (d) 𝑛 = 𝑎𝑛𝑦 𝑓𝑟𝑎𝑐𝑡𝑖𝑜𝑛𝑎𝑙 𝑣𝑎𝑙𝑢𝑒 20. (^) ∫ 𝒂𝟏^ 𝟐 𝒙𝒅𝒙 =

(a) (𝑎^2 − 𝑎)𝑙𝑛𝑎 (b) ✔ (𝑎^2 −𝑎) 𝑙𝑛𝑎 (c)^

(𝑎^2 −𝑎) log 𝑎 (d)^ (𝑎

(^2) − 𝑎)𝑙𝑛𝑎

21. ∫ 𝒆𝑻𝒂𝒏−𝟏𝒙 𝟏+𝒙𝟐^ 𝒅𝒙 = (a) 𝑒𝑇𝑎𝑛𝑥^ + 𝑐 (b) 1 2 𝑒

𝑇𝑎𝑛−1𝑥 (^) + 𝑐 (c) 𝑥 𝑒𝑇𝑎𝑛−1𝑥 (^) + 𝑐 (d) ✔ 𝑒𝑇𝑎𝑛−1𝑥 (^) + 𝑐

22. ∫ 𝒅𝒙 𝒙√𝒙𝟐−𝟏 =

(a) ✔𝑆𝑒𝑐−1𝑥 + 𝑐 (b) 𝑇𝑎𝑛−1𝑥 + 𝑐 (c) 𝐶𝑜𝑡−1𝑥 + 𝑐 (d) 𝑆𝑖𝑛−1𝑥 + 𝑐

23. ∫ 𝒔𝒊𝒏𝟑𝒙𝒅𝒙 is equal to

(a) 𝑐𝑜𝑠3𝑥 3 + 𝑐^ (b)^ ✔^ –^

𝑐𝑜𝑠3𝑥 3 + 𝑐^ (c)^ 3𝑐𝑜𝑠3𝑥 + 𝑐^ (d)^ −3 𝑐𝑜𝑠3𝑥 + 𝑐

24. If (^) ∫ 𝒇(𝒙)𝒅𝒙 = 𝟓, ∫ 𝒈(𝒙)𝒅𝒙 = 𝟒𝟐𝟏 𝟐^ 𝟏 𝒕𝒉𝒆𝒏 ∫−𝟐 𝟏𝟑𝒇(𝒙)𝒅𝒙 − ∫−𝟐 𝟏𝟐𝒈(𝒙)𝒅𝒙 =

(a) ✔ 7 (b) 9 (c) 12 (d) 8

25. (^) ∫ 𝒆𝒇(𝒙). 𝒇′(𝒙)𝒅𝒙 =

46. The general solution of differential equation 𝒅𝒚𝒅𝒙 = − 𝒚𝒙 is (a) 𝑥 𝑦 = 𝑐^ (b)^

𝑦 𝑥 = 𝑐^ (c)^ ✔^ 𝑥𝑦 = 𝑐^ (d)𝑥

(^2) 𝑦 (^2) = 𝑐

47. (^) ∫ 𝒙+𝟐𝒙+𝟏 𝒅𝒙 = (a) ln(𝑥 + 1) + 𝑐 (b) ln(𝑥 + 1) − 𝑥 + 𝑐 (c) ✔𝑥 + ln(𝑥 + 1) + 𝑐 (d) None 48. ∫ 𝒔𝒊𝒏𝟑𝒙𝒄𝒐𝒔𝒙𝒅𝒙 = (a) sin^3 𝑥 3 + 𝑐^ (b)^ ✔^

1 4 sin

(^4) 𝑥 + 𝑐 (c) − 1 4 sin

(^4) 𝑥 + 𝑐 (d) sin 4 𝑥 4 + 𝑐

49. ∫ 𝒙 𝒆𝒙𝒅𝒙 = (a) 𝑥 𝑒𝑥^ + 𝑥 + 𝑐 (b) ✔ 𝑥 𝑒𝑥^ − 𝑥 + 𝑐 (c) 𝑒𝑥^ − 𝑥 (d) None of these 50. (^) ∫ 𝒅𝒙 𝒙𝟐+𝟗 =

𝟑 𝟎 (a) 𝜋 4 (b) ✔ 12 𝜋 (c) 𝜋 2 (d) None of these

51. ∫ 𝒆𝒙^ [ 𝟏 𝒙 + 𝒍𝒏𝒙] = (a) 𝑒𝑥 1 𝑥 + 𝑐^ (b)^ – 𝑒

𝑥^1 𝑥 + 𝑐^ (c)^ ✔^ 𝑒

𝑥𝑙𝑛𝑥 + 𝑐 (d) – 𝑒𝑥𝑙𝑛𝑥 + 𝑐

52. (^) ∫ 𝒔𝒊𝒏𝒙𝒅𝒙 = −𝝅 𝝅 (a) ✔ 2 (b) -2 (c) 0 (d) - 53. (^) ∫ |𝒙|𝒅𝒙 = 𝟐 −𝟏 (a) 1 2 (b)^ −^

1 2 (c)^

5 2 (d)^ ✔^

3 2

54. ∫ (𝟒𝒙 + 𝒌)𝒅𝒙 = 𝟐 𝒕𝒉𝒆𝒏 𝒌 = 𝟏 𝟎 (a) 8 (b) -4 (c) ✔ 0 (d) - 55. ∫ 𝒆𝒙^ [ 𝟏 𝒙 −^

𝟏 𝒙𝟐] =

(a) ✔𝑒𝑥^1 𝑥 + 𝑐 (b) – 𝑒𝑥^1 𝑥 + 𝑐 (c) 𝑒𝑥𝑙𝑛𝑥 + 𝑐 (d) – 𝑒𝑥^ 𝑥^12 + 𝑐

56. Solution of the differential equation : 𝒅𝒚𝒅𝒙 = 𝟏 √𝟏−𝒙𝟐 (a) ✔𝑦 = sin−1^ 𝑥 + 𝑐 (b) 𝑦 = cos−1^ 𝑥 + 𝑐 (c) 𝑦 = tan−1^ 𝑥 + 𝑐 (d) None

UNIT # 04 Introduction to Analytic

Geometry

Each question has four possible answer. Tick the correct answer.

1. 𝑰𝒇 𝒙 < 0, 𝑦 < 0 then the point 𝑷(𝒙, 𝒚) lies in the quadrant (a) I (b) II (c) ✔ III (d) IV 2. The point P in the plane that corresponds to the ordered pair (𝒙, 𝒚) is called: (a) ✔𝑔𝑟𝑎𝑝ℎ 𝑜𝑓 (𝑥, 𝑦) (b) mid-point of 𝑥, 𝑦 (c) 𝑎𝑏𝑠𝑐𝑖𝑠𝑠𝑎 𝑜𝑓 𝑥, 𝑦 (d) ordinate of 𝑥, 𝑦 3. If 𝒙 < 0 , 𝑦 > 0 then the point 𝑷(−𝒙, −𝒚) lies in the quadrant (a) I (b) II (c) III (d) ✔ IV 4. The straight line which passes through one vertex and though the mid-point of the opposite side is called: (a) ✔Median (b) altitude (c) perpendicular bisector (d) normal 5. The straight line which passes through one vertex and perpendicular to opposite side is called: (a) Median (b) ✔ altitude (c) perpendicular bisector (d) normal 6. The point where the medians of a triangle intersect is called_________ of the triangle. (a) ✔Centroid (b) centre (c) orthocenter (d) circumference 7. The point where the altitudes of a triangle intersect is called_________ of the triangle. (a) Centroid (b) centre (c) ✔ orthocenter (d) circumference 8. The centroid of a triangle divides each median in the ration of (a) ✔2:1 (b) 1:2 (c) 1:1 (d) None of these 9. The point where the angle bisectors of a triangle intersect is called_________ of the triangle.

(a) Centroid (b) ✔in centre (c) orthocenter (d) circumference

10. If 𝒙 and 𝒚 have opposite signs then the point 𝑷(𝒙, 𝒚) lies the quadrants (a) I & II (b) I & III (c) ✔ II & IV (d) I & IV 11. A line bisecting 2nd^ and 4th^ quadrants has inclination: (a) 0° (b) 45° (c) ✔ 135° (d) ∞ 12. 𝒚 = 𝒙 is the straight line (a) ✔Bisecting I & III (b) parallel to 𝑥 − 𝑎𝑥𝑖𝑥 (c) bisecting II & IV (d) parallel to 𝑦 − 𝑎𝑥𝑖𝑠 13. If all the sides of four sided polygon are equal but the four angles are not equal to 𝟗𝟎° each then it is a (a) Kite (b) ✔ rhombus (c) ||gram (d) trapezoid 14. If 𝜶 is the inclination of a line 𝒍 then it must be true that (a) 0 ≤ 𝛼 ≤ 𝜋 2 (b) 𝜋 2 ≤ 𝛼 ≤ 𝜋 (c) ✔0 ≤ 𝛼 ≤ 𝜋 (d) 0 ≤ 𝛼 ≤ 2𝜋 15. The slope-intercept form of the equation of the straight line is (a) ✔𝑦 = 𝑚𝑥 + 𝑐 (b) 𝑦 − 𝑦 1 = 𝑚(𝑥 − 𝑥 1 ) (c) (^) 𝑎𝑥 + 𝑦𝑏 = 1 (d) 𝑥𝑐𝑜𝑠𝛼 + 𝑦𝑐𝑜𝑠𝛼 = 𝑝 16. The two intercepts form of the equation of the straight line is (a) 𝑦 = 𝑚𝑥 + 𝑐 (b) 𝑦 − 𝑦 1 = 𝑚(𝑥 − 𝑥 1 ) (c) ✔ (^) 𝑎𝑥 + 𝑦𝑏 = 1 (d) 𝑥𝑐𝑜𝑠𝛼 + 𝑦𝑐𝑜𝑠𝛼 = 𝑝 17. The Normal form of the equation of the straight line is (a) 𝑦 = 𝑚𝑥 + 𝑐 (b) 𝑦 − 𝑦 1 = 𝑚(𝑥 − 𝑥 1 ) (c) (^) 𝑎𝑥 + 𝑦𝑏 = 1 (d) ✔𝑥𝑐𝑜𝑠𝛼 + 𝑦𝑐𝑜𝑠𝛼 = 𝑝 18. In the normal form 𝒙𝒄𝒐𝒔𝜶 + 𝒚𝒄𝒐𝒔𝜶 = 𝒑 the value of 𝒑 is

(a) ✔Positive (b) Negative (c) positive or negative (d) Zero

19. If 𝜶 is the inclination of the line 𝒍 then 𝒙−𝒙𝟏 𝒄𝒐𝒔𝜶 =^

𝒚−𝒚𝟏 𝒔𝒊𝒏𝜶 = 𝒓(𝒔𝒂𝒚) (a) Point-slope form (b) normal form (c) ✔symmetric form (d) none of these

20. The slope of the line 𝒂𝒙 + 𝒃𝒚 + 𝒄 = 𝟎 is

(a) 𝑎𝑏 (b) ✔ – 𝑎𝑏 (c) 𝑏𝑎 (d) – 𝑏𝑎

21. The slope of the line perpendicular to 𝒂𝒙 + 𝒃𝒚 + 𝒄 = 𝟎

(a) 𝑎 𝑏 (b)^ –^

𝑎 𝑏 (c)^ ✔^

𝑏 𝑎 (d)^ –^

𝑏 𝑎

22. The general equation of the straight line in two variables 𝒙 and 𝒚 is

(a) ✔𝑎𝑥 + 𝑏𝑦 + 𝑐 = 0 (b) 𝑎𝑥^2 + 𝑏𝑦 + 𝑐 = 0 (c) 𝑎𝑥 + 𝑏𝑦^2 + 𝑐 = 0 (d) 𝑎𝑥^2 + 𝑏𝑦^2 + 𝑐 = 0

23. The 𝒙 − 𝒊𝒏𝒕𝒆𝒓𝒄𝒆𝒑𝒕 𝟒𝒙 + 𝟔𝒚 = 𝟏𝟐 is (a) 4 (b) 6 (c) ✔ 3 (d) 2 24. The lines 𝟐𝒙 + 𝒚 + 𝟐 = 𝟎 and 𝟔𝒙 + 𝟑𝒚 − 𝟖 = 𝟎 are

(a) ✔Parallel (b) perpendicular (c) neither (d) non coplanar

25. The point (−𝟐, 𝟒) lies ____ the line 𝟐𝒙 + 𝟓𝒚 − 𝟑 = 𝟎

(a) ✔Above (b) below (c) on (d) none of these

26. If three lines pass through one common point then the lines are called

(a) Parallel (b) coincident (c) ✔ concurrent (d) congruent

27. 𝟐𝒙 + 𝒚 + 𝒌 ( 𝒌 being a parameter) represents

(a) One line (b) two lines (c) ✔ family of lines (d) intersection lines

28. If the equations of the sides of a triangle are given then the intersection of any two lines in pairs gives ________ the triangles. (a) ✔Vertices (b) centre (c) mid-points of sides (d) centriod 29. A four sided polygon (quadrilateral) having two parallel and non-parallel sides is called (a) Square (b) rhombus (c) ✔ trapezium (d) ||gram 30. Equation of vertical line through (−𝟓, 𝟑) is (a) 𝑥 − 5 = 0 (b) ✔ 𝑥 + 5 = 0 (c) 𝑦 − 3 = 0 (d) 𝑦 + 3 = 0 31. Equation of horizontal line through (−𝟓, 𝟑) is (a) 𝑥 − 5 = 0 (b) 𝑥 + 5 = 0 (c) ✔ 𝑦 − 3 = 0 (d) 𝑦 + 3 = 0 32. Equation of line through (−𝟖, 𝟓) and having slope undefined is (a) ✔𝑥 + 8 = 0 (b) 𝑥 + 5 = 0 (c) 𝑦 − 5 = 0 (d) 𝑦 + 5 = 0 33. If 𝝋 be an angle between two lines 𝒍𝟏 and 𝒍𝟐 when slopes 𝒎𝟏 and 𝒎𝟐 , then angle from 𝒍𝟏 to 𝒍𝟐 (a) 𝑡𝑎𝑛𝜑 = (^) 1+𝑚𝑚^1 −𝑚^2 1 𝑚

(b) ✔ 𝑡𝑎𝑛𝜑 = (^) 1+𝑚𝑚^2 −𝑚^1 2 𝑚

(c) 𝑡𝑎𝑛𝜑 = (^) 1+𝑚𝑚^1 +𝑚^2 1 𝑚

(d) 𝑡𝑎𝑛𝜑 = (^) 1+𝑚𝑚^2 +𝑚^1 1 𝑚

59. The perpendicular distance of a line 𝟏𝟐𝒙 + 𝟓𝒚 = 𝟕 from (𝟎, 𝟎) is: (a) 131 (b) 137 (c) ✔ 137 (d) 13 60. Line passes through the point of intersection of two line 𝒍𝟏 and 𝒍𝟐 is (a) 𝑘 1 𝑙 1 = 𝑘 2 𝑙 2 (b) ✔ 𝑙 1 + 𝑘𝑙 2 = 0 (c) 𝑙 1 + 𝑘𝑙 2 = 1 (d) None 61. The coordinate 𝒂𝒙𝒆𝒔 divide the whole plane into ________ equal parts. (a) 2 (b) ✔ 4 (c) 8 (d) infinity many 62. If 𝟐𝒙 + 𝟓𝒚 + 𝒌 and 𝒌𝒙 + 𝟏𝟎𝒚 + 𝟑 = 𝟎 are parallel lines then 𝒌 (a) ✔ 25 (b) -25 (c) 2 (d) 3

UNIT # 05 Linear Inequalities and Linear

Programming

Each question has four possible answer. Tick the correct answer.

1. The solution of 𝒂𝒙 + 𝒃 < 𝑐 is (a) Closed half plane (b) ✔ open half plane (c) circle (d) parabola 2. A function which is to be maximized or minimized is called______ function (a) Subjective (b) ✔ objective (c) qualitative (d) quantitative 3. The number of variables in 𝒂𝒙 + 𝒃𝒚 ≤ 𝒄 are (a) 1 (b) ✔ 2 (c) 3 (d) 4 4. (0,0) is the solution of the inequality (a) 7𝑥 + 2𝑦 > 0 (b) 2𝑥 − 𝑦 > 0 (c) ✔ 𝑥 + 𝑦 ≥ 0 (d) 3𝑥 + 5𝑦 < 0 5. (0,0) is satisfied by (a) 𝑥 − 𝑦 < 10 (b) 2𝑥 + 5𝑦 > 10 (c) ✔ 𝑥 − 𝑦 ≥ 13 (d) None 6. The point where two boundary lines of a shaded region intersect is called _____ point. (a) Boundary (b) ✔ corner (c) stationary (d) feasible 7. If 𝒙 > 𝑏 then (a) – 𝑥 > −𝑏 (b) – 𝑥 < 𝑏 (c) 𝑥 < 𝑏 (d) ✔ – 𝑥 < −𝑏 8. The symbols used for inequality are (a) 1 (b) 2 (c) 3 (d) ✔ 4 9. A linear inequality contains at least _________ variables. (a) ✔One (b) two (c) three (d) more than three 10. An inequality with one or two variables has ________ solutions. (a) One (b) two (c) three (d) ✔infinitely many 11. 𝒂𝒙 + 𝒃𝒚 < 𝑐 is not a linear inequality if (a) ✔𝑎 = 0, 𝑏 = 0 (b) 𝑎 ≠ 0 , 𝑏 ≠ 0 (c) 𝑎 = 0, 𝑏 ≠ 0 (d) 𝑎 ≠ 0, 𝑏 = 0, 𝑐 = 0 12. The graph of corresponding linear equation of the linear inequality is a line called________ (a) ✔Boundary line (b) horizontal line (c) vertical line (d) inclined line 13. The graph of a linear equation of the form 𝒂𝒙 + 𝒃𝒚 = 𝒄 is a line which divides the whole plane into ______ disjoints parts. (a) ✔Two (b) four (c) more than four (d) infinitely many 14. The graph of the inequality 𝒙 ≤ 𝒃 is (a) Upper half plane (b) lower half plane (c) ✔ left half plane (d) right half plane 15. The graph of the inequality 𝒚 ≤ 𝒃 is (b) Upper half plane (b) ✔ lower half plane (c) left half plane (d) right half plane 16. The graph of the inequality 𝒂𝒙 + 𝒃𝒚 ≤ 𝒄 is _____ side of line 𝒂𝒙 + 𝒃𝒚 = 𝒄 (a) ✔Origin side (b) non-origin side (c) upper (d) lower 17. The graph of the inequality 𝒂𝒙 + 𝒃𝒚 ≥ 𝒄 is _____ side of line 𝒂𝒙 + 𝒃𝒚 = 𝒄 (b) Origin side (b) ✔ non-origin side (c) upper (d) left 18. The feasible solution which maximizes or minimizes the objective function is called (a) Exact solution (b) ✔ optimal solution (c) final solution (d) objective function 19. Solution space consisting of all feasible solutions of system of linear in inequalities is called

(a) Feasible solution (b) Optimal solution (c) ✔ Feasible region (d) General solution

20. Corner point is also called (a) Origin (b) Focus (c) ✔ Vertex (d) Test point 21. For feasible region: (a) ✔𝑥 ≥ 0, 𝑦 ≥ 0 (b) 𝑥 ≥ 0, 𝑦 ≤ 0 (c) 𝑥 ≤ 0, 𝑦 ≥ 0 (d) 𝑥 ≤ 0, 𝑦 ≤ 0 22. 𝒙 = 𝟎 is in the solution of the inequality (a) 𝑥 < 0 (b) 𝑥 + 4 < 0 (c) ✔2𝑥 + 3 > 0 (d)2𝑥 + 3 < 0 23. Linear inequality 𝟐𝒙 − 𝟕𝒚 > 3 is satisfied by the point (a) (5,1) (b) (-5,-1) (c) (0,0) (d) ✔ (1,-1) 24. The non-negative constraints are also called (a) ✔Decision variable (b) Convex variable (c) Decision constraints (d) concave variable 25. If the line segment obtained by joining any two points of a region lies entirely within the region , then the region is called (a) Feasible region (b) ✔ Convex region (c) Solution region (d) Concave region

UNIT # 06 Conic Section

Each question has four possible answer. Tick the correct answer.

1. The locus of a revolving line with one end fixed and other end on the circumference of a circle of a circle is called: (a) a sphere (b) a circle (c) ✔a cone (d) a conic 2. The set of points which are equal distance from a fixed point is called: (a) ✔Circle (b) Parabola (c) Ellipse (d) Hyperbola 3. The circle whose radius is zero is called: (a) Unit circle (b) ✔point circle (c) circumcircle (d) in-circle 4. The circle whose radius is 1 is called: (a) ✔Unit circle (b) point circle (c) circumcircle (d) in-circle 5. The equation 𝒙𝟐^ + 𝒚𝟐^ + 𝟐𝒈𝒙 + 𝟐𝒇𝒚 + 𝒄 = 𝟎 represents the circle with centre (a) (𝑔, 𝑓) (b) ✔ (−𝑔, −𝑓) (c) (−𝑓, −𝑔) (d) (𝑔, −𝑓) 6. The equation 𝒙𝟐^ + 𝒚𝟐^ + 𝟐𝒈𝒙 + 𝟐𝒇𝒚 + 𝒄 = 𝟎 represents the circle with centre

(a) ✔√𝑔^2 + 𝑓^2 − 𝑐 (b) √𝑔^2 + 𝑓^2 + 𝑐 (c) √𝑔^2 + 𝑐^2 − 𝑓 (d) √𝑔 + 𝑓 − 𝑐

7. The angle inscribed in semi-circle is: (a) ✔ 𝜋 2 (b)^

𝜋 3 (c)^

𝜋 4 (d) None of these

8. For any parabola in the standard form , if the directrix is 𝒙 = 𝒂 , then its equation is (a) 𝑦^2 = 4𝑎𝑥 (b) ✔ 𝑦^2 = −4𝑎𝑥 (c) 𝑥^2 = 4𝑎𝑦 (d) 𝑥^2 = −4𝑎𝑦 9. For any parabola in the standard form , if the directrix is 𝒙 = −𝒂 , then its equation is (a) ✔𝑦^2 = 4𝑎𝑥 (b) 𝑦^2 = −4𝑎𝑥 (c) 𝑥^2 = 4𝑎𝑦 (d) 𝑥^2 = −4𝑎𝑦 10. For any parabola in the standard form , if the directrix is 𝒚 = 𝒂 , then its equation is (a) 𝑦^2 = 4𝑎𝑥 (b) 𝑦^2 = −4𝑎𝑥 (c) 𝑥^2 = 4𝑎𝑦 (d) ✔ 𝑥^2 = −4𝑎𝑦 11. For any parabola in the standard form , if the directrix is 𝒚 = −𝒂 , then its equation is (a) 𝑦^2 = 4𝑎𝑥 (b) 𝑦^2 = −4𝑎𝑥 (c) ✔ 𝑥^2 = 4𝑎𝑦 (d) 𝑥^2 = −4𝑎𝑦 12. All lines through vertex and points on circle generate a (a) ✔Circle (b) Ellipse (c) Circular cone (d) None of these 13. The equation 𝒙𝟐^ + 𝒚𝟐^ = 𝟎 then circle is (a) ✔Point Circle (b) Unit Circle (c) Real circle (d) Imaginary Circle 14. The line perpendicular to the tangent at any point 𝑷(𝒙, 𝒚) is known as; (a) Tangent line (b) ✔ Normal at 𝑃 (c) Slope of tangent (d) None of these 15. The point 𝑷(−𝟓, 𝟔) lies __________ the circle 𝒙𝟐^ + 𝒚𝟐^ + 𝟒𝒙 − 𝟔𝒚 = 𝟏𝟐 (a) ✔Inside (b) Outside (c) On (d) None of these 16. The chord containing the centre of the circle is

(a) Radius of circle (b) ✔Diameter of circle (c) Area of circle (d) Tangent of circle

17. The ratio of the distance of a point from the focus to distance from the directrix is denoted by (a) ✔𝑟 (b) 𝑅 (c) 𝐸 (d) 𝑒

UNIT # 07 Vectors

Each question has four possible answer. Tick the correct answer.

1. The vector whose magnitude is 1 is called (a) Null vector (b) ✔ unit vector (c) free vector (d) scalar 2. If the terminal point 𝑩 of the vector 𝑨𝑩⃗⃗⃗⃗⃗⃗ coincides with its initial point 𝑨 , then |𝑨𝑩⃗⃗⃗⃗⃗⃗ | = |𝑩𝑩⃗⃗⃗⃗⃗⃗ | = (a) 1 (b) ✔ 0 (c) 2 (d) undefined 3. Two vectors are said to be negative of each other if they have the same magnitude and __________direction. (a) Same (b) ✔ opposite (c) negative (d) parallel 4. Parallelogram law of vector addition to describe the combined action of two forces, was used by (a) Cauchy (b) ✔ Aristotle (c) Alkhwarzmi (d) Leibnitz 5. The vector whose initial point is at the origin and terminal point is 𝑷 , is called (a) Null vector (b) unit vector (c) ✔position vector (d) normal vector 6. If 𝑹 be the set of real numbers, then the Cartesian plane is defined as (a) 𝑅^2 = {(𝑥^2 , 𝑦^2 ): 𝑥, 𝑦 ∈ 𝑅} (b) ✔ 𝑅^2 = {(𝑥, 𝑦): 𝑥, 𝑦 ∈ 𝑅} (c) 𝑅^2 = {(𝑥, 𝑦): 𝑥, 𝑦 ∈ 𝑅, 𝑥 = −𝑦} (d) 𝑅^2 = {(𝑥, 𝑦): 𝑥, 𝑦 ∈ 𝑅, 𝑥 = 𝑦} 7. The element (𝒙, 𝒚) ∈ 𝑹𝟐^ represents a (a) Space (b) ✔ point (c) vector (d) line 8. If 𝒖 = [𝒙, 𝒚] in 𝑹𝟐, then |𝒖| =? (a) 𝑥^2 + 𝑦^2 (b) ✔ √𝑥^2 + 𝑦^2 (c) ±√𝑥^2 + 𝑦^2 (d) 𝑥^2 − 𝑦^2 9. If |𝒖| = √𝒙𝟐^ + 𝒚𝟐^ = 𝟎 , then it must be true that (a) 𝑥 ≥ 0, 𝑦 ≥ 0 (b) 𝑥 ≤ 0, 𝑦 ≤ 0 (c) 𝑥 ≥ 0, 𝑦 ≤ 0 (d) ✔ 𝑥 = 0, 𝑦 = 0 10. Each vector [𝒙, 𝒚] in 𝑹𝟐^ can be uniquely represented as (a) 𝑥𝑖 − 𝑦𝑗 (b) ✔ 𝑥𝑖 + 𝑦𝑗 (c) 𝑥 + 𝑦 (d) √𝑥^2 + 𝑦^2 11. The lines joining the mid-points of any two sides of a triangle is always _____to the third side. (a) Equal (b) ✔ Parallel (c) perpendicular (d) base 12. A point P in space has __________ coordinates. (a) 1 (b) 2 (c) ✔ 3 (d) infinitely many 13. In space the vector 𝒊 can be written as (a) ✔ (1,0,0) (b) (0,1,0) (c) (0,0,1) (d) (1,0) 14. In space the vector 𝒋 can be written as (a) (1,0,0) (b) ✔ (0,1,0) (c) (0,0,1) (d) (1,0) 15. In space the vector 𝒌 can be written as (a) (1,0,0) (b) (0,1,0) (c) ✔ (0,0,1) (d) (1,0) 16. 𝒖 = 𝟐𝒊 + 𝟑𝒋 + 𝒌 , 𝒗 = −𝟔𝒊 − 𝟗𝒋 − 𝟑𝒌 are _________vectors. (a) ✔Parallel (b)perpendicular (c) reciprocal (d) negative 17. The angles 𝜶, 𝜷, 𝒂𝒏𝒅 𝜸 which a non-zero vector 𝒓 makes with 𝒙 − 𝒂𝒙𝒊𝒔 , 𝒚 − 𝒂𝒙𝒊𝒔 and 𝒛 − 𝒂𝒙𝒊𝒔 respectively are called_____________ of 𝒓. (a) Direction cosines (b) direction ratios (c) ✔ direction angles (d) inclinations 18. Measures of directions angles 𝜶, 𝜷 𝒂𝒏𝒅 𝜸 are (a) 𝛼 ≤ 0, 𝛽 ≤ 0, 𝛾 ≤ 0 (b) 0 ≤ 𝛼 ≤ 𝜋 2 , 0 ≤ 𝛽 ≤ 𝜋 2 , 0 ≤ 𝛾 ≤ 𝜋 2 (c) 𝛼 ≥ 0, 𝛽 ≥ 0, 𝛾 ≥ 0 (d)✔ 0 ≤ 𝛼 ≤ 𝜋, 0 ≤ 𝛽 ≤ 𝜋, 0 ≤ 𝛾 ≤ 𝜋 19. If 𝒖 = 𝟑𝒊 − 𝒋 + 𝟐𝒌 then [3,-1,2] are called ____________ of 𝒖. (a) Direction cosines (b) ✔ direction ratios (c) direction angles (d) elements 20. Which of the following can be the direction angles of some vector (a) 45°, 45°, 60° (b) 30°, 45°, 60° (c) ✔45°, 60°, 60° (d) obtuse Recall that here 𝒄𝒐𝒔𝟐𝜶 + 𝒄𝒐𝒔𝟐𝜷 + 𝒄𝒐𝒔𝟐𝜸 = 𝟏 should hold****.

21. Measure of angle 𝜽 between two vectors is always. (a) 0 < 𝜃 < 𝜋 (b) 0 ≤ 𝜃 ≤ 𝜋 2 (c) ✔0 ≤ 𝜃 ≤ 𝜋 (d) obtuse 22. If the dot product of two vectors is zero, then the vectors must be (a) Parallel (b) ✔ orthogonal (c) reciprocal (d) equal 23. If the cross product of two vectors is zero, then the vectors must be

(a) ✔ Parallel (b) orthogonal (c) reciprocal (d) Non coplanar

24. If 𝜽 be the angle between two vectors 𝒂 and 𝒃 , then 𝒄𝒐𝒔𝜽 =

(a) (^) |𝑎𝑎×𝑏||𝑏| (b) ✔ (^) |𝑎𝑎||𝑏.𝑏| (c) 𝑎 |𝑎.𝑏| (d) 𝑎 |𝑏.𝑏|

25. If 𝜽 be the angle between two vectors 𝒂 and 𝒃 , then projection of 𝒃 along 𝒂 is

(a) 𝑎×𝑏 |𝑎||𝑏|^ (b)^

𝑎.𝑏 |𝑎||𝑏|^ (c)^ ✔^

𝑎.𝑏 |𝑎| (d)^

𝑎.𝑏 |𝑏|

26. If 𝜽 be the angle between two vectors 𝒂 and 𝒃 , then projection of 𝒂 along 𝒃 is

(a) 𝑎×𝑏 |𝑎||𝑏| (b)^

𝑎.𝑏 |𝑎||𝑏| (c)^

𝑎.𝑏 |𝑎| (d)^ ✔^

𝑎.𝑏 |𝑏|

27. Let 𝒖 = 𝒂𝒊 + 𝒃𝒋 + 𝒄𝒌 then projection of 𝒖 along 𝒊 is

(a) ✔𝑎 (b) 𝑏 (c) 𝑐 (d) 𝑢

28. Let 𝒖 = 𝒂𝒊 + 𝒃𝒋 + 𝒄𝒌 then projection of 𝒖 along 𝒋 is

(a) 𝑎 (b) ✔ 𝑏 (c) 𝑐 (d) 𝑢

29. Let 𝒖 = 𝒂𝒊 + 𝒃𝒋 + 𝒄𝒌 then projection of 𝒖 along 𝒌 is

(a) 𝑎 (b) 𝑏 (c) ✔ 𝑐 (d) 𝑢

30. In any ∆𝑨𝑩𝑪 , the law of cosine is (a) ✔𝑎^2 = 𝑏^2 + 𝑐^2 − 2𝑏𝑐𝐶𝑜𝑠𝐴 (b) 𝑎 = 𝑏𝐶𝑜𝑠𝐶 + 𝑐𝐶𝑜𝑠𝐵 (c) 𝑎. 𝑏 = 0 (d) 𝑎 − 𝑏 = 0 31. In any ∆𝑨𝑩𝑪 , the law of projection is (a) 𝑎^2 = 𝑏^2 + 𝑐^2 − 2𝑏𝑐𝐶𝑜𝑠𝐴 (b) ✔ 𝑎 = 𝑏𝐶𝑜𝑠𝐶 + 𝑐𝐶𝑜𝑠𝐵 (c) 𝑎. 𝑏 = 0 (d) 𝑎 − 𝑏 = 0 32. If 𝒖 is a vector such that 𝒖. 𝒊 = 𝟎, 𝒖. 𝒋 = 𝟎, 𝒖. 𝒌 = 𝟎 then 𝒖 is called

(a) Unit vector (b) ✔ null vector (c) [𝑖, 𝑗, 𝑘] (d) none of these

33. Cross product or vector product is defined (a) In plane only (b) ✔in space only (c) everywhere (d) in vector field 34. If 𝒖 and 𝒗 are two vectors , then 𝒖 × 𝒗 is a vector

(a) Parallel to 𝑢and 𝑣 (b) parallel to 𝑢 (c) ✔ perpendicular to 𝑢 and 𝑣 (d) orthogonal to 𝑢

35. If 𝒖 and 𝒗 be any two vectors, along the adjacent sides of ||gram then the area of ||gram is

(a) 𝑢 × 𝑣 (b) ✔ |𝑢 × 𝑣| (c) 12 (𝑢 × 𝑣) (d) 12 |𝑢 × 𝑣 |

36. If 𝒖 and 𝒗 be any two vectors, along the adjacent sides of triangle then the area of triangle is

(a) 𝑢 × 𝑣 (b) |𝑢 × 𝑣| (c) 1 2 (𝑢^ × 𝑣)^ (d)^ ✔^

1 2 |𝑢^ × 𝑣^ |

37. The scalar triple product of 𝒂 , 𝒃 and 𝒄 is denoted by (a) 𝑎. 𝑏. 𝑐 (b) ✔ 𝑎. 𝑏 × 𝑐 (c) 𝑎 × 𝑏 × 𝑐 (d) (𝑎 + 𝑏) × 𝑐 38. The vector triple product of 𝒂 , 𝒃 and 𝒄 is denoted by (a) 𝑎. 𝑏. 𝑐 (b) 𝑎. 𝑏 × 𝑐 (c) ✔ 𝑎 × 𝑏 × 𝑐 (d) (𝑎 + 𝑏) × 𝑐 39. Notation for scalar triple product of 𝒂 , 𝒃 and 𝒄 is

(a) 𝑎. 𝑏 × 𝑐 (b) 𝑎 × 𝑏. 𝑐 (c)[ 𝑎. 𝑏. 𝑐] (d) ✔ all of these

40. If the scalar product of three vectors is zero, then vectors are (a) Collinear (b) ✔ coplanar (c) non coplanar (d) non-collinear 41. If 𝒂 and 𝒃 have same direction , then 𝒂. 𝒃 =

(a) ✔𝑎𝑏 (b) −𝑎𝑏 (c) 𝑎𝑏 sin 𝜃 (d) 𝑎 𝑏𝑡𝑎𝑛𝜃

42. For a vector 𝒂, 𝒂. 𝒂 =

(a) 2𝑎 (b) ✔ 𝑎^2 (c) 𝑎 2 (d)^

𝑎^2 2

43. If 𝒂 and 𝒃 have the opposite direction , then 𝒂. 𝒃 = (a) 𝑎𝑏 (b) ✔ – 𝑎. 𝑏 (c) 𝑎𝑏𝑠𝑖𝑛𝜃 (d) 𝑎𝑏𝑡𝑎𝑛𝜃 44. The angle in semi-circle is equal to:

(a) ✔ 𝜋 2 (b)^ 𝜋^ (c)^

𝜋 3 (d)^ 3𝜋

45. Two non zero vectors are perpendicular 𝒊𝒇𝒇