Inverse Trigonometric Functions - Class 12

Let $S$ be the set of all real values of $x$ satisfying the equation $\sin^{-1}(|x^2 - 2x|) + \cos^{-1}(|x| - 1) = \frac{\pi}{2}$. Find the sum of the elements of $S$.

Let $f(x) = \int_1^x \arcsin\left(\frac{2xt}{x^2+t^2}\right) dt$ for $x > 0$. Find the value of $f'\left(\frac{1}{2}\right) + f'(2)$.

Let $f(x) = \sin^{-1}(\sin x)$ and $g(x) = \cos^{-1}(\cos x)$. Define the function $\Delta(x)$ as the determinant $\Delta(x) = \begin{vmatrix} f(x) & g(x) & x \\ -1 & 10 & 2 \\ 0 & -1 & 1 \end{vmatrix}$ Let $S$ be the set of all $x \in (0, 2\pi)$ such that $\Delta(x) = 0$. Then the value of $\sum_{x \in S} x \Delta'(x)$ is equal to:

Let $z_k = \left( \frac{1 + i \tan \frac{\pi}{18}}{1 - i \tan \frac{\pi}{18}} \right)^k$ for $k \in \mathbb{N}$. Find the value of the following expression in degrees: $\cos^{-1}(\text{Re}(z_{13})) + \sin^{-1}(\text{Im}(z_{10})) + \tan^{-1}\left(\frac{\text{Im}(z_{16})}{\text{Re}(z_{16})}\right)$

Let $f: \mathbb{R} \to \mathbb{R}$ be defined as $f(x) = \sin^{-1}\left(\frac{2x}{1+x^2}\right) + \frac{x^3}{3} - x$. Find the total number of points of local extrema of $f(x)$.

Let $f(x) = \cos^{-1}(\cos x) + 3\left[\frac{x}{\pi}\right]\sin x$ on the interval $[0, 2\pi]$, where $[ \cdot ]$ denotes the greatest integer function. Let $k = 1$ if Lagrange's Mean Value Theorem is applicable to $f(x)$ on $[0, 2\pi]$, and $k = 0$ otherwise. Let $S$ be the set of all points $c \in (0, 2\pi)$ such that the tangent to the curve at $x = c$ is parallel to the chord joining the endpoints of the interval. Find the value of $k + \sum_{c \in S} c$.

Let $A$ be a $3 \times 3$ matrix defined as: $A = \begin{pmatrix} \ln(\tan \frac{\pi}{4}) & \cos^{-1}(\cos 5) & \tan^{-1}(\tan 4) \\ \sin^{-1}(\sin 5) & \sin^{-1}(\cos \frac{\pi}{2}) & \sec^{-1}(\sec 9) \\ \sin^{-1}(\sin 4) & \csc^{-1}(\csc 9) - \pi & \cos^{-1}(\sin \frac{\pi}{2}) \end{pmatrix}$ Let $S$ be the set of all column vectors $B \in \mathbb{R}^3$ such that the matrix equation $AX = B$ has at least one solution for $X \in \mathbb{R}^3$. Which of the following statements is true?

Let $x = \tan\left(\frac{17\pi}{20}\right)$. If $f(x) = \sin^{-1}\left(\frac{2x}{1+x^2}\right) + \cos^{-1}\left(\frac{1-x^2}{1+x^2}\right) + \tan^{-1}\left(\frac{2x}{1-x^2}\right) + \tan^{-1}\left(\frac{3x-x^3}{1-3x^2}\right)$ then find the value of $\left\{ \frac{7}{\pi} f(x) \right\}$, where $\{ \cdot \}$ denotes the fractional part function.

Evaluate the integral: $I = \int_{-\pi/2}^{3\pi/2} \frac{2|\sin^{-1}(\sin x)| + 4|\cos^{-1}(\cos x) - \frac{\pi}{2}|}{1 + 2024^{\cos^3 x}} \, dx$

Let $\vec{u} = \sin^{-1}(\sin 10)\hat{i} + \cos^{-1}(\cos 10)\hat{j} + \tan^{-1}(\tan 10)\hat{k}$. If the vectors $\vec{u}$, $\vec{v} = 2\hat{i} + \hat{j} + 3\hat{k}$, and $\vec{w} = \hat{i} + \lambda\hat{j} + 2\hat{k}$ are coplanar, then the value of $\lambda$ is:

Let $f(x) = \int_0^{\cos^{-1}(\cos x)} \left( \lfloor t \rfloor + \left\lfloor \frac{2t}{\pi} \right\rfloor + \left\lfloor \frac{3t}{\pi} \right\rfloor \right) dt$, where $\lfloor \cdot \rfloor$ denotes the greatest integer function. Find the value of $f'\left( \frac{22\pi}{7} \right)$.

Evaluate the integral: $I = \int_{-2\pi}^{2\pi} \left( \frac{x^5}{1 + x^2 + \cos^2 x} + \left| \left(\sin^{-1}(\sin x)\right)^2 - \frac{\pi^2}{16} \right| \right) dx$

Let $f(x) = \sin^{-1}(\sin x) + \cos^{-1}(\cos x) + \tan^{-1}(\tan x)$ for $x \in (0, \pi) \setminus \left\{\frac{\pi}{2}\right\}$. Let $L(x)$ be the linear approximation of $f(x)$ at $x = \frac{29\pi}{60}$. If $E$ is the absolute error when $L(x)$ is used to approximate the value of $f\left(\frac{31\pi}{60}\right)$, then $E$ equals:

Let a rectangle $OABC$ be constructed such that vertex $O$ is at the origin, vertex $A$ lies on the positive $x$-axis, vertex $C$ lies on the positive $y$-axis, and vertex $B$ lies on the curve $y = \frac{1}{2} \cos^{-1}(1 - 2x^2) + \sin^{-1}(2x\sqrt{1-x^2})$ in the first quadrant ($0 \le x \le 1$). If $P(x)$ denotes the perimeter of this rectangle, find the maximum value of $P(x)$.

Let $f(x) = \sin^{-1}\left(\frac{2x}{1+x^2}\right) + \cos^{-1}\left(\frac{1-x^2}{1+x^2}\right)$. Evaluate the integral $I = \int_{1/\sqrt{3}}^{\sqrt{3}} f(x) \, dx$

Let $m > 0$ be the real root of the equation $\sin^{-1}\left(\frac{2m}{1+m^2}\right) - \cos^{-1}\left(\frac{1-m^2}{1+m^2}\right) + \tan^{-1}\left(\frac{2m}{1-m^2}\right) = -\frac{2\pi}{3}$ Let $L_1$ be the tangent to the ellipse $\frac{x^2}{4} + \frac{y^2}{13} = 1$ having slope $m$ and a positive $y$-intercept. Let $L_2$ be the tangent to the circle $x^2 + y^2 - 2\sqrt{3}x - 4y - 9 = 0$ that is parallel to $L_1$ and furthest from it. The distance between $L_1$ and $L_2$ is:

Let $x = \tan \frac{\pi}{18}$, $y = \tan \frac{2\pi}{9}$, and $z = \tan \frac{7\pi}{18}$. Define the matrix $M$ as: $M = \begin{bmatrix} \frac{2x}{1+x^2} & \frac{1-x^2}{1+x^2} & 1 \\\\ \frac{2y}{1+y^2} & \frac{1-y^2}{1+y^2} & 1 \\\\ \frac{2z}{1+z^2} & \frac{1-z^2}{1+z^2} & 1 \end{bmatrix}$ If $N = \text{adj}(M)$, find the value of $\cos^{-1} \left( \frac{4}{3\sqrt{3}} \det(M N) \right)$.

Let $y = y(x)$ be a differentiable function on an interval containing $x = -1$, satisfying the implicit equation $y^2 - x^2 = 3 \exp\left( \frac{3}{\pi} \cos^{-1}\left(\frac{x}{|y|}\right) - 2 \right)$ If $y(-1) = -2$, then the value of $\frac{dy}{dx}$ at $x = -1$ is

A curve $C$ in $\mathbb{R}^3$ is defined by the intersection of the surfaces $x^3 + 3x = 4y$ and $y^3 + y = 2z$. Let $L$ be the tangent line to the curve $C$ at the point $P(1, 1, 1)$. If $\theta$ is the acute angle between the line $L$ and the plane $6x - 2y + 3z = 10$, find the value of $49 \sin \theta$.

Let $f: [-4, \infty) \to [1, \infty)$ be the unique continuous function satisfying $(f(x))^5 - 5f(x) = x$. Consider the integral function $I(t) = \int_{-4}^{t} (f(x) - 1)^2 \sin(t - x) \,dx$. The value of $\lim_{t \to -4^+} \frac{I(t)}{(t+4)^3}$ is:

Let $y(x)$ be the unique real-valued function satisfying the implicit equation $y^3 - 4y^2 - 7xy = \frac{18}{\pi} \text{Arg}\left( \sqrt{ \frac{x - \omega}{x - \bar{\omega}} } \right)$ where $\omega = e^{i 2\pi/3}$, $\text{Arg}(z) \in (-\pi, \pi]$ denotes the principal argument, and $\sqrt{z}$ denotes the principal square root. Find the value of $\frac{dy}{dx}$ at $x = -1$.

Let $f: [-1, 1] \to \mathbb{R}$ be defined as $f(x) = \sin^{-1} \left( \frac{2g(x)}{1 + (g(x))^2} \right)$ where $g(x) = (x^2 - 1)\text{sgn}(x)$ and $\text{sgn}(\cdot)$ denotes the signum function. Let $S$ be the set of all points $c \in (-1, 1)$ such that $f'(c) = 0$. The number of elements in $S$ is:

Let $P(x)$ be a polynomial of degree 3 satisfying $P(x) + P(-x) = 0$ for all $x \in \mathbb{R}$, $P(2) = 0$, and $\lim_{x \to 0} \frac{P(x)}{x} = -4$. Consider the function $f(x) = \sin^{-1} \left( \frac{P(|x|) + |P(x)|}{2x} \right)$ The domain of $f(x)$ is:

Let $f(x)$ be a real-valued function defined implicitly by the equation $\sin^3(f(x)) + 2\sin(f(x)) = x^4 - 8x^2 + 12$ where $f(x) \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$. Let $S$ be the set of all $x \in \mathbb{R}$ at which $f(x)$ attains either its global maximum or its global minimum. Find the value of $\sum_{x \in S} x^2$.

Let $A(x)$ be a $3 \times 3$ matrix with real entries such that $\det(A(x)) = \frac{1}{8} \left( \sin^{-1}(x-1) + \cos^{-1}(x-1) + \sin^{-1}(2x) \right)$. If $B(x) = 2A(x)$, find the domain of the real-valued function $f(x) = \sqrt{ \frac{2\pi}{3} - \det(B(x)) }$.

Let $L$ be the line in 3D space given by the symmetric equations: $\frac{x-2}{2} = \frac{y-1}{-1} = \frac{z-1}{1}$ Let $S$ be the set of all points $(x,y,z)$ on the line $L$ that satisfy the equation: $\sin^{-1}\left( \frac{(x-2)^2 + (y-1)^2 + (z-1)^2}{6} - 2 \right) + \cos^{-1}(z-1) = \frac{\pi}{2}$ Find the sum of the coordinates of all points in the set $S$.

Let $S = \{x \in [0, 4\pi] : \sin x \ge |\cos x|\}$. Evaluate the integral $I = \int_{S} x \left| \cos^{-1}(\cos x) - \sin^{-1}(\sin x) \right| dx$

Let $f(x) = \sin^{-1}(\sin x) + \cos^{-1}(\cos x)$. The sum of all solutions to the equation $f(x) = \frac{x}{4}$ in the interval $x \in [0, 20\pi]$ can be expressed as $\frac{p\pi}{q}$, where $p$ and $q$ are coprime positive integers. Find the value of $p+q$.

Let $A(x) = \begin{pmatrix} \frac{1}{\pi}\sin^{-1}(\sin x) & \frac{1}{\pi}\cos^{-1}(\sin x) & 0 \\ -\frac{1}{\pi}\cos^{-1}(\sin x) & \frac{1}{\pi}\sin^{-1}(\sin x) & 0 \\ 0 & 0 & 1 \end{pmatrix}$. If $S$ is the sum of all solutions to the equation $|\text{adj}(2A(x))| = 25$ in the interval $x \in [0, 10\pi]$, find the value of $S$.

Evaluate the integral: $I = \int_{0}^{\sqrt{3}} \left( \arctan\left(\frac{2x}{1-x^2}\right) + \frac{2x}{1+x^2} \right) dx$

Let $f(x) = \sin^{-1}(\sin x) \cdot \cos^{-1}(\cos x)$. Find the number of points of inflection of the curve $y = f(x)$ in the interval $(-2\pi, 2\pi)$.

Let $I = \int_{1}^{\sqrt{3}} \frac{1}{x} \sin^{-1}\left( \frac{2x}{1+x^2} \right) dx$. Then $I$ lies in the interval:

Let $A$ be the area of the region in the $xy$-plane bounded by the ellipse $4x^2 + 9y^2 = 36$, the positive $x$-axis, and the ray originating from the origin with polar angle $\phi = \frac{5\pi}{6}$. Find the exact value of $A$.

Let the vectors $\vec{a}(x)$, $\vec{b}(x)$, and $\vec{c}(x)$ be defined as: $\vec{a}(x) = \sin^{-1}(\sin x)\hat{i} + \cos^{-1}(\cos x)\hat{j} + \tan^{-1}(\tan x)\hat{k}$ $\vec{b}(x) = \cos^{-1}(\cos x)\hat{i} + \tan^{-1}(\tan x)\hat{j} + \sin^{-1}(\sin x)\hat{k}$ $\vec{c}(x) = \tan^{-1}(\tan x)\hat{i} + \sin^{-1}(\sin x)\hat{j} + \cos^{-1}(\cos x)\hat{k}$ For $x \in \left(\frac{\pi}{2}, \pi\right)$, let $\vec{a}', \vec{b}', \vec{c}'$ be the reciprocal system of vectors and $V'(x)$ be the volume of the parallelepiped formed by them. Evaluate the integral: $I = \int_{\pi/2}^{\pi} \frac{1}{x V'(x)} dx$

Consider the linear differential equation $\frac{dy}{dx} + P(x)y = Q(x)$ for $x \in (1, \sqrt{2})$, where $P(x) = \frac{2}{x\sqrt{x^2-1}(\sec^{-1}x - \csc^{-1}x)}$ Let $I(x) = e^{\int P(x) dx}$ be the integrating factor of this differential equation, where the constant of integration is taken to be zero. The value of $\int_{1}^{\sqrt{2}} I(x) dx$ is:

Evaluate the integral $I = \int_{0}^{1} 148 x^3 \sin^{-1}\left(2x\sqrt{1-x^2}\right) \,dx$

Let $S$ be the set of all complex numbers $z = x + iy$ where $x, y \in \mathbb{Z}$, satisfying the following two conditions:

(i) $\tan^{-1}(x) + \tan^{-1}(y) = \tan^{-1}\left(\frac{x+y}{1-xy}\right)$

(ii) $\cos^{-1}\left(\frac{x^2+y^2 - 5}{5}\right) \cdot (x^2+y^2 - 8) \le 0$

Find the number of elements in $S$.

Let $\vec{a}, \vec{b},$ and $\vec{c}$ be three unit vectors in $\mathbb{R}^3$. The angle $\theta$ between $\vec{a}$ and $\vec{b}$ is given by $\theta = \sin^{-1}\left(\frac{12}{13}\right) + \sin^{-1}\left(\frac{4}{5}\right)$. If $\vec{c} \cdot \vec{a} = \frac{4}{5}$ and $\vec{c} \cdot \vec{b} = \frac{36}{325}$, what is the volume of the parallelepiped formed by $\vec{a}, \vec{b},$ and $\vec{c}$?

Let $S(x) = \lim_{n \to \infty} \sum_{k=1}^n \tan^{-1}\left( \frac{x}{1+k(k+1)x^2} \right)$. Consider the function $f(x) = \int_0^x t^3 \left( \frac{2}{\pi} S(t) + 153 t^4 \sin\left(\frac{1}{t}\right) \right) dt$ for $x \neq 0$, and $f(0)=0$. Let $c$ be the unique point of inflection of $f(x)$ in the interval $\left(-\frac{1}{6}, \frac{1}{6}\right)$. Find the value of $\lim_{x \to c} \frac{f''(x) + 153 x^5 \cos\left(\frac{1}{x}\right)}{x |x|}$

Evaluate the limit: $\lim_{n \to \infty} \frac{1}{n} \sum_{r=1}^{2n} \left[ \sin^{-1}\left( \frac{2nr}{n^2 + r^2} \right) + \cos^{-1}\left( \frac{n^2 - r^2}{n^2 + r^2} \right) \right]$

Let $f(x) = \frac{x}{2x^2+2x+1}$ for $x > 0$. Let $\alpha$ be the unique point of inflection of the curve $y = f(x)$. Evaluate the limit: $L = \lim_{n \to \infty} \sum_{k=1}^{\lfloor \alpha n \rfloor} \arctan\left( \frac{k}{n^2 + 2nk + 2k^2} \right)$

Evaluate the sum of the infinite series $\sum_{n=3}^{\infty} \tan^{-1}\left( \frac{5}{n^2 - n + 25} \right)$

Let $A_n = [a_{ij}]$ be a $4 \times 4$ real skew-symmetric matrix for each $n \in \mathbb{N}$, such that its elements satisfy: $a_{12}a_{34} - a_{13}a_{24} + a_{14}a_{23} = e^{1/n^2}$ Let $B$ be a $4 \times 4$ real matrix satisfying $B^2 = I$ and $\text{Tr}(B) = 0$. Define the matrix sequence $M_n = (A_n^T B A_n)^{-1}$. Evaluate the infinite sum: $S = \sum_{n=1}^{\infty} \tan^{-1} \left( -\frac{1}{2} \ln(\det(M_n)) \right)$

Consider the following two sets of real numbers: $A = \left\{ x \in \mathbb{R} \setminus \{0\} : \sum_{n=1}^{\infty} \frac{x}{n^2 + x^2} < \int_0^{\infty} \frac{x}{t^2 + x^2} dt \right\}$ $B = \left\{ x \in \mathbb{R} \setminus \{0\} : \int_1^{\infty} \frac{x}{t^2 + x^2} dt = \arctan(x) \right\}$ Which of the following statements is correct?

Let $\omega(x) = \frac{1 + ix}{\sqrt{1+x^2}}$ for $x \in \mathbb{R}$. Define the function $f(x)$ by the infinite series: $f(x) = \sum_{n=1}^{\infty} \frac{1}{n} \text{Im}\left( \omega(x)^{2n} \right)$ Let $g(x) = \cos(f(x))$. Find the value of $125 \cdot g'\left(-\frac{3}{4}\right)$.

Evaluate the integral: $I = \int_{0}^{2\pi} \frac{x \left( \cos^{-1}(\cos x) + \left| \sin^{-1}(\sin x) \right| \right)}{\sin^4 x + \cos^4 x} \, dx$

Let $P(x_0, y_0, z_0)$ be a point in 3D space where $x_0 = \sin^{-1}\left(\sin \frac{17\pi}{5}\right)$, $y_0 = \cos^{-1}\left(\cos \frac{17\pi}{5}\right)$, and $z_0 = \tan^{-1}\left(\tan \frac{17\pi}{5}\right)$. If $Q(\alpha, \beta, \gamma)$ is the image of $P$ in the plane $x + y - z = 0$, then the coordinates of $Q$ are:

Let $C$ be the curve defined implicitly by $3xy - 9x - 4y^2 + 18y - 23 = 0$. Let $O'$ be the point of intersection of the asymptotes of $C$. A point $P(x_0, y_0)$ with $y_0 > 3$ lies on $C$ such that the normal to $C$ at $P$ passes through $O'$. If $L_1$ is the tangent to $C$ at $P$, and $L_2$ is a line parallel to $L_1$ passing through the origin $(0,0)$, let $d$ be the perpendicular distance between $L_1$ and $L_2$. Find the value of $10d^2$.

Let the function $f: (-\infty, -3] \to \left(0, \frac{\pi}{2}\right]$ be defined as $f(x) = \sin^{-1}\left( \frac{2|x+2|}{x^2 + 4x + 5} \right)$. If $f^{-1}(x)$ is the inverse function of $f(x)$, then $f^{-1}(x)$ is equal to:

Let $S$ be the set of all complex numbers $z$ satisfying $|z| = 1$ such that the equation $\sin^{-1}\left( z^4 + \frac{1}{z^4} \right) + \cos^{-1}\left( i\left(z^2 - \frac{1}{z^2}\right) \right) = \frac{\pi}{2}$ holds true. The value of $\sum_{z \in S} \arg(z)$, where $\arg(z) \in [0, 2\pi)$, is