2ND YEAR MATH (MCQ's), Essays (high school) of Mathematics

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MATHEMATICS

MULTIPLE CHOICE

QUESTIONS WITH

ANSWERS

M.Osama Ahmed

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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

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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,∞[

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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,∞)

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

𝑎 √1+(𝑎𝑥+𝑏)^2 (d)^ 𝑎 cosh

−1(𝑎𝑥 + 𝑏)

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34. If 𝒄𝒐𝒔𝒉−𝟏(𝒔𝒆𝒄𝒙), 𝒕𝒉𝒆𝒏 𝒅𝒚𝒅𝒙 = (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 𝒅𝒚 𝒅𝒙 = (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) + 𝒙

𝟐 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. 𝒅𝒚 𝒅𝒙 |(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^2 (c)^

3 √1−9𝑥^2 (d)^ ✔^ 3𝑥

51. (^) 𝒅𝒙𝒅 (𝟏𝟎𝒔𝒊𝒏𝒙) = (a) 10 𝑐𝑜𝑠𝑥^ (b) ✔ 10 𝑠𝑖𝑛𝑥. 𝑐𝑜𝑠𝑥. 𝑙𝑛10 (c) 10 𝑠𝑖𝑛𝑥. 𝑙𝑛10 (d) 10 𝑐𝑜𝑠𝑥. 𝑙𝑛 52. (^) 𝒅𝒙𝒅 (√𝒙 − 𝟏 √𝒙 )

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

10 | P a g e

(a) 𝑓′(𝑥)𝑔′(𝑥) (b) (𝑓𝑜𝑔)′(𝑥) (c) ✔ 𝑓′(𝑔(𝑥))𝑔′(𝑥) (d) 𝑓′(𝑔′(𝑥))

76. If 𝒚 = 𝒔𝒊𝒏𝒉−𝟏(𝒙𝟑) then 𝒅𝒚 𝒅𝒙 = (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. 𝒅 𝒅𝒙 (𝒕𝒂𝒏

−𝟏𝒙 − 𝒄𝒐𝒕−𝟏𝒙) =

(a) (^) √1+𝑥^22 (b) ✔ (^) 1+𝑥^22 (c) 0 (d) (^) 1+𝑥−2 2

87. If 𝒇 (𝟏𝒙) = 𝒕𝒂𝒏𝒙 , then 𝒇′^ (𝟏𝝅) = (a) 𝜋^2 (b) ✔ – 𝜋^2 (c) 1 (d) −1𝜋 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

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

11 | P a g e

4. (^) ∫ 𝐬𝐢𝐧(𝒂𝒙 + 𝒃) 𝒅𝒙 = (a) ✔−1𝑎 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) 𝑓𝑛(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. (^) ∫ (^) √𝒙+𝒂𝒅𝒙+√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. (^) ∫ (^) √𝒙𝒙𝟐 +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 𝒅

𝟐𝒚 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) 𝑒

𝑠𝑖𝑛𝑥 𝑐𝑜𝑠𝑥 (d)^

𝑒𝑐𝑜𝑠𝑥 𝑠𝑖𝑛𝑥

18. ∫(𝟐𝒙 + 𝟑)

𝟏 𝟐𝒅𝒙 = (a) 13 (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) 12 𝑒𝑇𝑎𝑛 −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 = 𝟒 𝟏 𝟐 𝒕𝒉𝒆𝒏 ∫^ 𝟑𝒇(𝒙)𝒅𝒙 − ∫^ 𝟐𝒈(𝒙)𝒅𝒙 =

𝟏 −𝟐

𝟏 −𝟐

𝟏 𝟐 (a) ✔ 7 (b) 9 (c) 12 (d) 8

13 | P a g e

46. The general solution of differential equation 𝒅𝒚 𝒅𝒙 = −^

𝒚 𝒙 is (a) 𝑥𝑦 = 𝑐 (b) 𝑦𝑥 = 𝑐 (c) ✔ 𝑥𝑦 = 𝑐 (d)𝑥^2 𝑦^2 = 𝑐

47. ∫ 𝒙+𝟐 𝒙+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) ✔ 14 sin^4 𝑥 + 𝑐 (c) − 14 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. (^) ∫𝟎 𝟑𝒙𝒅𝒙𝟐+𝟗 = (a) 𝜋 4 (b)^ ✔^

𝜋 12 (c)^

𝜋 2 (d) None of these

51. (^) ∫ 𝒆𝒙^ [𝟏𝒙 + 𝒍𝒏𝒙] = (a) 𝑒𝑥^1 𝑥 + 𝑐 (b) – 𝑒𝑥^1 𝑥 + 𝑐 (c) ✔ 𝑒𝑥𝑙𝑛𝑥 + 𝑐 (d) – 𝑒𝑥𝑙𝑛𝑥 + 𝑐 52. ∫ 𝒔𝒊𝒏𝒙𝒅𝒙 = −𝝅 𝝅 (a) ✔ 2 (b) -2 (c) 0 (d) - 53. (^) ∫−𝟏 𝟐|𝒙|𝒅𝒙 = (a) 12 (b) − 12 (c) 52 (d) ✔ (^32) 54. (^) ∫ (𝟒𝒙 + 𝒌)𝒅𝒙 = 𝟐 𝒕𝒉𝒆𝒏 𝒌 =𝟎^ 𝟏 (a) 8 (b) -4 (c) ✔ 0 (d) - 55. (^) ∫ 𝒆𝒙^ [𝟏𝒙 − (^) 𝒙𝟏𝟐] =

(a) ✔𝑒𝑥 1 𝑥 + 𝑐^ (b)^ – 𝑒

𝑥^1 𝑥 + 𝑐^ (c)^ 𝑒

𝑥𝑙𝑛𝑥 + 𝑐 (d) – 𝑒𝑥 1 𝑥^2 + 𝑐

56. Solution of the differential equation : 𝒅𝒚 𝒅𝒙 =^

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

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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) 𝑥 𝑎 +^

𝑦 𝑏 = 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) ✔ 𝑥 𝑎 +^

𝑦 𝑏 = 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) 𝑥 𝑎 +^

𝑦 𝑏 = 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−𝒙 𝒄𝒐𝒔𝜶𝟏 = 𝒚−𝒚 𝒔𝒊𝒏𝜶𝟏 = 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 = 𝑚 1 −𝑚 2 1+𝑚1 𝑚2^ (b)^ ✔^ 𝑡𝑎𝑛𝜑 =^

𝑚 2 −𝑚 1 1+𝑚2 𝑚1^ (c)^ 𝑡𝑎𝑛𝜑 =^

𝑚 1 +𝑚 2 1+𝑚1 𝑚2^ (d)^ 𝑡𝑎𝑛𝜑 =^

𝑚 2 +𝑚 1 1+𝑚1 𝑚

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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

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(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

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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****.

20 | P a g e

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