JEE Mathematics Inverse Circular Functions MCQs Set 01

Type 1 - One or more options may be correct

Question. The formula \(2\sin^{-1} x = \sin^{-1}(2x \sqrt{1-x^2})\) holds for
(a) \(x \in [0, 1]\)
(b) \(x \in [-\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}]\)
(c) \(x \in (-1, 0)\)
(d) \(x \in [-\frac{\sqrt{3}}{2}, \frac{\sqrt{3}}{2}]\)
Answer: (b) \(x \in [-\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}]\)

Question. \(\cos^{-1}(\cos x) = x\) is satisfied by
(a) \(x \in R\)
(b) \(x \in [0, \pi]\)
(c) \(x \in [-1, 1]\)
(d) None of the options
Answer: (b) \(x \in [0, \pi]\)

Question. If \(2\tan^{-1}x + \sin^{-1} \frac{2x}{1+x^2}\) is independent of \(x\) then
(a) \(x \in [1, \infty)\)
(b) \(x \in [-1, 1]\)
(c) \(x \in (-\infty, -1)\)
(d) None of the options
Answer: (a) \(x \in [1, \infty)\)

Question. If \(\tan^{-1}2, \tan^{-1}3\) are angles of a triangle then the third angle is
(a) \(\frac{\pi}{4}\)
(b) \(\frac{3\pi}{4}\)
(c) \(\frac{\pi}{2}\)
(d) None of the options
Answer: (a) \(\frac{\pi}{4}\)

Question. \(\sum_{n=1}^{3} \tan^{-1} \frac{1}{n}\) is equal to
(a) 0
(b) \(\pi\)
(c) \(\frac{\pi}{2}\)
(d) None of the options
Answer: (c) \(\frac{\pi}{2}\)

Question. The value of \(\tan \left\{ \cos^{-1} \frac{4}{5} + \sin^{-1} \frac{2}{\sqrt{13}} \right\}\) is
(a) \(\frac{7}{16}\)
(b) \(\frac{17}{6}\)
(c) \(\frac{6}{17}\)
(d) None of the options
Answer: (b) \(\frac{17}{6}\)

Question. If \(\sin^{-1} x + \cos^{-1} x = \frac{\pi}{6}\) then \(x\) is
(a) \(\frac{1}{2}\)
(b) \(\frac{\sqrt{3}}{2}\)
(c) \(\frac{\sqrt{3}}{2}\)
(d) None of the options
Answer: (b) \(\frac{\sqrt{3}}{2}\)

Question. The value of \(2\tan^{-1} \frac{\sqrt{1+x^2}-1}{x}\) is equal to
(a) \(\cot^{-1} x\)
(b) \(\sec^{-1} x\)
(c) \(\tan^{-1} x\)
(d) None of the options
Answer: (c) \(\tan^{-1} x\)

Question. The value of \(2\tan^{-1} \frac{1}{3} + \tan^{-1} \frac{1}{7}\) is
(a) \(\frac{\pi}{4}\)
(b) \(\frac{\pi}{2}\)
(c) \(\pi\)
(d) None of the options
Answer: (a) \(\frac{\pi}{4}\)

Question. The value of \(\cot^{-1} 3 + \text{cosec}^{-1} \sqrt{5}\) is
(a) \(\frac{\pi}{3}\)
(b) \(\frac{\pi}{2}\)
(c) \(\frac{\pi}{4}\)
(d) None of the options
Answer: (c) \(\frac{\pi}{4}\)

Question. \(\sin (\cot^{-1} (\tan (\cos^{-1} x)))\) is equal to
(a) \(x\)
(b) \(\sqrt{1-x^2}\)
(c) \(\frac{1}{x}\)
(d) None of the options
Answer: (a) \(x\)

Question. The value of \(\tan^{-1} \frac{1}{3} + \tan^{-1} \frac{1}{5} + \tan^{-1} \frac{1}{7} + \tan^{-1} \frac{1}{8}\) is
(a) \(\pi\)
(b) \(\frac{\pi}{4}\)
(c) \(\frac{3\pi}{4}\)
(d) None of the options
Answer: (b) \(\frac{\pi}{4}\)

Question. The value of \(\tan^2(\sec^{-1} 2) + \cot^2(\text{cosec}^{-1} 3)\) is
(a) 13
(b) 15
(c) 11
(d) None of the options
Answer: (c) 11

Question. \(\tan \left( \frac{\pi}{4} + \frac{1}{2} \cos^{-1} x \right) + \tan \left( \frac{\pi}{4} - \frac{1}{2} \cos^{-1} x \right), x \neq 0\), is equal to
(a) \(x\)
(b) \(2x\)
(c) \(\frac{2}{x}\)
(d) None of the options
Answer: (c) \(\frac{2}{x}\)

Question. The number of real solutions of the equation is \(\sqrt{1+\cos 2x} = \sqrt{2} \sin^{-1}(\sin x), -\pi \le x \le \pi\)
(a) 0
(b) 1
(c) 2
(d) infinite
Answer: (c) 2

Question. The number of real solutions of \(\tan^{-1} \sqrt{x(x+1)} + \sin^{-1} \sqrt{x^2+x+1} = \frac{\pi}{2}\) is
(a) zero
(b) one
(c) two
(d) infinite
Answer: (c) two

Question. The number of positive integral solutions of the equation is \(\tan^{-1} x + \cos^{-1} \frac{y}{\sqrt{1+y^2}} = \sin^{-1} \frac{3}{\sqrt{10}}\)
(a) one
(b) two
(c) zero
(d) None of the options
Answer: (b) two

Question. Considering principal values, the number of solutions of \(\tan^{-1} 2x + \tan^{-1} 3x = \frac{\pi}{4}\)
(a) two
(b) three
(c) one
(d) None of the options
Answer: (a) two

Question. The number of real solutions of \((x, y)\), where \(|y| = \sin x, y = \cos^{-1}(\cos x), -2\pi \le x \le 2\pi\), is
(a) 2
(b) 1
(c) 3
(d) 4
Answer: (c) 3

Question. If \(\cos^{-1} x > \sin^{-1} x\) then
(a) \(x < 0\)
(b) \(-1 < x < 0\)
(c) \(0 \le x < \frac{1}{\sqrt{2}}\)
(d) \(-1 \le x < \frac{1}{\sqrt{2}}\)
Answer: (d) \(-1 \le x < \frac{1}{\sqrt{2}}\)

Question. If \(\cot^{-1} \frac{n}{\pi} > \frac{\pi}{6}, n \in N\), then the maximum value of \(n\) is
(a) 1
(b) 5
(c) 9
(d) None of the options
Answer: (b) 5

Question. The set of values of \(k\) for which \(x^2 - kx + \sin^{-1}(\sin 4) > 0\) for all real \(x\) is
(a) \(\phi\)
(b) \((-2, 2)\)
(c) \(R\)
(d) None of the options
Answer: (a) \(\phi\)

Type 2 - One or more options may be correct

Question. Let \(f(x) = \sin^{-1} x + \cos^{-1} x\). Then \(\frac{\pi}{2}\) is equal to
(a) \(f(-\frac{1}{2})\)
(b) \(f(k^2 - 2k + 3), k\)
(c) \(f(\frac{1}{1+k^2}), k \in R\)
(d) \(f(-2)\)
Answer: (a) \(f(-\frac{1}{2})\), (c) \(f(\frac{1}{1+k^2}), k \in R\)

Question. At \(x = \frac{3}{2}\), the value is real for
(a) \(\tan^{-1} x\)
(b) \(\text{cosec}^{-1} x\)
(c) \(\cos^{-1} 2x\)
(d) None of the options
Answer: (a) \(\tan^{-1} x\), (b) \(\text{cosec}^{-1} x\)

Question. If \(\frac{1}{2} < |x| < 1\) then which of the following are real?
(a) \(\sin^{-1} x\)
(b) \(\tan^{-1} x\)
(c) \(\sec^{-1} x\)
(d) \(\cos^{-1} x\)
Answer: (a) \(\sin^{-1} x\), (b) \(\tan^{-1} x\), (d) \(\cos^{-1} x\)

Question. Let \(\tan^{-1} \left( \tan \frac{5\pi}{4} \right) = \alpha, \tan^{-1} \left( -\tan \frac{2\pi}{3} \right) = \beta\). Then
(a) \(\alpha < \beta\)
(b) \(4\alpha - 3\beta = 0\)
(c) \(\alpha + \beta = \frac{7\pi}{12}\)
(d) None of the options
Answer: (b) \(4\alpha - 3\beta = 0\), (c) \(\alpha + \beta = \frac{7\pi}{12}\)

Question. Let \(f(x) = e^{\cos^{-1}(\sin(x + \pi/3))}\). Then
(a) \(f(\frac{8\pi}{9}) = e^{5\pi/18}\)
(b) \(f(\frac{8\pi}{9}) = e^{13\pi/18}\)
(c) \(f(-\frac{7\pi}{4}) = e^{\pi/12}\)
(d) \(f(-\frac{7\pi}{4}) = e^{11\pi/12}\)
Answer: (b) \(f(\frac{8\pi}{9}) = e^{13\pi/18}\), (c) \(f(-\frac{7\pi}{4}) = e^{\pi/12}\)

Question. If \(f(x) = \cos^{-1} x + \cos^{-1} \left\{ \frac{x}{2} + \frac{1}{2}\sqrt{3-3x^2} \right\}\) then
(a) \(f(\frac{2}{3}) = \frac{\pi}{3}\)
(b) \(f(\frac{2}{3}) = 2\cos^{-1} \frac{2}{3} - \frac{\pi}{3}\)
(c) \(f(\frac{1}{3}) = \frac{\pi}{3}\)
(d) \(f(\frac{1}{3}) = 2\cos^{-1} \frac{1}{3} - \frac{\pi}{3}\)
Answer: (a) \(f(\frac{2}{3}) = \frac{\pi}{3}\), (d) \(f(\frac{1}{3}) = 2\cos^{-1} \frac{1}{3} - \frac{\pi}{3}\)