JEE Mathematics Inverse Circular Functions MCQs Set 02

Question. If \(\alpha\) satisfies the inequation \(x^2 - x - 2 > 0\) then a value exists for
(a) \(\sin^{-1} \alpha\)
(b) \(\sec^{-1} \alpha\)
(c) \(\cos^{-1} \alpha\)
(d) None of the options
Answer: (b) \(\sec^{-1} \alpha\)

Question. If \(\alpha, \beta\) are roots of the equation \(6x^2 - 11x + 3 = 0\) then
(a) both \(\cos^{-1} \alpha\) and \(\cos^{-1} \beta\) are real
(b) both \(\cos^{-1} \alpha\) and \(\cos^{-1} \beta\) are real
(c) both \(\cot^{-1} \alpha\) and \(\cot^{-1} \beta\) are real
(d) None of the options
Answer: (c) both \(\cot^{-1} \alpha\) and \(\cot^{-1} \beta\) are real

Question. Let \(f(x) = \sec^{-1}x + \tan^{-1}x\). Then \(f(x)\) is real for
(a) \(x \in [-1, 1]\)
(b) \(x \in R\)
(c) \(x \in (-\infty, -1] \cup [1, \infty)\)
(d) None of the options
Answer: (c) \(x \in (-\infty, -1] \cup [1, \infty)\)

Question. If \(\cos^{-1} x - \sin^{-1} x = 0\) then \(x\) is equal to
(a) \(\pm \frac{1}{\sqrt{2}}\)
(b) 1
(c) \(\sqrt{2}\)
(d) \(\frac{1}{\sqrt{2}}\)
Answer: (d) \(\frac{1}{\sqrt{2}}\)

Question. \(\text{cosec}^{-1}(\cos x)\) is real if
(a) \(x \in [-1, 1]\)
(b) \(x \in R\)
(c) \(x\) is an odd multiple of \(\frac{\pi}{2}\)
(d) \(x\) is a multiple of \(\pi\)
Answer: (d) \(x\) is a multiple of \(\pi\)

Question. The principal value of \(\sin^{-1} \left\{ \sin \frac{5\pi}{6} \right\}\) is
(a) \(\frac{\pi}{6}\)
(b) \(\frac{5\pi}{6}\)
(c) \(\frac{7\pi}{6}\)
(d) None of the options
Answer: (a) \(\frac{\pi}{6}\)

Question. The principal value of \(\cos^{-1} \left( -\sin \frac{7\pi}{6} \right)\) is
(a) \(\frac{5\pi}{3}\)
(b) \(\frac{7\pi}{6}\)
(c) \(\frac{\pi}{3}\)
(d) None of the options
Answer: (c) \(\frac{\pi}{3}\)

Question. The principal value of \(\sin^{-1} \left\{ \cos \left( \sin^{-1} \frac{\sqrt{3}}{2} \right) \right\}\) is
(a) \(\frac{\pi}{6}\)
(b) \(\frac{\pi}{3}\)
(c) \(-\frac{\pi}{3}\)
(d) None of the options
Answer: (a) \(\frac{\pi}{6}\)

Question. The principal value of \(\cos^{-1} \left\{ \frac{1}{\sqrt{2}} \left( \cos \frac{9\pi}{10} - \sin \frac{9\pi}{10} \right) \right\}\) is
(a) \(\frac{3\pi}{20}\)
(b) \(-\frac{1}{\sqrt{2}}\)
(c) 1
(d) None of the options
Answer: (d) None of the options

Question. The value of \(\cos \left\{ \tan^{-1} \left( \tan \frac{15\pi}{4} \right) \right\}\) is
(a) \(\frac{1}{\sqrt{2}}\)
(b) \(-\frac{1}{\sqrt{2}}\)
(c) 1
(d) None of the options
Answer: (a) \(\frac{1}{\sqrt{2}}\)

Question. \(-\frac{2\pi}{5}\) is the principal value of
(a) \(\cos^{-1} \left( \cos \frac{7\pi}{5} \right)\)
(b) \(\sin^{-1} \left( \sin \frac{7\pi}{5} \right)\)
(c) \(\sec^{-1} \left( \sec \frac{7\pi}{5} \right)\)
(d) None of the options
Answer: (b) \(\sin^{-1} \left( \sin \frac{7\pi}{5} \right)\)

Question. If \(\cos^{-1} \lambda + \cos^{-1} \mu + \cos^{-1} \nu = 3\pi\) then \(\lambda\mu + \mu\nu + \nu\lambda\) is equal to
(a) \(-3\)
(b) 0
(c) 3
(d) \(-1\)
Answer: (c) 3

Question. If \(\sum_{i=1}^{10} \cos^{-1} x_i = 0\) then \(\sum_{i=1}^{10} x_i\) is
(a) 0
(b) 10
(c) 5
(d) None of the options
Answer: (b) 10

Question. If \(\sum_{i=1}^{2n} \sin^{-1} x_i = n\pi\) then \(\sum_{i=1}^{2n} x_i\) is equal to
(a) \(n\)
(b) \(2n\)
(c) \(\frac{n(n+1)}{2}\)
(d) None of the options
Answer: (b) \(2n\)

Question. The value of \(\cos^{-1} \left( -\frac{1}{2} \right) + \sin^{-1} \left( \frac{\sqrt{3}}{2} \right)\) is
(a) \(\pi\)
(b) 0
(c) \(\frac{2\pi}{3}\)
(d) None of the options
Answer: (a) \(\pi\)

Question. The value of \(\tan \left\{ 2\tan^{-1} \frac{1}{5} - \frac{\pi}{4} \right\}\) is
(a) 0
(b) 1
(c) \(\frac{7}{17}\)
(d) None of the options
Answer: (d) None of the options

Question. The formula \(\cos^{-1} \frac{1-x^2}{1+x^2} = 2\tan^{-1} x\) holds only for
(a) \(x \in R\)
(b) \(|x| \le 1\)
(c) \(x \in (-1, 1]\)
(d) \(x \in [0, \infty)\)
Answer: (d) \(x \in [0, \infty)\)

Question. \(\tan^{-1} a + \tan^{-1} b\), where \(a > 0, b > 0, ab > 1\), is equal to
(a) \(\tan^{-1} \frac{a+b}{1-ab}\)
(b) \(\tan^{-1} \frac{a+b}{1-ab} - \pi\)
(c) \(\pi + \tan^{-1} \frac{a+b}{1-ab}\)
(d) None of the options
Answer: (c) \(\pi + \tan^{-1} \frac{a+b}{1-ab}\)

Question. The set of values of \(x\) for which \(\tan^{-1} \frac{x}{\sqrt{1-x^2}} = \sin^{-1} x\) holds is
(a) \(R\)
(b) \([-1, 1]\)
(c) \([0, 1]\)
(d) \([-1, 0]\)
Answer: (b) \([-1, 1]\)

Question. \(\cos^{-1} \left\{ \frac{1}{2}x^2 + \sqrt{1-x^2} \sqrt{1 - \frac{x^2}{4}} \right\} = \cos^{-1} \frac{x}{2} - \cos^{-1} x\) holds is
(a) \(|x| \le 1\)
(b) \(x \in R\)
(c) \(0 \le x < 1\)
(d) \(-1 \le x < 0\)
Answer: (c) \(0 \le x < 1\)

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

Type 2 - One or more options may be correct

Question. The value of \(\tan \left\{ \frac{1}{2} \sin^{-1} \frac{2x}{1+x^2} + \frac{1}{2} \cos^{-1} \frac{1-x^2}{1+x^2} \right\}\) is
(a) \(\frac{2x}{1-x^2}\) if \(0 \le x \le 1\)
(b) \(\frac{2x}{1-x^2}\) if \(x > 1\)
(c) not finite if \(x > 1\)
(d) None of the options
Answer: (a) \(\frac{2x}{1-x^2}\) if \(0 \le x \le 1\), (c) not finite if \(x > 1\)

Question. \(\alpha, \beta\) and \(\gamma\) are three angles given by \(\alpha = 2 \tan^{-1}(\sqrt{2}-1)\), \(\beta = \sin^{-1} \frac{1}{\sqrt{2}} + \sin^{-1}(-\frac{1}{2})\) and \(\gamma = \cos^{-1} \frac{1}{3}\). Then
(a) \(\alpha > \beta\)
(b) \(\beta < \gamma\)
(c) \(\alpha < \gamma\)
(d) None of the options
Answer: (b) \(\beta < \gamma\), (c) \(\alpha < \gamma\)

Question. If \(0 < x < 1\) then \(\tan^{-1} \frac{\sqrt{1-x^2}}{1+x}\) is equal to
(a) \(\frac{1}{2} \cos^{-1} x\)
(b) \(\cos^{-1} \sqrt{\frac{1+x}{2}}\)
(c) \(\sin^{-1} \sqrt{\frac{1-x}{2}}\)
(d) None of the options
Answer: (a) \(\frac{1}{2} \cos^{-1} x\), (b) \(\cos^{-1} \sqrt{\frac{1+x}{2}}\), (c) \(\sin^{-1} \sqrt{\frac{1-x}{2}}\)

Question. One of the values of \(x\) satisfying \(\tan(\sec^{-1} x) = \sin(\cos^{-1} \frac{1}{\sqrt{5}})\) is
(a) \(\frac{\sqrt{5}}{3}\)
(b) \(\frac{3}{\sqrt{5}}\)
(c) \(-\frac{\sqrt{5}}{3}\)
(d) \(-\frac{3}{\sqrt{5}}\)
Answer: (b) \(\frac{3}{\sqrt{5}}\), (d) \(-\frac{3}{\sqrt{5}}\)

Question. If \(\sin^{-1} x + \sin^{-1} y = \frac{2\pi}{3}\), \(\cos^{-1} x - \cos^{-1} y = \frac{\pi}{3}\) then the number of values of \((x, y)\) is
(a) two
(b) four
(c) zero
(d) None of the options
Answer: (d) None of the options

Question. The solution set of the equation \(\cos^{-1} x - \sin^{-1} x = \sin^{-1}(1 - x)\) is
(a) \([-1, 1]\)
(b) \([0, \frac{1}{2}]\)
(c) \([-1, 0]\)
(d) None of the options
Answer: (d) None of the options