Number of ways of arranging 8 identical books into 4 identical shelves where any number of shelves may remain empty is equal to

Let $$\mathrm{A}$$ be the point of intersection of the lines $$3 x+2 y=14,5 x-y=6$$ and $$\mathrm{B}$$ be the point of intersection of the lines $$4 x+3 y=8,6 x+y=5$$. The distance of the point $$P(5,-2)$$ from the line $$\mathrm{AB}$$ is

If $$\int \frac{\sin ^{\frac{3}{2}} x+\cos ^{\frac{3}{2}} x}{\sqrt{\sin ^3 x \cos ^3 x \sin (x-\theta)}} d x=A \sqrt{\cos \theta \tan x-\sin \theta}+B \sqrt{\cos \theta-\sin \theta \cot x}+C$$, where $$C$$ is the integration constant, then $$A B$$ is equal to

The distance of the point $$(2,3)$$ from the line $$2 x-3 y+28=0$$, measured parallel to the line $$\sqrt{3} x-y+1=0$$, is equal to

Let a unit vector $$\hat{u}=x \hat{i}+y \hat{j}+z \hat{k}$$ make angles $$\frac{\pi}{2}, \frac{\pi}{3}$$ and $$\frac{2 \pi}{3}$$ with the vectors $$\frac{1}{\sqrt{2}} \hat{i}+\frac{1}{\sqrt{2}} \hat{k}, \frac{1}{\sqrt{2}} \hat{j}+\frac{1}{\sqrt{2}} \hat{k}$$ and $$\frac{1}{\sqrt{2}} \hat{i}+\frac{1}{\sqrt{2}} \hat{j}$$ respectively. If $$\vec{v}=\frac{1}{\sqrt{2}} \hat{i}+\frac{1}{\sqrt{2}} \hat{j}+\frac{1}{\sqrt{2}} \hat{k}$$ then $$|\hat{u}-\vec{v}|^2$$ is equal to

The function $$f(x)=\frac{x}{x^2-6 x-16}, x \in \mathbb{R}-\{-2,8\}$$

If R is the smallest equivalence relation on the set $$\{1,2,3,4\}$$ such that $$\{(1,2),(1,3)\} \subset \mathrm{R}$$, then the number of elements in $$\mathrm{R}$$ is __________.

If the mean and variance of five observations are $$\frac{24}{5}$$ and $$\frac{194}{25}$$ respectively and the mean of the first four observations is $$\frac{7}{2}$$, then the variance of the first four observations in equal to

Let $$\mathrm{P}(3,2,3), \mathrm{Q}(4,6,2)$$ and $$\mathrm{R}(7,3,2)$$ be the vertices of $$\triangle \mathrm{PQR}$$. Then, the angle $$\angle \mathrm{QPR}$$ is

$$\text { Let } y=\log _e\left(\frac{1-x^2}{1+x^2}\right),-1 < x<1 \text {. Then at } x=\frac{1}{2} \text {, the value of } 225\left(y^{\prime}-y^{\prime \prime}\right) \text { is equal to }$$

If $$\log _e \mathrm{a}, \log _e \mathrm{~b}, \log _e \mathrm{c}$$ are in an A.P. and $$\log _e \mathrm{a}-\log _e 2 \mathrm{~b}, \log _e 2 \mathrm{~b}-\log _e 3 \mathrm{c}, \log _e 3 \mathrm{c} -\log _e$$ a are also in an A.P, then $$a: b: c$$ is equal to

If $$\sin \left(\frac{y}{x}\right)=\log _e|x|+\frac{\alpha}{2}$$ is the solution of the differential equation $$x \cos \left(\frac{y}{x}\right) \frac{d y}{d x}=y \cos \left(\frac{y}{x}\right)+x$$ and $$y(1)=\frac{\pi}{3}$$, then $$\alpha^2$$ is equal to

An integer is chosen at random from the integers $$1,2,3, \ldots, 50$$. The probability that the chosen integer is a multiple of atleast one of 4, 6 and 7 is

Let $$\overrightarrow{O A}=\vec{a}, \overrightarrow{O B}=12 \vec{a}+4 \vec{b} \text { and } \overrightarrow{O C}=\vec{b}$$, where O is the origin. If S is the parallelogram with adjacent sides OA and OC, then $$\mathrm{{{area\,of\,the\,quadrilateral\,OA\,BC} \over {area\,of\,S}}}$$ is equal to _________.

The function $$f(x)=2 x+3(x)^{\frac{2}{3}}, x \in \mathbb{R}$$, has

If each term of a geometric progression $$a_1, a_2, a_3, \ldots$$ with $$a_1=\frac{1}{8}$$ and $$a_2 \neq a_1$$, is the arithmetic mean of the next two terms and $$S_n=a_1+a_2+\ldots . .+a_n$$, then $$S_{20}-S_{18}$$ is equal to

Let $$\mathrm{r}$$ and $$\theta$$ respectively be the modulus and amplitude of the complex number $$z=2-i\left(2 \tan \frac{5 \pi}{8}\right)$$, then $$(\mathrm{r}, \theta)$$ is equal to

The sum of the solutions $$x \in \mathbb{R}$$ of the equation $$\frac{3 \cos 2 x+\cos ^3 2 x}{\cos ^6 x-\sin ^6 x}=x^3-x^2+6$$ is

Let $$x=\frac{m}{n}$$ ($$m, n$$ are co-prime natural numbers) be a solution of the equation $$\cos \left(2 \sin ^{-1} x\right)=\frac{1}{9}$$ and let $$\alpha, \beta(\alpha >\beta)$$ be the roots of the equation $$m x^2-n x-m+ n=0$$. Then the point $$(\alpha, \beta)$$ lies on the line

Let $$A=\left[\begin{array}{ccc}2 & 1 & 2 \\ 6 & 2 & 11 \\ 3 & 3 & 2\end{array}\right]$$ and $$P=\left[\begin{array}{lll}1 & 2 & 0 \\ 5 & 0 & 2 \\ 7 & 1 & 5\end{array}\right]$$. The sum of the prime factors of $$\left|P^{-1} A P-2 I\right|$$ is equal to

Let O be the origin, and M and $$\mathrm{N}$$ be the points on the lines $$\frac{x-5}{4}=\frac{y-4}{1}=\frac{z-5}{3}$$ and $$\frac{x+8}{12}=\frac{y+2}{5}=\frac{z+11}{9}$$ respectively such that $$\mathrm{MN}$$ is the shortest distance between the given lines. Then $$\overrightarrow{O M} \cdot \overrightarrow{O N}$$ is equal to _________.

Let the area of the region $$\left\{(x, y): 0 \leq x \leq 3,0 \leq y \leq \min \left\{x^2+2,2 x+2\right\}\right\}$$ be A. Then $$12 \mathrm{~A}$$ is equal to __________.

Let the set $$C=\left\{(x, y) \mid x^2-2^y=2023, x, y \in \mathbb{N}\right\}$$. Then $$\sum_\limits{(x, y) \in C}(x+y)$$ is equal to _________.

Let the slope of the line $$45 x+5 y+3=0$$ be $$27 r_1+\frac{9 r_2}{2}$$ for some $$r_1, r_2 \in \mathbb{R}$$. Then $$\lim _\limits{x \rightarrow 3}\left(\int_3^x \frac{8 t^2}{\frac{3 r_2 x}{2}-r_2 x^2-r_1 x^3-3 x} d t\right)$$ is equal to _________.

Let $$\alpha, \beta$$ be the roots of the equation $$x^2-\sqrt{6} x+3=0$$ such that $$\operatorname{Im}(\alpha)>\operatorname{Im}(\beta)$$. Let $$a, b$$ be integers not divisible by 3 and $$n$$ be a natural number such that $$\frac{\alpha^{99}}{\beta}+\alpha^{98}=3^n(a+i b), i=\sqrt{-1}$$. Then $$n+a+b$$ is equal to __________.

Let for any three distinct consecutive terms $$a, b, c$$ of an A.P, the lines $$a x+b y+c=0$$ be concurrent at the point $$P$$ and $$Q(\alpha, \beta)$$ be a point such that the system of equations

$$\begin{aligned} & x+y+z=6, \\ & 2 x+5 y+\alpha z=\beta \text { and } \end{aligned}$$

$$x+2 y+3 z=4$$, has infinitely many solutions. Then $$(P Q)^2$$ is equal to _________.

Let $$P(\alpha, \beta)$$ be a point on the parabola $$y^2=4 x$$. If $$P$$ also lies on the chord of the parabola $$x^2=8 y$$ whose mid point is $$\left(1, \frac{5}{4}\right)$$, then $$(\alpha-28)(\beta-8)$$ is equal to _________.

Remainder when $$64^{32^{32}}$$ is divided by 9 is equal to ________.

Let $$f(x)=\sqrt{\lim _\limits{r \rightarrow x}\left\{\frac{2 r^2\left[(f(r))^2-f(x) f(r)\right]}{r^2-x^2}-r^3 e^{\frac{f(r)}{r}}\right\}}$$ be differentiable in $$(-\infty, 0) \cup(0, \infty)$$ and $$f(1)=1$$. Then the value of ea, such that $$f(a)=0$$, is equal to _________.

If $$\int_\limits{\frac{\pi}{6}}^{\frac{\pi}{3}} \sqrt{1-\sin 2 x} d x=\alpha+\beta \sqrt{2}+\gamma \sqrt{3}$$, where $$\alpha, \beta$$ and $$\gamma$$ are rational numbers, then $$3 \alpha+4 \beta-\gamma$$ is equal to _________.