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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 `tan ^(-1) ((1-x)/(1+x))= (1)/(2) tan^(-1) x, x gt 0`, then x = ?A. `(1)/sqrt(3)`B. `-(1)/sqrt(3)`C. `-sqrt(3)`D. None of these |
| Answer» Correct Answer - A | |
| 52. |
Solve the following equation: `sin^(-1)x+sin^(-1)(1-x)=cos^(-1)x` |
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Answer» `sin^(-1) x + sin^(-1) (1-x) = cos^(-1)x ` `rArr " "sin[xsqrt(1-(1-x)^(2)) +(1-x)sqrt(1-x^(2))]= sin ^(-1)sqrt(1-x^(2))` `rArr xsqrt(2x-x^(2)) + sqrt(1-x)- xsqrt(1-x^(2)) = sqrt(1-x^(2))` `rArr " "xsqrt(2x-x^(2)) + xsqrt(1-x^(2)) = 0` `rArr" "x[sqrt(2x-x^(2)) -sqrt(1-x^(2))]=0` `rArr x = 0 or sqrt(2x-x^(2))-sqrt(1-x^(2))=0` Now `sqrt(2x-x^(2))-sqrt(1-x^(2)) = 0` `rArr" "sqrt(2x -x^(2)) = sqrt(1-x^(2))` `rArr" "2x -x^(2) = 1- x^(2)` `rArr " "2x =1` `rArr " " x=1//2` ` therefore " "x=0 or x = 1//2` |
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| 53. |
`sin^(- 1)(- 1/2)` |
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Answer» `sin^(-1)""(-(1)/(2))=-sin^(-1)""((1)/(2))` `= - sin^(-1)""(sin""(pi)/(6))=-(pi)/(6)` |
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| 54. |
If `sin ^(-1) x + sin ^(-1) y = (pi)/(6)`, then `cos^(-1) x + cos ^(-1) y =?`A. `(pi)/(6)`B. `(5pi)/(6)`C. `(pi)/(3)`D. `(2pi)/(3)` |
| Answer» Correct Answer - B | |
| 55. |
`sin[sin^(-1) (-(1)/(2))+ (pi)/(3)]=?`A. `-sqrt(3)/(2)`B. `-(1)/(2)`C. `(1)/(2)`D. `sqrt(3)/(2)` |
| Answer» Correct Answer - C | |
| 56. |
If `sin^(- 1)(x)+sin^(- 1)(2x)=pi/3` then `x=`A. `-(1)/(2)`B. `(1)/(2)`C. `pmsqrt((3)/(28))`D. None of these |
| Answer» Correct Answer - C | |
| 57. |
`sin(2tan^(-1) .(4)/(5))= ?`A. `(40)/(41)`B. `(9)/(41)`C. `(16)/(25)`D. None of these |
| Answer» Correct Answer - A | |
| 58. |
solve`:sin^-1x+sin^-1 2x=pi/3` |
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Answer» `sin^(-1)x + sin^(-1)2 x = (pi)/(3)` `rArr " "sin^(1)x + sin^(1)2x = sin ^(1).(sqrt(3))/(2) sin^(-1)2x = sin^(1).sqrt(3)/(2) -sin^(-1)x` `" "=sin^(1)[(sqrt(3))/(2) sqrt(1-x^(2) )-xsqrt(1-((sqrt(3))/(2))^(2))]` `rArr" "2x = sqrt(3)/(2)sqrt(1-x^(2))-(x)/(2)` `rArr" "(5x)/(2) =(sqrt(3))/(2) sqrt(1-x^(2))` `" "5x = sqrt(3) sqrt(1-x^(2))` `25x^(2) = 3(1-x^(2))=3-3x^(2)` `rArr " "28x^(2) =3` `rArr " "x^(2) =(3)/(28)` ` rArr" "x = pm (sqrt(3))/(2sqrt(7))` but gives equation is not satisfied by `x= - sqrt(3)/(2sqrt(7))` Therefore `x=sqrt(3)/(2sqrt(7))` |
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| 59. |
Solve the equations.`tan^(-1)(1-x)/(1+x)=1/2tan^(-1)x ,(x >0)` |
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Answer» `tan^(-1)""(1-x)/(1+x)=(1)/(2)tan^(-1)x, (x gt0)` `tan^(-1)1-tan^(-1)x=(1)/(2)tan^(-1)x` `implies (pi)/(4)=(1)/(2)tan^(-1)x+tan^(-1)x` `implies(3)/(2)tan^(-1)x=(pi)/(4)` `impliestan^(-1)x=(pi)/(4)` `implies tan^(-1)=(pi)/(6)` `implies x=tan""(pi)/(6)=(1)/(sqrt(3))` |
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| 60. |
Prove that:`2sin^(-1)3/5=tan^(-1)(24)/7` |
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Answer» LHS=`2 sin^(-1)""(3)/(5)=2tan^(-1)""((3)/(5))/(sqrt(1-((3)/(5))^(2)))` `( :. sin^(-1)x=tan^(-1)"(x)/(sqrt(1-x^(2))))` `=2tan^(-1)""(3)/(4)=tan^(-1)""(2xx(3)/(4))/(1-((3)/(4))^(2))` `=tan^(-1)""(3//2)/(7//16)=tan^(-1)""((3)/(2)xx(16)/(7))` `=tan^(-1)""((24)/(7))`=RHS. Hence Proved. |
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| 61. |
If ` tan ^(-1) 2x + tan ^(-1) 3 xx = (pi)/(4)`, then x = ?A. 1B. -1C. `-(1)/(6)`D. `(1)/(6)` |
| Answer» Correct Answer - D | |
| 62. |
Solve the equations.`2tan^(-1)(cosx)=tan^(-1)(2cose c x)` |
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Answer» `2tan^(-1)""(cosx)=tan^(-1)(2 " cosec " x)` `implies tan^(-2)((2 cosx)/(1-cos^(2)x))=tan^(-2)(2 " cosec " x)` `implies (2 cos x)/(sin^(2)x)=(2)/(sinx)` `implies cot x=1 impliesx=(pi)/(4)` |
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| 63. |
`tan^(- 1)(1/(sqrt(x^2-1))),|x|gt1` |
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Answer» `tan^(-1)""(1)/(sqrt(x^(2)-1)) " " Letx=cosec theta implies theta =cosec^(-1) x ` `=tan^(-1)""(1)/(sqrt("cosec"^(2)theta-1))=tan^(-1)((1)/(cot theta ))` `=tan^(-1)(tan theta)=theta=cosec^(-1)x` |
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| 64. |
Prove that: `tan^(-1)2/(11)+tan^(-1)7/(24)=tan^(-1)1/2` |
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Answer» `LHS=tan^(-1)""(2)/(11)+tan^(-1)""(7)/(24)` `=tan^(-1)""((2)/(11)+(7)/(24))/(1-(2)/(11)xx(7)/(24))` `tan^(-1)""((48+77)/(264))/((264-14)/(264))=tan^(-1)""(125)/(250)` `tan^(-1)""(1)/(2)=` RHS Hence Proved. |
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| 65. |
Solution of `tan ^(-1) (1 + x) + tan ^(-1) ( 1- x) = (pi)/(2)` is: |
| Answer» Correct Answer - A | |
| 66. |
Express each of thefollowing in the simplest form:`tan^(-1){sqrt((1-cosx)/(1+cosx))}, -pi |
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Answer» `tan^(-1)""(sqrt((1-cosx)/(1+cos x)))` `=tan^(-1)(sqrt((2sin^(2)""(x)/(2))/(2 cos^(2)""(x)/(2))))=tan^(-1)((sin""(x)/(2))/(cos""(x)/(2)))` `=tan^(-1)(tan"(x)/(2))=(x)/(2)` |
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| 67. |
Write the following function in the simplest form: `tan^(-1)(sqrt(1+x^2)-1)/x , x!=0` |
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Answer» `tan^(-1)""(sqrt(1+c^(2))-1)/(x) " " Let x=tan theta implies tan^(-1)x=theta` `=tan^(-1)""(sqrt(1+tan^(2))-1)/(tan theta)` `tan^(-1)((sec theta-1)/(tan theta))=tan^(-1)(((1)/(cos theta)-1)/((sin theta)/(cos theta)))` `=tan^(-1)((1-cos theta)/(sin theta))=tan^(-1)((2 sin^(2)""(theta)/(2))/(2 sin ""(theta)/(2) cos ""(theta)/(2)))` `tan^(-1)(tan""(theta)/(2))=(1)/(2) theta=(1)/(2) tan^(-1)x` |
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| 68. |
`tan^(-1)((cosx-sinx)/(cosx+sinx))=pi/4-x` |
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Answer» `tan^(-1)((cosx-sinx)/(cosx+sinx))` `=tan^(-1)""(((cosx)/(cosx)-(sinx)/(cosx))/((cosx)/(cosx)+(sinx)/(cosx)))` `tan^(-1)((1-tanx)/(1+tanx))` `= tan^(-1){tan((pi)/(4)-x)}=(pi)/(4)-x` |
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| 69. |
Write the following function in the simplest form: `tan^(-1)((3a^2x-x^3)/(a^3-3a x^2)), a >0;(-a)/(sqrt(3))lt=xlt=a/(sqrt(3))` |
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Answer» `tan^(-1)((3a^(2)x-x^(3))/(a^(3)-3ax^(2)))` `tan^(-1)((3a^(2)*a tan theta-a^(3)tan^(3)theta)/(a^(3)-3a*a^(2)tan^(2)theta))` `=tan^(-1)((3 tan theta-tan^(3)theta)/(1-3 tan^(2)theta))" " Let x = a tan theta implies (x)/(a)=tan thetaimplies theta = tan^(-1)""((x)/(a))` `=tan^(-1)(tan 3 theta)3 theta` `3 tan^(-1)""(x)/(a)` |
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| 70. |
For the principal values, evaluate each of the following:`tan^(-1){2cos(2s in^(-1)1/2)}``"cot"[sin^(-1){cos(tan^(-1)1)}]` |
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Answer» `tan^(-1)[2 cos ( 2 sin^(-1)""(1)/(2))]` `=tan^(-1)[2 cos(2 sin^(-1)sin""(pi)/(6))]` `=tan^(-1)[2 cos (2*(pi)/(6))]` `=tan^(-1)(2*(1)/(2))=tan^(-1)(1)` `=tan^(-1)(tan""(pi)/(4))=(pi)/(4)` |
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| 71. |
Prove that:`tan^(-1)sqrt(x)=1/2cos^(-1)((1-x)/(1+x)), x in [0,1]` |
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Answer» Let `sqrt(x)=tan thetaimpliestan^(2)theta` `:. RHS=(1)/(2) cos^(-1)""((1-x)/(1+x)), " " x in [0,1]` `=(1)/(2) cos^(-1)""((1-tan^(2) theta)/(1+tan^(2)theta))` `=(1)/(2)cos^(-1)(cos2theta)=(1)/(2)(2theta)` `=theta=tan^(-1)sqrt(x)=` LHS Hence Proved. |
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| 72. |
Prove that `cos^(-1)4/5cos^(-1)(12)/(13)=cos^(-1)(33)/(65)` |
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Answer» LHS`=cos^(-1)""(4)/(5)+cos^(-1)""(12)/(13)` `=cos^(-1)[(4)/(5)xx(12)/(13)-sqrt(1-((4)/(5))^(2))sqrt(1-((12)/(13))^(2))]` `=cos^(-1)[(48)/(65)-(3)/(5)xx(5)/(13)]=cos^(-1)((48)/(65)-(15)/(65))` `=cos^(-1)""(33)/(65)=` RHS. Hence Proved. |
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| 73. |
The value of `cos (2Cos^-1 0.8)` isA. 0.28B. 0.48C. 0.6D. None of these |
| Answer» Correct Answer - A | |
| 74. |
Prove that: `sin^(-1)((63)/(65))=sin^(-1)(5/(13))+cos^(-1)(3/5)` |
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Answer» RHS=`sin^(-1)"(5)/(13)+cos^(-1)""(3)/(5)` `=tan^(-1)""((5)/(13))/(sqrt(1-((5)/(13))^(2)))+tan^(-1)""(sqrt(1-((3)/(5))^(2)))/((3)/(5)) " "( :. sin^(-1)x=tan^(-1)""(x)/(sqrt(1-x^(2))) " and " cos^(-1)x=tan^(-1)""(sqrt(1-x^(2)))/(x))` `=tan^(-1)""(5)/(12)+tan^(-1)""(4)/(3)` `=tan^(-1)""((5)/(12)+(4)/(3))/(1-(5)/(12)xx(4)/(3))=tan^(-1)""((15+48)/(36))/((36-20)/(36))` `tan^(-1)""(63)/(16)`= LHS Hence Proved. |
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| 75. |
If `x_1,x_2, x_3, x_4` are the roots of the equation `x^4-x^3 sin2 beta+ x^2.cos2 beta-xcos beta-sin beta=0`, then `tan^-1x_1+tan^-1x_2+tan^-1x_3+tan^-1x_4` is equal toA. `alpha`B. `90^(@)- alpha`C. `180^(@)- alpha`D. `270^(@) - alpha` |
| Answer» Correct Answer - B | |