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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.
| 501. |
Statement-1: If `alpha,beta` roots of the equation `18(tan^(-1)x)^(2)-9pi tan^(-1)x+pi^(2)=0 then alpha+beta =(4)/sqrt(3)` Statement-2: `sec^(2)cos^(-1)(1/4)+cosec^(2)sin^(-1)(1/5)=41`A. Statement-1 is is True, Statement-2 is true, Statement-2 is a correct explanation for Statement-1.B. Statement-1 is True, Statement-2 is True, Statement-2 is not a correct explanation for Statement-1.C. Statement-1 is True, Statement-2 is False.D. Statement-1 is False, Statement-2 is True. |
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Answer» we have `(tan^(-1)x)^(2)-(pi)/(2)tan^(-1)x+(pi)^(2)/(18)=0` `rarr (tan^(-1)x)^(2)-((pi)/(6)+(pi)/(3))tan^(-1)x+(pi^(2))/(18)=0 ` `rarr tan^(-1)x=(pi)/(6),(pi)/(3) rarr tan^(-1) alpha =(pi)/(6) and tan^(-1) beta =-(pi)/(3)` `rarr alpha = tan (pi)/(6) =(1)sqrt(3) and beta =tan (pi)/(3)=sqrt(3) rarr alpha+beta =(4)/sqrt(3)` `so statement 1 is true `sec^(2)(cos^(-1)1/4)+cosec^(2)(sin^(-1)1/5)` `={sec(sec^(-1)4)}^(2)+(cosec^(-1)5)}^(2)=16+25=4` so statement 2 is true |
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| 502. |
If `1/2 le x le 1` then `sin^(-1) (3x-4x^(3))` equalsA. `3 sin^(-1) x `B. `pi -3 sin^(-1) x `C. `-pi -3 sin^(-1)x`D. none of these |
| Answer» Correct Answer - B | |
| 503. |
if `-1 le x le -1/2 then cos^(-1)(4x^(3)-3x)` equalsA. `3cos^(-1)x`B. `2pi -3cos^(-1)x`C. `-2pi +3 cos^(-1)x`D. none of these |
| Answer» Correct Answer - C | |
| 504. |
If` x in (1,oo) then tan^(-1)(2x)/(1-x^(2))` equalsA. `2 tan^(-1)x`B. `-pi+2tan^(-1)x`C. `pi+2 tan^(-1)x`none of theseD. |
| Answer» Correct Answer - B | |
| 505. |
If `x in [-1,1] then sin^(-1)((2x)/(1+x^(2)))` equalsA. `2 tan^(-1)x`B. `pi-2 tan^(-1)x`C. `-pi-2 tan^(-1)x`D. none of these |
| Answer» Correct Answer - A | |
| 506. |
If `x in (1,oo) then sin^(-1)((2x)/(1+x^(2)))` equalsA. `2 tan^(-1)x`B. `pi-2 tan^(-1)x`C. `-pi-2 tan^(-1)x`D. none of these |
| Answer» Correct Answer - B | |
| 507. |
If `x in (-oo,-1) then sin^(-1)((2x)/(1+x^(2)))` equalsA. `2 tan^(-1)x`B. `pi-2 tan^(-1)x`C. `-pi-2 tan^(-1)x`D. none of these |
| Answer» Correct Answer - C | |
| 508. |
If `0 le x lt oo, then cos^(-1)((1-x^(2))/(1+x^(2)))` equalsA. `2tan^(-1)x`B. `-2tan^(-1)x`C. `pi-2 tan^(-1)x`D. `pi+2 tan^(-1)x` |
| Answer» Correct Answer - A | |
| 509. |
If `-oo lt x le 0 then cos ^(-1)((1-x^(2))/(1+x^(2)))`equalsA. `2tan^(-1)x`B. `-2 tan^(-1)x`C. `pi-2 tan^(-1)x`D. `pi+2tan^(-1)x` |
| Answer» Correct Answer - B | |
| 510. |
If `1 lt x lt 1 then tan^(-1) (2x)/(1-x^(2))` equalsA. `2 tan^(-1)x`B. `-pi+2tan^(-1)x`C. `pi+2tan^(-1)x`D. none of these |
| Answer» Correct Answer - A | |
| 511. |
Statement -1: if `-1lexle1 then sin^(-1)(-x)=-sin^(-1)x and cos^(-1)(-x)=pi-cos^(-1)x` Statement-2: If `-1lexlex then cos^(-1)x=2sin^(-1)sqrt((1-x)/(2))= 2cos^(-1)sqrt((1+x)/(2))`A. Statement-1 is is True, Statement-2 is true, Statement-2 is a correct explanation for Statement-1.B. Statement-1 is True, Statement-2 is True, Statement-2 is not a correct explanation for Statement-1.C. Statement-1 is True, Statement-2 is False.D. Statement-1 is False, Statement-2 is True. |
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Answer» clearly statement -1 true putting `x =cos theta` we get `2 sin^(-1) sqrt(1-x)/(2)=2cos^(-1) sqrt(1+x)/(2)=theta =cos^(-1)x` so statement 2 is also true |
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| 512. |
If `x lt 0` then `tan^(-1)(1/x)` equalsA. `cot^(-1)x`B. `-cot^(-1)x`C. `-pi + cot^(-1)x`D. `-pi -cot^(-1) x ` |
| Answer» Correct Answer - C | |
| 513. |
The value of `sin[cot^(-1){cos(tan^(-1) x)}]` isA. `sqrt(x^(2)+2)/(sqrt(x^(2)+1)`B. `sqrt(x^(2)+1)/(sqrt(x^(2)+2)`C. `(x)/sqrt(x^(2)+2)`D. `(1)/sqrt(x^(2)+2)` |
| Answer» Correct Answer - B | |
| 514. |
If `x lt -(1)/sqrt(3) , then tan^(-1)(3x-x^(3))/(1-3x^(2))` equalsA. `3 tan^(-1)x`B. `-pi+3tan^(-1)x`C. `pi+3tan^(-1)x`D. none of these |
| Answer» Correct Answer - C | |
| 515. |
`1/2tan^(- 1)(12/5)` is equal toA. `tan^(-1)(3/2)`B. `tan^(-1)(2/3)`C. `tan^(-1)(3/4)`D. `tan^(-1)(7/17)` |
| Answer» Correct Answer - B | |
| 516. |
If `-(1)/sqrt(3) lt x lt (1)/sqrt(3), then tan^(-1) (3x-x^(3))/(1-3x^(2))` equalsA. `3tan^(-1)x`B. `-pi+3 tan^(-1)x`C. `pi+3 tan^(-1)x`D. none of these |
| Answer» Correct Answer - A | |
| 517. |
Prove that `sin(3sin^-1(1/3)) = 23/27` |
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Answer» LHS `sin(3sin^(-1)(1/3))` `Let sin^(-1)(1/2)=x` `sin(3x)=3sinx-4sinn^3x` `=3*1/3-4(1/3)^3` `=1-4/27` `=23/27` RHS. |
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| 518. |
If `sin^(-1)(2xsqrt(1-x^(2)))-2 sin^(-1) x=0` then x belongs to the intervalA. `[-1,1]`B. `[-1//sqrt(2),1sqrt(2)]`C. `[-1,-1//sqrt(2)]`D. `[1//sqrt(2),1]` |
| Answer» Correct Answer - B | |
| 519. |
`sin^(- 1)(4/5)+2tan^(- 1)(1/3)=`A. `tan^(-1)(3/4)`B. `tan^(-1)(4/3)`C. `tan^(-1)(3/sqrt10)`D. `pi/2` |
| Answer» Correct Answer - D | |
| 520. |
The value of `cot^(-1){(sqrt(1-sinx)+sqrt(1+sinx))/(sqrt(1-sinx) -sqrt(1+sinx))} is (0 lt x lt (pi)/(2))`A. `pi-(x)/(2)`B. `2pi-x`C. `(x)/(2)`D. `2pi-(x)/(2)` |
| Answer» Correct Answer - A | |
| 521. |
Prove the following: `cot^(-1) [(sqrt(1+sinx) + sqrt(1-sinx))/(sqrt(1+sinx)-sqrt(1-sinx))] = x/2 ; x in(0,pi/4)` |
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Answer» `sqrt(1+sinx)=sqrt(sin^2(x/2)+cos^2(x/2)+2sin(x/2)cos(x/2)` `=cos(x/2)+sin(x/2)` `sqrt(1-sinx)=cos(x/2)-sin(x/2)` `cot^(-1)((cos(x/2)+sin(x/2)+cos(x/2)-sin(x/2))/(cos(x/2)+sin(x/2)-cos(x/2)+sin(x/2)))` `cot^(-1)cot(x/2)=x/2`. |
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| 522. |
The value of x where `xgt0 and tan(sec^(-1)(1/x))=sin(tan^(-1)2)` isA. `sqrt(5)`B. `sqrt(5)/(3)`C. 1D. `2/3` |
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Answer» We have `tan(sec^(-1)(1)/(x))=sin(tan^(-1)2)` `rarr tan^(-1)sqrt(1-x^(2))/(x)=sin(sin^(-1))(2)/sqrt(5)` `rarr sqrt(1-x^(2))/(x)=(2)/sqrt(5)rarr(1)/(x^(2))-1=4/5` `rarr(1)/(x^(2))=9/5rarrx=sqrt(5)/(3)` |
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| 523. |
Express in terms of : `sin^(-1)(2xsqrt(1-x^(2)))` to `sin^(-1)x` for `1gexgt1/(sqrt(2))`A. `2sin^(-1)x`B. `2cos^(-1)x`C. `-2sin^(-1)x`D. `-2cos^(-1)x` |
| Answer» Correct Answer - B | |
| 524. |
If `y=cos^(-1)(cos10)` then y is equal to |
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Answer» We know , `cos^-1 in [0,pi).` So, we have to find a value from `0` to `pi`. Now, `cos(2npi-theta) = cos theta` For `n = 1, theta = 10`, `(2npi-theta) = (2pi -10) = (2**3.14 - 10) lt 0` For `n = 2, theta = 10`, `(2npi-theta) = (4pi -10) = (4**3.14 - 10) lt pi` So, it is in the range `[0,pi)`. `:. y = cos^-1(cos10) = cos^-1(cos(4pi-10)) = 4pi-10` `=> y = 4pi-10` |
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| 525. |
Find the value of expression: `sin(2tan^(-1)1/3)+cos(tan^(-1)2sqrt(2))`A. `14/15`B. `3/4`C. `(2sqrt2)/7`D. `(2sqrt2)/15` |
| Answer» Correct Answer - A | |
| 526. |
Find the value of expression: `sin(2tan^(-1)1/3)+cos(tan^(-1)2sqrt(2))`A. `12/13`B. `13/14`C. `14/15`D. none of these |
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Answer» `sin(2tan^(-1)1/3)+cos(tan^(-1)2sqrt(2))` `=sin(sin^(-1)3/5)+cos(cos^(-1)1/3)["before" 2tan^(-1)x=sin^(-1) (2x)/(1+x^(2))]` `=3/5+1/3=14/15` |
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| 527. |
Prove that:`cot^(-1)((sqrt(1+sinx)+sqrt(1-sinx))/(sqrt(1+sinx)-sqrt(1-sinx)))=x/2,x in (0,pi/4)` |
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Answer» `((sqrt(1+sinx)+sqrt(1-sinx))/(sqrt(1+sinx)-sqrt(1-sinx)))*((sqrt(1+sinx)+sqrt(1-sinx))/(sqrt(1+sinx)+sqrt(1-sinx)))` `((sqrt(1+sinx)+sqrt(1-sinx))^2/(1+sinx-1+sinx))` `(1+sinx+1-sinx+2sqrt((1+sinx)(1-sinx)))/(2sinx)` `(2+2sqrt(1-sin^2x))/(2sinx)` `(1+cosx)/sinx` `(1+2cos^2(x/2))/(2cos(x/2)sin(x/2))` `cot(x/2)` `cot^(-1)(cos(x/2))` `=x/2`. |
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| 528. |
Which of the following angles is greater?`theta_1=sin^(-1)+sin^(-1)1/3ortheta_2=cos^(-1)4/5+cos^(-1)1/3`A. `theta_(1) gt theta_(2)`B. `theta_(1)=theta_(2)`C. `theta_(1) lt theta_(2)`D. none of these |
| Answer» Correct Answer - C | |
| 529. |
`tan^(- 1)(1/4)+tan^(- 1)(2/9)=1/2tan^(- 1)(4/3)`A. `1/2cos^(-1)(3/5)`B. `1/2sin^(-1)(3/5)`C. `1/2tan^(-1)(3/5)`D. `tan^(-1)(1/2)` |
| Answer» Correct Answer - D | |
| 530. |
Find `tan^(-1)x/(sqrt(a^2-x^2))`in terms of `sin^(-1)`where `x in (0, a)dot` |
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Answer» `tan^(-1)(x/sqrt(a^2-x^2))=theta` `tantheta=x/sqrt(a^2-x^2)` `sintheta=x/a` `theta=sin^(-1)(x/a)`. |
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| 531. |
If `A=2tan^(-1)(2sqrt(2)-1)a n dB=3sin^(-1)(1/3)+sin^(-1)(3/5),`then which is greater.A. A=BB. `A lt B`C. `A gt B`D. none of these |
| Answer» Correct Answer - C | |
| 532. |
If `x=sin(2tan^(- 1)2), y=sin(1/2tan^(- 1)(4/3))` , then -A. `x=y^(2)`B. `y^(2)=1-x`C. `x^(2)=(y)/(2)`D. `y^(2)=1+x` |
| Answer» Correct Answer - B | |
| 533. |
Show that `sin^(-1)(3/5)-sin^(-1)(8/17)=cos^(-1)(84/85)` |
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Answer» `sin^(-1)(3/5)=x and sin^(-1)(8/17)=y` `3/5=sinx and 8/17=siny` cosx=4/5 and cosy=15/17 cos(x-y)=cosxcosy+sinxsiny =`(4/5)(15/17)+(3/5)(8/17)` cos(x-y)=84/85 `x-y=cos^(-1)(84/85)` `sin^(-1)(3/5)-sin^(-1)(8/17)=cos^(-1)(84/85)` Hence proved |
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| 534. |
If `2tan^(-1)x+sin^(-1)(2x)/(1+x^2)`is independent of `x ,`then`x >1`(b) `x |
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Answer» if|x|`<=`1 then `2tan^(-1)x=sin^(-1)((2x)/(1+x^2))` If x>1 then `2tan^(-1)x=pi-sin^(-1)((2x)/(1+x^2))` If x<-1, then `2tan^(-1)x=-pi-sin((2x)/(1+x^2))` x>1 or x<-1 Option A and B are correct. |
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| 535. |
If `2tan^(-1)x+sin^(-1)((2x)/(1+x^2) )` is independent of x then :A. `x gt 1`B. `x lt -1`C. `0 lt x lt 1`D. `-1 lt x lt 0` |
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Answer» Correct Answer - A::B We know that if `|x| le 1, " then " 2 tan^(-1) x = sin^(-1) ((2x)/(1 + x^(2)))` if `x gt 1, " then " 2 tan^(-1) x = pi - sin^(-1) ((2x)/(1 + x^(2)))` if `x lt -1, " then " 2 tan^(-1) x = - pi - sin^(-1) ((2x)/(1 + x^(2)))` Hence, the required values are `x lt -1 " or " x gt 1` |
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| 536. |
If `2tan^(-1)x+sin^(-1)((2x)/(1+x^2) )` is independent of x then :A. `x in [1,oo) in (-oo,-1)`B. `x in [-1,1]`C. `x in (-oo,1]`D. none of these |
| Answer» Correct Answer - A | |
| 537. |
Let `alpha=som^(-1)((36)/(85)),beta=cos^(-1)(4/5)a n dgamma=tan^(-1)(8/(15))`then`cotalpha+cotbeta+cotgamma=cotalphacotbetacotgamma``tanalphatanbeta+tanbetatangamma+tanalphatangamma=1``tanalpha+tanbeta+tangamma=tanalphatanbetatangamma``cotalphacotbeta+cotbetacotgamma+cotalphacotgamma=1`A. `cot alpha + cot beta + cot gamma = cot alpha cot beta cot gamma`B. `tan alpha tan beta + tan beta tan gamma + tan alpha tan gamma = 1`C. `tan alpha + tan beta + tan gamma = tan alpha tan beta tan gamma`D. `cot alpha cot beta + cot beta cot gamma + cot alpha cot gamma = 1` |
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Answer» Correct Answer - A::B `alpha = sin^(-1).(36)/(85) rArr alpha = (36)/(85) rArr tan alpha = (36)/(77)` `beta = cos^(-1) ((4)/(5)) rArr cos beta = (4)/(5) rArr tan beta = (3)/(4)` and `tan gamma = (8)/(15)` `:. tan (alpha + beta + gamma) = (tan alpha + tan beta + tan gamma - tan alpha tan beta tan gamma)/(1-tan alpha tan beta - tan beta tan gamma - tan gamma tan alpha)` `= ((36)/(77) + (3)/(4) + (8)/(15) -(36)/(77) .(3)/(4).(8)/(15))/(1-((36)/(77) xx (3)/(4) + (8)/(15) xx (3)/(4) + (8)/(15) xx (36)/(77)))` `rArr tan (alpha + beta + gamma) = oo` `rArr alpha + beta + gamma =(pi)/(2)` `rArr cot alpha + cot beta + cot gamma = cot alpha cot beeta cot gamma` `rArr tan alpha tan beta + tan beta tan gamma + tan alpha tan gamma = 1` |
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| 538. |
If `sin^(-1) ((4x)/(x^(2) + 4)) + 2 tan^(-1) (-(x)/(2))` is independent of x, find the value of x |
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Answer» `E = sin^(-1) ((4x)/(x^(2) + 4)) + 2 tan^(-1) (-(x)/(2))` `= sin^(-1) ((2 xx (x)/(2))/(((x)/(2))^(2) + 1)) -2 tan^(-1). (x)/(2)` `= 0` (For E to be independent of x) `rArr |(x)/(2)| le 1` or `|x| le 2 " or " -2 le x le 2` |
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| 539. |
Find the value of `2 cos^(-1).(3)/(sqrt13) + cot^(-1).(16)/(63) + (1)/(2) cos^(-1).(7)/(25)` |
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Answer» `E = 2 cos^(-1).(3)/(sqrt13) + cot^(-1).(16)/(63) + (1)/(2) cos^(-1).(7)/(25)` `= 2 tan^(-1).(2)/(3) + tan^(-1).(63)/(16) + (1)/(2) cos^(-1).(7)/(25)` Now, `2 tan^(-1).(2)/(3) = tan^(-1). (2 ((2)/(3)))/(1 - (4)/(9))` `= tan^(-1).(12)/(5)` Let `(1)/(2) cos^(-1).(7)/(25) = tan^(-1) x` `rArr cos^(-1).(7)/(25) = 2 tan^(-1) x` `rArr tan^(-1).(24)/(7) = tan^(-1).(2x)/(1 - x^(2))` `rArr (24)/(7) = (2x)/(1 - x^(2))` `rArr 12x^(2) + 7x - 12 = 0` `rArr (4x -3) (3x + 4) = 0` `rArr x = 3//4` `:. E = tan^(-1).(12)/(5) + tan^(-1).(63)/(16) + tan^(-1).(3)/(4)` `= pi + tan^(-1).((12)/(5) + (3)/(4))/(1-((12)/(5)) ((3)/(4))) + tan^(-1).(63)/(16)` `= pi + tan^(-1) (-(63)/(16)) + tan^(-1).(63)/(16)` `= pi` |
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| 540. |
If `cos^(-1)(x)/(2)+cos^(-1)(y)/(3)=theta` then the maximum of `9x^(2)-12xy costheta + 4y^(2)` isA. 18B. 30C. 24D. 36 |
| Answer» Correct Answer - D | |
| 541. |
If `(tan^(-1)x)^2+(cot^(-1)x)^2=(5pi^2)/8,`then find `xdot`A. `-1`B. 1C. 0D. none of these |
| Answer» Correct Answer - A | |
| 542. |
If `sin^(-1)(2x)/(1+x^2)=tan^(-1)(2x)/(1-x^2)`, then find the value of `xdot` |
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Answer» `sin^(-1).(2x)/(1 + x^(2)) = 2 tan^(-1) x, -1 le x le 1` `tan^(-1).(2x)/(1 - x^(2)) = 2 tan^(-1) x, -1 lt x lt 1` Therefore, equation is satisfied by `-1 lt x lt 1` |
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| 543. |
Find the principal value of `cot^(-1)(-sqrt3)` |
| Answer» Correct Answer - `(5pi)/6` | |
| 544. |
Find the principal value of ltrbgt (a) `cosec^(-1) (-1)` (b) `cot^(-1) (-(1)/(sqrt3))` |
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Answer» Correct Answer - (a) `-(pi)/(2)` (b) `(2pi)/(3)` (a) Let `theta` be the principal value of `cosec^(-1) (-1)`. `theta in [-pi//2, pi//2] - {0} and cosec^(-1) (-1) = theta` `:. theta = -(pi)/(2)` because `-(pi)/(2) in[-(pi)/(2), (pi)/(2)] - {0} and cosec (-(pi)/(2)) = -1` Therefore, principal value of `cosec^(-1) (-1) " is " -(pi)/(2)` (b) Let `theta` be the principal vlaue of `cot^(-1) (-(1)/(sqrt3))` `:. theta in(0, pi) and cot theta = - (1)/(sqrt3)` `:. theta = pi - (pi)/(3) = (2pi)/(3)`, because `(2pi)(3) in (0, pi) and cot(pi - (pi)/(3)) = - cot (pi)/(3) = -(1)/(sqrt3)` Therefore, principal value of `cot^(-1) (-(1)/(sqrt3)) = (2 pi)/(3)` |
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| 545. |
if `6sin^(-1)(x^2 -6x + 8.5)=pi` thenA. `x=1`B. `x=2`C. `x=3`D. `x=4` |
| Answer» Correct Answer - B::D | |
| 546. |
If `cos^(-1).(6x)/(1 + 9x^(2)) = -(pi)/(2) + 2 tan^(-1) 3x`, then find the values of x |
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Answer» `cos^(-1).(6x)/(1 + 9 x^(2)) = -(pi)/(2) + 2 tan^(-1) 3x` `(pi)/(2) - sin^(-1).(6x)/(1 + 9x^(2)) = -(pi)/(2) + 2 tan^(-1) 3x` `sin^(-1).(6x)/(1 + 9x^(2)) = pi - 2 tan^(-1) 3x` `sin^(-1).(2 xx 3x)/(1 + (3x)^(2)) = pi - 2 tan^(-1) 3x` It is true when `3x gt 1` or `x gt (1)/(3)` i.e., `x in ((1)/(3), oo)` |
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| 547. |
The principal value of `cot^(-1)x` lie inA. `(-pi/2,pi/2)`B. `[-pi/2,pi/2]`C. `(0,pi)`D. `[0,pi]` |
| Answer» Correct Answer - C | |
| 548. |
Find the principal value of `sec^(-1)(2/sqrt3)` |
| Answer» Correct Answer - `pi/6` | |
| 549. |
Find the principal value of `sin^(-1)(-sqrt3/2)` |
| Answer» Correct Answer - `-pi/3` | |
| 550. |
Find the principal value of the following (i) `sin^(-1).(1)/(2)` (ii) `tan ^(-1).(1)/sqrt(3)` (iii) `cot ^(-1)(-sqrt3)` |
| Answer» Correct Answer - `pi/6` | |