InterviewSolution
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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. |
Find the number of solution of `theta in [0,2pi]`satisfying the equation `((log)_(sqrt(3))tantheta(sqrt((log)_(tantheta)3+(log)_(sqrt(3))3sqrt(3)=-1)`A. `theta =(pi)/(6)`B. `(pi)/(3),(5pi)/(3)`C. has sum `(4pi)/(3)`D. `gt 2 ` |
| Answer» Correct Answer - A::C | |
| 2. |
If `xsina+ysin2a+zsin3a=sin4a``xsinb+ysin2b+zsin3b=sin4b``xsinc+ysin2c+zsin3c=sin4c`then the roots of the equation `t^3-(z/2)t^2-((y+2)/4)t+((z-x)/8)=0,a , b , c ,!=npi,`are`sina ,sinb ,sinc`(b) `cosa ,cosb ,cosc``sin2a ,sin2b ,sin2c`(d) `cos2a ,cos2bcos2c`A. sin a , sin b , sin cB. cos a, cos b , cos cC. sin 2a , sin 2b , sin 2cD. cos 2a , cos 2b , cos 2c |
| Answer» Correct Answer - B | |
| 3. |
Let `n`be a positive integer such that `sinpi/(2n)+cospi/(2n)=(sqrt(n))/2dot`Then`6lt=nlt=8`(b) `4A. `6 le n le 8`B. `4 lt n le 8`C. `6 lt n lt 8`D. `4 lt n lt 8` |
| Answer» Correct Answer - D | |
| 4. |
Let `log_(a)N=alpha + beta` where ` alpha ` is integer and ` beta =[0,1)`. Then , On the basis of above information , answer the following questions. If `N_1` is number of integers when a=2 and `alpha=2` and `N_2` is number of integers when ` alpha=1 and a=3` , then the minimum value of `(N_1 sec^(2) theta +N_(2) "cosec"^2 theta )A. `10+4sqrt(6)`B. `10+sqrt(6)`C. `10`D. 100 |
| Answer» Correct Answer - A | |
| 5. |
In the interval `[-pi/2,pi/2] `the equation `log_(sin theta)(cos 2 theta)=2` hasA. no solutionB. a unique solutionC. two solutionD. infinitely many solution |
| Answer» Correct Answer - B | |
| 6. |
If the equation `sin^4x-(k+2)s in^2x-(k+3)=0`has a solution then `k`must lie in the interval`(-4,-2)`(b) `[-3,2)``(-4,-3)`(d) `[-3,-2]`A. ` k gt -3 `B. `k lt -2 `C. `-3 le k le -2 `D. `k ` is any (+ve) value |
| Answer» Correct Answer - C | |
| 7. |
The solution of the equation ` sin 2x + sin 4x = 2 sin 3x ` isA. `x=(npi)/3`B. `x=npi `C. `x=2npi `D. None of the above |
| Answer» Correct Answer - A::C | |
| 8. |
The general solution of ` 4sin^(4) x + cos^(4) x=1 ` isA. `(2n+1)""(pi)/(2)`B. `npi`C. `npi pm sin^(-1)sqrt((2)/(5))`D. None of these |
| Answer» Correct Answer - B::C | |
| 9. |
General solution of `tan theta+tan 4 theta+tan 7 theta=tan theta tan 4 theta tan 7 theta` isA. `theta =(n pi )/4`B. `theta =(n pi )/(12) `C. `theta=(n pi )/12`D. None of these |
| Answer» Correct Answer - B | |
| 10. |
If number of solution and sum of solution of the equation ` 3sin^(2)x-7sinx +2=0, x in [0,2pi]` are respectively N and S and ` f_(n)(theta)=sin^(n)theta +cos^(n) theta`. On the basis of above information , answer the following questions. If ` alpha ` is solution of equation ` 3sin^(2)x-7sinx+2=0, x in [0,2pi]`, then the value of `f_(4)(alpha)` isA. `(97)/(81)`B. `(57)/(81)`C. `(65)/(81)`D. 0 |
| Answer» Correct Answer - C | |
| 11. |
If number of solution and sum of solution of the equation ` 3sin^(2)x-7sinx +2=0, x in [0,2pi]` are respectively N and S and ` f_(n)(theta)=sin^(n)theta +cos^(n) theta`. On the basis of above information , answer the following questions. Value of S isA. `(5pi)/(6)`B. `(7pi)/(6)`C. `2pi`D. `pi` |
| Answer» Correct Answer - D | |
| 12. |
If ` 3sin^(2)x-7sin x +2=0, x in [0(pi)/(2)] and f_(n)(theta)=sin^(n) theta + cos^(n) theta ` . On the basis of above information , answer the questions The value of ` ( sin 5 x + sin 4 x)/( 1+ 2 cos 3 x)` isA. `(3+ 2 sqrt(2))/(9)`B. `(3+ 4sqrt(2))/(9)`C. `(4sqrt(2)-2)/(9)`D. `(4sqrt(2)-3)/(9)` |
| Answer» Correct Answer - B | |
| 13. |
` cos 2x- 3 cos x + 1=(1)/((cot 2 x - cot x)sin (x-pi))` holds , ifA. cos x =0B. cos x =1C. `cos x=(5)/(2)`D. for no value of x |
| Answer» Correct Answer - A | |
| 14. |
If ` sin 3 alpha =4 sin alpha sin (x+alpha ) sin(x-alpha ) `, thenA. `x=npi pm (pi)/(3), n in I `B. `x=npi pm (pi)/(6), n in I `C. `x=npi pm (pi)/(2) , n in I `D. None of the above |
| Answer» Correct Answer - A | |
| 15. |
`lambda cos x - 3 sinx=lambda +1` is solvabel onlyl ,ifA. ` lambda in [0,5] `B. `lambda in [4,5]`C. lambda in (-oo,4]`D. None of these |
| Answer» Correct Answer - C | |
| 16. |
Which of the following is/are correct .A. `(tanx)^(In (sin x)) gt ( cot x )^(In sin x), AA x in (0,pi//4)`B. `4^(In "cosec"x) lt 5^(In "cosec"x),AA x in(0,pi//2)`C. `(1//2)^(In (cos x)) lt (1//3)^(In(cosx)),AA x in (0,pi//2) `D. `2^(In ( tan x )) gt 2^(In ( tan x )), AA x in (0,pi//2)` |
| Answer» Correct Answer - A::B::C::D | |
| 17. |
If `|k|=5 and 0^(@) le theta le 360^(@) ` , then the number of different solution of ` 3 cos theta + 4 sin theta =k` isA. ZeroB. TwoC. OneD. Infinite |
| Answer» Correct Answer - B | |
| 18. |
Total number of solutions of ` sinx=(|x|)/(10)` is equal toA. 4B. 6C. 7D. None of these |
| Answer» Correct Answer - B | |
| 19. |
The number of all possible 5-tuples `(a_(1),a_(2),a_(3),a_(4),a_(5))` such that ` a_(1)+a_(2) sin x+a_(3) cos x + a_(4) sin 2x +a_(5) cos 2 x =0` hold for all x isA. zeroB. 1C. 2D. infinite |
| Answer» Correct Answer - B | |
| 20. |
The value of ` theta ` , lying between ` theta =0 and theta=(pi)/(2)` and satisfying the equation . `|{:(1+ cos^(2 ) theta , sin^(2) theta , 4 sin 4 theta ),(cos^(2) theta , 1+sin^2 theta , 4 sin 4 theta ),(cos^(2) theta , sin^(2) theta , 1+ 4 sin 4 theta ):}|=0 , is `A. `(11pi)/(24)`B. `(7pi)/(24)`C. `(5pi)/(24)`D. None of these |
| Answer» Correct Answer - A::B | |
| 21. |
If number of solution and sum of solution of the equation ` 3sin^(2)x-7sinx +2=0, x in [0,2pi]` are respectively N and S and ` f_(n)(theta)=sin^(n)theta +cos^(n) theta`. On the basis of above information , answer the following questions. Value of N isA. 1B. 2C. 3D. 4 |
| Answer» Correct Answer - B | |
| 22. |
If `a , b in [0,2pi]`and the equation `x^2+4+3sin(a x+b)-2x=0`has at least one solution, then the value of `(a+b)`can be`(7pi)/2`(b) `(5pi)/2`(c) `(9pi)/2`(d) none of theseA. `(7pi)/2`B. `(3pi)/2`C. `(9pi)/2`D. None of these |
| Answer» Correct Answer - B | |
| 23. |
If [x] dnote the greatest integer less than or equal to x then the equation ` sin x=[1+sin x ]+[1-cos x ][` has no solution inA. `[-(pi)/(2),(pi)/(2)]`B. `[(pi)/(2),pi]`C. `[pi,(3pi)/(2)]`D. R |
| Answer» Correct Answer - A::B::C::D | |
| 24. |
`x_(1) and x_(2)` are two positive value of x for which ` 2 cos x,|cos x| and 3 sin^(2) x-2` are in GP. The minimum value of ` |x_(1)-x_(2)|` is equal toA. `(4pi)/(3)`B. `(pi)/3`C. `2 cos^(-1)""((2)/(3))`D. `cos^(-1)""((2)/(3))` |
| Answer» Correct Answer - C | |
| 25. |
Let the smallest positive value of x for which the function ` f(x)=sin""(x)/(3)+sin""(x)/(11), ( x in R )` achieves its maximum value be ` x_(0)`. Express `x_(0)` in degree i.e. ` x_(0)=alpha^(0)`. Then , the sum of the digits in ` alpha ` isA. 15B. 17C. 16D. 18 |
| Answer» Correct Answer - D | |
| 26. |
Let `log_(a)N=alpha + beta` where ` alpha ` is integer and ` beta =[0,1)`. Then , On the basis of above information , answer the following questions. The difference of largest and smallest integral value of N satisfying ` alpha =3 and a =5` , isA. 499B. 500C. 501D. 502 |
| Answer» Correct Answer - A | |
| 27. |
Statement I ` x =(kpi)/(13), k in I` does not represent the general solution of trigonometric equation. ` sin 13 x - sin 13 x cos 2x=0` Statement II Both ` x=r pi , r in I and x=(k pi )/(13) , k in I ` satisfies the trigonometric equation. `sin 13 x - sin 13 x cos 2x =0`A. Statement I is true , Statement II is true , Statement II is a correct explanation for Statement I.B. Statement I is true , Statement II is true , Statement II is not a correct explanation for Statement II.C. Statement I is true , Statement II is falseD. Statement I is false , Statement II true . |
| Answer» Correct Answer - D | |
| 28. |
The equation ` 2 sin""(x)/(2) cos^(2)x-2 sin""(x)/(2) sin^(2) x =cos^(2)x-sin^(2)x ` has a root for which the false statement isA. sin 2x =1B. ` cos x=(1)/(2)`C. `cos 2x=-(1)/(2)`D. `cos x =1 ` |
| Answer» Correct Answer - D | |
| 29. |
Find the number of solution of the equation `e^(sinx)-e^(-sinx)-4=0`A. x=0B. `x=sin^(-1)[log(2-sqrt(5))]`C. no real solutionD. None of the above |
| Answer» Correct Answer - C | |
| 30. |
If ` sin^(100) theta - cos^(100) theta =1` , then ` theta ` isA. `2n pi +(pi)/(3) , n in I `B. `n pi +(pi)/(2) , n in I `C. `n pi +(pi)/(4) , n in I `D. `2n pi -(pi)/(3) , n in I ` |
| Answer» Correct Answer - B | |
| 31. |
`alpha ` is a root of equation `( 2 sin x - cos x ) (1+ cos x)=sin^2 x , beta ` is a root of the equation ` 3 cos 2x - 10 cos x +3 =0 and gamma ` is a root of the equation `1-sin2 x = cos x- sin x : 0 le alpha , beta, gamma , le pi//2 ` ` cos alpha + cos beta + cos gamma ` can be equal toA. `(3sqrt(6)+2sqrt(2)+6)/(6sqrt(2))`B. `(3sqrt(3)-8)/(6)`C. `(3sqrt(3)+2)/(6)`D. None of these |
| Answer» Correct Answer - A | |
| 32. |
If ` sqrt(3) sin x - cos x ="min"_(alpha in R){2,e^(2),pi, alpha^(2)-4alpha +7}`, thenA. ` x =2n pi, n in I `B. `x=2n pi +(2pi)/(3) , n in I `C. `x=npi +(-1)^(n)""(pi)/(4)+(pi)/(6), n in I `D. `x=npi +(-1)^(n)""(pi)/(4)-(pi)/(3), n in I ` |
| Answer» Correct Answer - B | |
| 33. |
If the equation ` cos 3 x cos^(3) x + sin 3 x sin^(3) x =0` , then x is equal toA. `(2n+1)""(pi)/(4)`B. `(2n-1)""(pi)/(4)`C. `(npi)/(4)`D. None of these |
| Answer» Correct Answer - A | |
| 34. |
If ` cot (alpha + beta )=0 `, then ` sin(alpha+2beta )`=A. `sin alpha `B. `cos alpha `C. `sin beta `D. ` cos 2 beta ` |
| Answer» Correct Answer - A | |
| 35. |
If `: cot theta + cot((pi)/4 + theta) = 2`, then : `theta` =A. `2 n pi pm(pi)/(6)`B. `2n pi pm (pi)/3`C. `npi pm (pi)/3`D. `npi pm (pi)/6` |
| Answer» Correct Answer - D | |
| 36. |
The value of the determinants `|{:(1,a,a^(2)),(cos(n-1)x,cos nx , cos(n+1)x),(sin(n-1)x , sin nx , sin(n+1)x):}|` is zero ifA. ` x = n pi `B. `x = n pi //2 `C. `x=(2n+1)pi//2`D. `x=(1+a^(2))/(2a)n in I ` |
| Answer» Correct Answer - A | |
| 37. |
If an angle and a side of a right angle triangle is known , then rest of the sides and angles can be found as follows In `DeltaABC `(figure ), if `angleB=90^(@),angleC=theta and BC=x`, then `AB=x tan theta and AC=x sec theta ` . Now, consider in isosceles triangle PQR (Figure 2), Where PQ=PR and `20N=sqrt(3)` On the basis of the above information answer the question Area of circumcircle of quadrilateral PLOM isA. `pi`B. `(pi)/4`C. `(3pi)/4`D. `3pi` |
| Answer» Correct Answer - B | |
| 38. |
The expression ` nsin^(2) theta + 2 n cos( theta + alpha ) ` `sin alpha sin theta + cos2(alpha + theta ) ` is independent of ` theta ` , the value of n isA. 1B. 2C. 3D. 4 |
| Answer» Correct Answer - B | |
| 39. |
ABC is a triangle such that `sin(2A+B)=sin(C-A)=-sin(B+2C)=1/2`. If A,B, and C are in AP. then the value of A,B and C are..A. `45^(@),60^(@),75^(@)`B. `30^(@), 60^(@), 90^(@)`C. `20^(@) , 60^(@), 100^(@)`D. None of these |
| Answer» Correct Answer - A | |
| 40. |
Find the number of integral value of `n`so that `sinx(sinx+cosx)=n`has at least one solution.A. 2B. 1C. 3D. zero |
| Answer» Correct Answer - A | |
| 41. |
The number of solutions of the equations `y=1/3[sin theta+[sin theta +[sin theta]]] and [y+[y]]=2 cos theta ` [ where , [.] denote the greatest integer function ] is/are |
| Answer» Correct Answer - A | |
| 42. |
Which of the following set of values of x satisfies the equation `2^((2 sin^(2) x-3 +1))+2^((2-2 sin^(2)x+3 sin x))=9` ?A. `x =npi pm (pi)/6 , n in I `B. `x = n pi pm (pi)/3, n in I `C. `x = n pi , n in I `D. `x=2n pi +(pi)/(2), n in I ` |
| Answer» Correct Answer - A::D | |
| 43. |
The number of solutions of the equation `sin((pix)/(2sqrt3))=x^2-2sqrt3x+4` |
| Answer» Correct Answer - D | |
| 44. |
find the range of values of t for which `2sint=(1-2x+5x^2)/(3x^2-2x-1)`A. `[-(pi)/(2),(pi)/(2)]`B. `[(3pi)/(10),(pi)/(2)]`C. `[-(pi)/(2),(pi)/(10)] cup[(3pi)/(10),(pi)/(2)]`D. All of these |
| Answer» Correct Answer - A::B::C::D | |
| 45. |
Solve `(sqrt(5)-1)/(sinx)+(sqrt(10+2sqrt(5)))/(cosx)=8,x in (0,pi/2)`A. `(pi)/(10)`B. `(3pi)/(10)`C. `(9pi)/(10)`D. `(7pi)/(10)` |
| Answer» Correct Answer - A::B | |
| 46. |
`0 lt a lt 2 pi , sin^(-1)(sin a ) lt x^(2)-2x ` for all ` x in I` then ` a in `A. `(0,pi +1)`B. `(pi+1,(3pi)/(2))`C. `((3pi)/(2), 2 pi -1)`D. `(2pi-1, 2 pi )` |
| Answer» Correct Answer - B::C | |
| 47. |
The maximum value of the expression`|sqrt(sin^2x+2a^2)-sqrt(2a^2-1-cos^2x)|`, where `aa n dx`are real numbers, is`sqrt(3)`(b) `sqrt(2)`(c) `1`(d) `sqrt(5)`A. 1B. 2C. `sqrt(2)`D. `sqrt(3)` |
| Answer» Correct Answer - C | |
| 48. |
If `(sin alpha)^(x)+(cos alpha)^(x) ge 1,0 lt a lt (pi)/(2)` thenA. `n in [2,oo)`B. `(-oo,2]`C. `n in [-1,1]`D. None of these |
| Answer» Correct Answer - B | |
| 49. |
The number of integral values of a for which the system of linear equations ` x sin theta -2 y cos theta -az=0`, `x+2y+z=0,-x+y+z=0` may have non-trivial solutions, thenA. at a=2 the given system will have finite solutions for ` theta in R `B. number of possible integral values of a is 3C. for a=1 the sytem will have finite solutionsD. for a=3 the sytem will have unique solution |
| Answer» Correct Answer - B::C::D | |
| 50. |
If `|1-(|sin x |)/(1+ sin x )| ge (2)/(3)`, then sin x lies inA. `(-oo,(-1)/(2)] cup [(1)/(2),oo)`B. `(-(1)/(2),(1)/(2))`C. `[-(1)/(2),(1)/(2)]`D. None of the above |
| Answer» Correct Answer - C | |