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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.
| 1. |
Axis of a parabola lies along x-axis. If its vertex and focus are at distances 2 and 4, respectively, from the origin on the positive x-axis, then which of the following points does not lie on it ?A. (4, -4)B. `(5, 2sqrt(6))`C. (8, 6)D. `(6, 4 sqrt(2))` |
| Answer» Correct Answer - C | |
| 2. |
If two vertices of a triangle are `(0,2)` and `(4,3)` and its orthocentre is `(0,0)` then the third vertex of the triangle lies in (a) `I^(st)` quadrant (b) `2^(nd)quadrant (c) `3^(rd)quadrant (d) `4^(th)quadrantA. FourthB. SecondC. ThirdD. First |
| Answer» Correct Answer - B | |
| 3. |
If `Sigma_(i=1)^(20) ((""^(20)C_(i-1))/(""^(20)C_(i)+""^(20)C_(i-1)))^(3)=(k)/(21)`, then k equalsA. 200B. 50C. 100D. 400 |
| Answer» Correct Answer - C | |
| 4. |
If `dy/dx+3/cos^2xy=1/cos^2x,x in((-pi)/3,pi/3)and y(pi/4)=4/3," then "y(-pi/4)` equalsA. `1/3+e^6`B. `1/3`C. `-1/4`D. `1/3+e^3` |
| Answer» Correct Answer - A | |
| 5. |
The integral `int_(pi//6)^(pi//4)(dx)/(sin2x(tan^(5)x+cot^(5)x))` equalsA. `1/10(pi/4-tan^(-1)(1/(9sqrt3)))`B. `1/5(pi/4-tan^(-1)(1/(3sqrt3)))`C. `pi/10`D. `1/20-tan^(-1)(1/(9sqrt3))` |
| Answer» Correct Answer - A | |
| 6. |
If the system fo equations x+y+z = 5 x + 2y + 3z = 9 `x + 3y + alphaz = beta` has infinitely many solution, then `beta - alpha` equalsA. 5B. 18C. 21D. 8 |
| Answer» Correct Answer - D | |
| 7. |
If two vertices of a triangle are `(0,2)` and `(4,3)` and its orthocentre is `(0,0)` then the third vertex of the triangle lies in (a) `I^(st)` quadrant (b) `2^(nd)quadrant (c) `3^(rd)quadrant (d) `4^(th)quadrantA. FourthB. SecondC. Third D. First |
| Answer» Correct Answer - B | |
| 8. |
Two sides of a parallelogram are along the lines x+y=3 and x=y+3. If its diagonals intersect at (2, 4) , then one of its vertices isA. (2, 6)B. (2,1)C. (3, 5)D. (3, 6) |
| Answer» Correct Answer - D | |
| 9. |
If the straight line 2x-3y+17-0 is perpendicular to the line passing through the points (7, 17) and `(15, beta)`, then `beta` equalsA. `-5`B. `-(35)/(3)`C. `(35)/(3)`D. 5 |
| Answer» Correct Answer - D | |
| 10. |
Consider the set of all lines px+qy+r=0 such that 3p+2q+4r=0. Which one of the following statements is true ?A. The lines are all parallel.B. Each line passes through the origin.C. The lines are not concurrent.D. The lines are concurrent at the point `((3)/(4),(1)/(2))`. |
| Answer» Correct Answer - D | |
| 11. |
An urn contains 5 red and 2 green balls. A ball is drawn at random from the urn. If the drawn ball is green, then a red ball is added to the urn and if the drawn ball is red, then a green ball is added to the urn, the original ball is not returned to the urn. Now, a second ball is drawn at random from it. The probability that the second ball is red isA. `(26)/(49)`B. `(32)/(49)`C. `(27)/(49)`D. `(21)/(49)` |
| Answer» Correct Answer - B | |
| 12. |
In a class 140 students numbered 1 to 140, all even numbered students opted Mathematics course, those whose number is divisible by 3 opted Physics course and those whose number is divisible by 5 opted Chemistry course. Then the number of students who did not opt for any of the three courses is (a) 38 (b) 1 (c)42 (d) 102A. 102B. 42C. 1D. 38 |
| Answer» Correct Answer - D | |
| 13. |
The length of the common chord of the two circles `x^2+y^2-4y=0` and `x^2+y^2-8x-4y+11=0` isA. `2sqrt(11)`B. `3sqrt(2)`C. `6sqrt(3)`D. `8sqrt(2)` |
| Answer» Correct Answer - C | |
| 14. |
The tangent to the curve `y=x^2-5x+5.` parallel to the line `2y=4x+1,` also passes through the point :A. `((1)/(4),(7)/(2))`B. `((7)/(2),(1)/(4))`C. `(-(1)/(8),7)`D. `((1)/(8),-7)` |
| Answer» Correct Answer - D | |
| 15. |
Let `Z_0` is the root of equation `x^2+x+1=0` and `Z=3+6i(Z_0)^(81)-3i(Z_0)^(93)` Then arg `(Z)` is equal to (a) `(pi)/(4)` (b) `(pi)/(3)` (c) `pi` (d) `(pi)/(6)`A. `(pi)/(4)`B. `(pi)/(3)`C. 0D. `(pi)/(6)` |
| Answer» Correct Answer - A | |
| 16. |
Let N be the set of numbers and two functions f and g be defined as `f,g:N to N` such that `f(n)={((n+1)/(2), ,"if n is odd"),((n)/(2),,"if n is even"):}` and `g(n)=n-(-1)^(n)`. Then, fog isA. both one-one and ontoB. one-one but not ontoC. neither one-one nor ontoD. onto but not one-one |
| Answer» Correct Answer - D | |
| 17. |
Let `f: (-1,1)toR` be a function defind by f(x) =max. `{-absx,-sqrt(1-x^2)}`. If K is the set of all points at which f is not differentiable, then K has set of all points at which f is not differentible, then K has exactlyA. three elementsB. one elementC. five elementsD. two elements |
| Answer» Correct Answer - A | |
| 18. |
Let `A={theta in (-pi /2,pi):(3+2i sin theta )/(1-2 sin theta )` is purely imaginary } Then the sum of the elements in A isA. `(5pi)/(6)`B. `(2pi)/(3)`C. `(3pi)/(4)`D. `pi` |
| Answer» Correct Answer - B | |
| 19. |
Let `(z-alpha)/(z+alpha)` is purely imaginary and `|z|=2, alphaepsilonR` then `alpha` is equal to (A) `2` (B) `1` (C) `sqrt2` (D) `sqrt3`A. 1B. 2C. `sqrt(2)`D. `(1)/(2)` |
| Answer» Correct Answer - B | |
| 20. |
Let `Z_(1)` and `Z_(2)` be two complex numbers satisfying `|Z_(1)|=9` and `|Z_(2)-3-4i|=4`. Then the minimum value of `|Z_(1)-Z_(2)|` is |
| Answer» Correct Answer - A | |
| 21. |
Consider the statement : `" P(n) : n^(2)-n+41` is prime." Then, which one of the following is true?A. P(5) is false but P(3) is trueB. Both P(3) and P(5) are falseC. P(3) is false but P(5) is trueD. Both P(3) and P(5) are true |
| Answer» Correct Answer - D | |
| 22. |
if `theta` denotes the acute angle between the curves, `y = 10-x^2" and " y=2+x^2` at a point of their intersection, then `abstantheta` is equal toA. `4//9`B. `7//17`C. `8//17`D. `8//15` |
| Answer» Correct Answer - D | |
| 23. |
Let S be the set of all points in `(-pi, pi)` at which the Then, S is a subset of which of the following?A. `{-(3pi)/4,-pi/4,(3pi)/4,pi/4}`B. `{-(3pi)/4,-pi/2,(pi)/2,(3pi)/4}`C. `{-(pi)/2,-pi/4,(pi)/4,(pi)/2}`D. `{-pi/4,0,pi/4}` |
| Answer» Correct Answer - A | |
| 24. |
Let K be the set of all values of x, where the function ` f(x) = sin |x| - |x| + 2(x-pi) cos |x| ` is not differentiable. Then, the set K is equal toA. `{pi}`B. `{0}`C. `phi`(an empty set)D. `{0,pi}` |
| Answer» Correct Answer - C | |
| 25. |
If the probability of hitting a target by a shooter, in any shot is 1/3, then the minimum number of independent shots at the target required by him so that the probability of hitting the target at least once is greater than `(5)/(6)` isA. 6B. 5C. 4D. 3 |
| Answer» Correct Answer - B | |
| 26. |
If 5, 5r and `5r^(2)` are the lengths of the sides of a triangle, then r cannot be equal toA. `(3)/(2)`B. `(3)/(4)`C. `(5)/(4)`D. `(7)/(4)` |
| Answer» Correct Answer - D | |
| 27. |
If `.^(n)C_(4),.^(n)C_(5), .^(n)C_(6)` are in A.P., then find the value of n.A. 14B. 11C. 9D. 12 |
| Answer» Correct Answer - A | |
| 28. |
If `19^(th)` term of a non-zero A.P. is zero, then (`49^(th)` term) : (`29^(th)` term) isA. `3 : 1`B. `4 : 1`C. `2 : 1`D. `1 : 3` |
| Answer» Correct Answer - A | |
| 29. |
Let `S_(k) = (1+2+3+...+k)/(k)`. If `S_(1)^(2) + s_(2)^(2) +...+S_(10)^(2) = (5)/(12)A`, then A is equal toA. 303B. 283C. 156D. 301 |
| Answer» Correct Answer - A | |
| 30. |
The sum of all two-digit positive numbers which when divided by 7 yield 2 or 5 as remainder isA. 1365B. 1256C. 1465D. 1356 |
| Answer» Correct Answer - D | |
| 31. |
Let y = y(x) be the solution of the differential equation `x dy/dx+y=xlog_ex,(xgt1)." If " 2y(2)=log_e4-1," then "y(e)` is equal toA. `e^2/4`B. `e/4`C. `-e/2`D. `-e^2/2` |
| Answer» Correct Answer - B | |
| 32. |
Let a, b and c be the 7th, 11th and 13th terms, respectively, of a non-constant A.P.. If these are also the three consecutive terms of a G.P., then `(a)/(c )` is equal toA. `1//2`B. 4C. 2D. `7//13` |
| Answer» Correct Answer - B | |
| 33. |
Let x, y be positive real numbers and m, n be positive integers, The maximum value of the expression `(x^(m)y^(n))/((1+x^(2m))(1+y^(2n)))` isA. `(1)/(2)`B. `(1)/(4)`C. `(m+n)/(6mn)`D. 1 |
| Answer» Correct Answer - B | |
| 34. |
The number of integral values of m for which the quadratic expression `(1 + 2m)x^(2) - 2(1 + 3m)x + 4(1 + m), x in R`, is always positive isA. 8B. 7C. 6D. 3 |
| Answer» Correct Answer - B | |
| 35. |
the mirror image of point `(3,1,7)` with respect to the plane `x-y+z=3` is `P`. then equation plane which is passes through the point `P` and contains the line `x/1=y/2=z/1`.A. `x+y-3z=0`B. `3x+z=0`C. `x-4y+7z=0`D. `2x-y=0` |
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Answer» Correct Answer - C Let image (x,y,z) of `therefore (x-3)/(1)=(y-1)/(-1)=(z-7)/(1)=-2((3-1+7-3)/(1^(2)+1^(2)+1^(2)))=-4` `P(x,y,z)=(-1,5,3)` plane passing through p(-1,5,3)is `a(x+1)+b(y-5)+c(z-3)=0` also plane contains line `(x)/(1)=(y)/(2)=(z)/(1)` `therefore (0,0,0)`satisfy `implies a-5b-3c=0` `and a+2b+c=0` from (2) and (3) pin in `(1)(X+1)-4(y-5)+7(z-3)=0` `or x-4y+7z=0` |
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| 36. |
The logical statement `[~(~pvvq)vv(p^^r)]^^(~q^^r)` is equivalent to (a) `(~p^^~q)^^r` (b) `~p vv r` (c) `(p^^r)^^~q` (d) `(p^^~q)vvr`A. ` (p wedge r) wedge ~q`B. `(~p wedge ~q)wedge r`C. `~p vee r`D. `(p wedge~q)vee r` |
| Answer» Correct Answer - A | |
| 37. |
If the Boolean expression `(p oplusq)wedge (~p Theta q)` is equivalent to `p wedge q`, where `oplus, Theta in {vee, wedge}`, then the ordered pair (oplus, Theta)` isA. `( wedge, vee)`B. `( vee, vee)`C. `( wedge, wedge)`D. `(vee, wedge)` |
| Answer» Correct Answer - A | |
| 38. |
The maximum volume (in cu.m) of the right circular cone having slant height 3 m isA. `3sqrt3 pi`B. `6pi`C. `2sqrt3pi`D. `4/3pi` |
| Answer» Correct Answer - C | |
| 39. |
The shortest distance between the point `((3)/(2),0)` and the curve `y=sqrt(x),(x gt 0)`, isA. `sqrt5/2`B. `5/4`C. `3/2`D. `sqrt3/2` |
| Answer» Correct Answer - A | |
| 40. |
Let `A=[(2,b,1),(b,b^(2)+1,b),(1,b,2)]` where `b gt 0`. Then the minimum value of `("det.(A)")/(b)` isA. `sqrt(3)`B. `-sqrt(3)`C. `-2sqrt(3)`D. `2sqrt(3)` |
| Answer» Correct Answer - D | |
| 41. |
If `q` is false and `(p^^q)`harr r` is also true then which of the following are tautology (A) `(pvvr)-gt(p^^r)` (B) `(pvvr)` (C) `(p^^r)-gt(pvvr)` (D) `p^^r`A. `(p vee r) to(p wedge r)`B. `p vee r`C. `p wedge r`D. `(p wedge r) rarr (p vee r)` |
| Answer» Correct Answer - D | |
| 42. |
`(~pvv~q)` is logically equivalent toA. ` ~ p wedge ~ q`B. `p wedge q`C. `~ p wedge q`D. `p wedge ~ q` |
| Answer» Correct Answer - A | |
| 43. |
If a curve passes through the point (1, -2) and has slope of the tangent at any point (x,y) on it as `(x^2-2y)/x`, then the curve also passes through the pointA. `(-sqrt2,1)`B. `(sqrt3, 0)`C. `(-1,2)`D. (3, 0) |
| Answer» Correct Answer - B | |
| 44. |
In `R^(3)`, consider the planes `P_(1):y=0` and `P_(2),x+z=1.` Let `P_(3)` be a plane, different from `P_(1)` and `P_(2)` which passes through the intersection of `P_(1)` and `P_(2)`, If the distance of the point (0,1,0) from `P_(3)` is 1 and the distance of a point `(alpha,beta,gamma)` from `P_(3)` is 2, then which of the following relation(s) is/are true?A. `2alpha+beta+2gamma+2=0`B. `2alpha+beta+2gamma+4=0`C. `2alpha+beta+2gamma-10=0`D. `2alpha+beta+2gamma-8=0` |
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Answer» Correct Answer - B::D b.,d Clearly , planr `P_(3) is P_(2) + lamdaP_(1) =0` . `implies x+ lamday+z-1=0` Distance of this from point `(0,1,0)`is 1, `=(0+lamda+0-1)/(sqrt(1+lamda^(2)+1))=+_1` `therefore lamda =-(1)/(2)` thus , equation of `P_(3)is 2x-y+2z-2=0` DIstance of this plane from point `(alpha, beta,gamma)`is 2. `=|(2alpha-beta+2gamma-2)/(3)|=2` `implies 2alpha-beta+2gamma=2+-6` thus options (b) and (d) are correct. |
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| 45. |
Mean and variance of five observations are `4` and `5.2` respectively. If three of these observations are `3, 4, 4` then find absolute difference between the other two observations (A) `3` (B) `7` (C) `2` (D) `5`A. 1B. 3C. 7D. 5 |
| Answer» Correct Answer - C | |
| 46. |
let L be a straight line passing through the origin. Suppose that all the points on L are at a constant distance from the two planes `P_1 : x + 2y-z +1 = 0` and `P_2 : 2x-y + z-1 = 0`, Let M be the locus of the feet of the perpendiculars drawn from the points on L to the plane `P_1`. Which of the following points lie(s) on M?A. `(0,-(5)/(6),-(2)/(3))`B. `(-(1)/(6),-(1)/(3),(1)/(6))`C. `(-(5)/(6),0,(1)/(6))`D. `(-(1)/(3),0,(2)/(3))` |
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Answer» Correct Answer - A::B a.b . Let `vecv` be the vetor along I. `P_(1):x+2y-Z+1=0 and P_(2):2x-y+z-1=0` ` therefore vecc=|{:(hati,hatj,hatk),( 1,2,-1),( 2,-1,1)}|= hati-3hatj-5hatk` since I. is through origin any point on line I. is `A(lamda,3lamda ,-5lamda).`Foot of perpendicular from A to `P_(1)` , is `(h- lamda)/( 1)`=(k+3lamda)/(-1)=-(lamda-6lamda+5lamda+1))/(1+4+1)=-(1)/(6)` `therefore h= lammda-(1)/(6) ,k =- 3lamda -(1)/(3) ,il=-5 lamda+(1)/(6)` so, foot ro rpoint on lovus M is `(lamda-(1)/(6),-3lamda-(1)/(3)-(1)/(3),-5lamda+(1)/(6))` So points (a) and (b) lie on this loucs . |
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| 47. |
The equation of the plane passing through the point 1,1,1) andperpendicular to the planes `2x+y-2z=5a n d3x-6y-2z=7,`is`14 x+2y+15 z=3``14 x+2y-15 z=1``14 x+2y+15 z=31``14 x-2y+15 z=27`A. `14x+2y+15x=31`B. `14x+2y-15z=1`C. `14x+2y+15x=3`D. `14x-2y+15z=27` |
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Answer» Correct Answer - A (a) given that required plane is perpendicular to the given two planes , therefore , the ormal vector of required plane I sperpendicular to the normal to the given planes . therefore , the normal vector of required plane is parallel to the vector : `|{:(hati,hatj,hatk),(2,1,-2),(3,-6,-2):}|=-14hati-2hatj-15hatk` thus , the equation of required of required plane passing through (1,1,1) will be : `-14(x-1)-2(y-1)-15(z-1)=0` `implies 14x+2y+15z=31` |
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| 48. |
Average height & variance of `5` students in a class is `150` and `18` respectively. A new student whose height is `156 cm` is added to the group. Find new variance. (a) `20` (b) `22` (c) `16` (d) `14`A. 22B. 20C. 16D. 18 |
| Answer» Correct Answer - B | |
| 49. |
The value of `int_(0)^(pi)abscosx^3dx` isA. `2//3`B. 0C. `-4//3`D. `4//3` |
| Answer» Correct Answer - D | |
| 50. |
`int(3x^(13)+2x^(11))/((2x^4+3x^2+1)^4)dx`A. `x^4/((2x^4+3x^2+1)^3)+C`B. `x^12/(6(2x^4+3x^2+1)^3)+C`C. `x^4/(6(2x^4+3x^2+1)^3)+C`D. `x^12/((2x^4+3x^2+1)^3)+C` |
| Answer» Correct Answer - B | |