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InterviewSolution
This section includes InterviewSolutions, each offering curated multiple-choice questions to sharpen your knowledge and support exam preparation. Choose a topic below to get started.
| 51. |
If `int (x+1)/(sqrt(2x-1))dx = f(x)sqrt(2x-1)+C`, where C is a constant of integration, then f(x) is equal toA. `1/3(x+4)`B. `1/3(x+1)`C. `2/3(x+2)`D. `2/3(x-4)` |
| Answer» Correct Answer - A | |
| 52. |
The integral `intcos(log_(e)x)dx` is equal to: (where C is a constant of integration)A. `x/2[sin(log_ex-cos(log_ex)]+C`B. `x/2[cos(log_ex+sin(log_ex)]+C`C. `x[cos(log_ex+sin(log_ex)]+C`D. `x[cos(log_ex-sin(log_ex)]+C` |
| Answer» Correct Answer - B | |
| 53. |
If `int(sqrt(1-x^2))/x^4dx=A(x) (sqrt(1-x^2))^m+C`,for a suitable chosen integer m and a function A(x), where C is a constant of integration, then `(A(x))^m` equalsA. `(-1)/(3x^3)`B. `(-1)/(27x^9)`C. `1/(9x^4)`D. `1/(27x^6)` |
| Answer» Correct Answer - B | |
| 54. |
Let `f(x)={(-,1, -2lexlt0),(x^2,-1,0lexlt2):}` if `g(x)=|f(x)|+f(|x|)` then `g(x)` in `(-2,2)` (A) not continuous (B) not differential at one point (C) differential at all points (D) not differential at two pointsA. Differentiable at all pointsB. not differentiable at two pointsC. Not continuousD. not differentiable at one point |
| Answer» Correct Answer - D | |
| 55. |
The solution of the differential equation, `dy/dx=(x-y)^(2)`, when `y(1)=1,` isA. `log_eabs((2-y)/(2-x))=2(y-1)`B. `log_eabs((2-x)/(2-y))=x-y`C. `-log_eabs((1+x-y)/(1-x+y))=x+y-2`D. `-log_eabs((1-x+y)/(1+x-y))=2(x-1)` |
| Answer» Correct Answer - D | |
| 56. |
If y(x) is the the solution of the differntial equation `dy/dx+((2x+1)/x)y=e^(-2x),xgt0," where "y(1)=1/2e^(-2)`, thenA. y(x) is decreasing in (0,1)B. y(x) is decreasing in `(1/2,1)`C. `y(log_e2)=(log_e2)/4`D. `y(log_e2)=log_2 4` |
| Answer» Correct Answer - B | |
| 57. |
Let `n ge 2` be a natural number and `0 lt theta lt (pi)/(2)`, Then, `int ((sin^(n)theta - sin theta)^(1/n) cos theta)/(sin^(n+1) theta)d theta` is equal to (where C is a constant of integration)A. `n/(n^2-1)(1-1/(sin^(n+1)theta))^((n+1)/n)+C`B. `n/(n^2+1)(1-1/(sin^(n-1)theta))^((n+1)/n)+C`C. `n/(n^2-1)(1-1/(sin^(n-1)theta))^((n+1)/n)+C`D. `n/(n^2-1)(1+1/(sin^(n-1)theta))^((n+1)/n)+C` |
| Answer» Correct Answer - C | |
| 58. |
Let `alpha` and `beta` be the roots of the quadratic equation `x^(2)` sin `theta - x (sin theta cos theta + 1) + cos theta = 0 (0 lt theta lt 45^(@))`, and `alpha lt beta`. Then `Sigma_(n=0)^(oo) (alpha^(n) + ((-1)^(n))/(beta^(n)))` is equal toA. `(1)/(1 - cos theta) + (1)/(1 + sin theta)`B. `(1)/(1 + cos theta) + (1)/(1 - sin theta)`C. `(1)/(1 - cos theta) - (1)/(1 + sin theta)`D. `(1)/(1 + cos theta) - (1)/(1 - sin theta)` |
| Answer» Correct Answer - A | |
| 59. |
The values of `lambda` such that sum of the squares of the roots of the quadratic equation `x^(2) + (3 - lambda) x + 2 = lambda` has the least value isA. 2B. `(4)/(9)`C. `(15)/(8)`D. 1 |
| Answer» Correct Answer - A | |
| 60. |
If normals are drawn to the ellipse `x^2 + 2y^2 = 2` from the point `(2, 3).` then the co-normal points lie on the curveA. `(x^(2))/(2)+(y^(2))/(4)=1`B. `(x^(2))/(4)+(y^(2))/(2)=1`C. `(1)/(2x^(2))+(1)/(4y^(2))=1`D. `(1)/(4x^(2))+(1)/(2y^(2))=1` |
| Answer» Correct Answer - C | |
| 61. |
The area (in sq. units) of the region bounded by the parabola `y=x^2+2" and the lines " y=x+1, x=0 " and " x=3`, isA. `15/4`B. `15/2`C. `21/2`D. `17/4` |
| Answer» Correct Answer - B | |
| 62. |
Let `f(x)=x^3-3(a-2)x^2+3ax+7` and `f(x)` is increasing in `(0,1]` and decreasing is `[1,5)`, then roots of the equation `(f(x)-14)/((x-1)^2)=0` is (A) `1` (B) `3` (C) `7` (D) `-2`A. 6B. 5C. 7D. -7 |
| Answer» Correct Answer - C | |
| 63. |
If both the roots of the quadratic equation `x^(2) - mx + 4 = 0` are real and distinct and they lie in the interval [1, 5], then m lies in the intervalA. (4, 5]B. (3, 4)C. (5, 6)D. `(-5, -4)` |
| Answer» Correct Answer - A | |
| 64. |
Let a parabola be `y=12-x^2`. Find the maximum area of rectangle whose base lie on x-axis and two points lie on parabola. (A) `8` (B) `4` (C) `32` (D) `34`A. `20sqrt2`B. `18sqrt2`C. 32D. 36 |
| Answer» Correct Answer - C | |
| 65. |
If the vertices of the parabola be at `(-2,0)` and `(2,0)` and one of the foci be at `(-3,0)` then which one of the following points does not lie on the hyperbola? (a) `(-6, 2sqrt(10))` (b) `(2sqrt6,5)` (c) `(4, sqrt(15))` (d) `(6, 5sqrt2)`A. `( 4, sqrt(15))`B. `(-6,2sqrt(10))`C. `(6,5sqrt(2))`D. `(2, sqrt(6),5)` |
| Answer» Correct Answer - B | |
| 66. |
If a hyperbola has length of its conjugate axis equal to 5 and the distance between its foci is 13, then the eccentricity of the hyperbola isA. 2B. `(13)/(6)`C. `(13)/(8)`D. `(13)/(12)` |
| Answer» Correct Answer - D | |
| 67. |
If eccentricity of the hyperbola `(x^(2))/(cos^(2) theta)-(y^(2))/(sin^(2) theta)=1` is move than `2` when `theta in (0,(pi)/2)`. Find the possible values of length of latus rectum (a) `(3,oo)` (b) `1,3//2)` (c) `(2,3)` (d) `(-3,-2)`A. (2,3)B. `(3, oo)`C. `(3//2, 2)`D. `(1, 3//2)` |
| Answer» Correct Answer - B | |
| 68. |
The equation of tangent to hyperbola `4x^2-5y^2=20` which is parallel to `x-y=2` is (a) `x-y+3=0` (b) `x-y+1=0` (c) `x-y=0` (d) `x-y-3=0`A. x-y+9=0B. x-y+7=0C. x-y+1=0D. x-y-3=0 |
| Answer» Correct Answer - C | |
| 69. |
A square is incribed in a circle `x^2+y^2-6x+8y-103=0` such that its sides are parallel to co-ordinate axis then the distance of the nearest vertex to origin, is equal to (A) `13` (B) `sqrt127` (C) `sqrt41` (D) `1`A. 13B. `sqrt(137)`C. `6`D. `sqrt(41)` |
| Answer» Correct Answer - D | |
| 70. |
If the fractional part of the number `(2^(403))/(15)` is `(k)/(15)` then k is equal toA. 14B. 6C. 4D. 8 |
| Answer» Correct Answer - D | |
| 71. |
There are m men and two women participating in a chess tournament. Each participant plays two games with every other participant. If the number of games played by the men between themselves exceeds the number of games played between the men and the women by 84, then teh value of m isA. 9B. 11C. 12D. 7 |
| Answer» Correct Answer - C | |
| 72. |
Given three statements P: 5 is a prime number, Q:7 is a factor of 192, R:The LCM of 5 & 7 is 35 Then which of the following statements are true (a) `Pv(~Q^^R)` (b) `~P^^(~Q^^R)` (c) `(PvQ)^^~R` (d) `~P^^(~Q^^R)`A. `(p wedgeq)vee (~r)`B. `(~p)wedge(~q wedge r)`C. `(~p) vee (q wedge r)`D. `p vee(~q wedge r)` |
| Answer» Correct Answer - D | |
| 73. |
for `x > 1` if `(2x)^(2y)=4e^(2x-2y)` then `(1+log_e 2x)^2 (dy)/(dx)`A. `log_e 2x`B. `(xlog_e2x+log_e2)/x`C. `xlog_e2x`D. `(xlog_e2x-log_e2)/x` |
| Answer» Correct Answer - D | |
| 74. |
`f(x)={(5, xle1),(a+bx,1A. continuous if a =5 and b = 5B. continuous if a =-5 and b = 10C. continuous if a = 0 and b = 5D. not continuous for any values of a and b |
| Answer» Correct Answer - D | |
| 75. |
Let `f(x){{:(max.{absx,x^2}","" "absxle2),(" "8-2absx","" "2ltabsxle4):}`.Let S be the set of points in the intercal (-4,4) at which f is not differentible. Then SA. is an empty setB. equals {-2,-1,1,2}C. equals {-2,-1,0,1,2}D. equals {-2,2} |
| Answer» Correct Answer - C | |
| 76. |
Let `f:RtoR` be defined by `f(x)=x/(1+x^2),x inR`. Then the range of f isA. (-1,1)-{0}B. `[- 1/2,1/2]`C. `R-[- 1/2,1/2]`D. R-[-1,1] |
| Answer» Correct Answer - B | |
| 77. |
Two circles with equal radii are intersecting at the points (0, 1) and (0,-1). The tangent at the point (0,1) to one of the circles passes through the centre of the other circle. Then the distance between the centres of these circles is.A. 1B. `sqrt (2)`C. `2sqrt(2)`D. `2` |
| Answer» Correct Answer - D | |
| 78. |
The tangent to the curve `y=xe^(x^2)` passing through the point (1,e) also passes through the pointA. `(4/3,2e)`B. (2, 3e)C. `(5/3,2e)`D. (3, 6e) |
| Answer» Correct Answer - A | |
| 79. |
Let `S_n=1+q+q^2 +?+q^n` and `T_n =1+((q+1)/2)+((q+1)/2)^2+?+((q+1)/2)^n` If `alpha T_100=^101C_1 +^101C_2 xS_1 +^101C_101 xS_100,` then the value of `alpha` is equal to (A) `2^99` (B) `2^101` (C) `2^100` (D) `-2^100`A. `2^(100)`B. 200C. `2^(99)`D. 202 |
| Answer» Correct Answer - A | |
| 80. |
Let `vecx,vecy and vecz` be three vector each of magnitude `sqrt(2)` and the angle between each pair of them is `(pi)/(3).` if `veca` is a non - zero vector perpendicular to `vecx and vecy xxvecz and vecb` is a non-zero vector perpendicular to `vecy and vecz xx vecx,` thenA. `vecb=(vecb.vecz)(vecz-vecx)`B. `veca=(veca.vecy)(vecy-vecz)`C. `veca.vecb=-(veca.vecy)(vecb.vecz)`D. `veca=(veca.vecy)(vecz-vecy)` |
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Answer» Correct Answer - A::B::C a,,b., c. According to the question `vecx.vecz=vecx.vecy=vecy.vecz=sqrt(2).sqrt(2). cos ""(pi)/(3)=1` Given `veca` is perpendicular to `vecx and vecyxxvecz` `therefore veca =lamda_(1)(vecx xx(vecyxx vecz))` `implies veca=lamda_(1) ((vecx.vecz)vecY-(vecx . vecy)vecz)` `implies veca=lamda_(1) (vecy-vecz)` Now` veca. vecy = lamda _(1) ( vecy. vecy- vecy.vecz) = lamda_(1) (2-1) ` ` implies lamda_(1) = veca . vecy` From (1) and (2) , `veca= ( veca. vecy) ( vecy- vexz)` Similarly , ` vecb = ( vecb. vecz)( vecz - vecx) ` Now , `veca. vecb =(vecb. vecz) [ ( vecy-vecz).( vecz- vecx)]` `= ( veca . vecy) (vecb. vecz) [1-1-2+1]` ` =- ( veca. vecy) ( vecb. vecz) ` |
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| 81. |
The positive value of `lambda` for which the coefficient of `x^(2)` in the expression `x^(2) (sqrt(2) + (lambda)/(x^(2)))^(10)` is 720 isA. `sqrt(5)`B. 4C. `2sqrt(2)`D. 3 |
| Answer» Correct Answer - B | |
| 82. |
The sum of series `1+6+(9(1^(2)+2^(2)+3^(2)))/(7) + (12(1^(2)+2^(2)+3^(2)+4^(2)))/(9)+(15(1^(2)+2^(2)+...+5^(2)))/(11)+...` up to 15 terms isA. 7820B. 7830C. 7520D. 7510 |
| Answer» Correct Answer - A | |
| 83. |
Let `P=[[1,0,0],[4,1,0],[16,4,1]]`and `I` be the identity matrix of order `3`. If `Q = [q_()ij ]` is a matrix, such that `P^(50)-Q=I`, then `(q_(31)+q_(32))/q_(21)` equalsA. 15B. 9C. 135D. 10 |
| Answer» Correct Answer - D | |
| 84. |
The value of r for which `.^(20)C_(r ), .^(20)C_(r - 1) .^(20)C_(1) + .^(20)C_(2) + …… + .^(20)C_(0) .^(20)C_(r )` is maximum, isA. 20B. 15C. 11D. 10 |
| Answer» Correct Answer - A | |
| 85. |
If the middle term of the expansion of `(x^3/3+3/x)^8` is `5670` then sum of all real values of `x` is equal to (A) `6` (B) `3` (C) `0` (D) `2`A. 6B. 8C. 0D. 4 |
| Answer» Correct Answer - C | |
| 86. |
If the sum of the first 15 terms of the series `((3)/(4))^(3)+(1(1)/(2))^(3)+(2(1)/(4))^(3)+3^(3)+(3(3)/(4))^(3)+...` is equal to 225k, then k is equal toA. 9B. 27C. 108D. 54 |
| Answer» Correct Answer - B | |
| 87. |
The sum of an infinite geometric series with positive terms is 3 and the sums of the cubes of its terms is `(27)/(19)`. Then the common ratio of this series isA. `(4)/(9)`B. `(2)/(9)`C. `(2)/(3)`D. `(1)/(3)` |
| Answer» Correct Answer - C | |
| 88. |
Number of irrational terms in expansion of `(2^(1/5) + 3^(1/10))^60` isA. 55B. 49C. 48D. 54 |
| Answer» Correct Answer - D | |
| 89. |
In a random experiment, a fair die is rolled until two fours are obtained in succession.The probability that the experiment will end in the fifth throw of the die is equal toA. `(150)/(6^(5))`B. `(175)/(6^(5))`C. `(200)/(6^(5))`D. `(225)/(6^(5))` |
| Answer» Correct Answer - B | |
| 90. |
Two integers are selected at random from the set {1, 2, …, 11}. Given that the sum of selected numbers is even, the conditional probability that both the numbers are even isA. `(2)/(5)`B. `(1)/(2)`C. `(3)/(5)`D. `(7)/(10)` |
| Answer» Correct Answer - A | |
| 91. |
Let `S={1,2,...,20}` A subset `B` of S is said to be `"nice"`, if the sum of the elements of `B` is 203. Then the probability that a randomly chosen subset of `S` is `"nice"` is: (a) `7/(2^20)` (b) `5/(2^20)` (c) `4/(2^20)` (d) `6/(2^20)`A. `(6)/(2^(20))`B. `(5)/(2^(20))`C. `(4)/(2^(20))`D. `(7)/(2^(20))` |
| Answer» Correct Answer - B | |
| 92. |
The system of linear equations x + y + z = 2 2x + 3y + 2z = 5 `2x + 3y + (a^(2) - 1)z = a + 1`A. has infinitely many solutions for a = 4B. is inconsisten when `|a| = sqrt(3)`C. is inconsistent when a = 4D. has a unique solution for `|a| = sqrt(3)` |
| Answer» Correct Answer - B | |
| 93. |
The Boolean expression `~(pvvq)vv(~p^^q)` is equivalent to (1) `~p` (2) `p` (3) `q` (4) `~q`A. `pwedge (~q)`B. `p vee(~q)`C. `(~p) wedge (~q)`D. `p wedge q` |
| Answer» Correct Answer - C | |
| 94. |
Let `I=int_(a)^(b) (x^4-2x^2)dx`. If is minimum, then the ordered pair (a, b) isA. `(-sqrt2, 0)`B. `(-sqrt2, sqrt2)`C. `(0,sqrt2)`D. `(sqrt2,-sqrt2)` |
| Answer» Correct Answer - B | |
| 95. |
For `x in RR - {0, 1},` let `f_1(x) =1/x, f_2(x) = 1-x and f_3(x) = 1/(1-x)` be three given functions. If a function, `J(x)` satisfies `(f_2oJ_of_1)(x) = f_3(x)` then `J(x)` is equal to :A. `f_3(x)`B. `f_1(x)`C. `f_2(x)`D. `1/x f_3(x)` |
| Answer» Correct Answer - A | |
| 96. |
The mean and standart deviation of five observations `x_1,x_2,x_3,x_4,x_5` and are 10 and 3 respectively, then variance of the observation `x_1,x_2,x_3,x_4,x_5,-50` is equal to (a) 437.5 (b) 507.5 (c) 537.5 (d) 487.5A. 582.5B. 507.5C. 586.5D. 509.5 |
| Answer» Correct Answer - B | |
| 97. |
Matrix`=[[e^t,e^-t(sint-2cost),e^-t(-2sint-cost)],[e^t,-e^-t(2sint+cost),e^-t(sint-2cost)],[e^t,e^tcost,e^-tsint]]` is invertible. (a) only if `t=(pi)/(2)` (b) only `y=pi` (c) `tepsilonR` (d) `t!inR`A. invertible only if `t = (pi)/(2)`B. not invertible for any `t in R`C. invertible for all `t in R`D. invertible only if `t = pi` |
| Answer» Correct Answer - C | |
| 98. |
`lim_(ntoinfty) (n/(n^2+1^2)+n/(n^2+2^2)+n/(n^2+3^2)+...+1/(2n))` is equal toA. `pi/4`B. `tan^(-1)(2)`C. `tan^(-1)(3)`D. `pi/2` |
| Answer» Correct Answer - B | |
| 99. |
`lim_(xto 1^-) (sqrtpi-sqrt(2sin^-1x))/(sqrt(1-x))` is equal toA. `1/(sqrt(2pi))`B. `sqrtpi/2`C. `sqrt2/pi`D. `sqrtpi` |
| Answer» Correct Answer - C | |
| 100. |
` lim_(xto pi/4) (cot^3x-tanx)/(cos(x+pi/4))` isA. 4B. `8sqrt2`C. 8D. `4sqrt2` |
| Answer» Correct Answer - C | |