Inverse Trigonometric Functions PYQs Set 4 (PDF)

Set 4 of 6 · Questions 91-120 of 152

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Inverse Trigonometric Functions · Set 4 (Q91-120)
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91. $If the functions are defined as f (x) = x and g (x) = 1 - x, then what is the common domain of the following functions: f + g, f - g, f / g, g / f, g - f where (f \pm g) (x) = f (x) \pm g (x), (f / g) (x) = \frac{f ( x )}{g ( x )}$ (1) $0\leqx\leq1$ (2) $0\leqx<1$ (3) $0 < x < 1$ (4) $0<x\leq1$ 92. $sin (2 sin^{- 1} \frac{63}{65}) =$ (1) $\frac{8 63}{65}$ (2) $\frac{63}{65}$ (3) $\frac{2 126}{65}$ (4) $\frac{4 65}{65}$ 93. $If cos^{- 1} (\frac{2}{3 x}) + cos^{- 1} (\frac{3}{4 x}) = \frac{π}{2} (x > \frac{3}{4}), then x is equal to:$ (1) $\frac{145}{12}$ (2) $\frac{145}{10}$ (3) $\frac{146}{12}$ (4) $\frac{145}{11}$ 94. $sin^{- 1} \frac{3}{2} + sin^{- 1} \frac{2}{3} =$ (1) $sin^{- 1} \frac{3 + 2}{2 3}$ (2) $π - sin^{- 1} (\frac{3 + 2}{2 3})$ (3) $- π - sin^{- 1} (\frac{3 + 2}{2 3})$ (4) $π + sin^{- 1} (\frac{3 + 2}{2 3})$ 95. $Let M and m respectively be the maximum and minimum values of the function f (x) = tan^{- 1} (sin x + cos x) in (0, \frac{π}{2}) . Then the value of tan (M - m) is equal to:$ (1) $2 + 3$ (2) $2 - 3$ (3) $3 + 22$ (4) $3 - 22$ 96. $If sin^{- 1} x + sin^{- 1} y = \frac{π}{2}, then x^{2} is equal to$ (1) $1 - y^{2}$ (2) $y^{2}$ (3) $0$ (4) $1 - y$ 97. $If a = sin^{- 1} (sin (5)) and b = cos^{- 1} (cos (5)), then a^{2} + b^{2} is equal to$ (1) $4 π^{2} + 25$ (2) $8 π^{2} - 40 π + 50$ (3) $4 π^{2} - 20 π + 50$ (4) $25$ 98. $The number of real roots of the equation tan^{- 1} x (x + 1) + sin^{- 1} x^{2} + x + 1 = \frac{π}{4} is$ (1) $1$ (2) $2$ (3) $4$ (4) $0$ 99. $If x = sin (2 tan^{- 1} 2), y = cos (2 tan^{- 1} 3) and z = sec (3 tan^{- 1} 4), then$ (1) $x < y < z$ (2) $y < z < x$ (3) $z < x < y$ (4) $z < y < x$ 100. $If x = sin^{- 1} (sin 10) and y = cos^{- 1} (cos 10), then y - x is equal to :$ (1) $0$ (2) $10$ (3) $7 π$ (4) $π$ 101. $If \frac{( x + 1 ) ^{2}}{x ^{3} + x} = \frac{A}{x} + \frac{Bx + C}{x ^{2} + 1}, then cosec^{- 1} (\frac{1}{A}) + cot^{- 1} (\frac{1}{B}) + sec^{- 1} C =$ (1) $0$ (2) $\frac{π}{2}$ (3) $\frac{5 π}{6}$ (4) $\frac{π}{6}$ 102. $The value of cot^{- 1} (\frac{1 + tan ^{2} ( 2 ) - 1}{ta n ( 2 )}) - cot^{- 1} (\frac{1 + tan ^{2} ( \frac{1}{2} ) + 1}{ta n ( \frac{1}{2} )}) is equal to$ (1) $π - \frac{3}{2}$ (2) $π + \frac{5}{2}$ (3) $π - \frac{5}{4}$ (4) $π + \frac{3}{2}$ 103. $If x \in (\frac{- π}{2}, \frac{π}{2}), then lo g se c x =$ (1) $2 cosech^{- 1} (cot^{2} \frac{x}{2} - 1)$ (2) $2 cosech^{- 1} (cot^{2} \frac{x}{2} + 1)$ (3) $2 coth^{- 1} (cosec^{2} \frac{x}{2} - 1)$ (4) $2 coth^{- 1} (cosec^{2} \frac{x}{2} + 1)$ 104. $The domain of the function f (x) = sin^{- 1} (\frac{3 x ^{2} + x - 1}{( x - 1 ) ^{2}}) + cos^{- 1} (\frac{x - 1}{x + 1}) is :$ (1) $(0, \frac{1}{4})$ (2) $(- 2, 0) \cup (\frac{1}{4}, \frac{1}{2})$ (3) $(\frac{1}{4}, \frac{1}{2}) \cup (0)$ (4) $(0, \frac{1}{2})$ 105. $Considering only the principal values of inverse functions, the set A = (x \geq 0 : tan^{- 1} (2 x) + tan^{- 1} (3 x) = \frac{π}{4})$ (1) $contains two elements$ (2) $contains more than two elements$ (3) $is a singleton$ (4) $is an empty set$ 106. $The value of sin (2 cos^{- 1} \frac{5}{3}) is$ (1) $\frac{5}{3}$ (2) $\frac{2 5}{3}$ (3) $\frac{4 5}{9}$ (4) $\frac{2 5}{9}$ 107. $The value of tan^{- 1} \frac{1 + 3}{3 + 3} + sec^{- 1} (\frac{8 + 4 3}{6 + 3 3})^{1/2} is$ (1) $\frac{π}{4}$ (2) $\frac{π}{2}$ (3) $\frac{π}{3}$ (4) $\frac{π}{6}$ 108. $For a > 0, if f (x) = ax + b is an onto function from [- 1, 1] to [0, 2], then cot (tan^{- 1} \frac{1}{7} + tan^{- 1} \frac{1}{8} + tan^{- 1} \frac{1}{5}) =$ (1) $f (- 1)$ (2) $f (1)$ (3) $f (0)$ (4) $f (2)$ 109. $c os (cot^{- 1} (- 3) + \frac{π}{6}) =$ (1) $0$ (2) $1$ (3) $\frac{1}{2}$ (4) $-1$ 110. $2 tan^{- 1} (\frac{1}{3}) + tan^{- 1} (\frac{1}{7}) is equal to$ (1) $tan^{- 1} (\frac{49}{29})$ (2) $\frac{π}{2}$ (3) $0$ (4) $\frac{π}{4}$ 111. $If cos^{- 1} x - cos^{- 1} \frac{y}{2} = α, where - 1 \leq x \leq 1, - 2 \leq y \leq 2, x \leq \frac{y}{2}, then for all x, y, 4 x^{2} - 4 xy cos α + y^{2} is equal to :$ (1) $4 sin^{2} α$ (2) $2 sin^{2} α$ (3) $2 sin^{2} α - 2 x^{2} y^{2}$ (4) $4 cos^{2} α + 2 x^{2} y^{2}$ 112. $If the domain of the function f (x) = cos^{- 1} (\frac{2 x - 5}{11 - 3 x}) + sin^{- 1} (2 x^{2} - 3 x + 1) is the interval [α, β], then α + 2 β is equal to :$ (1) $3$ (2) $5$ (3) $1$ (4) $2$ 113. $cosec (2 cot^{- 1} (5) + cos^{- 1} (\frac{4}{5})) is equal to$ (1) $\frac{75}{56}$ (2) $\frac{65}{56}$ (3) $\frac{56}{33}$ (4) $\frac{65}{33}$ 114. $Considering only the principal values of inverse trigonometric functions, the number of positive real values of x satisfying tan^{- 1} (x) + tan^{- 1} (2 x) = \frac{π}{4} is :$ (1) $More than 2$ (2) $1$ (3) $2$ (4) $0$ 115. $If \frac{( x + 1 ) ^{2}}{x ^{3} + x} = \frac{A}{x} + \frac{Bx + C}{x ^{2} + 1}, then sin^{- 1} A + tan^{- 1} B + sec^{- 1} C =$ (1) $\frac{π}{2}$ (2) $\frac{π}{6}$ (3) $0$ (4) $\frac{5 π}{6}$ 116. $cot h^{- 1} (2) + cosec h^{- 1} (- 22) is equal to$ (1) $log \frac{3}{2}$ (2) $log 6$ (3) $log \frac{3}{2}$ (4) $log \frac{3}{2}$ 117. $If tan^{- 1} \frac{1}{5} + \frac{1}{2} sec^{- 1} x + tan^{- 1} \frac{1}{8} = \frac{π}{8}, then x^{2} =$ (1) $\frac{12}{7}$ (2) $\frac{50}{49}$ (3) $\frac{13}{12}$ (4) $\frac{1}{2}$ 118. $Given that the inverse trigonometric function assumes principal values only. Let x, y be any two real numbers in [- 1, 1] such that cos^{- 1} x - sin^{- 1} y = α, \frac{- π}{2} \leq α \leq π . Then, the minimum value of x^{2} + y^{2} + 2 xy sin α is$ (1) $0$ (2) $-1$ (3) $\frac{1}{2}$ (4) $- \frac{1}{2}$ 119. $For n \in N, if cot^{- 1} 3 + cot^{- 1} 4 + cot^{- 1} 5 + cot^{- 1} n = \frac{π}{4}, then n is equal to______$ 120. $Let the maximum value of (sin^{- 1} x)^{2} + (cos^{- 1} x)^{2} for x \in (- \frac{3}{2}, \frac{1}{2}) be \frac{m}{n} π^{2}, where gcd (m, n) = 1 . Then m + n is equal to____.$