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The remainder when $(11)^{1011}+(1011)^{11}$ is divided by 9 is

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The number of bijective functions $f:\{1,3,5,7, \ldots, 99\} \rightarrow\{2,4,6,8, \ldots . ., 100\}$ such that $f(3) \geq f(9) \geq f(15)...

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Let $\stackrel{\mathbb{J}}{\mathrm{a}}=4 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}$ and $\stackrel{\mathrm{b}}{\mathrm{b}}=3 \hat{\mathrm{i}}-4 \hat{\mathrm{j}}+5 \hat{\mathrm{k}}$ and $\stackrel{n}{\mathrm{c}}$ is a vector such that $\stackrel{\mathbb{Z}}{\mathrm{c}} \cdot(\mathrm{a} \times \mathrm{b})+25=0,...

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Let $\alpha_1, \alpha_2, \ldots, \alpha_7$ be the roots of the equation $x^7+3 x^5-13 x^3-15 x=0$ and $\left|\alpha_1\right| \geq\left|\alpha_2\right| \geq \ldots...

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Let $\alpha=8-14 i, \quad A=\left\{z \in C: \frac{\alpha z-\bar{\alpha} \bar{z}}{z^2-(\bar{z})^2-112 i}=1\right\}$ and $B=\{z \in C:|z+3 i|=4\}$ Then $\sum_{\varepsilon \in A \subset...

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Let $x=\{11,12,13, \ldots, 40,41\}$ and $Y=\{61,62,63, \ldots, 90,91\}$ be the two sets of observations. If $\bar{x}$ and $\bar{y}$ are their...

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Let $\left\{a_k\right\}$ and $\left\{b_k\right\}, k \in N$, be two G.P.s with common ratio $r_1$ and $r_2$ respectively such that $a_1=b_1=4$...

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Let $A$ be a symmetric matrix such that $|A|=2$ and $\left[\begin{array}{ll}2 & 1 \\ 3 & \frac{3}{2}\end{array}\right] A=\left[\begin{array}{cc}1 & 2...

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If the equation of the normal to the curve $y=\frac{x-a}{(x+b)(x-2)}$ at the point $(1,-3)$ is $x-4 y=13$, then the value...

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Let $a_1=b_1=1$ and $a_n=a_{n-1}+(n-1), b_n=b_{n-1}+a_{n-1}, \forall n \geq 2$. If $\sum_{n=1}^{10} \frac{b_n}{2^n}$ and $T=\sum_{n=1}^8 \frac{n}{2^{n-1}}$, then $2^7(2 S-$ T )...

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A circle with centre $(2,3)$ and radius 4 intersects the line $x+y=3$ at the points $P$ and $Q$. If the...

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A triangle is formed by the tangents at the point $(2,2)$ on the curves $y^2=2 x$ and $x^2+y^2=4 x$, and...

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The total number of 4-digit numbers whose greatest common divisor with 54 is 2 , is $\_\_\_\_$ .

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Consider a function $f: N \rightarrow R$, satisfying $f(1)+2 f(2)+3 f(3)+\ldots+x f(x)=x(x+1) f(x) ; x \geq 2$ with $f(1)=1$. Then $\frac{1}{f(2022)}+\frac{1}{f(2028)}$...

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Let $R$ be a relation defined on $N$ as a $R b$ is $2 a+3 b$ is a multiple of...

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The number of 3 digit numbers, that are divisible by either 3 or 4 but not divisible by 48 ,...

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Let $y=y(x)$ be the solution of the differential equation $x \log _e x^{\frac{d y}{d x}}+y=x^2 \log _e x,(x>1)$. If $y(2)=2$,...

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If the tangent at a point $P$ on the parabola $y^2=3 x$ is parallel to the line $x+2 y=1$ and...

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The value of the integral $\int_{1 / 2}^2 \frac{\tan ^{-1} x}{x} d x \quad$ is equal to

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If the lines $\frac{x-1}{1}=\frac{y-2}{2}=\frac{z+3}{1}$ and $\frac{x-a}{2}=\frac{y+2}{3}=\frac{z-3}{1}$ intersects at the point $P$, then the distance of the point $P$ from the...

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The plane $2 x-y+z=4$ intersects the line segment joining the points $A(a,-2,4)$ and $B(2, b,-3)$ at the point $C$ in...

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The letters of the word OUGHT are written in all possible ways and these words are arranged as in a...

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The set of all values of $\lambda$ for which the equation $\cos ^2 2 x-2 \sin ^4 x-2 \cos ^2...

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The area of the region $$ A=\left\{(x, y):|\cos x-\sin x| \leq y \leq \sin x, 0 \leq x \leq \frac{\pi}{2}\right\} $$

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The set of all values of $t \in R$, for which the matrix

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Let $f$ and $g$ be twice differentiable functions on $R$ such that $$ \begin{aligned} & f^{\prime \prime}(x)=g^{\prime \prime}(x)+6 x \\ & f^{\prime}(1)=4 g^{\prime}(1)-3=9 \\ &...

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Let $K$ be the sum of the coefficients of the odd powers of $x$ in the expansion of $(1+x)^{99}$. Let...

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The value of the integral $\int_1^2\left(\frac{t^4+1}{t^6+1}\right) d t$ is:

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Let $S=\left(w_1, w_2, \ldots\right\}$ be the sample space associated to a random experiment. Let $P\left(w_n\right)=\frac{P\left(w_{n-1}\right)}{2}, n \geq$ 2. Let $A=\{2...

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is equal to

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Shortest distance between the lines $\frac{x-1}{2}=\frac{y+8}{-7}=\frac{z-4}{5}$ and $\frac{x-1}{2}=\frac{y-2}{1}=\frac{z-6}{-3}$ is

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Let $f$ be a polynomial function such that $\log _2(f(x))=\left(\log _2\left(2+\frac{2}{3}+\frac{2}{9}+\ldots \ldots \infty\right)\right) \cdot \log _3\left(1+\frac{f(x)}{f(1 / x)}\right), x>0 \quad$...

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Consider the circle C : $x^2+y^2-6 x-8 y-11=0$. Let a variable chord AB of the circle C subtend a right...

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Let a line $L_1$ pass through the origin and be perpendicular to the lines $$ \begin{aligned} & \mathrm{L}_2: \overrightarrow{\mathrm{r}}=(3+\mathrm{t}) \hat{i}+(2 \mathrm{t}-1) \hat{j}+(2 \mathrm{t}+4)...

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If $\int_{\pi / 6}^{\pi / 4}\left(\cot \left(x-\frac{\pi}{3}\right) \cot \left(x+\frac{\pi}{3}\right)+1\right) d x=\alpha \log _{\mathrm{e}}(\sqrt{3}-1)$, then $9 \alpha^2$ is equal to

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The sum of squares of all the real solutions of the equation $\log _{(x+1)}\left(2 x^2+5 x+3\right)=4-\log _{(2 x+3)}\left(x^2+2 x+1\right)$ is...

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Let $\frac{x^2}{f\left(a^2+7 a+3\right)}+\frac{y^2}{f(3 a+15)}=1$ represent an ellipse with major axis along $y$-axis, where $f$ is a strictly decreasing positive function...

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Let $f(x)=\left\{\begin{array}{cc}\frac{1}{3} & , x \leq \pi / 2 \\ \frac{\mathrm{~b}(1-\sin x)}{(\pi-2 x)^2} & , x>\pi / 2\end{array}\right.$. If $f$...

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Let $f:(1, \infty) \rightarrow \mathbf{R}$ be a function defined as $f(x)=\frac{x-1}{x+1}$. Let $f^{i+1}(x)=f\left(f^i(x)\right), i=1,2, \ldots, 25$, where $f^1(x)=f(x)$. If $g(x)+f^{26}(x)=0,...

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Let $y=y(x)$ be the solution of the differential equation $x \sqrt{1-x^2} d y+\left(y \sqrt{1-x^2}-x \cos ^{-1} x\right) d x=0, x...

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The value of the integral $\int_0^2 \frac{\sqrt{x\left(x^2+x+1\right)}}{(\sqrt{x+1})\left(\sqrt{x^4+x^2+1}\right)} \mathrm{d} x$ is equal to:

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Let the foot of perpendicular from the point $(\lambda, 2,3)$ on the line $\frac{x-4}{1}=\frac{y-9}{2}=\frac{z-5}{1}$ be the point $(1, \mu, 2)$....

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Let $\overrightarrow{\mathrm{a}}=4 \hat{i}-\hat{j}+3 \hat{k}, \overrightarrow{\mathrm{~b}}=10 \hat{i}+2 \hat{j}-\hat{k}$ and a vector $\overrightarrow{\mathrm{c}}$ be such that $2(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}})+3(\overrightarrow{\mathrm{~b}} \times \overrightarrow{\mathrm{c}})=\overrightarrow{0}$. If...

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For the function $f(x)=\mathrm{e}^{\sin |x|}-|x|, x \in \mathbf{R}$, consider the following statements: Statement I : $f$ is differentiable for all $x...

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Let $\alpha=3 \sin ^{-1}\left(\frac{6}{11}\right)$ and $\beta=3 \cos ^{-1}\left(\frac{4}{9}\right)$, where inverse trigonometric functions take only the principal values. Given below are two...

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Let $O$ be the vertex of the parabola $y^2=4 x$ and its chords $O P$ and $O Q$ are perpendicular...

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If a straight line drawn through the point of intersection of the lines $4 x+3 y-1=0$ and $3 x+4 y-1=0$,...

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A person has three different bags and four different books. The number of ways, in which he can put these books...

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If $26\left(\frac{2^3}{3}\left({ }^{12} \mathrm{C}_2\right)+\frac{2^5}{5}\left({ }^{12} \mathrm{C}_4\right)+\frac{2^7}{7}\left({ }^{12} \mathrm{C}_6\right)+\cdots+\frac{2^{13}}{13}\left({ }^{12} \mathrm{C}_{12}\right)\right)=3^{13}-\alpha$, then $\alpha$ is equal to :

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A set of four observations has mean 1 and variance 13 . Another set of six observations has mean 2...

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