class 12 inverse trigonometric functions multiple choice

Multiple choice (Inverse Trigonometry)

class 12 inverse trigonometric functions multiple choice

choose the correct option in the following question:(class 12 inverse trigonometric functions multiple choice)

Question 1: The value of  \tan^{-1}(\sqrt{3})+\cos^{-1}(-\frac{1}{2}) corresponding to principal branches is

(a) -π/12                         (b) 0

(c) π                                 (d) π/3

Answer (c)

Explanation: \tan^{-1}(\sqrt{3})+\cos^{-1}(-\frac{1}{2})

\Rightarrow \pi/6 + (\pi - \cos^{-1}(\frac{1}{2})

\Rightarrow \pi/6 + (\pi - \pi/6)

\Rightarrow \pi/6  + \pi - \pi/6

\Rightarrow \pi

Question 2: The value of \cot(\sin^{-1}x) is

(a) \frac{\sqrt{1+x^2}}{x}           (b)  \frac{x}{\sqrt{1+x^2}}

(c) 1/x                                                  (d) \frac{\sqrt{1-x^2}}{x}

Answer (d)

Explanation: \cot(\sin^{-1}x)

Let \sin^{-1} x = y

\Rightarrow \sin y = x

\Rightarrow \cos^2y = 1 - \sin^2 y

\Rightarrow \cos^2 y = 1 - x^2

\Rightarrow \cos y = \sqrt{1-x^2}

\Rightarrow \sin^{-1} x = y = \cos^{-1}\sqrt{1-x^2}

Since, \cot y = \dfrac{\cos y}{\sin y}

y= \cot^{-1}[\dfrac{\sqrt{1-x^2}}{x}]

\sin^{-1} x = y= \cot^{-1}[\dfrac{\sqrt{1-x^2}}{x}]

\cot(\sin^{-1}x) = \cot( \cot^{-1}(\dfrac{\sqrt{1-x^2}}{x}))

= \dfrac{\sqrt{1-x^2}}{x}

Question 3: The value of \sin^{-1}(\cos\frac{\pi}{9}) is

(a) π/9                       (b) 5π/9

(c) -5π/9                    (d)  7π/18

Answer (d)

Explanation: \sin^{-1}(\cos\frac{\pi}{9})

=\sin^{-1}(\sin(\frac{\pi}{2}-\frac{\pi}{9}))

= \dfrac{7\pi}{18}

Question 4: Let \theta = \sin^{-1}(\sin(-600)), Then value of θ is

(a) π/3                        (b) π/2

(c) 2π/3                       (d) – 2π/3

Answer (a)

Explanation: \theta = \sin^{-1}(\sin(-600))

\theta = \sin^{-1}(\sin(-10\pi/3))

\Rightarrow \theta = \sin^{-1}(\sin(-10\pi/3+4\pi))

\Rightarrow \theta = \sin^{-1}(\sin(2\pi/3))

\Rightarrow \theta = \sin^{-1}(\sin(\pi - 2\pi/3))

\Rightarrow \theta = \sin^{-1}(\sin(\pi/3))

\Rightarrow \theta = \pi/3

Question 5: The principal value of \tan^{-1}(\tan 3\pi/5) is

(a) 2π/5                    (b) -2π/5

(c) 3π/5                    (d) -3π/5

Answer: (b) -2π/5

Explanation: \tan^{-1} (\tan 3\pi/5)

\tan^{-1} (\tan 3\pi/5) = \tan^{-1} (\tan[\pi - 2\pi/5])

= \tan^{-1} (- \tan 2\pi/5) {since \tan(\pi - x) = -\tan x}

= –\tan^{-1} (\tan 2\pi/2)

= –2π/5

Question 6: \sin[\pi/3 - \sin^{-1}(-½)] is equal to:

(a) 1/2                (b) 1/3

(c) -1                   (d) 1

Answer: (d) 1

Explanation: \sin[\pi/3 - \sin^{-1}(-½)]

= \sin[\pi/3 - \sin^{-1}[\sin (-\pi/6))]

= sin[π/3 – (-π/6)]

= sin(π/3 + π/6)

= sin (π/2)

= 1

Question 7: The domain of \sin^{-1}(2x) is

(a) [0, 1]                         (b) [– 1, 1]

(c) [-1/2, 1/2]                (d) [–2, 2]

Answer: (c) [-1/2, 1/2]

Explanation: Let, \sin^{-1}(2x) = \theta.

\Rightarrow 2x = sin θ.

Since, – 1 ≤ sin θ ≤ 1

\Rightarrow– 1 ≤ 2x ≤ 1\Rightarrow-1/2 \le x \le 1/2.  Therefore, the domain of\sin^{-1}(2x)is [-½, ½].  <strong>Question 8:</strong> If\sin^{–1}x + \sin^{–1}y = \pi/2, then value of\cos^{–1}x + \cos^{–1}yis  (a) π/2                    (b) π  (c) 0                         (d) 2π/3  <strong>Answer: (a) π/2</strong>  <strong>Explanation:</strong> Given, \sin^{–1} x + \sin^{–1} y = \pi/2\Rightarrow [(\pi/2) – \cos^{-1}x] + [(\pi/2) – \cos^{-1}y] = \pi/2(π/2) + (π/2) - (π/2) =\cos^{-1}x + \cos^{-1}yTherefore,\cos^{-1}x + \cos^{-1}y= π/2.  <strong>Question 9:</strong> Which of the following is the principal value branch of\cos^{–1}x?  (a) [-π/2, π/2]              (b) (0, π)  (c) [0, π]                         (d) (0, π) - {π/2}  <strong>Answer: (c) [0, π]</strong>  Explanation: The principal value branch of\cos^{-1}xis [0, π].  <strong>Question 10:</strong> The value of the expression\sin [\cot^{–1} (\cos (\tan^{–1} 1))]is  (a) 0              (b) 1  (c) 1/√3        (d) √(2/3)  <strong>Answer: (d) √(2/3)</strong>  <strong>Explanation:</strong>\sin [\cot^{–1} (\cos (\tan^{–1} 1))]=\sin [\cot^{–1} (\cos (\tan^{–1} (\tan \pi/4))]{since tan π/4 = 1}= \sin[\cot^{-1} (\cos \pi/4)]= \sin[\cot^{-1}(1/√2)]= \sin [\sin^{-1}(√(⅔))]= √(2/3)  <strong>Question 11:</strong> The domain ofy = \cos^{–1} (x^2 – 4)is  (a) [3, 5]                                  (b) [0, π]  (c) [-√5, -√3] ∩ [-√5, √3]    (d) [-√5, -√3] ∪ [√3, √5]  <strong>Answer: (d)</strong>  <strong>Explanation:</strong> Given,y = \cos^{–1} (x^2 – 4 )\Rightarrow \cos y = x^2 – 4Since, -1 \le cos y \le 1  So, - 1 \lex^2 – 4\le 1  ⇒ 3 \lex^2\le 5  ⇒ √3 \le x \le √5  ⇒ x∈ [-√5, -√3] ∪ [√3, √5]  <strong>Question 12:</strong> If α \le2 \sin^{–1}x + \cos^{–1}x\le β, then  (a) α = -π/2, β = π/2  (b) α = 0, β = π  (c) α = -π/2, β = 3π/2  (d) α = 0, β = 2π  <strong>Answer: (b) α = 0, β = π</strong>  <strong>Explanation:</strong> Given,  α \le2 \sin^{–1}x + \cos^{–1}x\le β  We know that,  -π/2 \le\sin^{–1}x \le π/2  ⇒ (-π/2) + (π/2) \le\sin^{–1}x+ (π/2) \le (π/2) + (π/2)  ⇒ 0 \le\sin^{–1}x + (\sin^{–1}x + \cos^{–1}x)\le π  ⇒ 0 \le2 \sin^{–1}x + \cos{–1}x\le π  By comparing with α \le2 \sin^{–1}x + \cos{–1}x\le β, we get α = 0, β = π.  <strong>Question 13:</strong> The value of\sin (2 \tan^{–1} (.75))is equal to  (a) .75                       (b) 1.5  (c) .96                       (d)\sin 1.5<strong>Answer: (c) .96</strong>  <strong>Explanation:</strong>\sin (2\tan^{–1} (.75))Let,\tan^{–1} (.75)= θ\Rightarrow \tan \theta = 0.75\Rightarrow \tan \theta = 3/4Now, sin θ = 3/5 and cos θ = 4/5.\sin (2\tan^{–1} (.75)) = \sin 2θ\theta{astan^{-1}(.75) = \theta}  = 2 sin θ cos θ  = 2 × (3/5) × (4/5)  = 24/25  = 0.96  Therefore,\sin (2\tan^{–1} (.75)) = .96.  <strong>Question 14:</strong>\sin(\tan^{-1} x), where |x| < 1, is equal to:  (a)x/√(1 – x^2)(b)1/√(1 – x^2)(c)1/√(1 + x^2)(d)x/√(1 + x^2)<strong>Answer: (d)</strong>x/√(1 + x^2)<strong>Explanation:</strong> Let\tan^{-1}x =θ.  So, tan θ = x = x/1  From this, we can write the sin θ and cos θ values as:\sin \theta = x/√(1 + x^2)\cos \theta = 1/√(1 + x^2)Now,\sin(\tan^{-1} x) = \sin \theta = x/√(1 + x^2).  <strong>Question 15: The value of the expression</strong>2\sec^{-1}2+\sin^{-1}(\frac{1}{2})is  (a) π/6                        (b) 5π/6  (c) 7π/6                       (d) 1  <strong>Answer (b)</strong>  <strong>Explanation: </strong>2\sec^{-1}2+\sin^{-1}(\frac{1}{2})= \frac{2\pi}{3} + \frac{\pi}{6}=\frac{5\pi}{6}<strong>Question 16: The value of</strong>\tan^2(\sec^{-1}2) +\cot^2(\opratorname{cosec}^{-1}3)is  (a) 5                              (b) 11  (c)  13                            (d) 15  <strong>Answer (b)</strong>  <strong>Question 17: The domain of the function defined by</strong>f(x) =\sin^{-1}\sqrt{1-x}is  (a) [1, 2]                     (b) [-1, 1]  (c) [0, 1]                      (d) None of these  <strong>Answer (a)</strong>  <strong>Question 18: The value of</strong> \cot[\cos^{-1}(\frac{7}{25})]<strong>is</strong>  (a) 25/24                    (b) 25/7  (c) 24/25                     (d) 7/24  <strong>Answer (d)</strong>  <strong>Question 19: </strong>\sin(\tan^{-1}x), |x|<1<strong>is equal to</strong>  (a)\frac{x}{\sqrt{1-x^2}}               (b)\frac{1}{\sqrt{1-x^2}}(c)\frac{1}{\sqrt{1+x^2}}               (d)\frac{x}{\sqrt{1+x^2}}<strong>Answer (d)</strong>  <strong>Question 20:  If</strong>\cos^{-1}\alpha+\cos^{-1}\beta+\cos^{-1}\gamma = 3\pithen\alpha(\beta +\gamma)+\beta(\gamma+\alpha)+\gamma(\alpha+\beta)<strong>equals</strong>  (a) 0                            (b) 1  (c) 6                             (d) 12  <strong>Answer (c)</strong>  <strong>Question 21: Which of the following is the principal value branch of</strong>\cos^{-1}x?  (a) [-π/2, π/2]             (b) (0, π)  (c) [0, π]                        (d) (o, π) - {π/2}  <strong>Answer (c)</strong>  <strong>Question 22: The value of </strong>\sin^{-1}[\cos(33π/5)]is  (a) 3π/5                         (b) -7π/5  (c) π/10                         (d) -π/10  <strong>Answer (d)</strong>  <strong>Question 23: The domain of the function</strong>\cos^{-1}(2x-1)is  (a) [0, 1]                        (b) [-1, 1]  (c) (-1, 1)                        (c) [0, π]  <strong>Answer (a)</strong>  <strong>Question 24:Which of the following is the principal value branch of</strong>\operatorname{cosec}^{-1}x?  (a) ( -π/2, π/2)             (b) (0, π) - { π/2}  (c) {- π/2, π/2}             (d) [- π/2, π/2] - {0}  <strong>Answer (d)</strong>  <strong>Question 25: If</strong>\cos(\sin^{-1}\frac{3}{5}+\cos^{-1})= 0, <strong>Then x is equal to</strong>  (a) 1/5                    (b) 3/5  (c) 0                         (d) 1  <strong>Answer (b)</strong>  <strong>Question 26: The value of</strong>\cos^{-1}(2x^2-1),0 \leq x\lea 1<strong>is equal to</strong>  (a)2\cos^{-1}x            (b)\sin^{-1}x(c)\pi – 2\cos^{-1}x        (d)\pi + 2\cos^{-1}x<strong>Answer (a)</strong>  <strong>Question 27: The number of real solutions of the equation</strong>\sqrt{1+\cos 2x} = \sqrt{2}\cos^{-1}(\cos x)in [ π/2, π] is  (a) 0                       (b) 1  (c) 2                        (d) ∞  <strong>Answer (a)</strong>  <strong>Question 28: The value of</strong> 2\cos^{-1}(\frac{-1}{2})+2\sin^{-1}(\frac{-1}{2})-\cos^{-1}(-1)<strong>is</strong>  (a) 0                      (b) π/2  (c) π                       (d) 2π  <strong>Answer (a)</strong>  <strong>Question 29: The value of</strong>\cot[\frac{1}{2}\sin^{-1}\frac{\sqrt{3}}{2}]<strong>is</strong>  (a) 1                         (b) 1/√3  (c) √3                       (d) 0  <strong>Answer (c)</strong>  <strong>Question 30:</strong>\sin(\cot^{-1}x)<strong>is equal to</strong>  (a)\sqrt{1+x^2}            (b) x  (c)(1+x^2)^{-3/2}            (d)(1+x^2)^{-1/2}$

Answer (d)

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