Uploaded byKarunaGupta1982
DOCX, PDF112 views

Mcq exemplar class 12

The document contains 30 multiple choice questions related to various mathematical concepts such as relations, functions, matrices, determinants, and trigonometric functions. The questions test knowledge of properties such as equivalence, injectivity, surjectivity, inverse functions, composition of functions, matrix operations, and evaluating determinants.

Embed presentation

Download to read offline
MULTIPLE QUESTIONS (EXEMPLAR PROBLEMS) 1. Let R be a relationon the setN of natural numbersdefinedbynRm if n dividesm. thenR is a. Reflexive andsymmetric b. Transitive and symmetric c. Equivalence d. Reflexive,transitive butnot symmetric 2. Let N be the set of natural numbers and the function f : N  be definedby f(n) = 2n + 3  n  N. thenf is a. Surjective b. Injective c. Biijective d. None of these 3. Set A has 3 elementsandthe setB has 4 elements.Thenthe number ofinjective mappings that can be definedfromA to B is a. 144 b. 12 c. 24 d. 64 4. Let f : R  R be definedbyf(x) = sin x and g : R R be definedbyg(x) = x,then f o g is a. x2 sinx b. (sinx)2 c. sin x2 d. 𝒔𝒊𝒏 𝒙 𝒙 𝟐 5. Let f : R  R be definedbyf(x) = 3x – 4. Then f-1 (x) isgiven by a. 𝒙+𝟒 𝟑 b. 𝒙 𝟑 - 4 c. 3x – 4 d. none of these 6. The maximumnumber of equivalence relationsonthe set A = {1, 2, 3} are a. 1 b. 2 c. 3 d. 5 7. Let us define a relationR in R as aRb if a b. thenR is a. An equivalence relation b. Reflexive,transitive butnot symmetric c. Symmetric,transitive but not reflexive d. Neithertransitive nor reflexive butsymmetric. 8. The identityelementforthe binary operation  definedinQ  {0} as a  b= 𝒂𝒃 𝟐  a, b  Q  {0} is a. 1 b. 2 c. 0 d. none of these 9. Let A = {1, 2, 3,….., n} and B = {a, b}. thenthe numbersof surjections from A intoB is a. n P2 b. 2n – 2 c. 2n – 1 d. none of these 10. Let f : R  R be definedbyf(x) = 3x2 - 5 and g : R  R by g(x) = 𝒙 𝒙 𝟐+ 𝟏 .theng o f is a. 𝟑 𝒙 𝟐− 𝟓 𝟗𝒙 𝟒− 𝟑𝟎𝒙 𝟐+ 𝟐𝟔 b. 𝟑 𝒙 𝟐− 𝟓 𝟗𝒙 𝟒− 𝟔𝒙 𝟐+ 𝟐𝟔 c. 𝟑 𝒙 𝟐 𝒙 𝟒+𝟐𝒙 𝟐− 𝟒 d. 𝟑 𝒙 𝟐 𝟗𝒙 𝟒+ 𝟑𝟎𝒙 𝟐− 𝟐 11. Whichof the followingfunctionsfromZ into Z are bijections? a. f(x) = x3 b. f(x) = x + 2 c. f(x) = 2x + 1 d. f(x) = x2 + 1 12. Let f : R  R be the functionsdefinedbyf(x) = x3 + 5. Then f-1 (x) is a. (𝒙 + 𝟓) 𝟏 𝟑 b. (𝒙 − 𝟓) 𝟏 𝟑 c. (𝟓 − 𝒙) 𝟏 𝟑 d. 5 – x 13. Let f : R - { 𝟑 𝟓 }  R be definedbyf(x) = 𝟑𝒙+𝟐 𝟓𝒙−𝟑 . Then
a. f-1 (x) = f(x) b. f-1 (x) = - f(x) c. (fo f) x = - x d. f-1 (x) = 𝟏 𝟏𝟗 f(x) 14. Let f : [2, )  Rbe the functiondefined byf(x) = x2 - 4x + 5, then the range of f is a. R b. [1, ) c. [4, ) d. [5, ) 15. Let f : R  R be givenby f(x) = tan x. Then f-1 (1) is a. 𝝅 𝟒 b. { n  + 𝝅 𝟒 : n  Z} c. does not exist d. none of these 16. Let the relationR be definedinN by aRb if2a + 3b = 30. Then R = ……………………. 17. The principal value branch of sec-1 is a. [− 𝝅 𝟐 , 𝝅 𝟐 ] - {0} b. [0, ]- { 𝝅 𝟐 } c. (0, ) d. (− 𝝅 𝟐 , 𝝅 𝟐 ) 18. The value of 𝐬𝐢𝐧−𝟏 ( 𝒄𝒐𝒔( 𝟒𝟑  𝟓 )) is a. 𝟑𝝅 𝟓 b. −𝟕𝝅 𝟓 c. 𝝅 𝟏𝟎 d. - 𝝅 𝟏𝟎 19. The principal value ofthe expressioncos-1 [cos (-6800 )]is a. 𝟐𝝅 𝟗 b. −𝟐𝝅 𝟗 c. 𝟑𝟒𝝅 𝟗 d. 𝝅 𝟗 20. If tan-1 x = 𝝅 𝟏𝟎 for some x  R, thenthe value of cot-1 x is a. 𝝅 𝟓 b. 𝟐𝝅 𝟓 c. 𝟑𝝅 𝟓 d. 𝟒𝝅 𝟓 21. The principal value ofsin-1 ( −√ 𝟑 𝟐 ) is a. − 𝟐𝝅 𝟑 b. − 𝝅 𝟑 c. 𝟒𝝅 𝟑 d. 𝟓𝝅 𝟑 22. The greatestand leastvaluesof (sin-1 x)2 + (cos-1 x)2 are respectively a. 𝟓𝝅 𝟐 𝟒 𝒂𝒏𝒅 𝝅 𝟐 𝟖 b. 𝝅 𝟐 𝒂𝒏𝒅 −𝝅 𝟐 c. 𝝅 𝟐 𝟒 𝒂𝒏𝒅 −𝝅 𝟐 𝟒 d. 𝝅 𝟐 𝟒 𝒂𝒏𝒅 𝟎. 23. The value of sin (2 sin-1 (.6)) is a. .48 b. .96 c. 1.2 d. sin 1.2 24. If sin-1 x + sin-1 y =  𝟐 , then value of cos-1 x + cos-1 y is a. 𝝅 𝟐 b.  c. 0 d. 𝟐𝝅 𝟑 25. The value of the expressionsin[cot-1 (cos (tan-1 1))]is a. 0 b. 1 c. 𝟏 √ 𝟑 d. √ 𝟐 𝟑 26. Whichof the followingisthe principal value branch ofcosec-1 x? a. [− 𝝅 𝟐 , 𝝅 𝟐 ] b. (0, ) - { 𝝅 𝟐 } c. (− 𝝅 𝟐 , 𝝅 𝟐 ) d. [− 𝝅 𝟐 , 𝝅 𝟐 ] - {0} 27. If 3tan-1 x + cot-1 x = ,thenx equals a. 0 b. 1 c. -1 d. ½ 28. The value of sin-1 𝐬𝐢𝐧−𝟏 ( 𝒄𝒐𝒔( 𝟑𝟑  𝟓 )) is a. 𝟑𝝅 𝟓 b. −𝟕𝝅 𝟓 c. 𝝅 𝟏𝟎 d. - 𝝅 𝟏𝟎
29. If cos (𝐬𝐢𝐧−𝟏 𝟐 𝟓 + 𝐜𝐨𝐬−𝟏 𝒙)= 0, thenx is equal to a. 𝟏 𝟓 b. 𝟐 𝟓 c. 0 d. 1 30. The value of cos-1 (𝒄𝒐𝒔 𝟑𝝅 𝟐 )is equal to a. 𝟑𝝅 𝟐 b. 𝟓𝝅 𝟐 c. 𝝅 𝟐 d. 𝟕𝝅 𝟐 31. The value of the expression2sec-1 2 + sin-1 ( 𝟏 𝟐 ) is a. 𝝅 𝟔 b. 𝟓𝝅 𝟔 c. 1 d. 𝟕𝝅 𝟔 32. If sin-1 ( 𝟐𝒂 𝟏+ 𝒂 𝟐 ) + cos-1 ( 𝟏− 𝒂 𝟐 𝟏+ 𝒂 𝟐 ) = tan-1 ( 𝟐𝒙 𝟏− 𝒙 𝟐 ), where a, x  ]0, 1, then the value of x is a. 0 b. 𝒂 𝟐 c. a d. 𝟐𝒂 𝟏− 𝒂 𝟐 33. The value of the expressiontan ( 𝟏 𝟐 𝐜𝐨𝐬−𝟏 𝟐 √ 𝟓 ) is a. 2 + √ 𝟓 b. √ 𝟓 – 2 c. 5 + √ 𝟐 d. √ 𝟓+ 𝟐 𝟐 34. If A = [ 𝟐 −𝟏 𝟑 −𝟒 𝟓 𝟏 ] 𝒂𝒏𝒅 𝑩 = [ 𝟐 𝟑 𝟒 −𝟐 𝟏 𝟓 ] , then a. Only AB is defined b. Only BA is defined c. AB and BA both are defined d. AB and BA both are not defined. 35. If A and B are two matrices of the order 3  m and 3  n. respectively,andm = n, then the order of matrix (5A – 2B) is a. m  3 b. 3  3 c. m  n d. 3  n 36. if A = [ 𝟎 𝟏 𝟏 𝟎 ] , thenA2 isequal to a. [ 𝟎 𝟏 𝟏 𝟎 ] b. [ 𝟏 𝟎 𝟏 𝟎 ] c. [ 𝟎 𝟏 𝟎 𝟏 ] d. [ 𝟏 𝟎 𝟎 𝟏 ] 37. If matrix A = [ 𝒂𝒊𝒋]2  2 , where aij = 1 if i  j= 0 if i = j a. I b. A c. 0 d. none of these 38. The matrix [ 𝟏 𝟎 𝟎 𝟎 𝟐 𝟎 𝟎 𝟎 𝟒 ]is a a. Identitymatrix b. symmetric matrix c. skew symmetricmatrix d. none of these 39. If A is matrix of order m  n and B is a matrix such that AB and BAare both defined,thenorderof matrix B is a. m  m b. n  n c. m  n d. n  m 40. if A and B are matrices ofsame order, then(AB - BA) isa a. skew symmetricmatrix b. null matrix
c. symmetricmatrix d. unit matrix 41. If A is a square matrix such that A2 = I, then(A – I)3 + (A + I)3 - 7A is equal to a. A b. I – A c. I + A d. 3A 42. For any two matrices A and B, we have a. AB = BA b. AB  BA c. AB= 0 d. none of these 43. If x , y  R, thenthe determinant  = | 𝒄𝒐𝒔 𝒙 −𝒔𝒊𝒏 𝒙 𝟏 𝒔𝒊𝒏 𝒙 𝒄𝒐𝒔 𝒙 𝟏 𝒄𝒐𝒔(𝒙 + 𝒚) −𝒔𝒊𝒏(𝒙 + 𝒚) 𝟎 | liesinthe interval a. [- √ 𝟐, √ 𝟐] b. [-1, 1] c. [-√ 𝟐, 1] d. [-1. -√ 𝟐] 44. The value of determinant | 𝒂 − 𝒃 𝒃 + 𝒄 𝒂 𝒃 − 𝒂 𝒄 + 𝒂 𝒃 𝒄 − 𝒂 𝒂 + 𝒃 𝒄 | a. a3 + b3 + c3 b. 3 bc c. a3 + b3 + c3 – 3abc d. none of these 45. the determinants | 𝒃 𝟐 − 𝒂𝒃 𝒃 − 𝒄 𝒃𝒄 − 𝒂𝒄 𝒂𝒃 − 𝒂 𝟐 𝒂 − 𝒃 𝒃 𝟐 − 𝒂𝒃 𝒃𝒄 − 𝒂𝒄 𝒄 − 𝒂 𝒂𝒃 − 𝒂 𝟐 | equals a. abc(b-c) (c-a) (a-b) b. (b-c) (c-a) (a-b) c. (a+b+c)(b-c) (c-a) (a-b) d. None ofthese 46. the maximumvalue of ∆ = | 𝟏 𝟏 𝟏 𝟏 𝟏 + 𝒔𝒊𝒏 𝟏 𝟏 + 𝒄𝒐𝒔 𝟏 𝟏 | is ( is a real number) a. 𝟏 𝟐 b. √ 𝟑 𝟐 c. √ 𝟐 d. 𝟐√ 𝟑 𝟒 47. If A = [ 𝟐  −𝟑 𝟎 𝟐 𝟓 𝟏 𝟏 𝟑 ] , then A-1 existsif a.  = 2 b.   2 c.   -2 d. none of these 48. If x, y, z are all differentfromzero and | 𝟏 + 𝒙 𝟏 𝟏 𝟏 𝟏 + 𝒚 𝟏 𝟏 𝟏 𝟏 + 𝒛 | = 0, thenvalue of x-1 + y-1 + z-1 is a. x y z b. x-1 y-1 z-1 c. –x-y-z d. -1 49. The functionf(x) = { 𝒔𝒊𝒏𝒙 𝒙 + 𝒄𝒐𝒔 𝒙, 𝒊𝒇 𝒙 ≠ 𝟎 𝒌, 𝒊𝒇 𝒙 = 𝟎 iscontinuous at x = 0, then the value of k is a. 3 b. 1 c. 2 d. 1.5 50. The functionf(x) = [x] denotesthe greatestintegerfunction,continuous at a. 4 b. -2 c. 1 d. 1.5 51. The functionf(x) = | 𝒙| + | 𝒙 + 𝟏| is a. Continuousat x = 0 as well as x = 1 b. Continuousat x = 1 but not at x = 0 c. Discontinuousat x = 0 as well as at x = 1 d. Continuousat x = 0 but not at x = 1
52. The value of k which makes the functiondefinedby f(x) = { 𝒔𝒊𝒏 𝟏 𝒙 , 𝒊𝒇 𝒙 ≠ 𝟎 𝒌, 𝒊𝒇 𝒙 = 𝟎 , continuose at x = 0 is a. 8 b. 1 c. -1 d. none of these 53. If u = sin-1 ( 𝟐𝒙 𝟏+ 𝒙 𝟐 ) and y = tan-1 ( 𝟐𝒙 𝟏− 𝒙 𝟐 ) , then 𝒅𝒖 𝒅𝒗 is a. 𝟏 𝟐 b. x c. 𝟏− 𝒙 𝟐 𝟏+ 𝒙 𝟐 d. 1 54. The value of c inMean value theoremfor the functionf(x) = x(x – 2), x  [1. 2] is a. 𝟑 𝟐 b. 𝟐 𝟑 c. 𝟏 𝟐 d. 𝟑 𝟐 55. The functionf(x) = 𝟒− 𝒙 𝟐 𝟒𝒙− 𝒙 𝟑 is a. Discontinuousat onlyone point b. Discontinuousat exactlytwo point c. Discontinuousat exactlythree point d. None of these 56. The setof points where the functionf givenby f(x) = | 𝟐𝒙 − 𝟏| isdifferentiable is a. R b. R - { 𝟏 𝟐 } c. (0,  ) d. none of these 57. If y = log ( 𝟏− 𝒙 𝟐 𝟏+ 𝒙 𝟐 ) , thenfind 𝒅𝒚 𝒅𝒙 is equal to a. 𝟒𝒙 𝟑 𝟏− 𝒙 𝟒 b. −𝟒𝒙 𝟏− 𝒙 𝟒 c. 𝟏 𝟒− 𝒙 𝟒 d. −𝟒𝒙 𝟑 𝟏− 𝒙 𝟒 58. If x = t2 , y = t3 , then 𝒅 𝟐 𝒚 𝒅𝒙 𝟐 is a. 𝟑 𝟐 b. 𝟑 𝟒𝒕 c. 𝟑 𝟐𝒕 d. 𝟑 𝟐𝒕 59. The value of c on Rolle’stheoremforthe functionf(x) = x3 - 3x in the interval [0, √ 𝟑] is a. 1 b. -1 c. 𝟑 𝟐 d. 𝟏 𝟑 60. For the functionf(x) = x + 𝟏 𝒙 , x  [1, 3], the value ofc for mean value theoremis a. 1 b. 2 c. √ 𝟑 d. none of these 61. The two curves x3 -3xy2 + 2 = 0 and 3x2 y – y3 = 2 a. Touch each other b. Cut at right angle c. Cut an angle 𝝅 𝟑 d. Cut an angle 𝝅 𝟒 62. The tangent to the curve givenby x = et . cost, y =et . sin t at t = 𝝅 𝟒 makes withx– axis an angle : a. 0 b. 𝝅 𝟒 c. 𝝅 𝟑 d. 𝝅 𝟐 63. The equationof the normal to the curve y = sin x at (0, 0) is: a. x = 0 b. y = 0 c. x + y = 0 d. x – y = 0
64. the pointon the curve y2 = x, where the tangentmakes an angle of 𝝅 𝟒 with x – axis is a. ( 𝟏 𝟐 , 𝟏 𝟒 ) b. ( 𝟏 𝟒 , 𝟏 𝟐 ) c. (4, 2) d. (1, 1) 65. Minimumvalue of f if f(x) = sin x [ −𝝅 𝟐 , 𝝅 𝟐 ] is………………….. 66. The sidesof an equilateral triangle are increasingat the ratio of 2 cm/sec. the rate at which the area increases,whenside is 10 cm is: a. 10 cm2 /s b. √ 𝟑cm2 /s c. 10 √ 𝟑cm2 /s d. 𝟏𝟎 𝟑 cm2 /s 67. The curve y = 𝒙 𝟏 𝟓 has at (0, 0) a. A vertical tangent (parallel to y 0 axis) b. A horizontal tangent(parallel to x – axis) c. An oblique tangent d. No tangent 68. The equationof normal to the curve 3x2 - y2 = 8 which isparallel to the line x + 3y = 8 is a. 3x – y = 8 b. 3x + y + 8 = 0 c. x + 3y ±8 = 0 d. x + 3y = 0 69. If y = x4 – 10 and ifx changes from 2 to 1.99, what is the change in y a. .32 b. .032 c. 5.68 d. 5.968 70. The pointsat which the tangents to the curve y = x3 - 12x + 18 are parallel to x – axisare: a. (2, -2), (-2,-34) b. (2, 34), (-2,0) c. (0, 34), (-2, 0) d. (2,2), (-2, 34) 71. The two curves x3 - 3xy2 + 2 = 0 and 3x2 y – y3 - 2= 0 intersectat an angle of a. 𝝅 𝟒 b. 𝝅 𝟑 c. 𝝅 𝟐 d. 𝝅 𝟔 72. y = x (x – 3)2 decreasesforthe valuesof x givenby : a. 1 < x< 3 b. x < 0 c. x > 0 d. 0< x < 𝟑 𝟐 73. Whichof the followingfunctionsisdecreasingon (𝟎, 𝝅 𝟐 ) a. sin 2x b. tan x c. cos x d. cos 3x 74. if x isreal, the minimumvalue of x2 - 8x + 17 is a. -1 b. 0 c. 1 d. 2 75. The smallestvalue of the polynomial x3 – 18x2 + 96x in [0,9] is a. 126 b. 0 c. 135 d. 160 76. The functionf(x) = 2x3 – 3x2 - 12x + 4, has a. Two points of local maximum b. Two points of local minimum c. One maximumand one minimum d. No maximumor minimum 77. Maximumslope of the curve y = -x3 + 3x2 + 9x – 27 is:
a. 0 b. 12 c. 16 d. 32 78. ∫ 𝒆 𝒙 (𝒄𝒐𝒔 𝒙 − 𝒔𝒊𝒏 𝒙 )dx is equal to a. 𝒆 𝒙 cos x + C b. 𝒆 𝒙 sin x + C c. -𝒆 𝒙 cos x + C d. -𝒆 𝒙 sin x + C 79. ∫ 𝒅𝒙 𝒔𝒊𝒏 𝟐 𝒙 𝒄𝒐𝒔 𝟐 𝒙 is equal to a. Tan x + cot x + C b. (tan x + cit x)2 + C c. tan x – cot x+C d. (tan x – cot x)2 + C 80. If ∫ 𝟑𝒆 𝒙− 𝟓𝒆−𝒙 𝟒𝒆 𝒙+ 𝟓𝒆−𝒙 dx = ax + b log | 𝟒𝒆 𝒙 + 𝟓𝒆−𝒙|+ 𝑪 , then a. a = −𝟏 𝟖 , 𝒃 = 𝟕 𝟖 b. a = 𝟏 𝟖 , 𝒃 = 𝟕 𝟖 c. a = −𝟏 𝟖 , 𝒃 = −𝟕 𝟖 d. a = 𝟏 𝟖 , 𝒃 = −𝟕 𝟖 81. if f and g are continuous functionsin [0, 1] satisfy f(x) = f(a – x) and g = (x) + g (a – x), then∫ 𝒇( 𝒙). 𝒈(𝒙) 𝒂 𝟎 dx is equal to a. 𝒂 𝟐 b. 𝒂 𝟐 ∫ 𝒇(𝒙) 𝒂 𝟎 dx c. ∫ 𝒇(𝒙) 𝒂 𝟎 dx d. a ∫ 𝒇(𝒙) 𝒂 𝟎 dx 82. If ∫ 𝒆 𝒕 𝟏+𝒕 𝒅𝒕 = 𝒂, 𝒕𝒉𝒆𝒏 ∫ 𝒆 𝒕 ( 𝟏+𝒕) 𝟐 𝟏 𝟎 𝟏 𝟎 dt is equal to a. a – 1 + 𝒆 𝟐 b. a + 1 - 𝒆 𝟐 c. a – 1 - 𝒆 𝟐 d. a + 1 + 𝒆 𝟐 83. ∫ | 𝒙 𝒄𝒐𝒔 𝝅𝒙| 𝒅𝒙 𝟏 𝟎 is equal to a. 𝟖 𝝅 b. 𝟒 𝝅 c. 𝟐 𝝅 d. 𝟏 𝝅 84. ∫ 𝒄𝒐𝒔 𝟐𝒙−𝒄𝒐𝒔 𝟐 𝒄𝒐𝒔 𝒙−𝒄𝒐𝒔 dx is equal to a. 2(sinx+ x cos  ) + C b. 2(sinx - x cos  ) + C c. 2(sinx + 2xcos  ) + C d. 2(sinx - 2xcos  ) + C 85. ∫ 𝒅𝒙 𝒔𝒊𝒏 ( 𝒙−𝒂) 𝒔𝒊𝒏 (𝒙−𝒃) is equal to a. sin (b– a) log | 𝒔𝒊𝒏 (𝒙−𝒃) 𝒔𝒊𝒏 (𝒙−𝒂) |+ C b. cosec (b– a) log | 𝒔𝒊𝒏 (𝒙−𝒂) 𝒔𝒊𝒏 (𝒙−𝒃) |+ C c. cosec (b– a) log | 𝒔𝒊𝒏 (𝒙−𝒃) 𝒔𝒊𝒏 (𝒙−𝒂) |+ C d. sin (b– a) log | 𝒔𝒊𝒏 (𝒙−𝒂) 𝒔𝒊𝒏 (𝒙−𝒃) |+ C 86. ∫ 𝐭𝐚𝐧−𝟏 √ 𝒙 dx is equal to a. (x + 1) tan-1 √ 𝒙 - √ 𝒙 + C b. x tan-1 √ 𝒙 - √ 𝒙 + C c. √ 𝒙 - x tan-1 √ 𝒙 + C d. √ 𝒙 – ( + 1) x tan-1 √ 𝒙 + C 87. ∫ 𝒆 𝒙 ( 𝟏−𝒙 𝟏+ 𝒙 𝟐 ) 𝟐 is equal to a. 𝒆 𝒙 𝟏+ 𝒙 𝟐 + C b. −𝒆 𝒙 𝟏+ 𝒙 𝟐 + C c. 𝒆 𝒙 ( 𝟏+ 𝒙 𝟐) 𝟐 + C d. −𝒆 𝒙 ( 𝟏+ 𝒙 𝟐) 𝟐 + C 88. ∫ 𝒙 𝟗 ( 𝟒𝒙 𝟐+ 𝟏) 𝟔 dx is equal to
a. 𝟏 𝟓𝒙 (𝟒 − 𝟏 𝒙 𝟐 ) −𝟓 + 𝑪 b. 𝟏 𝟓 (𝟒 + 𝟏 𝒙 𝟐 ) −𝟓 + 𝑪 c. 𝟏 𝟏𝟎𝒙 ( 𝟏 + 𝟒)−𝟓 + 𝑪 d. 𝟏 𝟏𝟎 ( 𝟏 𝒙 𝟐 + 𝟒) −𝟓 + 𝑪 89. If ∫ 𝒅𝒙 ( 𝒙+𝟐)(𝒙 𝟐+ 𝟏) = 𝒂 𝒍𝒐𝒈| 𝟏+ 𝒙 𝟐| + 𝒃 𝐭𝐚𝐧−𝟏 𝒙 + 𝟏 𝟓 𝒍𝒐𝒈 | 𝒙 + 𝟐| + 𝑪, then a. a = −𝟏 𝟏𝟎 , 𝒃 = −𝟐 𝟓 b. a = 𝟏 𝟏𝟎 , 𝒃 = − 𝟐 𝟓 c. a = 𝟏𝟎 , 𝒃 = 𝟐 𝟓 d. a = 𝟏 𝟏𝟎 , 𝒃 = 𝟐 𝟓 90. ∫ 𝒙 𝟑 𝒙+𝟏 is equal to a. x + 𝒙 𝟐 𝟐 + 𝒙 𝟑 𝟑 − 𝒍𝒐𝒈| 𝟏− 𝒙| + 𝑪 b. x + 𝒙 𝟐 𝟐 − 𝒙 𝟑 𝟑 − 𝒍𝒐𝒈| 𝟏− 𝒙| + 𝑪 c. x - 𝒙 𝟐 𝟐 − 𝒙 𝟑 𝟑 − 𝒍𝒐𝒈 | 𝟏+ 𝒙| + 𝑪 d. x - 𝒙 𝟐 𝟐 + 𝒙 𝟑 𝟑 − 𝒍𝒐𝒈 | 𝟏+ 𝒙| + 𝑪 91. If ∫ 𝒙 𝟑 𝒅𝒙 √ 𝟏+ 𝒙 𝟐 = 𝒂( 𝟏 + 𝒙 𝟐) 𝟑 𝟐 + 𝒃√ 𝟏 + 𝒙 𝟐 + C, then a. a = 𝟏 𝟑 , b = 1 b. a = −𝟏 𝟑 , b = 1 c. a = −𝟏 𝟑 , b = -1 d. a = 𝟏 𝟑 , b = -1 92. ∫ 𝒅𝒙 𝟏+𝒄𝒐𝒔 𝟐𝒙 𝝅 𝟒 −𝝅 𝟒 is equal to a. 1 b. 2 c. 3 d. 4 93. ∫ √ 𝟏 − 𝒔𝒊𝒏 𝟐𝒙 𝝅 𝟐 𝟎 dx is equal to a. 2√ 𝟐 b. 2 (√ 𝟐+ 1) c. 2 d. 2(√ 𝟐- 1) 94. The area enclosedby the circle x2 + y2 = 2 is equal to a. 4 sq unit b. 2 √ 𝟐 sq unit c. 42 sq unit d. 2 sq unit 95. The area enclosedby the ellipse 𝒙 𝟐 𝒂 𝟐 + 𝒚 𝟐 𝒃 𝟐 = 1 is equal to a. 2 ab b.  ab c.  a2 b d. ab2 96. The area of the regionboundedby the curve y = x2 and the line y = 16 a. 𝟑𝟐 𝟑 b. 𝟐𝟓𝟔 𝟑 c. 𝟔𝟒 𝟑 d. 𝟏𝟐𝟖 𝟑 97. The area of the regionboundedby the y – axis,y = cos x and y = sinx,0 ≤ 𝒙 ≤ 𝝅 𝟐 is
a. √ 𝟐sq unit b. (√ 𝟐+ 1) sq unit c. (√ 𝟐- 1) sq unit d. (2√ 𝟐- 1) sq unit 98. The area of the regionboundedby the curve y = √𝟏𝟔 − 𝒙 𝟐 and x – axis is a. 8 sq unit b. 20  sq unit c. 16  squnit d. 256  squnit 99. Area of the regionboundedby the curve y = cos x betweenx= 0 and x =  is a. 2 sq unit b. 4 sq unit c. 3 sq unit d. 1 sq unit 100. The area of the regionboundedby parabola y2 = x and the straight line 2y = x is a. 𝟒 𝟑 sq unit b. 1 sq unit c. 𝟏 𝟑 sq unit d. 𝟐 𝟑 sq unit 101. The area of the regionboundedby the curve y = x + 1 and the linesx = 2 and x = 3 is a. 2 sq unit b.  sq unit c. 3 sq unit d. 4  squnit 102. The area of the regionboundedby the curve y = x + 1 and the linesx = 2 and x = 3 is a. 𝟕 𝟐 sq unit b. 𝟗 𝟐 sq unit c. 𝟏𝟏 𝟐 sq unit d. 𝟏𝟑 𝟐 sq unit 103. The degree ofthe differential equation(1+ 𝒅𝒚 𝒅𝒙 )3 = ( 𝒅 𝟐 𝒚 𝒅𝒙 𝟐 )2 is a. 1 b. 2 c. 3 d. 4 104. The degree ofthe differential equation 𝒅 𝟐 𝒚 𝒅𝒙 𝟐 + 𝟑 ( 𝒅𝒚 𝒅𝒙 ) 𝟐 = 𝒙 𝟐 𝒍𝒐𝒈 ( 𝒅 𝟐 𝒚 𝒅𝒙 𝟐 ) is a. 1 b. 2 c. 3 d. not defined 105. The order and degree ofthe differential equation [ 𝟏+ ( 𝒅𝒚 𝒅𝒙 ) 𝟐 ] 𝟐 = 𝒅 𝟐 𝒚 𝒅𝒙 𝟐 respectively,are a. 1, 2 b. 2, 2 c. 2, 1 d. 4, 2 106. The solutionof the differential equation2x. 𝒅𝒚 𝒅𝒙 – y = 3 representsa familyof a. Straight lines b. Circles c. Parabolas d. Ellipses 107. The integratingfactor ofthe differential equation 𝒅𝒚 𝒅𝒙 (x log x) + y = 2 logx is a. ex b. log xc. log (logx) d. x 108. A solutionof the differential equation( 𝒅𝒚 𝒅𝒙 )2 – x 𝒅𝒚 𝒅𝒙 + y = 0 is a. y = 2 b. y = 2x c. y = 2x – 4 d. y = 2x2 – 4 109. Whichof the followingisnot a homogeneousfunctionof x and y. a. x2 + 2xyb. 2x – y c. cos2 ( ) + d. sinx – cos y 110. Solutionof the differential equation 𝒅𝒚 𝒙 + 𝒅𝒚 𝒚 = 0 is a. = 𝟏 𝒙 + 𝟏 𝒚 = 𝒄 b. log x . log y = c c. xy = c d. x + y = c 111. The solutionof the differential equationx + 2y = x2 is
a. y = 𝒙 𝟐+ 𝒄 𝟒𝒙 𝟐 b. y = 𝒙 𝟐 𝟒 + c c. y = 𝒙 𝟐+ 𝒄 𝒙 𝟐 d. y = 𝒙 𝟐+ 𝒄 𝟒𝒙 𝟐 112. The degree ofthe differential equation[1 + ( 𝒅𝒚 𝒅𝒙 )2 ] = 𝒅 𝟐 𝒚 𝒅𝒙 𝟐 is a. 4 b. 𝟑 𝟐 c. not defined d. 2 113. The order and degree ofthe differential equation 𝒅 𝟐 𝒚 𝒅𝒙 𝟐 + ( 𝒅𝒚 𝒅𝒙 ) 𝟏 𝟒 + 𝒙 𝟏 𝟓= 0, respectivelyare a. 2 and not defined b. 2 and 2 c. 2 and 3 d. 3 and 3 114. Integrating factor of the differential equationcosx 𝒅𝒚 𝒅𝒙 + y sin x = 1 is: a. cos x b. tan xc. sec x d. sin x 115. Familyy = Ax + A3 of curvesis representedbythe differential equationofdegree: a. 1 b. 2 c. 3 d. 4 116. Solutionof 𝒅𝒚 𝒅𝒙 - y = 1, y(0) = 1 isgiven by a. xy = -ex b. xy = -e-x c. xy = -1 d. y = 2 ex – 1 117. The numberof solutions of 𝒅𝒚 𝒅𝒙 = 𝒚+𝟏 𝒙−𝟏 when y(1) = 2 is: a. None b. one c. two d. infinite 118. Integrating factor of the differential equation(1– x2 ) 𝒅𝒚 𝒅𝒙 - xy = 1 is a. –x b. 𝒙 𝟏+ 𝒙 𝟐 c.√𝟏 − 𝒙 𝟐 d. ½ log(1 – x2 ) 119. The general solutionof ex cos y dx – ex siny dy = 0 is: a. ex cos y = k b. ex sin y = k c. ex = k cos y d. ex = k siny 120. The solutionof the differential equation 𝒅𝒚 𝒅𝒙 = 𝟏+ 𝒚 𝟐 𝟏+ 𝒙 𝟐 is: a. y = tan-1 x b. y-x = k (1 =xy) c. x = tan-1 y d. tan (xy) = k 121. The integratingfactor ofthe differential equation 𝒅𝒚 𝒅𝒙 + 𝒚 = 𝟏+𝒚 𝒙 is: a. 𝒙 𝒆 𝒙 b. 𝒆 𝒙 𝒙 c. xex d. ex 122. The solutionof the differential equationcosx sin y dx + sinx cos y dy = 0 is: a. 𝒔𝒊𝒏𝒙 𝒔𝒊𝒏 𝒚 = c b. sin x siny = c c. sin x + siny = c d. cos x cos y = c 123. The solutionof x 𝒅𝒚 𝒅𝒙 + y = ex is: a. y = 𝒆 𝒙 𝒙 + 𝒌 𝒙 b. y = xex + cx c. y = xex + k d. x = 𝒆 𝒚 𝒚 + 𝒌 𝒚 124. The differential equationofthe familyof curves x2 + y2 – 2ay = 0, where a isarbitrary constant, is: a. (x2 – y2 ) 𝒅𝒚 𝒅𝒙 = 2xy b. 2(x2 + y2 ) 𝒅𝒚 𝒅𝒙 = xy c. 2(x2 - y2 ) 𝒅𝒚 𝒅𝒙 = xy d. (x2 +y2 ) 𝒅𝒚 𝒅𝒙 = 2xy 125. The general solutionof 𝒅𝒚 𝒅𝒙 = 2x 𝒆 𝒙 𝟐− 𝒚 is: a. 𝒆 𝒙 𝟐− 𝒚 = 𝒄 b. 𝒆−𝒚 + 𝒆 𝒙 𝟐 = 𝒄 c. 𝒆 𝒚 + 𝒆 𝒙 𝟐 = 𝒄 d. 𝒆 𝒙 𝟐+ 𝒚 = 𝒄 126. The general solutionof the differential equation 𝒅𝒚 𝒅𝒙 + xy is:
a. y = c𝒆 −𝒙 𝟐 𝟐 b. y = c𝒆 𝒙 𝟐 𝟐 c. y = (x + c) 𝒆 𝒙 𝟐 𝟐 d. y = (c-x) 𝒆 𝒙 𝟐 𝟐 127. the solutionof the equation(2y – 1) dx– (2x + 3) dy = 0 is: a. 𝟐𝒙−𝟏 𝟐𝒚+𝟑 = k b. 𝟐𝒚+ 𝟏 𝟐𝒙− 𝟑 = k c. 𝟐𝒙+𝟑 𝟐𝒚−𝟏 = k d. 𝟐𝒙−𝟏 𝟐𝒚−𝟏 = k 128. The solutionof 𝒅𝒚 𝒅𝒙 + y = e-x ,y (0) = 0 is: a. y = e-x (x– 1) b. y = xex c. y = xe-x + 1 d. y = xe-x 129. The order and degree ofthe differential equation ( 𝒅 𝟑 𝒚 𝒅𝒙 𝟑 ) 𝟐 − 𝟑 𝒅 𝟐 𝒚 𝒅𝒙 𝟐 + 𝟐 ( 𝒅𝒚 𝒅𝒙 ) 𝟒 = 𝒚 𝟒 are: a. 1, 4 b. 3, 4 c. 2, 4 d. 3, 2 130. The order and degree ofthe differential equation [ 𝟏+ ( 𝒅𝒚 𝒅𝒙 ) 𝟐 ] = 𝒅 𝟐 𝒚 𝒅𝒙 𝟐 are: a. 2, 𝟑 𝟐 b. 2, 3 . 2, 1 d. 3, 4 131. Whichof the followingisthe general solutionof 𝒅 𝟐 𝒚 𝒅𝒙 𝟐 − 𝒅𝒚 𝒅𝒙 𝟐 + y = 0? a. y = (Ax+ B)ex b. y = (Ax + B) e-x c. y = Aex + Be-x d. y = Acos x + B sinx 132. Solutionof the differential equation 𝒅𝒚 𝒅𝒙 + 𝒚 𝒙 = sinx is: a. x (y + cos x) = sin x + c b. x (y – cos x 0 = sin x + c c. xy cos x = sin x + c d. x (y + cos x) = cos x + c 133. The solutionof the differential equation 𝒅𝒚 𝒅𝒙 = ex – y + x2 e-y is: a. y = ex – y – x2 e-y + c b. ey – ex = + c c. ex + ey = + c d. ex – ey = 𝒙 𝟑 𝟑 + c 134. The magnitude of the vector6𝒊̂ + 𝟐𝒋̂ + 𝟑𝒌̂ is a. 5 b. 7 c. 12 d. 1 135. The positionvector of the point which dividedthe joinof points withpositionvectors 𝒂⃗⃗ + 𝒃⃗⃗ and 2𝒂⃗⃗ − 𝒃⃗⃗ in the ratio 1 : 2 is a. 𝟑𝒂⃗⃗ + 𝟐𝒃⃗⃗ 𝟑 b. 𝒂⃗⃗ c. 𝟓𝒂⃗⃗ − 𝒃⃗⃗ 𝟑 d. 𝟒𝒂⃗⃗ + 𝒃⃗⃗ 𝟑 136. The vector with initial pointP(2, -3, 5) and terminal point Q(3, -4, 7) is a. 𝒊̂ − 𝒋̂ + 𝟐𝒌̂ b. 𝟓𝒊̂ − 𝟕𝒋̂ + 𝟏𝟐𝒌̂ c. −𝒊̂ + 𝒋̂ − 𝟐𝒌̂ d. None of these 137. The angle betweenthe vectors 𝒊̂ − 𝒋̂ 𝒂𝒏𝒅 𝒋̂− 𝒌̂ is a. 𝝅 𝟑 b. 𝟐𝝅 𝟑 c. −𝝅 𝟑 d. 𝟓𝝅 𝟔 138. The value of  for whichthe two vectors 2𝒊̂ − 𝒋̂ + 𝟐𝒌̂ 𝒂𝒏𝒅 𝟑𝒊̂ +  𝒋̂+ 𝒌̂ are perpendicularis
a. 2 b. 4 c. 6 d. 8 139. The area of the parallelogramwhose adjacent sidesare 𝒊̂ + 𝒌̂ and 𝟐𝒊̂ + 𝒋̂ + 𝒌̂ is a. 3 b. 4 c. √ 𝟐 d. √ 𝟑 140. If | 𝒂⃗⃗ | = 𝟖, | 𝒃⃗⃗ | = 𝟑 𝒂𝒏𝒅 | 𝒂⃗⃗ × 𝒃⃗⃗ | = 𝟏𝟐 , thenvalue of is a. 6√ 𝟑 b. 8√ 𝟑 c. 12√ 𝟑 d. none of these 141. The 2 vector 𝒋̂ + 𝒌̂ and 3𝒊̂ − 𝒋̂ + 𝟒𝒌̂ representsthe two sidesAB and AC,respectivelyofa ABC . the length of the medianthrough A is a. √ 𝟑𝟒 𝟐 b. √ 𝟒𝟖 𝟐 c. √ 𝟏𝟖 d. None of these 142. The projectionof vector 𝒂⃗⃗ = 𝟐𝒊̂ − 𝒋̂ + 𝒌̂ along 𝒃⃗⃗ = 𝒊̂ + 𝟐𝒋̂ + 𝟐𝒌̂ is a. 2 b. √ 𝟔 c. 𝟐 𝟑 d. 𝟏 𝟑 143. If 𝒂⃗⃗ and 𝒃⃗⃗ are unit vectors,then what is the angle between 𝒂⃗⃗ and 𝒃⃗⃗ for √ 𝟑 𝒂⃗⃗ and 𝒃⃗⃗ to be a unit vector? a. 30o b. 45o c. 60o d. 90o 144. The unit vector perpendiculartothe vectors 𝒊̂ − 𝒋̂ 𝒂𝒏𝒅 𝒊̂ + 𝒋̂ forming a right handedsystemis a. 𝒌̂ b. -𝒌̂ c. 𝒊̂− 𝒋̂ √ 𝟐 d. 𝒊̂+ 𝒋̂ √ 𝟐 145. If | 𝒂⃗⃗ | = 𝟑 and -1  k  2 , then| 𝒌𝒂⃗⃗ | liesinthe interval a. [0,6] b. [-3,6] c. [3,6] d. [1,2] 146. The vector in the directionof the vector 𝒊̂ − 𝟐𝒋̂ + 𝟐𝒌̂ that has magnitude 9 is a. 𝒊̂ − 𝟐𝒋̂ + 𝟐𝒌̂ b. 𝒊̂− 𝟐𝒋̂+ 𝟐𝒌̂ 𝟑 c. (𝒊̂ − 𝟐𝒋̂ + 𝟐𝒌̂ ) d. 9( 𝒊̂ − 𝟐𝒋̂ + 𝟐𝒌̂) 147. The angle betweentwo vectors 𝒂⃗⃗ and 𝒃⃗⃗ with magnitude √ 𝟑 and 4, respectively,and 𝒂⃗⃗ . 𝒃⃗⃗ = 2√ 𝟑 is a. 𝝅 𝟔 b. 𝝅 𝟑 c. 𝝅 𝟐 d. 𝟓𝝅 𝟐 148. Findthe value of  such that the vectors 𝒂⃗⃗ = 𝟐𝒊̂ +  𝒋̂+ 𝒌̂ and 𝒃⃗⃗ = 𝒊̂ + 𝟐𝒋̂ + 𝟑𝒌̂ are orthogonal a. 0 b. 1 c. 𝟑 𝟐 d. −𝟓 𝟐 149. the value of  for which the vectors 𝟑𝒊̂ − 𝟔𝒋̂ + 𝒌̂ 𝒂𝒏𝒅 𝟐𝒊̂ − 𝟒𝒋̂ +  𝒌̂ are parallel is a. 𝟐 𝟓 b. 𝟐 𝟑 c. 𝟑 𝟐 d. 𝟓 𝟐 150. For any vector 𝒂⃗⃗ , the value of ( 𝒂⃗⃗ × 𝒊̂) 𝟐 + ( 𝒂⃗⃗ × 𝒋̂) 𝟐 + ( 𝒂⃗⃗ × 𝒌̂) 𝟐 is equal to a. 𝒂⃗⃗ 𝟐 b. 3 𝒂⃗⃗ 𝟐 c. 4 𝒂⃗⃗ 𝟐 d. 2𝒂⃗⃗ 𝟐 151. If | 𝒂⃗⃗ | = 𝟏𝟎,| 𝒃⃗⃗ | = 𝟐 𝒂𝒏𝒅 𝒂⃗⃗ . 𝒃⃗⃗ = 𝟏𝟐, 𝐭𝐡𝐞𝐧 𝐯𝐚𝐥𝐮𝐞 𝐨𝐟 | 𝒂⃗⃗ × 𝒃⃗⃗ |, is a. 5 b. 10 c. 14 c. 16 152. If 𝒂⃗⃗ , 𝒃⃗⃗ , 𝒄⃗ are unit vectorssuch that 𝒂⃗⃗ + 𝒃⃗⃗ + 𝒄⃗ = 𝟎⃗⃗ , thenvalue of 𝒂⃗⃗ . 𝒃⃗⃗ + 𝒃⃗⃗ . 𝒄⃗ + 𝒂⃗⃗ . 𝒄⃗ a. 1 b. 3 c. - 𝟑 𝟐 d. none of these 153. Projectionvector of 𝒂⃗⃗ on 𝒃⃗⃗ is
a. ( 𝒂⃗⃗ .𝒃⃗⃗ | 𝒃⃗⃗ | 𝟐) 𝒃⃗⃗ b. 𝒂⃗⃗ .𝒃⃗⃗ | 𝒃⃗⃗ | c. 𝒂⃗⃗ .𝒃⃗⃗ | 𝒂⃗⃗ | d. ( 𝒂⃗⃗ .𝒃⃗⃗ | 𝒂⃗⃗ | 𝟐 ) 𝒃⃗⃗ 154. If 𝒂⃗⃗ , 𝒃⃗⃗ , 𝒄⃗ are three vectors such that 𝒂⃗⃗ + 𝒃⃗⃗ + 𝒄⃗ = 𝟎⃗⃗ and | 𝒂⃗⃗ | = 𝟐, | 𝒃⃗⃗ | = 𝟑 , | 𝒄⃗ | = 𝟓, then value of 𝒂⃗⃗ . 𝒃⃗⃗ + 𝒃⃗⃗ . 𝒄⃗ + 𝒂⃗⃗ . 𝒄⃗ a. 0 b. 1 c. -19 d. 38 155. P is a point on the line segmentjoiningthe points(3, 2, -1) and (6, 2, -2). If x co-ordinate of P is5, thenits y co- ordinate is a. 2 b. 1 c. -1 d. -2 156. If ,, are the anglesthat a line makes withthe positive directionofx, y, z axis, respectively,thenthe directioncosinesof the line are: a. sin,sin,sin b. cos,cos,cos c. tan,tan,tan d. cos2 ,cos2 ,cos2  157. The equationsof x – axis in space are a. x = 0, y = 0 b. x = 0, z = 0 c. x = 0 d. y = 0, z = 0 158. If the directionscosinesofa line are k, k, k then a. k > 0 b. 0 < k< 1 c. k = 1 d. k = 𝟏 √ 𝟑 𝒐𝒓 − 𝟏 √ 𝟑 159. The distance of the plane 𝒓⃗ . ( 𝟐 𝟕 𝒊̂ + 𝟑 𝟕 𝒋̂ − 𝟔 𝟕 𝒌̂) = 1 from the originis a. 1 b. 7 c. 𝟏 𝟕 d. none of these 160. The area of the quadrilateral ABCD, where A(0,4,1), b(2,3,-1),c(4,5,0) and D(2, 6, 2) is equal to a. 9 sq unit b. 18 sq unit c. 27 sq unit d. 81 sq unit 161. The unit vector normal to the plane x + 2y + 3z – 6 = 0 𝟏 √ 𝟏𝟒 𝒊̂ + 𝟐 √ 𝟏𝟒 𝒋̂ + 𝟑 √ 𝟏𝟒 𝒌̂ . 162. The corner pointsof the feasible regiondeterminedbythe systemof linearconstraints are (0, 10), (5,5), (15, 150), (0, 20). Let Z = px + qy, where p, q > 0. Conditionon p and q so that the maximumof Z occurs at both the points(15, 15) and (0, 20) is a. p = q b. p = 2q c. q = 2p d. q = 3p 163. Feasible region(shaded) fora LPP isshown in the Fig.Minimumof Z = 4x + 3y occurs at the point a. (0, 80 b. (2, 5 ) c. (4, 3) d. (9, 0) 164. The corner points of the feasible regiondeterminedbythe system oflinear constraints are (0,0), (0, 40), (20, 40), (60, 20), (60, 0). The objective functionis Z = 4x + 3y. Compare the quantity inColumn A and ColumnB ColumnA ColumnB Maximumof Z 325 a. The quantity in columnA is greater b. The quantity in columnB is greater c. The two quantitiesare equal
d. The relationshipcan not be determinedonthe basis of the informationsupplied. 165. Corner pointsof the feasible regionforan LPP are (0, 2), (3, 0), (6,0), (6, 8) and (0, 5). Let F = 4x + 6y be the objective function.The minimum value of F occurs at a. (0, 2) only b. (3, 0) only c. The mid pointof the line segmentjoiningthe points(0, 2) and (3, 0) only d. Any point on the line segmentjoiningthe points(0, 20 and (3, 0). 166. Let A and B be two events.If P (A) = 0.2, P(B) = 0.4, P(A B) = 0.6, then P(AIB) is equal to a. 0.8 b. 0.5 c. 0.3 d. 0 167. Let A and B be two eventssuch that P(A) = 0.6, P(B) = 0.2 and P(AIB) = 0.5. thenP(AIB) equals a. 𝟏 𝟏𝟎 b. 𝟑 𝟏𝟎 c. 𝟑 𝟖 d. 𝟔 𝟕 168. If A and B are independenteventssuchthat 0 < P(A) < 1 and 0 <P(B) < 1 then whichof the followingisnot correct? a. A and B are mutually exclusive b. A and B are independent c. A and B are independent d. A and B are independent 169. If P(A) = 𝟒 𝟓 , and P(A  B) = 𝟕 𝟏𝟎 , thenP(B I A) is equal to a. 𝟏 𝟏𝟎 b. 𝟏 𝟏𝟖 c. 𝟕 𝟖 d. 𝟏𝟕 𝟐𝟎 170. If P(A  B) = 𝟕 𝟏𝟎 and P(B) = 𝟏𝟕 𝟐𝟎 , then P(B I A) is equal to a. 𝟏𝟒 𝟏𝟕 b. 𝟏 𝟖 c. 𝟕 𝟖 d. 𝟏𝟕 𝟐𝟎 171. If P(A) = 𝟑 𝟏𝟎 , P(B) = 𝟐 𝟓 and P(A  B) = 𝟑 𝟓 , thenP(B I A) + P(AI B) is equal to a. 𝟏 𝟒 b. 𝟏 𝟑 c. 𝟓 𝟏𝟐 d. 𝟕 𝟐 172. If P(A) = 𝟐 𝟓 , P(B) = 𝟑 𝟏𝟎 and P(A  B) = 𝟏 𝟓 , thenP(B I A) . P(A IB) is equal to a. 𝟓 𝟔 b. 𝟓 𝟕 c. 𝟐𝟓 𝟒𝟐 d. 1 173. If A and B are two eventssuch that P(A) = ½, P(B) = 𝟏 𝟑 , P(A / B) = 𝟏 𝟒 , thenP(A  B) equals a. 𝟏 𝟏𝟐 b. 𝟑 𝟒 c. 𝟏 𝟒 d. 𝟑 𝟏𝟔 174. If P(A) = 0.4, P(B) = 0.8 and P(B I A) = 0.6, thenP(A  B) is equal to a. 0.24 b. 0.3 c. 0.48 d. 0.96 175. You are giventhat A and B are two eventssuch that P(B) = 𝟑 𝟓 , P(A I B) = ½ and P(A  B) = 𝟒 𝟓 , thenP(A) equals a. 𝟑 𝟏𝟎 b. 𝟏 𝟐 c. 𝟏 𝟓 d. 𝟑 𝟓
176. If P(B) = 𝟑 𝟓 , P(AIB) = ½ and P(A  B) = 𝟒 𝟓 , then P(A  B) + P(A  B) = a. 𝟒 𝟓 b. 𝟏 𝟐 c. 𝟏 𝟓 d. 1 177. Let A and B be two eventssuch that P(A) = 𝟑 𝟖 , P(B) = 𝟓 𝟖 and thenP(A I B) . P(A IB) isequal to a. 𝟐 𝟓 b. 𝟑 𝟖 c. 𝟑 𝟐𝟎 d. 𝟔 𝟐𝟓 178. If A and B are such eventsthat P(A) > 0 and P(B)  1, thenP(A I B) equals a. 1 – P(A I B) b. 1 – P(A IB) c. 𝟏−𝑷(𝑨∪𝑩) 𝑷(𝑩′) d. P(A) IP(B ) 179. A bag contains5 red and 3 blue balls. If3 balls are drawn at random without replacementthe probabilityof gettingexactly one red ball is a. 𝟒𝟓 𝟏𝟗𝟔 b. 𝟏𝟑𝟓 𝟑𝟗𝟐 c. 𝟏𝟓 𝟓𝟔 d. 𝟏𝟓 𝟐𝟗 180. Assume that in a family,each child isequallylikelyto be or a girl.A family withthree childrenischosen at random. The probabilitythat the eldestchildisa girl given that the familyhas at least one girl is a. 𝟒 𝟕 b. 𝟏 𝟐 c. 𝟏 𝟑 d. 𝟐 𝟑 181. A die is thrown and a card is selectedatrandom from a deck of 52 playing cards. The probabilityof gettingan evennumberon the die and a spade card is a. 𝟑 𝟒 b. 𝟏 𝟐 c. 𝟏 𝟒 d. 𝟏 𝟖 182. A box contains 3 orange balls,3 greenballsand 2 blue balls. Three balls are drawn at random from the box without replacement.The probabilityof drawing 2 greenballsand one blue ball is a. 𝟑 𝟖 b. 𝟐 𝟐𝟏 c. 𝟏 𝟐𝟖 d. 𝟏𝟔𝟕 𝟏𝟔𝟖 183. Eight coins are tossedtogether.The probabilityof gettingexactly 3 headsis a. 𝟏 𝟐𝟓𝟔 b. 𝟕 𝟑𝟐 c. 𝟓 𝟑𝟐 d. 𝟑 𝟑𝟐 184. In a college,30% studentsfail in physics,255 fail in mathematics and 10% fail in both. One studentis chosen at random. The probabilitythat she failsin physicsif she has failedin mathematics is a. 𝟏 𝟏𝟎 b. 𝟐 𝟓 c. 𝟏 𝟑 d. 𝟗 𝟐𝟎 185. The probabilitydistributionof a discrete random variable X is givenbelow: X 2 3 4 5 P(X) 𝟓 𝒌 𝟕 𝒌 𝟗 𝒌 𝟏𝟏 𝒌 The value of k is a. 8 b. 16 c. 32 d. 48

More Related Content

PDF
Quadratic Function Presentation
byRyanWatt
 
PDF
Algebra formulas
byMatthew McKenzie
 
PPTX
4.6 radical equations
bymath123b
 
PDF
Algebra formulae
byTamizhmuhil
 
PPT
9 determinant and cross product
bymath267
 
DOCX
Practice questions( calculus ) xii
byindu psthakur
 
DOCX
Class xii practice questions
byindu psthakur
 
PDF
5 work sheet( for manasthali) class 11
byKarunaGupta1982
 
Quadratic Function Presentation
byRyanWatt
 
Algebra formulas
byMatthew McKenzie
 
4.6 radical equations
bymath123b
 
Algebra formulae
byTamizhmuhil
 
9 determinant and cross product
bymath267
 
Practice questions( calculus ) xii
byindu psthakur
 
Class xii practice questions
byindu psthakur
 
5 work sheet( for manasthali) class 11
byKarunaGupta1982
 

What's hot

PDF
(Www.entrance exam.net)-sail placement sample paper 5
bySAMEER NAIK
 
PDF
Additional Mathematics form 4 (formula)
byFatini Adnan
 
PPTX
5 2 solving 2nd degree equations
bymath123b
 
DOCX
Assessments for class xi
byindu psthakur
 
PDF
Finite elements : basis functions
byTarun Gehlot
 
PPTX
The Quadratic Function Derived From Zeros of the Equation (SNSD Theme)
bySNSDTaeyeon
 
PPT
Integration
bysuefee
 
PPTX
Mathematics 9 Lesson 1-D: System of Equations Involving Quadratic Equations
byJuan Miguel Palero
 
PDF
brain gate
byPrince Jairaj
 
PPT
15 multi variable functions
bymath267
 
PPTX
5 4 equations that may be reduced to quadratics
bymath123b
 
(Www.entrance exam.net)-sail placement sample paper 5
bySAMEER NAIK
 
Additional Mathematics form 4 (formula)
byFatini Adnan
 
5 2 solving 2nd degree equations
bymath123b
 
Assessments for class xi
byindu psthakur
 
Finite elements : basis functions
byTarun Gehlot
 
The Quadratic Function Derived From Zeros of the Equation (SNSD Theme)
bySNSDTaeyeon
 
Integration
bysuefee
 
Mathematics 9 Lesson 1-D: System of Equations Involving Quadratic Equations
byJuan Miguel Palero
 
brain gate
byPrince Jairaj
 
15 multi variable functions
bymath267
 
5 4 equations that may be reduced to quadratics
bymath123b
 

Similar to Mcq exemplar class 12

PDF
Mcq for manavsthali( 7 worksheets)
byKarunaGupta1982
 
PDF
Aieee 2003 maths solved paper by fiitjee
byMr_KevinShah
 
PDF
MFMTQP_MAT_nda question paper for nda class 12
byEMALLIKARJUNAREDDY
 
PDF
ISI MSQE Entrance Question Paper (2008)
byCrackDSE
 
PDF
Nbhm m. a. and m.sc. scholarship test 2007
byMD Kutubuddin Sardar
 
PDF
01. Functions-Exercise. Module-4 pdf
byRajuSingh806014
 
DOCX
Mathsclass xii (exampler problems)
bynitishguptamaps
 
PDF
ISI MSQE Entrance Question Paper (2010)
byCrackDSE
 
PDF
ISI MSQE Entrance Question Paper (2004)
byCrackDSE
 
PDF
Changed pattern of CBSE Class XII Mathematics -2016-17-with_marking_scheme
byPratima Nayak ,Kendriya Vidyalaya Sangathan
 
PDF
TMUA 2021 Paper 1 Solutions (Handwritten).pdf
byssuser625c41
 
PDF
ISI MSQE Entrance Question Paper (2012)
byCrackDSE
 
DOCX
some important questions for practice clas 12
bynitishguptamaps
 
DOC
12th mcq
bynitishguptamaps
 
DOC
12th mcq
bynitishguptamaps
 
DOCX
Please be sure to save at least once every 15 minutes. If you leav.docx
byrandymartin91030
 
DOCX
Assignment of class 12 (chapters 2 to 9)
byKarunaGupta1982
 
PDF
Maieee03
byAshish Yadav
 
PDF
Maths imp questions for 2018
byKarunaGupta1982
 
PDF
Nbhm m. a. and m.sc. scholarship test 2005
byMD Kutubuddin Sardar
 
Mcq for manavsthali( 7 worksheets)
byKarunaGupta1982
 
Aieee 2003 maths solved paper by fiitjee
byMr_KevinShah
 
MFMTQP_MAT_nda question paper for nda class 12
byEMALLIKARJUNAREDDY
 
ISI MSQE Entrance Question Paper (2008)
byCrackDSE
 
Nbhm m. a. and m.sc. scholarship test 2007
byMD Kutubuddin Sardar
 
01. Functions-Exercise. Module-4 pdf
byRajuSingh806014
 
Mathsclass xii (exampler problems)
bynitishguptamaps
 
ISI MSQE Entrance Question Paper (2010)
byCrackDSE
 
ISI MSQE Entrance Question Paper (2004)
byCrackDSE
 
Changed pattern of CBSE Class XII Mathematics -2016-17-with_marking_scheme
byPratima Nayak ,Kendriya Vidyalaya Sangathan
 
TMUA 2021 Paper 1 Solutions (Handwritten).pdf
byssuser625c41
 
ISI MSQE Entrance Question Paper (2012)
byCrackDSE
 
some important questions for practice clas 12
bynitishguptamaps
 
12th mcq
bynitishguptamaps
 
12th mcq
bynitishguptamaps
 
Please be sure to save at least once every 15 minutes. If you leav.docx
byrandymartin91030
 
Assignment of class 12 (chapters 2 to 9)
byKarunaGupta1982
 
Maieee03
byAshish Yadav
 
Maths imp questions for 2018
byKarunaGupta1982
 
Nbhm m. a. and m.sc. scholarship test 2005
byMD Kutubuddin Sardar
 

More from KarunaGupta1982

DOCX
Mcq exemplar class 12
byKarunaGupta1982
 
PDF
Assignment chapters 3 to 7
byKarunaGupta1982
 
PDF
Class xii probability
byKarunaGupta1982
 
PDF
Continuity and differentiability
byKarunaGupta1982
 
PDF
Application of integrals
byKarunaGupta1982
 
DOCX
Class 11 chapters 9, 10, 11
byKarunaGupta1982
 
DOCX
Three dimensional geometry
byKarunaGupta1982
 
DOCX
Assignment of straight lines and conic section
byKarunaGupta1982
 
DOCX
Three dimensional geometry
byKarunaGupta1982
 
DOCX
Class xii worksheet (chapters 2,5,6)
byKarunaGupta1982
 
DOCX
Class X1 assignment of chapters 1,3,4,5
byKarunaGupta1982
 
DOCX
Differential equation
byKarunaGupta1982
 
DOCX
Surface chemistry notes
byKarunaGupta1982
 
DOCX
Assignment of solution
byKarunaGupta1982
 
DOCX
Notes of Matrices and Determinants
byKarunaGupta1982
 
DOCX
Class xi worksheet (9,10,11) two levels
byKarunaGupta1982
 
DOCX
Maxima and minima
byKarunaGupta1982
 
DOCX
Sample paper class XII MATHEMATICS
byKarunaGupta1982
 
DOCX
Class xii assignment chapters (1 to 9)
byKarunaGupta1982
 
DOCX
Integrals formula and short questions
byKarunaGupta1982
 
Mcq exemplar class 12
byKarunaGupta1982
 
Assignment chapters 3 to 7
byKarunaGupta1982
 
Class xii probability
byKarunaGupta1982
 
Continuity and differentiability
byKarunaGupta1982
 
Application of integrals
byKarunaGupta1982
 
Class 11 chapters 9, 10, 11
byKarunaGupta1982
 
Three dimensional geometry
byKarunaGupta1982
 
Assignment of straight lines and conic section
byKarunaGupta1982
 
Three dimensional geometry
byKarunaGupta1982
 
Class xii worksheet (chapters 2,5,6)
byKarunaGupta1982
 
Class X1 assignment of chapters 1,3,4,5
byKarunaGupta1982
 
Differential equation
byKarunaGupta1982
 
Surface chemistry notes
byKarunaGupta1982
 
Assignment of solution
byKarunaGupta1982
 
Notes of Matrices and Determinants
byKarunaGupta1982
 
Class xi worksheet (9,10,11) two levels
byKarunaGupta1982
 
Maxima and minima
byKarunaGupta1982
 
Sample paper class XII MATHEMATICS
byKarunaGupta1982
 
Class xii assignment chapters (1 to 9)
byKarunaGupta1982
 
Integrals formula and short questions
byKarunaGupta1982
 

Recently uploaded

PPTX
CHAPTER NO. 09 HCP BY GG LABORATORY CLINICAL TEST
byGanu Gavade
 
PDF
Digital Transformation, AI, and ML in Pharma R&D and Manufacturing Workflows ...
byAjaz Hussain
 
PDF
First Semester BCA Exam 2025 – Constitution Values – 1 Solved Answers.pdf
byKuvempu University
 
PDF
Application of ICT Lecture 7 Ethical Considerations in Use of ICT Platforms a...
byKhalil Khan Marwat
 
PPTX
SPECIAL STSAINS USED IN HISTOPATHOLOGY.pptx
byKISMAT ALI
 
PDF
Application of ICT Lecture 3 ICT in Education.pdf
byKhalil Khan Marwat
 
PDF
Homebound (2025): A Critical Analysis of Social Realism, Systemic Apathy, and...
bykrutivyas2005
 
PPTX
Shipping Policies & Picking Policies in Odoo 18 Inventory
byCeline George
 
PPTX
How to Manage Sorting Cart by Category in Odoo 18 POS
byCeline George
 
PPTX
CHAPTER NO. 05 DIURETICS PHARMACOGNOSY.pptx
byGanu Gavade
 
PPTX
Road Safety as a shared responsibility, 2026
bySougata Pal
 
PPTX
CARDIOTONIC CHAPTER NO. 5 PHARMACOGNOSY DIGITALIS AND ARJUNA BARK
byGanu Gavade
 
PPTX
DIGITAL TRANSFORMATION MODULE KEY POINTS
byLisa Harris
 
PPTX
ENGLISH 3-Q3-W7.pptx 2025-2026..................................................
byWindyDizonMiranda
 
PPTX
How to Configure Discount Code in Odoo 18 Sales
byCeline George
 
PPTX
English 8 Q3 Wk8.pptx opinion editorial article( Writing , Revising, Editing...
byjacquelyncastre03
 
PPTX
No Storage Needed! Odoo Cross-Docking
byCeline George
 
PPTX
How to Create a Search Panel in Odoo 18
byCeline George
 
PPTX
Auto Plan Feature in Odoo 18 Planning Module
byCeline George
 
PPTX
How to use sorted() method in Odoo 18
byCeline George
 
CHAPTER NO. 09 HCP BY GG LABORATORY CLINICAL TEST
byGanu Gavade
 
Digital Transformation, AI, and ML in Pharma R&D and Manufacturing Workflows ...
byAjaz Hussain
 
First Semester BCA Exam 2025 – Constitution Values – 1 Solved Answers.pdf
byKuvempu University
 
Application of ICT Lecture 7 Ethical Considerations in Use of ICT Platforms a...
byKhalil Khan Marwat
 
SPECIAL STSAINS USED IN HISTOPATHOLOGY.pptx
byKISMAT ALI
 
Application of ICT Lecture 3 ICT in Education.pdf
byKhalil Khan Marwat
 
Homebound (2025): A Critical Analysis of Social Realism, Systemic Apathy, and...
bykrutivyas2005
 
Shipping Policies & Picking Policies in Odoo 18 Inventory
byCeline George
 
How to Manage Sorting Cart by Category in Odoo 18 POS
byCeline George
 
CHAPTER NO. 05 DIURETICS PHARMACOGNOSY.pptx
byGanu Gavade
 
Road Safety as a shared responsibility, 2026
bySougata Pal
 
CARDIOTONIC CHAPTER NO. 5 PHARMACOGNOSY DIGITALIS AND ARJUNA BARK
byGanu Gavade
 
DIGITAL TRANSFORMATION MODULE KEY POINTS
byLisa Harris
 
ENGLISH 3-Q3-W7.pptx 2025-2026..................................................
byWindyDizonMiranda
 
How to Configure Discount Code in Odoo 18 Sales
byCeline George
 
English 8 Q3 Wk8.pptx opinion editorial article( Writing , Revising, Editing...
byjacquelyncastre03
 
No Storage Needed! Odoo Cross-Docking
byCeline George
 
How to Create a Search Panel in Odoo 18
byCeline George
 
Auto Plan Feature in Odoo 18 Planning Module
byCeline George
 
How to use sorted() method in Odoo 18
byCeline George
 

Mcq exemplar class 12

  • 1.
    MULTIPLE QUESTIONS (EXEMPLARPROBLEMS) 1. Let R be a relationon the setN of natural numbersdefinedbynRm if n dividesm. thenR is a. Reflexive andsymmetric b. Transitive and symmetric c. Equivalence d. Reflexive,transitive butnot symmetric 2. Let N be the set of natural numbers and the function f : N  be definedby f(n) = 2n + 3  n  N. thenf is a. Surjective b. Injective c. Biijective d. None of these 3. Set A has 3 elementsandthe setB has 4 elements.Thenthe number ofinjective mappings that can be definedfromA to B is a. 144 b. 12 c. 24 d. 64 4. Let f : R  R be definedbyf(x) = sin x and g : R R be definedbyg(x) = x,then f o g is a. x2 sinx b. (sinx)2 c. sin x2 d. 𝒔𝒊𝒏 𝒙 𝒙 𝟐 5. Let f : R  R be definedbyf(x) = 3x – 4. Then f-1 (x) isgiven by a. 𝒙+𝟒 𝟑 b. 𝒙 𝟑 - 4 c. 3x – 4 d. none of these 6. The maximumnumber of equivalence relationsonthe set A = {1, 2, 3} are a. 1 b. 2 c. 3 d. 5 7. Let us define a relationR in R as aRb if a b. thenR is a. An equivalence relation b. Reflexive,transitive butnot symmetric c. Symmetric,transitive but not reflexive d. Neithertransitive nor reflexive butsymmetric. 8. The identityelementforthe binary operation  definedinQ  {0} as a  b= 𝒂𝒃 𝟐  a, b  Q  {0} is a. 1 b. 2 c. 0 d. none of these 9. Let A = {1, 2, 3,….., n} and B = {a, b}. thenthe numbersof surjections from A intoB is a. n P2 b. 2n – 2 c. 2n – 1 d. none of these 10. Let f : R  R be definedbyf(x) = 3x2 - 5 and g : R  R by g(x) = 𝒙 𝒙 𝟐+ 𝟏 .theng o f is a. 𝟑 𝒙 𝟐− 𝟓 𝟗𝒙 𝟒− 𝟑𝟎𝒙 𝟐+ 𝟐𝟔 b. 𝟑 𝒙 𝟐− 𝟓 𝟗𝒙 𝟒− 𝟔𝒙 𝟐+ 𝟐𝟔 c. 𝟑 𝒙 𝟐 𝒙 𝟒+𝟐𝒙 𝟐− 𝟒 d. 𝟑 𝒙 𝟐 𝟗𝒙 𝟒+ 𝟑𝟎𝒙 𝟐− 𝟐 11. Whichof the followingfunctionsfromZ into Z are bijections? a. f(x) = x3 b. f(x) = x + 2 c. f(x) = 2x + 1 d. f(x) = x2 + 1 12. Let f : R  R be the functionsdefinedbyf(x) = x3 + 5. Then f-1 (x) is a. (𝒙 + 𝟓) 𝟏 𝟑 b. (𝒙 − 𝟓) 𝟏 𝟑 c. (𝟓 − 𝒙) 𝟏 𝟑 d. 5 – x 13. Let f : R - { 𝟑 𝟓 }  R be definedbyf(x) = 𝟑𝒙+𝟐 𝟓𝒙−𝟑 . Then
  • 2.
    a. f-1 (x) =f(x) b. f-1 (x) = - f(x) c. (fo f) x = - x d. f-1 (x) = 𝟏 𝟏𝟗 f(x) 14. Let f : [2, )  Rbe the functiondefined byf(x) = x2 - 4x + 5, then the range of f is a. R b. [1, ) c. [4, ) d. [5, ) 15. Let f : R  R be givenby f(x) = tan x. Then f-1 (1) is a. 𝝅 𝟒 b. { n  + 𝝅 𝟒 : n  Z} c. does not exist d. none of these 16. Let the relationR be definedinN by aRb if2a + 3b = 30. Then R = ……………………. 17. The principal value branch of sec-1 is a. [− 𝝅 𝟐 , 𝝅 𝟐 ] - {0} b. [0, ]- { 𝝅 𝟐 } c. (0, ) d. (− 𝝅 𝟐 , 𝝅 𝟐 ) 18. The value of 𝐬𝐢𝐧−𝟏 ( 𝒄𝒐𝒔( 𝟒𝟑  𝟓 )) is a. 𝟑𝝅 𝟓 b. −𝟕𝝅 𝟓 c. 𝝅 𝟏𝟎 d. - 𝝅 𝟏𝟎 19. The principal value ofthe expressioncos-1 [cos (-6800 )]is a. 𝟐𝝅 𝟗 b. −𝟐𝝅 𝟗 c. 𝟑𝟒𝝅 𝟗 d. 𝝅 𝟗 20. If tan-1 x = 𝝅 𝟏𝟎 for some x  R, thenthe value of cot-1 x is a. 𝝅 𝟓 b. 𝟐𝝅 𝟓 c. 𝟑𝝅 𝟓 d. 𝟒𝝅 𝟓 21. The principal value ofsin-1 ( −√ 𝟑 𝟐 ) is a. − 𝟐𝝅 𝟑 b. − 𝝅 𝟑 c. 𝟒𝝅 𝟑 d. 𝟓𝝅 𝟑 22. The greatestand leastvaluesof (sin-1 x)2 + (cos-1 x)2 are respectively a. 𝟓𝝅 𝟐 𝟒 𝒂𝒏𝒅 𝝅 𝟐 𝟖 b. 𝝅 𝟐 𝒂𝒏𝒅 −𝝅 𝟐 c. 𝝅 𝟐 𝟒 𝒂𝒏𝒅 −𝝅 𝟐 𝟒 d. 𝝅 𝟐 𝟒 𝒂𝒏𝒅 𝟎. 23. The value of sin (2 sin-1 (.6)) is a. .48 b. .96 c. 1.2 d. sin 1.2 24. If sin-1 x + sin-1 y =  𝟐 , then value of cos-1 x + cos-1 y is a. 𝝅 𝟐 b.  c. 0 d. 𝟐𝝅 𝟑 25. The value of the expressionsin[cot-1 (cos (tan-1 1))]is a. 0 b. 1 c. 𝟏 √ 𝟑 d. √ 𝟐 𝟑 26. Whichof the followingisthe principal value branch ofcosec-1 x? a. [− 𝝅 𝟐 , 𝝅 𝟐 ] b. (0, ) - { 𝝅 𝟐 } c. (− 𝝅 𝟐 , 𝝅 𝟐 ) d. [− 𝝅 𝟐 , 𝝅 𝟐 ] - {0} 27. If 3tan-1 x + cot-1 x = ,thenx equals a. 0 b. 1 c. -1 d. ½ 28. The value of sin-1 𝐬𝐢𝐧−𝟏 ( 𝒄𝒐𝒔( 𝟑𝟑  𝟓 )) is a. 𝟑𝝅 𝟓 b. −𝟕𝝅 𝟓 c. 𝝅 𝟏𝟎 d. - 𝝅 𝟏𝟎
  • 3.
    29. If cos(𝐬𝐢𝐧−𝟏 𝟐 𝟓 + 𝐜𝐨𝐬−𝟏 𝒙)= 0, thenx is equal to a. 𝟏 𝟓 b. 𝟐 𝟓 c. 0 d. 1 30. The value of cos-1 (𝒄𝒐𝒔 𝟑𝝅 𝟐 )is equal to a. 𝟑𝝅 𝟐 b. 𝟓𝝅 𝟐 c. 𝝅 𝟐 d. 𝟕𝝅 𝟐 31. The value of the expression2sec-1 2 + sin-1 ( 𝟏 𝟐 ) is a. 𝝅 𝟔 b. 𝟓𝝅 𝟔 c. 1 d. 𝟕𝝅 𝟔 32. If sin-1 ( 𝟐𝒂 𝟏+ 𝒂 𝟐 ) + cos-1 ( 𝟏− 𝒂 𝟐 𝟏+ 𝒂 𝟐 ) = tan-1 ( 𝟐𝒙 𝟏− 𝒙 𝟐 ), where a, x  ]0, 1, then the value of x is a. 0 b. 𝒂 𝟐 c. a d. 𝟐𝒂 𝟏− 𝒂 𝟐 33. The value of the expressiontan ( 𝟏 𝟐 𝐜𝐨𝐬−𝟏 𝟐 √ 𝟓 ) is a. 2 + √ 𝟓 b. √ 𝟓 – 2 c. 5 + √ 𝟐 d. √ 𝟓+ 𝟐 𝟐 34. If A = [ 𝟐 −𝟏 𝟑 −𝟒 𝟓 𝟏 ] 𝒂𝒏𝒅 𝑩 = [ 𝟐 𝟑 𝟒 −𝟐 𝟏 𝟓 ] , then a. Only AB is defined b. Only BA is defined c. AB and BA both are defined d. AB and BA both are not defined. 35. If A and B are two matrices of the order 3  m and 3  n. respectively,andm = n, then the order of matrix (5A – 2B) is a. m  3 b. 3  3 c. m  n d. 3  n 36. if A = [ 𝟎 𝟏 𝟏 𝟎 ] , thenA2 isequal to a. [ 𝟎 𝟏 𝟏 𝟎 ] b. [ 𝟏 𝟎 𝟏 𝟎 ] c. [ 𝟎 𝟏 𝟎 𝟏 ] d. [ 𝟏 𝟎 𝟎 𝟏 ] 37. If matrix A = [ 𝒂𝒊𝒋]2  2 , where aij = 1 if i  j= 0 if i = j a. I b. A c. 0 d. none of these 38. The matrix [ 𝟏 𝟎 𝟎 𝟎 𝟐 𝟎 𝟎 𝟎 𝟒 ]is a a. Identitymatrix b. symmetric matrix c. skew symmetricmatrix d. none of these 39. If A is matrix of order m  n and B is a matrix such that AB and BAare both defined,thenorderof matrix B is a. m  m b. n  n c. m  n d. n  m 40. if A and B are matrices ofsame order, then(AB - BA) isa a. skew symmetricmatrix b. null matrix
  • 4.
    c. symmetricmatrix d. unitmatrix 41. If A is a square matrix such that A2 = I, then(A – I)3 + (A + I)3 - 7A is equal to a. A b. I – A c. I + A d. 3A 42. For any two matrices A and B, we have a. AB = BA b. AB  BA c. AB= 0 d. none of these 43. If x , y  R, thenthe determinant  = | 𝒄𝒐𝒔 𝒙 −𝒔𝒊𝒏 𝒙 𝟏 𝒔𝒊𝒏 𝒙 𝒄𝒐𝒔 𝒙 𝟏 𝒄𝒐𝒔(𝒙 + 𝒚) −𝒔𝒊𝒏(𝒙 + 𝒚) 𝟎 | liesinthe interval a. [- √ 𝟐, √ 𝟐] b. [-1, 1] c. [-√ 𝟐, 1] d. [-1. -√ 𝟐] 44. The value of determinant | 𝒂 − 𝒃 𝒃 + 𝒄 𝒂 𝒃 − 𝒂 𝒄 + 𝒂 𝒃 𝒄 − 𝒂 𝒂 + 𝒃 𝒄 | a. a3 + b3 + c3 b. 3 bc c. a3 + b3 + c3 – 3abc d. none of these 45. the determinants | 𝒃 𝟐 − 𝒂𝒃 𝒃 − 𝒄 𝒃𝒄 − 𝒂𝒄 𝒂𝒃 − 𝒂 𝟐 𝒂 − 𝒃 𝒃 𝟐 − 𝒂𝒃 𝒃𝒄 − 𝒂𝒄 𝒄 − 𝒂 𝒂𝒃 − 𝒂 𝟐 | equals a. abc(b-c) (c-a) (a-b) b. (b-c) (c-a) (a-b) c. (a+b+c)(b-c) (c-a) (a-b) d. None ofthese 46. the maximumvalue of ∆ = | 𝟏 𝟏 𝟏 𝟏 𝟏 + 𝒔𝒊𝒏 𝟏 𝟏 + 𝒄𝒐𝒔 𝟏 𝟏 | is ( is a real number) a. 𝟏 𝟐 b. √ 𝟑 𝟐 c. √ 𝟐 d. 𝟐√ 𝟑 𝟒 47. If A = [ 𝟐  −𝟑 𝟎 𝟐 𝟓 𝟏 𝟏 𝟑 ] , then A-1 existsif a.  = 2 b.   2 c.   -2 d. none of these 48. If x, y, z are all differentfromzero and | 𝟏 + 𝒙 𝟏 𝟏 𝟏 𝟏 + 𝒚 𝟏 𝟏 𝟏 𝟏 + 𝒛 | = 0, thenvalue of x-1 + y-1 + z-1 is a. x y z b. x-1 y-1 z-1 c. –x-y-z d. -1 49. The functionf(x) = { 𝒔𝒊𝒏𝒙 𝒙 + 𝒄𝒐𝒔 𝒙, 𝒊𝒇 𝒙 ≠ 𝟎 𝒌, 𝒊𝒇 𝒙 = 𝟎 iscontinuous at x = 0, then the value of k is a. 3 b. 1 c. 2 d. 1.5 50. The functionf(x) = [x] denotesthe greatestintegerfunction,continuous at a. 4 b. -2 c. 1 d. 1.5 51. The functionf(x) = | 𝒙| + | 𝒙 + 𝟏| is a. Continuousat x = 0 as well as x = 1 b. Continuousat x = 1 but not at x = 0 c. Discontinuousat x = 0 as well as at x = 1 d. Continuousat x = 0 but not at x = 1
  • 5.
    52. The valueof k which makes the functiondefinedby f(x) = { 𝒔𝒊𝒏 𝟏 𝒙 , 𝒊𝒇 𝒙 ≠ 𝟎 𝒌, 𝒊𝒇 𝒙 = 𝟎 , continuose at x = 0 is a. 8 b. 1 c. -1 d. none of these 53. If u = sin-1 ( 𝟐𝒙 𝟏+ 𝒙 𝟐 ) and y = tan-1 ( 𝟐𝒙 𝟏− 𝒙 𝟐 ) , then 𝒅𝒖 𝒅𝒗 is a. 𝟏 𝟐 b. x c. 𝟏− 𝒙 𝟐 𝟏+ 𝒙 𝟐 d. 1 54. The value of c inMean value theoremfor the functionf(x) = x(x – 2), x  [1. 2] is a. 𝟑 𝟐 b. 𝟐 𝟑 c. 𝟏 𝟐 d. 𝟑 𝟐 55. The functionf(x) = 𝟒− 𝒙 𝟐 𝟒𝒙− 𝒙 𝟑 is a. Discontinuousat onlyone point b. Discontinuousat exactlytwo point c. Discontinuousat exactlythree point d. None of these 56. The setof points where the functionf givenby f(x) = | 𝟐𝒙 − 𝟏| isdifferentiable is a. R b. R - { 𝟏 𝟐 } c. (0,  ) d. none of these 57. If y = log ( 𝟏− 𝒙 𝟐 𝟏+ 𝒙 𝟐 ) , thenfind 𝒅𝒚 𝒅𝒙 is equal to a. 𝟒𝒙 𝟑 𝟏− 𝒙 𝟒 b. −𝟒𝒙 𝟏− 𝒙 𝟒 c. 𝟏 𝟒− 𝒙 𝟒 d. −𝟒𝒙 𝟑 𝟏− 𝒙 𝟒 58. If x = t2 , y = t3 , then 𝒅 𝟐 𝒚 𝒅𝒙 𝟐 is a. 𝟑 𝟐 b. 𝟑 𝟒𝒕 c. 𝟑 𝟐𝒕 d. 𝟑 𝟐𝒕 59. The value of c on Rolle’stheoremforthe functionf(x) = x3 - 3x in the interval [0, √ 𝟑] is a. 1 b. -1 c. 𝟑 𝟐 d. 𝟏 𝟑 60. For the functionf(x) = x + 𝟏 𝒙 , x  [1, 3], the value ofc for mean value theoremis a. 1 b. 2 c. √ 𝟑 d. none of these 61. The two curves x3 -3xy2 + 2 = 0 and 3x2 y – y3 = 2 a. Touch each other b. Cut at right angle c. Cut an angle 𝝅 𝟑 d. Cut an angle 𝝅 𝟒 62. The tangent to the curve givenby x = et . cost, y =et . sin t at t = 𝝅 𝟒 makes withx– axis an angle : a. 0 b. 𝝅 𝟒 c. 𝝅 𝟑 d. 𝝅 𝟐 63. The equationof the normal to the curve y = sin x at (0, 0) is: a. x = 0 b. y = 0 c. x + y = 0 d. x – y = 0
  • 6.
    64. the pointonthe curve y2 = x, where the tangentmakes an angle of 𝝅 𝟒 with x – axis is a. ( 𝟏 𝟐 , 𝟏 𝟒 ) b. ( 𝟏 𝟒 , 𝟏 𝟐 ) c. (4, 2) d. (1, 1) 65. Minimumvalue of f if f(x) = sin x [ −𝝅 𝟐 , 𝝅 𝟐 ] is………………….. 66. The sidesof an equilateral triangle are increasingat the ratio of 2 cm/sec. the rate at which the area increases,whenside is 10 cm is: a. 10 cm2 /s b. √ 𝟑cm2 /s c. 10 √ 𝟑cm2 /s d. 𝟏𝟎 𝟑 cm2 /s 67. The curve y = 𝒙 𝟏 𝟓 has at (0, 0) a. A vertical tangent (parallel to y 0 axis) b. A horizontal tangent(parallel to x – axis) c. An oblique tangent d. No tangent 68. The equationof normal to the curve 3x2 - y2 = 8 which isparallel to the line x + 3y = 8 is a. 3x – y = 8 b. 3x + y + 8 = 0 c. x + 3y ±8 = 0 d. x + 3y = 0 69. If y = x4 – 10 and ifx changes from 2 to 1.99, what is the change in y a. .32 b. .032 c. 5.68 d. 5.968 70. The pointsat which the tangents to the curve y = x3 - 12x + 18 are parallel to x – axisare: a. (2, -2), (-2,-34) b. (2, 34), (-2,0) c. (0, 34), (-2, 0) d. (2,2), (-2, 34) 71. The two curves x3 - 3xy2 + 2 = 0 and 3x2 y – y3 - 2= 0 intersectat an angle of a. 𝝅 𝟒 b. 𝝅 𝟑 c. 𝝅 𝟐 d. 𝝅 𝟔 72. y = x (x – 3)2 decreasesforthe valuesof x givenby : a. 1 < x< 3 b. x < 0 c. x > 0 d. 0< x < 𝟑 𝟐 73. Whichof the followingfunctionsisdecreasingon (𝟎, 𝝅 𝟐 ) a. sin 2x b. tan x c. cos x d. cos 3x 74. if x isreal, the minimumvalue of x2 - 8x + 17 is a. -1 b. 0 c. 1 d. 2 75. The smallestvalue of the polynomial x3 – 18x2 + 96x in [0,9] is a. 126 b. 0 c. 135 d. 160 76. The functionf(x) = 2x3 – 3x2 - 12x + 4, has a. Two points of local maximum b. Two points of local minimum c. One maximumand one minimum d. No maximumor minimum 77. Maximumslope of the curve y = -x3 + 3x2 + 9x – 27 is:
  • 7.
    a. 0 b.12 c. 16 d. 32 78. ∫ 𝒆 𝒙 (𝒄𝒐𝒔 𝒙 − 𝒔𝒊𝒏 𝒙 )dx is equal to a. 𝒆 𝒙 cos x + C b. 𝒆 𝒙 sin x + C c. -𝒆 𝒙 cos x + C d. -𝒆 𝒙 sin x + C 79. ∫ 𝒅𝒙 𝒔𝒊𝒏 𝟐 𝒙 𝒄𝒐𝒔 𝟐 𝒙 is equal to a. Tan x + cot x + C b. (tan x + cit x)2 + C c. tan x – cot x+C d. (tan x – cot x)2 + C 80. If ∫ 𝟑𝒆 𝒙− 𝟓𝒆−𝒙 𝟒𝒆 𝒙+ 𝟓𝒆−𝒙 dx = ax + b log | 𝟒𝒆 𝒙 + 𝟓𝒆−𝒙|+ 𝑪 , then a. a = −𝟏 𝟖 , 𝒃 = 𝟕 𝟖 b. a = 𝟏 𝟖 , 𝒃 = 𝟕 𝟖 c. a = −𝟏 𝟖 , 𝒃 = −𝟕 𝟖 d. a = 𝟏 𝟖 , 𝒃 = −𝟕 𝟖 81. if f and g are continuous functionsin [0, 1] satisfy f(x) = f(a – x) and g = (x) + g (a – x), then∫ 𝒇( 𝒙). 𝒈(𝒙) 𝒂 𝟎 dx is equal to a. 𝒂 𝟐 b. 𝒂 𝟐 ∫ 𝒇(𝒙) 𝒂 𝟎 dx c. ∫ 𝒇(𝒙) 𝒂 𝟎 dx d. a ∫ 𝒇(𝒙) 𝒂 𝟎 dx 82. If ∫ 𝒆 𝒕 𝟏+𝒕 𝒅𝒕 = 𝒂, 𝒕𝒉𝒆𝒏 ∫ 𝒆 𝒕 ( 𝟏+𝒕) 𝟐 𝟏 𝟎 𝟏 𝟎 dt is equal to a. a – 1 + 𝒆 𝟐 b. a + 1 - 𝒆 𝟐 c. a – 1 - 𝒆 𝟐 d. a + 1 + 𝒆 𝟐 83. ∫ | 𝒙 𝒄𝒐𝒔 𝝅𝒙| 𝒅𝒙 𝟏 𝟎 is equal to a. 𝟖 𝝅 b. 𝟒 𝝅 c. 𝟐 𝝅 d. 𝟏 𝝅 84. ∫ 𝒄𝒐𝒔 𝟐𝒙−𝒄𝒐𝒔 𝟐 𝒄𝒐𝒔 𝒙−𝒄𝒐𝒔 dx is equal to a. 2(sinx+ x cos  ) + C b. 2(sinx - x cos  ) + C c. 2(sinx + 2xcos  ) + C d. 2(sinx - 2xcos  ) + C 85. ∫ 𝒅𝒙 𝒔𝒊𝒏 ( 𝒙−𝒂) 𝒔𝒊𝒏 (𝒙−𝒃) is equal to a. sin (b– a) log | 𝒔𝒊𝒏 (𝒙−𝒃) 𝒔𝒊𝒏 (𝒙−𝒂) |+ C b. cosec (b– a) log | 𝒔𝒊𝒏 (𝒙−𝒂) 𝒔𝒊𝒏 (𝒙−𝒃) |+ C c. cosec (b– a) log | 𝒔𝒊𝒏 (𝒙−𝒃) 𝒔𝒊𝒏 (𝒙−𝒂) |+ C d. sin (b– a) log | 𝒔𝒊𝒏 (𝒙−𝒂) 𝒔𝒊𝒏 (𝒙−𝒃) |+ C 86. ∫ 𝐭𝐚𝐧−𝟏 √ 𝒙 dx is equal to a. (x + 1) tan-1 √ 𝒙 - √ 𝒙 + C b. x tan-1 √ 𝒙 - √ 𝒙 + C c. √ 𝒙 - x tan-1 √ 𝒙 + C d. √ 𝒙 – ( + 1) x tan-1 √ 𝒙 + C 87. ∫ 𝒆 𝒙 ( 𝟏−𝒙 𝟏+ 𝒙 𝟐 ) 𝟐 is equal to a. 𝒆 𝒙 𝟏+ 𝒙 𝟐 + C b. −𝒆 𝒙 𝟏+ 𝒙 𝟐 + C c. 𝒆 𝒙 ( 𝟏+ 𝒙 𝟐) 𝟐 + C d. −𝒆 𝒙 ( 𝟏+ 𝒙 𝟐) 𝟐 + C 88. ∫ 𝒙 𝟗 ( 𝟒𝒙 𝟐+ 𝟏) 𝟔 dx is equal to
  • 8.
    a. 𝟏 𝟓𝒙 (𝟒 − 𝟏 𝒙 𝟐 ) −𝟓 +𝑪 b. 𝟏 𝟓 (𝟒 + 𝟏 𝒙 𝟐 ) −𝟓 + 𝑪 c. 𝟏 𝟏𝟎𝒙 ( 𝟏 + 𝟒)−𝟓 + 𝑪 d. 𝟏 𝟏𝟎 ( 𝟏 𝒙 𝟐 + 𝟒) −𝟓 + 𝑪 89. If ∫ 𝒅𝒙 ( 𝒙+𝟐)(𝒙 𝟐+ 𝟏) = 𝒂 𝒍𝒐𝒈| 𝟏+ 𝒙 𝟐| + 𝒃 𝐭𝐚𝐧−𝟏 𝒙 + 𝟏 𝟓 𝒍𝒐𝒈 | 𝒙 + 𝟐| + 𝑪, then a. a = −𝟏 𝟏𝟎 , 𝒃 = −𝟐 𝟓 b. a = 𝟏 𝟏𝟎 , 𝒃 = − 𝟐 𝟓 c. a = 𝟏𝟎 , 𝒃 = 𝟐 𝟓 d. a = 𝟏 𝟏𝟎 , 𝒃 = 𝟐 𝟓 90. ∫ 𝒙 𝟑 𝒙+𝟏 is equal to a. x + 𝒙 𝟐 𝟐 + 𝒙 𝟑 𝟑 − 𝒍𝒐𝒈| 𝟏− 𝒙| + 𝑪 b. x + 𝒙 𝟐 𝟐 − 𝒙 𝟑 𝟑 − 𝒍𝒐𝒈| 𝟏− 𝒙| + 𝑪 c. x - 𝒙 𝟐 𝟐 − 𝒙 𝟑 𝟑 − 𝒍𝒐𝒈 | 𝟏+ 𝒙| + 𝑪 d. x - 𝒙 𝟐 𝟐 + 𝒙 𝟑 𝟑 − 𝒍𝒐𝒈 | 𝟏+ 𝒙| + 𝑪 91. If ∫ 𝒙 𝟑 𝒅𝒙 √ 𝟏+ 𝒙 𝟐 = 𝒂( 𝟏 + 𝒙 𝟐) 𝟑 𝟐 + 𝒃√ 𝟏 + 𝒙 𝟐 + C, then a. a = 𝟏 𝟑 , b = 1 b. a = −𝟏 𝟑 , b = 1 c. a = −𝟏 𝟑 , b = -1 d. a = 𝟏 𝟑 , b = -1 92. ∫ 𝒅𝒙 𝟏+𝒄𝒐𝒔 𝟐𝒙 𝝅 𝟒 −𝝅 𝟒 is equal to a. 1 b. 2 c. 3 d. 4 93. ∫ √ 𝟏 − 𝒔𝒊𝒏 𝟐𝒙 𝝅 𝟐 𝟎 dx is equal to a. 2√ 𝟐 b. 2 (√ 𝟐+ 1) c. 2 d. 2(√ 𝟐- 1) 94. The area enclosedby the circle x2 + y2 = 2 is equal to a. 4 sq unit b. 2 √ 𝟐 sq unit c. 42 sq unit d. 2 sq unit 95. The area enclosedby the ellipse 𝒙 𝟐 𝒂 𝟐 + 𝒚 𝟐 𝒃 𝟐 = 1 is equal to a. 2 ab b.  ab c.  a2 b d. ab2 96. The area of the regionboundedby the curve y = x2 and the line y = 16 a. 𝟑𝟐 𝟑 b. 𝟐𝟓𝟔 𝟑 c. 𝟔𝟒 𝟑 d. 𝟏𝟐𝟖 𝟑 97. The area of the regionboundedby the y – axis,y = cos x and y = sinx,0 ≤ 𝒙 ≤ 𝝅 𝟐 is
  • 9.
    a. √ 𝟐squnit b. (√ 𝟐+ 1) sq unit c. (√ 𝟐- 1) sq unit d. (2√ 𝟐- 1) sq unit 98. The area of the regionboundedby the curve y = √𝟏𝟔 − 𝒙 𝟐 and x – axis is a. 8 sq unit b. 20  sq unit c. 16  squnit d. 256  squnit 99. Area of the regionboundedby the curve y = cos x betweenx= 0 and x =  is a. 2 sq unit b. 4 sq unit c. 3 sq unit d. 1 sq unit 100. The area of the regionboundedby parabola y2 = x and the straight line 2y = x is a. 𝟒 𝟑 sq unit b. 1 sq unit c. 𝟏 𝟑 sq unit d. 𝟐 𝟑 sq unit 101. The area of the regionboundedby the curve y = x + 1 and the linesx = 2 and x = 3 is a. 2 sq unit b.  sq unit c. 3 sq unit d. 4  squnit 102. The area of the regionboundedby the curve y = x + 1 and the linesx = 2 and x = 3 is a. 𝟕 𝟐 sq unit b. 𝟗 𝟐 sq unit c. 𝟏𝟏 𝟐 sq unit d. 𝟏𝟑 𝟐 sq unit 103. The degree ofthe differential equation(1+ 𝒅𝒚 𝒅𝒙 )3 = ( 𝒅 𝟐 𝒚 𝒅𝒙 𝟐 )2 is a. 1 b. 2 c. 3 d. 4 104. The degree ofthe differential equation 𝒅 𝟐 𝒚 𝒅𝒙 𝟐 + 𝟑 ( 𝒅𝒚 𝒅𝒙 ) 𝟐 = 𝒙 𝟐 𝒍𝒐𝒈 ( 𝒅 𝟐 𝒚 𝒅𝒙 𝟐 ) is a. 1 b. 2 c. 3 d. not defined 105. The order and degree ofthe differential equation [ 𝟏+ ( 𝒅𝒚 𝒅𝒙 ) 𝟐 ] 𝟐 = 𝒅 𝟐 𝒚 𝒅𝒙 𝟐 respectively,are a. 1, 2 b. 2, 2 c. 2, 1 d. 4, 2 106. The solutionof the differential equation2x. 𝒅𝒚 𝒅𝒙 – y = 3 representsa familyof a. Straight lines b. Circles c. Parabolas d. Ellipses 107. The integratingfactor ofthe differential equation 𝒅𝒚 𝒅𝒙 (x log x) + y = 2 logx is a. ex b. log xc. log (logx) d. x 108. A solutionof the differential equation( 𝒅𝒚 𝒅𝒙 )2 – x 𝒅𝒚 𝒅𝒙 + y = 0 is a. y = 2 b. y = 2x c. y = 2x – 4 d. y = 2x2 – 4 109. Whichof the followingisnot a homogeneousfunctionof x and y. a. x2 + 2xyb. 2x – y c. cos2 ( ) + d. sinx – cos y 110. Solutionof the differential equation 𝒅𝒚 𝒙 + 𝒅𝒚 𝒚 = 0 is a. = 𝟏 𝒙 + 𝟏 𝒚 = 𝒄 b. log x . log y = c c. xy = c d. x + y = c 111. The solutionof the differential equationx + 2y = x2 is
  • 10.
    a. y = 𝒙𝟐+ 𝒄 𝟒𝒙 𝟐 b. y = 𝒙 𝟐 𝟒 + c c. y = 𝒙 𝟐+ 𝒄 𝒙 𝟐 d. y = 𝒙 𝟐+ 𝒄 𝟒𝒙 𝟐 112. The degree ofthe differential equation[1 + ( 𝒅𝒚 𝒅𝒙 )2 ] = 𝒅 𝟐 𝒚 𝒅𝒙 𝟐 is a. 4 b. 𝟑 𝟐 c. not defined d. 2 113. The order and degree ofthe differential equation 𝒅 𝟐 𝒚 𝒅𝒙 𝟐 + ( 𝒅𝒚 𝒅𝒙 ) 𝟏 𝟒 + 𝒙 𝟏 𝟓= 0, respectivelyare a. 2 and not defined b. 2 and 2 c. 2 and 3 d. 3 and 3 114. Integrating factor of the differential equationcosx 𝒅𝒚 𝒅𝒙 + y sin x = 1 is: a. cos x b. tan xc. sec x d. sin x 115. Familyy = Ax + A3 of curvesis representedbythe differential equationofdegree: a. 1 b. 2 c. 3 d. 4 116. Solutionof 𝒅𝒚 𝒅𝒙 - y = 1, y(0) = 1 isgiven by a. xy = -ex b. xy = -e-x c. xy = -1 d. y = 2 ex – 1 117. The numberof solutions of 𝒅𝒚 𝒅𝒙 = 𝒚+𝟏 𝒙−𝟏 when y(1) = 2 is: a. None b. one c. two d. infinite 118. Integrating factor of the differential equation(1– x2 ) 𝒅𝒚 𝒅𝒙 - xy = 1 is a. –x b. 𝒙 𝟏+ 𝒙 𝟐 c.√𝟏 − 𝒙 𝟐 d. ½ log(1 – x2 ) 119. The general solutionof ex cos y dx – ex siny dy = 0 is: a. ex cos y = k b. ex sin y = k c. ex = k cos y d. ex = k siny 120. The solutionof the differential equation 𝒅𝒚 𝒅𝒙 = 𝟏+ 𝒚 𝟐 𝟏+ 𝒙 𝟐 is: a. y = tan-1 x b. y-x = k (1 =xy) c. x = tan-1 y d. tan (xy) = k 121. The integratingfactor ofthe differential equation 𝒅𝒚 𝒅𝒙 + 𝒚 = 𝟏+𝒚 𝒙 is: a. 𝒙 𝒆 𝒙 b. 𝒆 𝒙 𝒙 c. xex d. ex 122. The solutionof the differential equationcosx sin y dx + sinx cos y dy = 0 is: a. 𝒔𝒊𝒏𝒙 𝒔𝒊𝒏 𝒚 = c b. sin x siny = c c. sin x + siny = c d. cos x cos y = c 123. The solutionof x 𝒅𝒚 𝒅𝒙 + y = ex is: a. y = 𝒆 𝒙 𝒙 + 𝒌 𝒙 b. y = xex + cx c. y = xex + k d. x = 𝒆 𝒚 𝒚 + 𝒌 𝒚 124. The differential equationofthe familyof curves x2 + y2 – 2ay = 0, where a isarbitrary constant, is: a. (x2 – y2 ) 𝒅𝒚 𝒅𝒙 = 2xy b. 2(x2 + y2 ) 𝒅𝒚 𝒅𝒙 = xy c. 2(x2 - y2 ) 𝒅𝒚 𝒅𝒙 = xy d. (x2 +y2 ) 𝒅𝒚 𝒅𝒙 = 2xy 125. The general solutionof 𝒅𝒚 𝒅𝒙 = 2x 𝒆 𝒙 𝟐− 𝒚 is: a. 𝒆 𝒙 𝟐− 𝒚 = 𝒄 b. 𝒆−𝒚 + 𝒆 𝒙 𝟐 = 𝒄 c. 𝒆 𝒚 + 𝒆 𝒙 𝟐 = 𝒄 d. 𝒆 𝒙 𝟐+ 𝒚 = 𝒄 126. The general solutionof the differential equation 𝒅𝒚 𝒅𝒙 + xy is:
  • 11.
    a. y =c𝒆 −𝒙 𝟐 𝟐 b. y = c𝒆 𝒙 𝟐 𝟐 c. y = (x + c) 𝒆 𝒙 𝟐 𝟐 d. y = (c-x) 𝒆 𝒙 𝟐 𝟐 127. the solutionof the equation(2y – 1) dx– (2x + 3) dy = 0 is: a. 𝟐𝒙−𝟏 𝟐𝒚+𝟑 = k b. 𝟐𝒚+ 𝟏 𝟐𝒙− 𝟑 = k c. 𝟐𝒙+𝟑 𝟐𝒚−𝟏 = k d. 𝟐𝒙−𝟏 𝟐𝒚−𝟏 = k 128. The solutionof 𝒅𝒚 𝒅𝒙 + y = e-x ,y (0) = 0 is: a. y = e-x (x– 1) b. y = xex c. y = xe-x + 1 d. y = xe-x 129. The order and degree ofthe differential equation ( 𝒅 𝟑 𝒚 𝒅𝒙 𝟑 ) 𝟐 − 𝟑 𝒅 𝟐 𝒚 𝒅𝒙 𝟐 + 𝟐 ( 𝒅𝒚 𝒅𝒙 ) 𝟒 = 𝒚 𝟒 are: a. 1, 4 b. 3, 4 c. 2, 4 d. 3, 2 130. The order and degree ofthe differential equation [ 𝟏+ ( 𝒅𝒚 𝒅𝒙 ) 𝟐 ] = 𝒅 𝟐 𝒚 𝒅𝒙 𝟐 are: a. 2, 𝟑 𝟐 b. 2, 3 . 2, 1 d. 3, 4 131. Whichof the followingisthe general solutionof 𝒅 𝟐 𝒚 𝒅𝒙 𝟐 − 𝒅𝒚 𝒅𝒙 𝟐 + y = 0? a. y = (Ax+ B)ex b. y = (Ax + B) e-x c. y = Aex + Be-x d. y = Acos x + B sinx 132. Solutionof the differential equation 𝒅𝒚 𝒅𝒙 + 𝒚 𝒙 = sinx is: a. x (y + cos x) = sin x + c b. x (y – cos x 0 = sin x + c c. xy cos x = sin x + c d. x (y + cos x) = cos x + c 133. The solutionof the differential equation 𝒅𝒚 𝒅𝒙 = ex – y + x2 e-y is: a. y = ex – y – x2 e-y + c b. ey – ex = + c c. ex + ey = + c d. ex – ey = 𝒙 𝟑 𝟑 + c 134. The magnitude of the vector6𝒊̂ + 𝟐𝒋̂ + 𝟑𝒌̂ is a. 5 b. 7 c. 12 d. 1 135. The positionvector of the point which dividedthe joinof points withpositionvectors 𝒂⃗⃗ + 𝒃⃗⃗ and 2𝒂⃗⃗ − 𝒃⃗⃗ in the ratio 1 : 2 is a. 𝟑𝒂⃗⃗ + 𝟐𝒃⃗⃗ 𝟑 b. 𝒂⃗⃗ c. 𝟓𝒂⃗⃗ − 𝒃⃗⃗ 𝟑 d. 𝟒𝒂⃗⃗ + 𝒃⃗⃗ 𝟑 136. The vector with initial pointP(2, -3, 5) and terminal point Q(3, -4, 7) is a. 𝒊̂ − 𝒋̂ + 𝟐𝒌̂ b. 𝟓𝒊̂ − 𝟕𝒋̂ + 𝟏𝟐𝒌̂ c. −𝒊̂ + 𝒋̂ − 𝟐𝒌̂ d. None of these 137. The angle betweenthe vectors 𝒊̂ − 𝒋̂ 𝒂𝒏𝒅 𝒋̂− 𝒌̂ is a. 𝝅 𝟑 b. 𝟐𝝅 𝟑 c. −𝝅 𝟑 d. 𝟓𝝅 𝟔 138. The value of  for whichthe two vectors 2𝒊̂ − 𝒋̂ + 𝟐𝒌̂ 𝒂𝒏𝒅 𝟑𝒊̂ +  𝒋̂+ 𝒌̂ are perpendicularis
  • 12.
    a. 2 b.4 c. 6 d. 8 139. The area of the parallelogramwhose adjacent sidesare 𝒊̂ + 𝒌̂ and 𝟐𝒊̂ + 𝒋̂ + 𝒌̂ is a. 3 b. 4 c. √ 𝟐 d. √ 𝟑 140. If | 𝒂⃗⃗ | = 𝟖, | 𝒃⃗⃗ | = 𝟑 𝒂𝒏𝒅 | 𝒂⃗⃗ × 𝒃⃗⃗ | = 𝟏𝟐 , thenvalue of is a. 6√ 𝟑 b. 8√ 𝟑 c. 12√ 𝟑 d. none of these 141. The 2 vector 𝒋̂ + 𝒌̂ and 3𝒊̂ − 𝒋̂ + 𝟒𝒌̂ representsthe two sidesAB and AC,respectivelyofa ABC . the length of the medianthrough A is a. √ 𝟑𝟒 𝟐 b. √ 𝟒𝟖 𝟐 c. √ 𝟏𝟖 d. None of these 142. The projectionof vector 𝒂⃗⃗ = 𝟐𝒊̂ − 𝒋̂ + 𝒌̂ along 𝒃⃗⃗ = 𝒊̂ + 𝟐𝒋̂ + 𝟐𝒌̂ is a. 2 b. √ 𝟔 c. 𝟐 𝟑 d. 𝟏 𝟑 143. If 𝒂⃗⃗ and 𝒃⃗⃗ are unit vectors,then what is the angle between 𝒂⃗⃗ and 𝒃⃗⃗ for √ 𝟑 𝒂⃗⃗ and 𝒃⃗⃗ to be a unit vector? a. 30o b. 45o c. 60o d. 90o 144. The unit vector perpendiculartothe vectors 𝒊̂ − 𝒋̂ 𝒂𝒏𝒅 𝒊̂ + 𝒋̂ forming a right handedsystemis a. 𝒌̂ b. -𝒌̂ c. 𝒊̂− 𝒋̂ √ 𝟐 d. 𝒊̂+ 𝒋̂ √ 𝟐 145. If | 𝒂⃗⃗ | = 𝟑 and -1  k  2 , then| 𝒌𝒂⃗⃗ | liesinthe interval a. [0,6] b. [-3,6] c. [3,6] d. [1,2] 146. The vector in the directionof the vector 𝒊̂ − 𝟐𝒋̂ + 𝟐𝒌̂ that has magnitude 9 is a. 𝒊̂ − 𝟐𝒋̂ + 𝟐𝒌̂ b. 𝒊̂− 𝟐𝒋̂+ 𝟐𝒌̂ 𝟑 c. (𝒊̂ − 𝟐𝒋̂ + 𝟐𝒌̂ ) d. 9( 𝒊̂ − 𝟐𝒋̂ + 𝟐𝒌̂) 147. The angle betweentwo vectors 𝒂⃗⃗ and 𝒃⃗⃗ with magnitude √ 𝟑 and 4, respectively,and 𝒂⃗⃗ . 𝒃⃗⃗ = 2√ 𝟑 is a. 𝝅 𝟔 b. 𝝅 𝟑 c. 𝝅 𝟐 d. 𝟓𝝅 𝟐 148. Findthe value of  such that the vectors 𝒂⃗⃗ = 𝟐𝒊̂ +  𝒋̂+ 𝒌̂ and 𝒃⃗⃗ = 𝒊̂ + 𝟐𝒋̂ + 𝟑𝒌̂ are orthogonal a. 0 b. 1 c. 𝟑 𝟐 d. −𝟓 𝟐 149. the value of  for which the vectors 𝟑𝒊̂ − 𝟔𝒋̂ + 𝒌̂ 𝒂𝒏𝒅 𝟐𝒊̂ − 𝟒𝒋̂ +  𝒌̂ are parallel is a. 𝟐 𝟓 b. 𝟐 𝟑 c. 𝟑 𝟐 d. 𝟓 𝟐 150. For any vector 𝒂⃗⃗ , the value of ( 𝒂⃗⃗ × 𝒊̂) 𝟐 + ( 𝒂⃗⃗ × 𝒋̂) 𝟐 + ( 𝒂⃗⃗ × 𝒌̂) 𝟐 is equal to a. 𝒂⃗⃗ 𝟐 b. 3 𝒂⃗⃗ 𝟐 c. 4 𝒂⃗⃗ 𝟐 d. 2𝒂⃗⃗ 𝟐 151. If | 𝒂⃗⃗ | = 𝟏𝟎,| 𝒃⃗⃗ | = 𝟐 𝒂𝒏𝒅 𝒂⃗⃗ . 𝒃⃗⃗ = 𝟏𝟐, 𝐭𝐡𝐞𝐧 𝐯𝐚𝐥𝐮𝐞 𝐨𝐟 | 𝒂⃗⃗ × 𝒃⃗⃗ |, is a. 5 b. 10 c. 14 c. 16 152. If 𝒂⃗⃗ , 𝒃⃗⃗ , 𝒄⃗ are unit vectorssuch that 𝒂⃗⃗ + 𝒃⃗⃗ + 𝒄⃗ = 𝟎⃗⃗ , thenvalue of 𝒂⃗⃗ . 𝒃⃗⃗ + 𝒃⃗⃗ . 𝒄⃗ + 𝒂⃗⃗ . 𝒄⃗ a. 1 b. 3 c. - 𝟑 𝟐 d. none of these 153. Projectionvector of 𝒂⃗⃗ on 𝒃⃗⃗ is
  • 13.
    a. ( 𝒂⃗⃗ .𝒃⃗⃗ |𝒃⃗⃗ | 𝟐) 𝒃⃗⃗ b. 𝒂⃗⃗ .𝒃⃗⃗ | 𝒃⃗⃗ | c. 𝒂⃗⃗ .𝒃⃗⃗ | 𝒂⃗⃗ | d. ( 𝒂⃗⃗ .𝒃⃗⃗ | 𝒂⃗⃗ | 𝟐 ) 𝒃⃗⃗ 154. If 𝒂⃗⃗ , 𝒃⃗⃗ , 𝒄⃗ are three vectors such that 𝒂⃗⃗ + 𝒃⃗⃗ + 𝒄⃗ = 𝟎⃗⃗ and | 𝒂⃗⃗ | = 𝟐, | 𝒃⃗⃗ | = 𝟑 , | 𝒄⃗ | = 𝟓, then value of 𝒂⃗⃗ . 𝒃⃗⃗ + 𝒃⃗⃗ . 𝒄⃗ + 𝒂⃗⃗ . 𝒄⃗ a. 0 b. 1 c. -19 d. 38 155. P is a point on the line segmentjoiningthe points(3, 2, -1) and (6, 2, -2). If x co-ordinate of P is5, thenits y co- ordinate is a. 2 b. 1 c. -1 d. -2 156. If ,, are the anglesthat a line makes withthe positive directionofx, y, z axis, respectively,thenthe directioncosinesof the line are: a. sin,sin,sin b. cos,cos,cos c. tan,tan,tan d. cos2 ,cos2 ,cos2  157. The equationsof x – axis in space are a. x = 0, y = 0 b. x = 0, z = 0 c. x = 0 d. y = 0, z = 0 158. If the directionscosinesofa line are k, k, k then a. k > 0 b. 0 < k< 1 c. k = 1 d. k = 𝟏 √ 𝟑 𝒐𝒓 − 𝟏 √ 𝟑 159. The distance of the plane 𝒓⃗ . ( 𝟐 𝟕 𝒊̂ + 𝟑 𝟕 𝒋̂ − 𝟔 𝟕 𝒌̂) = 1 from the originis a. 1 b. 7 c. 𝟏 𝟕 d. none of these 160. The area of the quadrilateral ABCD, where A(0,4,1), b(2,3,-1),c(4,5,0) and D(2, 6, 2) is equal to a. 9 sq unit b. 18 sq unit c. 27 sq unit d. 81 sq unit 161. The unit vector normal to the plane x + 2y + 3z – 6 = 0 𝟏 √ 𝟏𝟒 𝒊̂ + 𝟐 √ 𝟏𝟒 𝒋̂ + 𝟑 √ 𝟏𝟒 𝒌̂ . 162. The corner pointsof the feasible regiondeterminedbythe systemof linearconstraints are (0, 10), (5,5), (15, 150), (0, 20). Let Z = px + qy, where p, q > 0. Conditionon p and q so that the maximumof Z occurs at both the points(15, 15) and (0, 20) is a. p = q b. p = 2q c. q = 2p d. q = 3p 163. Feasible region(shaded) fora LPP isshown in the Fig.Minimumof Z = 4x + 3y occurs at the point a. (0, 80 b. (2, 5 ) c. (4, 3) d. (9, 0) 164. The corner points of the feasible regiondeterminedbythe system oflinear constraints are (0,0), (0, 40), (20, 40), (60, 20), (60, 0). The objective functionis Z = 4x + 3y. Compare the quantity inColumn A and ColumnB ColumnA ColumnB Maximumof Z 325 a. The quantity in columnA is greater b. The quantity in columnB is greater c. The two quantitiesare equal
  • 14.
    d. The relationshipcannot be determinedonthe basis of the informationsupplied. 165. Corner pointsof the feasible regionforan LPP are (0, 2), (3, 0), (6,0), (6, 8) and (0, 5). Let F = 4x + 6y be the objective function.The minimum value of F occurs at a. (0, 2) only b. (3, 0) only c. The mid pointof the line segmentjoiningthe points(0, 2) and (3, 0) only d. Any point on the line segmentjoiningthe points(0, 20 and (3, 0). 166. Let A and B be two events.If P (A) = 0.2, P(B) = 0.4, P(A B) = 0.6, then P(AIB) is equal to a. 0.8 b. 0.5 c. 0.3 d. 0 167. Let A and B be two eventssuch that P(A) = 0.6, P(B) = 0.2 and P(AIB) = 0.5. thenP(AIB) equals a. 𝟏 𝟏𝟎 b. 𝟑 𝟏𝟎 c. 𝟑 𝟖 d. 𝟔 𝟕 168. If A and B are independenteventssuchthat 0 < P(A) < 1 and 0 <P(B) < 1 then whichof the followingisnot correct? a. A and B are mutually exclusive b. A and B are independent c. A and B are independent d. A and B are independent 169. If P(A) = 𝟒 𝟓 , and P(A  B) = 𝟕 𝟏𝟎 , thenP(B I A) is equal to a. 𝟏 𝟏𝟎 b. 𝟏 𝟏𝟖 c. 𝟕 𝟖 d. 𝟏𝟕 𝟐𝟎 170. If P(A  B) = 𝟕 𝟏𝟎 and P(B) = 𝟏𝟕 𝟐𝟎 , then P(B I A) is equal to a. 𝟏𝟒 𝟏𝟕 b. 𝟏 𝟖 c. 𝟕 𝟖 d. 𝟏𝟕 𝟐𝟎 171. If P(A) = 𝟑 𝟏𝟎 , P(B) = 𝟐 𝟓 and P(A  B) = 𝟑 𝟓 , thenP(B I A) + P(AI B) is equal to a. 𝟏 𝟒 b. 𝟏 𝟑 c. 𝟓 𝟏𝟐 d. 𝟕 𝟐 172. If P(A) = 𝟐 𝟓 , P(B) = 𝟑 𝟏𝟎 and P(A  B) = 𝟏 𝟓 , thenP(B I A) . P(A IB) is equal to a. 𝟓 𝟔 b. 𝟓 𝟕 c. 𝟐𝟓 𝟒𝟐 d. 1 173. If A and B are two eventssuch that P(A) = ½, P(B) = 𝟏 𝟑 , P(A / B) = 𝟏 𝟒 , thenP(A  B) equals a. 𝟏 𝟏𝟐 b. 𝟑 𝟒 c. 𝟏 𝟒 d. 𝟑 𝟏𝟔 174. If P(A) = 0.4, P(B) = 0.8 and P(B I A) = 0.6, thenP(A  B) is equal to a. 0.24 b. 0.3 c. 0.48 d. 0.96 175. You are giventhat A and B are two eventssuch that P(B) = 𝟑 𝟓 , P(A I B) = ½ and P(A  B) = 𝟒 𝟓 , thenP(A) equals a. 𝟑 𝟏𝟎 b. 𝟏 𝟐 c. 𝟏 𝟓 d. 𝟑 𝟓
  • 15.
    176. If P(B)= 𝟑 𝟓 , P(AIB) = ½ and P(A  B) = 𝟒 𝟓 , then P(A  B) + P(A  B) = a. 𝟒 𝟓 b. 𝟏 𝟐 c. 𝟏 𝟓 d. 1 177. Let A and B be two eventssuch that P(A) = 𝟑 𝟖 , P(B) = 𝟓 𝟖 and thenP(A I B) . P(A IB) isequal to a. 𝟐 𝟓 b. 𝟑 𝟖 c. 𝟑 𝟐𝟎 d. 𝟔 𝟐𝟓 178. If A and B are such eventsthat P(A) > 0 and P(B)  1, thenP(A I B) equals a. 1 – P(A I B) b. 1 – P(A IB) c. 𝟏−𝑷(𝑨∪𝑩) 𝑷(𝑩′) d. P(A) IP(B ) 179. A bag contains5 red and 3 blue balls. If3 balls are drawn at random without replacementthe probabilityof gettingexactly one red ball is a. 𝟒𝟓 𝟏𝟗𝟔 b. 𝟏𝟑𝟓 𝟑𝟗𝟐 c. 𝟏𝟓 𝟓𝟔 d. 𝟏𝟓 𝟐𝟗 180. Assume that in a family,each child isequallylikelyto be or a girl.A family withthree childrenischosen at random. The probabilitythat the eldestchildisa girl given that the familyhas at least one girl is a. 𝟒 𝟕 b. 𝟏 𝟐 c. 𝟏 𝟑 d. 𝟐 𝟑 181. A die is thrown and a card is selectedatrandom from a deck of 52 playing cards. The probabilityof gettingan evennumberon the die and a spade card is a. 𝟑 𝟒 b. 𝟏 𝟐 c. 𝟏 𝟒 d. 𝟏 𝟖 182. A box contains 3 orange balls,3 greenballsand 2 blue balls. Three balls are drawn at random from the box without replacement.The probabilityof drawing 2 greenballsand one blue ball is a. 𝟑 𝟖 b. 𝟐 𝟐𝟏 c. 𝟏 𝟐𝟖 d. 𝟏𝟔𝟕 𝟏𝟔𝟖 183. Eight coins are tossedtogether.The probabilityof gettingexactly 3 headsis a. 𝟏 𝟐𝟓𝟔 b. 𝟕 𝟑𝟐 c. 𝟓 𝟑𝟐 d. 𝟑 𝟑𝟐 184. In a college,30% studentsfail in physics,255 fail in mathematics and 10% fail in both. One studentis chosen at random. The probabilitythat she failsin physicsif she has failedin mathematics is a. 𝟏 𝟏𝟎 b. 𝟐 𝟓 c. 𝟏 𝟑 d. 𝟗 𝟐𝟎 185. The probabilitydistributionof a discrete random variable X is givenbelow: X 2 3 4 5 P(X) 𝟓 𝒌 𝟕 𝒌 𝟗 𝒌 𝟏𝟏 𝒌 The value of k is a. 8 b. 16 c. 32 d. 48