Trigonometric Functions - Class 11

Evaluate the following limit: $\lim_{n \to \infty} \frac{1}{n} \sum_{k=1}^{n} \left( \left[ \left( \frac{180 k}{n} \right)^\circ \right] + \left[ \left( \frac{360 k}{n} \right)^\circ \right] \right)$ where $[x]$ denotes the greatest integer less than or equal to $x$.

Let $\arg(z)$ denote the principal argument of a complex number $z$, where $-\pi < \arg(z) \le \pi$. Consider the region $S$ in the complex plane defined by $S = \left\{ z \in \mathbb{C} : |z| \le 6, \ \arg(z^2) \ge \frac{\pi}{2}, \ \cos(\arg(z)) \le 0 \right\}$ Find the area of the region $S$.

Let $L_1$ be a line passing through the origin with direction angles $\alpha, \beta, \gamma$ such that $\sin \alpha = \frac{3\sqrt{5}}{7}$ for $\alpha \in \left(\frac{\pi}{2}, \pi\right)$, $\tan \beta = -\frac{2\sqrt{10}}{3}$ for $\beta \in \left(\frac{\pi}{2}, \pi\right)$, and $\gamma \in \left(0, \frac{\pi}{2}\right)$. Let $L_2$ be the line passing through the origin with direction ratios $(3, 6, -2)$. If $\Pi$ is the plane passing through the origin which is perpendicular to the acute angle bisector of $L_1$ and $L_2$, find the coordinates of the image of the point $P(-2, -3, 6)$ in the plane $\Pi$.

Let $f(x) = \int_0^x \left( \sin\left(4t - \frac{\pi}{6}\right) + \sin\left(2t + \frac{\pi}{6}\right) \right) dt$. Let $M$ and $m$ denote the number of local maxima and local minima of $f(x)$ in the interval $(0, 2\pi)$, respectively. Find the value of $10M + m$.

Let $A = \begin{pmatrix} a & -\frac{1}{2} & c \\ d & -\frac{\sqrt{3}}{2} & f \\ -\frac{1}{2} & 0 & h \end{pmatrix}$ be a real orthogonal matrix with $\det(A) = 1$. If $\text{Tr}(A) < -\frac{\sqrt{3}}{2}$, find the value of $4(a + f)$.

Let the complex numbers $z_1$ and $z_2$ represent two vectors in the Argand plane, given by: $z_1 = \sin\left(\frac{25\pi}{2} + \theta\right) + i \cos\left(\frac{23\pi}{2} - \theta\right)$ $z_2 = \cos\left(\frac{17\pi}{2} - 2\theta\right) + i \sin\left(\frac{19\pi}{2} + 2\theta\right)$ If the scalar projection of the vector represented by $z_2$ onto the vector represented by $z_1$ is $-\frac{\sqrt{3}}{2}$ and $\theta \in (0, \pi)$, then the sum of all possible values of $\theta$ is:

Let $a_k = \binom{3}{k} \cos\left(\frac{k\pi}{2}\right)$ and $b_k = \binom{3}{k} \sin\left(\frac{k\pi}{2}\right)$ for $k=0,1,2,3$. A continuous curve $C$ in the first quadrant ($x > 0, y > 0$) satisfies the differential equation: $x \frac{dy}{dx} - y = x \frac{\sum_{k=0}^{3} b_k x^{3-k} y^k}{\sum_{k=0}^{3} a_k x^{3-k} y^k}$ If $C$ passes through the point $(1, 1)$, which of the following is a possible value of the slope of the ray from the origin to a point on $C$?

Let $y(x)$ be a differentiable function satisfying the differential inequality $\frac{dy}{dx} - \left( \frac{\cos 2x}{1 + \sin 2x} \right) y < -\cos\left(x - \frac{\pi}{4}\right)$ for $x \in \left[ \frac{5\pi}{6}, \pi \right]$. If $y(\pi) = 0$, then which of the following inequalities is necessarily true?

Let $f(x) = 3 \sin^{-1}\left( \frac{3\sin x + 4\cos x}{5} \right) - 2 \cos^{-1}\left( \frac{5\cos x - 12\sin x}{13} \right)$. Find the value of $f'\left( \frac{3\pi}{4} \right)$.

Let $z = x + iy$ be a complex number and $\vec{v} = x\hat{i} + y\hat{j} + 0\hat{k}$ be its corresponding position vector in $\mathbb{R}^3$. Let $S$ be the set of all complex numbers $z$ satisfying the following system of conditions:

1. $|z(2\sqrt{2}-i) - 1|^2 - |z - (2\sqrt{2}+i)|^2 \ge 24$
2. $|\vec{v} \times \vec{u}|^2 + (\vec{v} \cdot \vec{u})^2 \le 400$, where $\vec{u} = 3\hat{i} + 4\hat{j} + 0\hat{k}$
3. $\operatorname{Im}(z^4) \ge 0$

Find the area of the region $S$ in the complex plane.

Let $M(\theta) = \begin{pmatrix} 4\cos^3\theta - 3\cos\theta & 3\sin\theta - 4\sin^3\theta \\ 3\sin\theta - 4\sin^3\theta & 3\cos\theta - 4\cos^3\theta \end{pmatrix}$ and $N(\theta) = \begin{pmatrix} 2\cos^2\theta - 1 & 2\sin\theta\cos\theta \\ 2\sin\theta\cos\theta & 1 - 2\cos^2\theta \end{pmatrix}$.

Let $S$ be the set of all $\theta \in [0, 2\pi]$ satisfying $(M(\theta) N(\theta))^3 = \begin{pmatrix} -\frac{1}{2} & -\frac{\sqrt{3}}{2} \\ \frac{\sqrt{3}}{2} & -\frac{1}{2} \end{pmatrix}$ Find the sum of all elements in $S$.

Let $E$ be the ellipse $\frac{x^2}{3} + y^2 = 1$. For a point $P$ on $E$, let $\theta \in [0, \pi]$ be its eccentric angle. If $(x_P, y_P)$ are the coordinates of $P$, then the value of the integral $I = \int_{0}^{\pi} [x_P + y_P] \, d\theta$ where $[ \cdot ]$ denotes the greatest integer function, is:

Let $y(x)$ be a real-valued continuous function on $[1, \infty)$ satisfying the implicit relation: $y^5 + 2y^3 + 3y = \sin^{-1}\left(\frac{2x}{1+x^2}\right) + \cos^{-1}\left(\frac{1-x^2}{1+x^2}\right) + 2x^2 + 4 - \pi$ Evaluate the limit: $\lim_{x \to 1^+} \frac{y(x) - 1}{x - 1}$

Let $N$ be the exponent of the highest power of $5$ that divides $970!$. Let $\theta_k$ ($k \in \mathbb{N}$) be the sequence of positive solutions to the equation $\tan \theta = \sqrt{3}$, arranged in strictly increasing order. Consider the polynomial $P(x) = \prod_{k=1}^{N} \left( 1 + 2\cos(\theta_k) x \right)$ If $C$ is the coefficient of $x^{100}$ in $P(x)$, find the exponent of the highest power of $3$ that divides $|C|$.

Let $S(x) = \lim_{n \to \infty} \sum_{k=1}^n \frac{1}{2^k} \tan\left(\frac{x}{2^k}\right)$ for $x \in (0, \pi)$. Evaluate the limit: $\lim_{x \to 0} \frac{3x S(x) - x^2}{x^4}$

Evaluate the definite integral $I = \int_{-1}^{-1/2} \left( \cos^{-1}(4x^3 - 3x) + 3\cos^{-1}(-x) \right) dx$

Evaluate the definite integral: $I = \int_0^{\frac{11\pi}{2}} \frac{10}{5 - 3 \cos x} \, dx$

Evaluate the integral: $I = \int_{-1}^{1} \text{Im}\left( e^{i\pi/3} \frac{1 + i\sqrt{x^2}}{1 - i\sqrt{x^2}} \right) dx$

Let $I = \int_0^{2\pi} \frac{dx}{2 + \cos x + |\sin x|}$. Which of the following inequalities is true?

Let $B$ be a $3 \times 3$ matrix with real entries and $I$ be the $3 \times 3$ identity matrix. Define the matrix-valued function $A(x) = B \cos x + I |\sin x|$ and let $f(x) = \det(A(x))$. Which of the following conditions on $B$ is both necessary and sufficient to guarantee that $f(x)$ is differentiable at $x = \pi$?

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

Let $f(x) = \sqrt{2 + \sqrt{2 + 2\cos 4x}}$ and $g(x) = \sqrt{2 - \sqrt{2 + 2\cos 4x}}$. Let $A$ be the area of the region bounded by the curves $y = f(x)$ and $y = g(x)$ for $x \in [0, 37\pi]$. If $N$ is the number of points in the interval $(0, 37\pi)$ where $f(x)$ is non-differentiable, find the value of $A + N$.

Let $I = \int_{0}^{\pi} \frac{\sin\left(2x + \frac{\pi}{6}\right) \cos\left(x + \frac{\pi}{6}\right) - \cos\left(2x + \frac{\pi}{6}\right) \sin\left(x + \frac{\pi}{6}\right)}{3 + \cos(2x)} \left\{ \frac{2x}{\pi} \right\} \, dx$ where $\{t\}$ denotes the fractional part of $t$. Find the value of $I$.

Let $f(x) = \begin{vmatrix} \sin x & \cos x & \sin^3 x \\ -\sin x & -\cos x & 6\sin x - 9\sin^3 x \\ \cos x & -\sin x & 3\sin^2 x \cos x \end{vmatrix}$. Find the value of $f''\left(\frac{\pi}{12}\right)$.

Let $f(x) = |\sin(\sin^{-1}x + \cos^{-1}(x-1))| + |\cos(\sin^{-1}x + \cos^{-1}(x-1))|$. If $M$ is the maximum value of $f(x)$ over its domain of definition, then $M$ equals:

A curve is defined implicitly by the equation $x = 3\sqrt{2}y + |\sin 3y - \sin y|$ for $y \in \left(0, \frac{\pi}{2}\right)$. Let $y = g(x)$ be the continuous function defined by this curve. If $x_0 = \frac{3\sqrt{2}\pi}{4}$, and the left-hand and right-hand derivatives of $g(x)$ at $x_0$ are denoted by $L = \lim_{x \to x_0^-} g'(x)$ and $R = \lim_{x \to x_0^+} g'(x)$ respectively, find the value of $\frac{1}{L \cdot R}$.

In a triangle $ABC$, it is given that $\cos(3A) + \cos(3B) + \cos(3C) = 1$ and $\angle B > \angle C$. If $\sin(3B) + \cos(3B) = 0$, then the value of $\frac{2}{\pi} \sum_{k \in \{A,B,C\}} \left( \arcsin(\sin 3k) + \arccos(\cos 3k) \right)$ is equal to:

Find the sum of all real values of $x$ in the interval $[0, 1000]$ that satisfy the equation: $\sin^{-1}(\sin x^\circ) - \cos^{-1}(\cos x^\circ) = -\frac{\pi}{6}$

Let $\vec{u} = \sin(x^\circ)\hat{i} + \cos(x^\circ)\hat{j}$ and $\vec{v} = \sin(x)\hat{i} + \cos(x)\hat{j}$, where $x > 0$ is a real number. A line $L$ with direction cosines $l, m, n$ is orthogonal to $\vec{u}$ and makes an angle of $\frac{\pi}{4}$ with the positive z-axis. If the acute angle between $L$ and $\vec{v}$ is $\frac{\pi}{3}$, find the smallest positive value of $x$.

Let $f(x) = \cos x |\cos x| - \sin x |\sin x|$ and $F(x) = \int_0^x f(t) dt$. If $M$ and $m$ are the global maximum and global minimum values of $F(x)$ on the interval $[0, 10\pi]$, find the value of $M - m$.

Let $A$ be the solution set of the inequality $\left( \sum_{j=1}^{8} \cos((2j-1)x) \right) \sin x \ (\cos 6x - 1) \ge 0$ for $x \in (0, 2\pi)$. Let $I$ be the subset of $A$ consisting of all isolated points (i.e., points in $A$ that do not belong to any open interval contained in $A$). Find the sum of all elements in $I$.

Let $f(x) = \frac{\sqrt{1 - \sin^2 x} + \cos x}{\sin(2x)}$. If $L_1 = \lim_{x \to \frac{\pi}{2}^-} f(x)$ and $L_2 = \lim_{x \to \frac{\pi}{2}^+} f(x)$, then the value of $L_1^2 + L_2^2$ is:

Let $S$ be the set of all $\theta \in (0, \pi)$ satisfying $\cos(7\theta) = 0$ and $\cos\theta \neq 0$. Evaluate the sum: $\sum_{\theta \in S} \left( \sec^4\theta + \csc^4\theta \right)$

Let $A, B, C$ be the angles of a triangle. Consider the matrix $M = \begin{pmatrix} -1 & \cos C & \cos B \\ \cos C & -1 & \cos A \\ \cos B & \cos A & -1 \end{pmatrix}$. Let $\lambda_1, \lambda_2, \lambda_3$ be the eigenvalues of $M$. If $X = \begin{pmatrix} \csc A \\ \csc B \\ \csc C \end{pmatrix}$, find the value of $X^T \text{adj}(M) X + \lambda_1 + \lambda_2 + \lambda_3$.

Let $y = f(x)$ be a twice-differentiable function for $x \ge 1$ such that $f(1) = \frac{3\pi}{2}$. Let $m_N(t)$ denote the slope of the normal to the curve at the point $(t, f(t))$. If $f(x)$ satisfies the integral equation $\int_{1}^{x} \frac{t + m_N(t) f(t)}{t^2 m_N(t)} \sqrt{1 - \cos^2\left(\frac{f(t)}{t}\right)} \, dt = 1 - \frac{1}{x}$ for all $x \ge 1$, then the value of $f(2)$ is:

Consider the ellipse $E: \frac{(x-1)^2}{9} + \frac{(y-2)^2}{25} = 1$. A tangent drawn from the point $P\left(-2, \frac{9}{2}\right)$ to $E$ intersects the upper latus rectum of $E$ at a point $Q$. The sum of all possible values of the abscissa of $Q$ is:

Let $f(\theta) = \begin{vmatrix} \cos\theta & \sin\theta & 1 \\ \sin\theta & 1 & \cos\theta \\ 1 & \cos\theta & \sin\theta \end{vmatrix}$. Find the sum of all values of $\theta \in [0, 2\pi)$ such that $f(\theta) = 0$.

Let $\theta$ be an angle whose measure in degrees is $x$ and in radians is $y$. Find the sum of all valid values of $y \in [0, 2\pi]$ that satisfy the equation: $\frac{\cos 3x + \cos 2x + \cos x}{\text{sgn}\left( 12y^2 - 11\pi y + 2\pi^2 \right)} = 0$

Evaluate the integral: $I = \int_{0}^{2\pi} \frac{x \sqrt{1 - \cos 2x}}{1 + \cos^2 x} \, dx$

Let $\Delta(x) = \begin{vmatrix} \sin\left(x+\frac{\pi}{4}\right) & \cos\left(x+\frac{\pi}{4}\right) & \ln(1+\sin^2 x) \\ \sin\left(x-\frac{\pi}{4}\right) & \cos\left(x-\frac{\pi}{4}\right) & \sin 2x \\ \sin\left(x+\frac{3\pi}{4}\right) & \cos\left(x+\frac{3\pi}{4}\right) & \cos 2x \end{vmatrix}$ and $f(x) = [\Delta(x)] \left( \Delta^2(x) - 1 \right)$, where $[ \cdot ]$ denotes the greatest integer function. If $L$ and $R$ are the left-hand and right-hand derivatives of $f(x)$ at $x = \frac{\pi}{4}$ respectively, then the value of $L^2 + R^2$ is

Let $I(x) = \int_{0}^{x} \ln \left( \frac{\sin 2t + \sin 4t + \sin 6t}{\cos 2t + \cos 4t + \cos 6t} \right)^2 dt$. Find the sum of all $x \in [0, \pi]$ such that $I'(x) = \ln 3$.

Let $\alpha_1, \alpha_2, \alpha_3$ be the roots of the equation $x^3 - 233x^2 + 17x - 5 = 0$ and $\beta_1, \beta_2, \beta_3$ be the roots of the equation $x^3 + 233x^2 - 19x + 7 = 0$. A $3 \times 3$ matrix $M$ is defined such that its $(i, j)$-th entry is given by $m_{ij} = \sin^2(\alpha_i + \beta_j) + \cos^2(\alpha_i - \beta_j)$ for $1 \le i, j \le 3$. Find the value of $\det(M)$.

Let $\alpha = \left( \sum_{r=0}^{11} \binom{11}{r} \right) \left( \prod_{k=1}^{10} \cos \frac{2^{k-1} \pi}{1025} \right)$ and $\beta = \sum_{k=0}^{5} \binom{5}{k} (-1)^k \left(\frac{3}{2}\right)^k \left(\frac{1}{2}\right)^{5-k}$. Find the sum of all integers $x$ satisfying the inequality $(x - \alpha)^2 (x - \beta)^2 (x - 4) (x - 6) \le 0$

Evaluate the integral: $I = \int_0^\pi \frac{|\sin 2x| + 8\cos^2 x}{5 + 4\cos 2x} dx$

Let $L$ be a line passing through the point $P(3, 4, 5)$ with direction cosines $(l, m, n)$. The line $L$ intersects the line $L_1: x-1 = y-2 = z-3$ at exactly one point, and also intersects the line $L_2: \frac{x}{1} = \frac{y}{2} = \frac{z}{3}$ at exactly one point. If $S$ is the set of all possible values of $m$, then $S$ is equal to:

Let $f(x) = \exp\left( \sqrt{2} \int_{0}^{x} \frac{dt}{\sin t - \cos t} \right)$. Evaluate the limit: $L = \lim_{x \to \frac{\pi}{4}} \frac{f(x)}{\left| x - \frac{\pi}{4} \right| + \left(x - \frac{\pi}{4}\right)^2 \sin\left(\frac{1}{x - \frac{\pi}{4}}\right)}$

Let $A$ be the set of all positive integers which are multiples of $2$ or $3$. Consider the function $S(\theta) = \sum_{n \in A} (-1)^n \sin^n \theta$ for $\theta \in \left( \frac{\pi}{2}, \pi \right)$. If $I = \int_{2\pi/3}^{5\pi/6} \sqrt{ S(\theta) + \frac{\sin^3 \theta}{1-\sin^6 \theta} } \, d\theta$ then the value of $e^I$ is

Let $P_1$ and $P_2$ be two planes defined by the equations: $P_1: x \sqrt{1-\sin 2\theta} + y \sqrt{1+\sin 2\theta} + z = \sqrt{2}$ $P_2: x \sqrt{1+\sin 2\theta} + y \sqrt{1-\sin 2\theta} - z = \sqrt{2}$ where $\theta \in \left[0, \frac{\pi}{2}\right]$. Let $L(\theta)$ be the line of intersection of the planes $P_1$ and $P_2$, and let $\phi(\theta)$ be the acute angle between the line $L(\theta)$ and the $xy$-plane. The value of the integral $\int_0^{\pi/2} \tan^2 \phi(\theta) \, d\theta$ is:

Let $f(x) = \sin([x^2])$, where $[ \cdot ]$ denotes the greatest integer function. Which of the following statements is TRUE?

Find the complete set of all real values of $a$ for which the equation $|\sqrt{3} \sin x - \cos x| + |\sin x + \sqrt{3} \cos x| = a x^2 - 2a x + a + 2\sqrt{2}$ has at least one real solution.