Inverse Trigonometric Functions PYQs Set 5 (PDF)

Set 5 of 6 · Questions 121-150 of 152

In the print window choose “Save as PDF”. The file prints without a web address or date line.

Mere Teacher · Previous Year Questions JEE Main Mathematics · Inverse Trigonometric Functions · Set 5 (Q121-150)
Scan or visit mereteacher.com for free PYQ practice, timed CBT tests, step-by-step solutions and BroBot AI. Answers and solutions are free with a sign-up.
121. $The integral \int_{1/4}^{3/4} cos (2 cot^{- 1} \frac{1 - x}{1 + x}) dx is equal to$ (1) $1/211$ (2) $- 1/2$ (3) $- 1/4$ (4) $1/4$ 122. $A possible value of tan (\frac{1}{4} sin^{- 1} \frac{63}{8}) is$ (1) $\frac{1}{2 2}$ (2) $\frac{1}{7}$ (3) $7 - 1$ (4) $22 - 1$ 123. $If the inverse trigonometric functions take principal values, then cos^{- 1} (\frac{3}{10} cos (tan^{- 1} (\frac{4}{3})) + \frac{2}{5} sin (tan^{- 1} (\frac{4}{3}))) is equal to:$ (1) $0$ (2) $\frac{π}{4}$ (3) $\frac{π}{3}$ (4) $\frac{π}{6}$ 124. $sin^{- 1} (\frac{12}{13}) + cos^{- 1} (\frac{4}{5}) + tan^{- 1} (\frac{63}{16}) =$ (1) $2 π$ (2) $π$ (3) $0$ (4) $- π$ 125. $c os (2 cos^{- 1} \frac{1}{5} + sin^{- 1} \frac{1}{5}) =$ (1) $\frac{1}{5}$ (2) $\frac{- 2 6}{5}$ (3) $- \frac{1}{5}$ (4) $\frac{6}{5}$ 126. $The number of roots of the equation, (81)^{sin^{2} x} + (81)^{cos^{2} x} = 30 in the interval [0, π] is equal to :$ (1) $3$ (2) $4$ (3) $8$ (4) $2$ 127. $All x satisfying the inequality (cot^{- 1} x)^{2} - 7 (cot^{- 1} x) + 10 > 0, lies in the interval:$ (1) $(- \infty, cot 5) \cup (cot 4, cot 2)$ (2) $(cot 2, \infty)$ (3) $(- \infty, cot 5) \cup (cot 2, \infty)$ (4) $(cot 5, cot 4)$ 128. $The domain of the function f (x) = sec^{- 1} (3 x - 4) + tanh^{- 1} (\frac{x + 3}{5}) is$ (1) $(- 8, 1) \cup (\frac{5}{3}, 2)$ (2) $(1, \frac{5}{3})$ (3) $[- 8, 1] \cup (\frac{5}{3}, 2)$ (4) $(- 8, 1] \cup (\frac{5}{3}, 2)$ 129. $Let f : R - 3 \to R - 1 be defined by (x) = \frac{x - 2}{x - 3} . Let R \to R be given as g (x) = 2 x - 3 . Then, the sum of all the values of x for which f^{- 1} (x) + g - 1 (x) = \frac{13}{2} is equal to$ (1) $7$ (2) $2$ (3) $5$ (4) $3$ 130. $Let S_{k} = \sum_{r = 1}^{k} tan^{- 1} (\frac{6 ^{r}}{2 ^{2 r + 1} + 3 ^{2 r + 1}}) . Then lim_{k \to \infty} S_{k} is equal to:$ (1) $tan^{- 1} (\frac{3}{2})$ (2) $\frac{π}{2}$ (3) $cot^{- 1} (\frac{3}{2})$ (4) $tan^{- 1} (3)$ 131. $For any positive integer n, define f_{n} : (0, \infty) \to R as f_{n} (x) = \sum_{j = 1}^{n} tan^{- 1} (\frac{1}{1 + ( x + j ) ( x + j - 1 )}) for all x \in (0, \infty) . (Here, the inverse trigonometric function tan^{- 1} x assumes values in (- π /2, π /2) .) Then, which of the following statement(s) is (are) TRUE? (JEE Advanced 2018)$ (1) $\sum_{j = 1}^{5} tan^{2} (f_{j} (0)) = 55$ (2) $\sum_{j = 1}^{10} (1 + f_{j}^{'} (0)) sec^{2} (f_{j} (0)) = 10$ (3) $For any fixed positive integer n, lim x \to \infty tan (f_{n} (x)) = \frac{1}{n}$ (4) $For any fixed positive integer n, lim x \to \infty sec^{2} (f_{n} (x)) = 1$ 132. $The range of sec^{- 1} x is$ (1) $[0, π]$ (2) $[0, π] - {\frac{π}{2}}$ (3) $(\frac{- π}{2}, \frac{π}{2})$ (4) $-π3,π3$ 133. $The value of cot (\sum_{n = 1}^{23} cot^{- 1} (1 + \sum_{k = 1}^{n} 2 k)) is (JEE Advanced 2013)$ (1) $\frac{23}{25}$ (2) $\frac{25}{23}$ (3) $\frac{23}{24}$ (4) $\frac{24}{23}$ 134. $The set of all values of k for which (tan^{- 1} x)^{3} + (cot^{- 1} x)^{3} = k π^{3}, x \in R, is the interval:$ (1) $[\frac{1}{32}, \frac{7}{8})$ (2) $(\frac{1}{24}, \frac{13}{16})$ (3) $(\frac{1}{48}, \frac{13}{16})$ (4) $[\frac{1}{32}, \frac{9}{8})$ 135. $Let (a, b) \subset (0, 2 π) 0 be the largest interval for which sin^{- 1} (sin θ) - cos^{- 1} (sin θ) > 0, θ ϵ (0, 2 π) holds. If (α x^{2} + βx + sin^{- 1} (x^{2} - 6 x + 10) + cos^{- 1} (x^{2} - 6 x + 10) = 0 and α - β = b = a, the α is equal to$ (1) $\frac{π}{48}$ (2) $\frac{π}{16}$ (3) $\frac{π}{8}$ (4) $\frac{π}{12}$ 136. $Let x * y = x^{2} + y^{3} and (x * 1) * 1 = x * (1 * 1) . Then a value of 2 sin^{- 1} (\frac{x ^{4} + x ^{2} - 2}{x ^{4} + x ^{2} + 2}) is$ (1) $\frac{π}{4}$ (2) $\frac{π}{3}$ (3) $\frac{π}{2}$ (4) $\frac{π}{6}$ 137. $Let S = {x \in R : 0 < x < 1 and 2 tan^{- 1} (\frac{1 - x}{1 + x}) = cos^{- 1} (\frac{1 - x ^{2}}{1 + x ^{2}})} . If n (S) denotes the number of elements in S then$ (1) $nS=2 and only one element in S is less then \frac{1}{2} .$ (2) $nS=1 and the element in S is more han \frac{1}{2}$ (3) $nS=1 and the element in S is less than \frac{1}{2}$ (4) $n (S) = 0$ 138. $If \frac{s i n ^{- 1} x}{a} = \frac{c o s ^{- 1} x}{b} = \frac{t a n ^{- 1} y}{c}; 0 < x < 1 then the value of (\frac{π c}{a + b}) is$ (1) $\frac{1 - y ^{2}}{2 y}$ (2) $\frac{1 - y ^{2}}{1 + y ^{2}}$ (3) $1 - y^{2}$ (4) $\frac{1 - y ^{2}}{y y}$ 139. $2 π - (sin^{- 1} \frac{4}{5} + sin^{- 1} \frac{5}{13} + sin^{- 1} \frac{16}{65}) is equal to:$ (1) $\frac{π}{2}$ (2) $\frac{7 π}{4}$ (3) $\frac{3 π}{2}$ (4) $\frac{5 π}{4}$ 140. $The domain of the function cos e c^{- 1} (\frac{1 + x}{x}) is :$ (1) $(- 1, - \frac{1}{2}] \cup (0, \infty)$ (2) $[- \frac{1}{2}, 0) \cup [1, \infty)$ (3) $(- \frac{1}{2}, \infty) - {0}$ (4) $[- \frac{1}{2}, \infty) - {0}$ 141. $Let a_{1} = 1, a_{2}, a_{3}, a_{4} \dots be consecutive natural numbers. Then tan^{- 1} (\frac{1}{1 + a _{1} + a _{2}}) + tan^{- 1} (\frac{1}{1 + a ) 2 a _{3}}) + \dots + tan^{- 1} (\frac{1}{1 + a _{2021} a _{2022}}) is equal to$ (1) $\frac{π}{4} - cot^{- 1} (2022)$ (2) $cot^{- 1} (2022) - \frac{π}{4}$ (3) $tan^{- 1} (2022) - \frac{π}{4}$ (4) $\frac{π}{4} - tan^{- 1} (2022)$ 142. $The value of cot (\sum_{n = 1}^{50} tan^{- 1} (\frac{1}{1 + n + n ^{2}})) is$ (1) $\frac{26}{25}$ (2) $\frac{25}{26}$ (3) $\frac{50}{51}$ (4) $\frac{52}{51}$ 143. $The inverse of y = 5^{log x} is$ (1) $x = (1/ y)^{log 5}$ (2) $x = y^{\frac{1}{log 5}}$ (3) $x = 5^{log y}$ (4) $x = 5^{\frac{1}{log y}}$ 144. $For any positive integer n, let S_{n} : (0, \infty) \to R be defined by S_{n} (x) = \sum_{k = 1}^{n} cot^{- 1} (\frac{1 + k ( k + 1 ) x ^{2}}{x}) where any x \in R, cot^{- 1} (x) \in (0, π) and tan^{- 1} (x) \in (- \frac{π}{2}, \frac{π}{2}) . Then which of the following statements is (are) TRUE ? (JEE Advanced 2021)$ (1) $S_{10} (x) = \frac{π}{2} - tan^{- 1} (\frac{1 + 11 x ^{2}}{10 x}), for all x>0$ (2) $lim_{n \to \infty} cot (S_{n} (x)) = x, for all x > 0$ (3) $The equation S_{3} (x) = \frac{π}{4} has a root in (0, \infty)$ (4) $tanSn(x)\leq12, for all n\geq1 and x>0$ 145. $Match the items of List-I with those of the items of List-II List-I A. Range of sec^{- 1} (1 + cos^{2} x), [.] denote greatest integer function B. Domain of f (x), where f (x + \frac{1}{x}) = x^{2} + \frac{1}{x ^{2}} C. f (x + y) = f (x) + f (y); f (1) = 5 D. sin^{- 1} x - cos^{- 1} x + sin^{- 1} (1 - x) = 0 \Rightarrow x \in List-II I. odd function II. (0, \frac{1}{2}) III. (sec^{- 1} 5, sec^{- 1} 4) IV. R - (- 2, 2) V. (sec^{- 1} 1, sec^{- 1} 2) A B C D$ (1) $V IV I II$ (2) $III IV II I$ (3) $V II III IV$ (4) $III II I IV$ 146. $Let tan^{- 1} (x) \in (- \frac{π}{2}, \frac{π}{2}), for x \in R . Then, the number of real solutions of the equation 1 + cos (2 x) = 2 tan^{- 1} (tanx) in the set (\frac{- 3 π}{2}, \frac{- π}{2}) \cup (- \frac{π}{2}, \frac{π}{2}) \cup (\frac{π}{2}, \frac{3 π}{2}) is equal to [2023 Adv.]$ 147. $Let f : [0, 4 π] \to [0, π] be defined by f (x) = cos^{- 1} (cosx) . The number of points x \in [0, 4 π] satisfying the equation f (x) = \frac{10 - x}{10}, is [JEE (Adv) 2014]$ 148. $The value of sec^{- 1} (\frac{1}{4} \sum_{k = 0}^{10} sec (\frac{7 π}{12} + \frac{k π}{2}) sec (\frac{7 π}{12} + \frac{( k + 1 ) π}{2})) in the interval (- \frac{π}{4}, \frac{3 π}{4}) equals . (JEE Advanced 2019)$ 149. $Let x = sin (2 tan^{- 1} α) and t = sin (\frac{1}{2} tan^{- 1} \frac{4}{3}) . If S = {α \in R : t^{2} = 1 - x}, then \sum_{α \in S} 16 α^{3} is equal to _____$ 150. $For k \in R, let the solutions of the equation cos (sin^{- 1} (x cot (tan^{- 1} (cos (sin^{- 1} x))))) = k, 0 < (x) < \frac{1}{2} be α and β, where the inverse trigonometric functions take only principal values. If the solutions of the equation x^{2} - bx - 5 = 0 are \frac{1}{α ^{2}} + \frac{1}{β ^{2}} and \frac{α}{β}, then \frac{b}{k ^{2}} is equal to______.$