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Inverse Trigonometric Functions Set-4 MCQ Online Practice Test | Class 12 | Quiz Tweek

Inverse Trigonometric Functions Set-4

Practice Important MCQs for Inverse Trigonometric Functions — Mathematics, Class 12.

Inverse trigonometric functions are essential in Class 12 because they connect trigonometry with algebra through principal values and well-defined ranges. Mastery of identities, branch conditions, and limits involving $\sin^{-1}$ , $\cos^{-1}$ , and $\tan^{-1}$ is frequently tested in board exams and is also crucial for competitive exams where correct domain/range handling distinguishes top scorers.

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Q1. If $y=\arcsin\!\left(\dfrac{3}{5}\right)$ , then the value of $\cos 2y$ is:

(A)

$-\dfrac{7}{25}$

(B)

$\dfrac{7}{25}$

(C)

$1$

(D)

$\dfrac{9}{25}$

Q2. Evaluate $\arctan 1+\arctan 2+\arctan 3$ .

(A)

$\dfrac{3\pi}{4}$

(B)

$\dfrac{\pi}{2}$

(C)

$\dfrac{\pi}{4}$

(D)

$\pi$

Q3. If $\arcsin x+\arccos(2x)=\dfrac{\pi}{2}$ , then $x$ equals:

(A)

$0$

(B)

$\dfrac{1}{2}$

(C)

$-\dfrac{1}{2}$

(D)

$1$

Q4. Evaluate

\lim_{x\to 0}\frac{\arcsin x-\arctan x}{x^3}.

(A)

$0$

(B)

$\dfrac{1}{6}$

(C)

$\dfrac{1}{2}$

(D)

$-\dfrac{1}{2}$

Q5. For real $x\neq\pm 1$ , the equality $\displaystyle \arctan\!\left(\frac{2x}{1-x^2}\right)=2\arctan x$ holds for which values of $x$ ?

(A)

$x>1$

(B)

$-1<x<1$

(C)

$x<-1$

(D)

$(-\infty,-1)\cup(1,\infty)$

Q6. Let $\theta=\sin^{-1}\!\left(\frac{3}{5}\right)$ and $\phi=\cos^{-1}\!\left(\frac{5}{13}\right)$ . Compute $\sin(\theta+\phi)$ .

(A)

$\displaystyle \frac{9}{13}$

(B)

$\displaystyle \frac{63}{65}$

(C)

$\displaystyle \frac{7}{13}$

(D)

$\displaystyle \frac{29}{65}$

Q7. Solve for real $x$ :

\sin^{-1}\!\left(\frac{2x}{1+x^{2}}\right)=\cos^{-1}\!\left(\frac{1-x^{2}}{1+x^{2}}\right).

(A)

$(-1,1)$

(B)

$\{0,1\}$

(C)

$\mathbb{R}$

(D)

$[0,1]$

Q8. Find all real solutions of

\sin^{-1}\!\left(\frac{2x}{1+x^{2}}\right)+\tan^{-1}x=\frac{\pi}{2}.

(A)

$\displaystyle \frac{1}{\sqrt{3}}$

(B)

$-\displaystyle \frac{1}{\sqrt{3}}$

(C)

$\sqrt{3}$

(D)

No real solution

Q9. The equation

\tan^{-1}x+\tan^{-1}\!\left(\frac{1-x}{1+x}\right)=\frac{\pi}{4}

is to be solved for real $x\neq -1$ . The solution set is:

(A)

$(-\infty,-1)$

(B)

$(-1,1)$

(C)

$(-1,\infty)$

(D)

No real solution

Q10. Determine the range of the function

f(x)=\sin^{-1}\!\left(\frac{2x}{1+x^{2}}\right)+\cos^{-1}\!\left(\frac{1-x^{2}}{1+x^{2}}\right),\qquad x\in\mathbb{R}.

(A)

$(0,\pi)$

(B)

$[0,\pi]$

(C)

$[0,\tfrac{\pi}{2}]$

(D)

$[\tfrac{\pi}{2},\pi]$

Q11. Evaluate $\tan\!\left(\sin^{-1}\frac{3}{5}+\cos^{-1}\frac{12}{13}\right)$

(A)

$\displaystyle \frac{56}{31}$

(B)

$-\displaystyle \frac{56}{33}$

(C)

$\displaystyle \frac{56}{33}$

(D)

$\displaystyle \frac{14}{11}$

Q12. Solve for $x\in[-1,1]$ : $\sin^{-1}x+\sin^{-1}(2x)=\dfrac{\pi}{2}$

(A)

$\displaystyle \frac{1}{\sqrt{5}}$

(B)

$-\displaystyle \frac{1}{\sqrt{5}}$

(C)

$0$

(D)

$\displaystyle \frac{1}{2}$

Q13. Find the range of the function $f(x)=\tan^{-1}\!\left(\dfrac{2x}{1-x^{2}}\right)$ for real $x$ .

(A)

$\displaystyle \left[-\frac{\pi}{2},\frac{\pi}{2}\right]$

(B)

$\displaystyle \left(-\frac{\pi}{2},\frac{\pi}{2}\right)$

(C)

$\displaystyle (0,\pi)$

(D)

$\displaystyle (-\pi,\pi)$

Q14. Solve the equation $\cos^{-1}\!\left(\dfrac{1-x^{2}}{1+x^{2}}\right)=2\tan^{-1}x$ for real $x$ .

(A)

$x>0$

(B)

$x\le 0$

(C)

$x=0$

(D)

$x\ge 0$

Q15. Evaluate $\displaystyle \lim_{x\to 0}\dfrac{\tan^{-1}x-\sin^{-1}x}{x^{3}}$

(A)

$-\dfrac{1}{2}$

(B)

$-\dfrac{1}{3}$

(C)

$0$

(D)

$\dfrac{1}{6}$

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