Let $$f: \mathbb{R} \rightarrow \mathbb{R}$$ be a function defined by $$f(x)=\frac{x}{\left(1+x^4\right)^{1 / 4}}$$, and $$g(x)=f(f(f(f(x))))$$. Then, $$18 \int_0^{\sqrt{2 \sqrt{5}}} x^2 g(x) d x$$ is equal to

Let $$a$$ and $$b$$ be be two distinct positive real numbers. Let $$11^{\text {th }}$$ term of a GP, whose first term is $$a$$ and third term is $$b$$, is equal to $$p^{\text {th }}$$ term of another GP, whose first term is $$a$$ and fifth term is $$b$$. Then $$p$$ is equal to

Let $$y=f(x)$$ be a thrice differentiable function in $$(-5,5)$$. Let the tangents to the curve $$y=f(x)$$ at $$(1, f(1))$$ and $$(3, f(3))$$ make angles $$\pi / 6$$ and $$\pi / 4$$, respectively with positive $$x$$-axis. If $$27 \int_\limits1^3\left(\left(f^{\prime}(t)\right)^2+1\right) f^{\prime \prime}(t) d t=\alpha+\beta \sqrt{3}$$ where $$\alpha, \beta$$ are integers, then the value of $$\alpha+\beta$$ equals

For $$\alpha, \beta \in(0, \pi / 2)$$, let $$3 \sin (\alpha+\beta)=2 \sin (\alpha-\beta)$$ and a real number $$k$$ be such that $$\tan \alpha=k \tan \beta$$. Then, the value of $$k$$ is equal to

If $$z$$ is a complex number, then the number of common roots of the equations $$z^{1985}+z^{100}+1=0$$ and $$z^3+2 z^2+2 z+1=0$$, is equal to

Let $$f: \mathbb{R}-\{0\} \rightarrow \mathbb{R}$$ be a function satisfying $$f\left(\frac{x}{y}\right)=\frac{f(x)}{f(y)}$$ for all $$x, y, f(y) \neq 0$$. If $$f^{\prime}(1)=2024$$, then

Let $$R=\left(\begin{array}{ccc}x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z\end{array}\right)$$ be a non-zero $$3 \times 3$$ matrix, where $$x \sin \theta=y \sin \left(\theta+\frac{2 \pi}{3}\right)=z \sin \left(\theta+\frac{4 \pi}{3}\right) \neq 0, \theta \in(0,2 \pi)$$. For a square matrix $$M$$, let trace $$(M)$$ denote the sum of all the diagonal entries of $$M$$. Then, among the statements:

(I) Trace $$(R)=0$$

(II) If trace $$(\operatorname{adj}(\operatorname{adj}(R))=0$$, then $$R$$ has exactly one non-zero entry.

If $$x^2-y^2+2 h x y+2 g x+2 f y+c=0$$ is the locus of a point, which moves such that it is always equidistant from the lines $$x+2 y+7=0$$ and $$2 x-y+8=0$$, then the value of $$g+c+h-f$$ equals

Let $$a$$ and $$b$$ be real constants such that the function $$f$$ defined by $$f(x)=\left\{\begin{array}{ll}x^2+3 x+a & , x \leq 1 \\ b x+2 & , x>1\end{array}\right.$$ be differentiable on $$\mathbb{R}$$. Then, the value of $$\int_\limits{-2}^2 f(x) d x$$ equals

Let $$\mathrm{f}: \mathbb{R} \rightarrow \mathbb{R}$$ be defined as $$f(x)=a e^{2 x}+b e^x+c x$$. If $$f(0)=-1, f^{\prime}\left(\log _e 2\right)=21$$ and $$\int_0^{\log _e 4}(f(x)-c x) d x=\frac{39}{2}$$, then the value of $$|a+b+c|$$ equals

Let $$L_1: \vec{r}=(\hat{i}-\hat{j}+2 \hat{k})+\lambda(\hat{i}-\hat{j}+2 \hat{k}), \lambda \in \mathbb{R}$$,

$$L_2: \vec{r}=(\hat{j}-\hat{k})+\mu(3 \hat{i}+\hat{j}+p \hat{k}), \mu \in \mathbb{R} \text {, and } L_3: \vec{r}=\delta(\ell \hat{i}+m \hat{j}+n \hat{k}), \delta \in \mathbb{R}$$

be three lines such that $$L_1$$ is perpendicular to $$L_2$$ and $$L_3$$ is perpendicular to both $$L_1$$ and $$L_2$$. Then, the point which lies on $$L_3$$ is

Let $$\vec{a}=\hat{i}+\alpha \hat{j}+\beta \hat{k}, \alpha, \beta \in \mathbb{R}$$. Let a vector $$\vec{b}$$ be such that the angle between $$\vec{a}$$ and $$\vec{b}$$ is $$\frac{\pi}{4}$$ and $$|\vec{b}|^2=6$$. If $$\vec{a} \cdot \vec{b}=3 \sqrt{2}$$, then the value of $$\left(\alpha^2+\beta^2\right)|\vec{a} \times \vec{b}|^2$$ is equal to

Let $$P$$ be a point on the hyperbola $$H: \frac{x^2}{9}-\frac{y^2}{4}=1$$, in the first quadrant such that the area of triangle formed by $$P$$ and the two foci of $$H$$ is $$2 \sqrt{13}$$. Then, the square of the distance of $$P$$ from the origin is

Let $$f(x)=(x+3)^2(x-2)^3, x \in[-4,4]$$. If $$M$$ and $$m$$ are the maximum and minimum values of $$f$$, respectively in $$[-4,4]$$, then the value of $$M-m$$ is

Suppose $$2-p, p, 2-\alpha, \alpha$$ are the coefficients of four consecutive terms in the expansion of $$(1+x)^n$$. Then the value of $$p^2-\alpha^2+6 \alpha+2 p$$ equals

Consider the system of linear equations $$x+y+z=5, x+2 y+\lambda^2 z=9, x+3 y+\lambda z=\mu$$, where $$\lambda, \mu \in \mathbb{R}$$. Then, which of the following statement is NOT correct?

Let $$\vec{a}$$ and $$\vec{b}$$ be two vectors such that $$|\vec{b}|=1$$ and $$|\vec{b} \times \vec{a}|=2$$. Then $$|(\vec{b} \times \vec{a})-\vec{b}|^2$$ is equal to

If the domain of the function $$f(x)=\log _e\left(\frac{2 x+3}{4 x^2+x-3}\right)+\cos ^{-1}\left(\frac{2 x-1}{x+2}\right)$$ is $$(\alpha, \beta]$$, then the value of $$5 \beta-4 \alpha$$ is equal to

Bag A contains 3 white, 7 red balls and Bag B contains 3 white, 2 red balls. One bag is selected at random and a ball is drawn from it. The probability of drawing the ball from the bag A, if the ball drawn is white, is

Let $$A(\alpha, 0)$$ and $$B(0, \beta)$$ be the points on the line $$5 x+7 y=50$$. Let the point $$P$$ divide the line segment $$A B$$ internally in the ratio $$7:3$$. Let $$3 x-25=0$$ be a directrix of the ellipse $$E: \frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$ and the corresponding focus be $$S$$. If from $$S$$, the perpendicular on the $$x$$-axis passes through $$P$$, then the length of the latus rectum of $$E$$ is equal to,

In an examination of Mathematics paper, there are 20 questions of equal marks and the question paper is divided into three sections : $$A, B$$ and $$C$$. A student is required to attempt total 15 questions taking at least 4 questions from each section. If section $$A$$ has 8 questions, section $$B$$ has 6 questions and section $$C$$ has 6 questions, then the total number of ways a student can select 15 questions is __________.

Let $$Y=Y(X)$$ be a curve lying in the first quadrant such that the area enclosed by the line $$Y-y=Y^{\prime}(x)(X-x)$$ and the co-ordinate axes, where $$(x, y)$$ is any point on the curve, is always $$\frac{-y^2}{2 Y^{\prime}(x)}+1, Y^{\prime}(x) \neq 0$$. If $$Y(1)=1$$, then $$12 Y(2)$$ equals __________.

The number of real solutions of the equation $$x\left(x^2+3|x|+5|x-1|+6|x-2|\right)=0$$ is _________.

Consider two circles $$C_1: x^2+y^2=25$$ and $$C_2:(x-\alpha)^2+y^2=16$$, where $$\alpha \in(5,9)$$. Let the angle between the two radii (one to each circle) drawn from one of the intersection points of $$C_1$$ and $$C_2$$ be $$\sin ^{-1}\left(\frac{\sqrt{63}}{8}\right)$$. If the length of common chord of $$C_1$$ and $$C_2$$ is $$\beta$$, then the value of $$(\alpha \beta)^2$$ equals _______.

Let a line passing through the point $$(-1,2,3)$$ intersect the lines $$L_1: \frac{x-1}{3}=\frac{y-2}{2}=\frac{z+1}{-2}$$ at $$M(\alpha, \beta, \gamma)$$ and $$L_2: \frac{x+2}{-3}=\frac{y-2}{-2}=\frac{z-1}{4}$$ at $$N(a, b, c)$$. Then, the value of $$\frac{(\alpha+\beta+\gamma)^2}{(a+b+c)^2}$$ equals __________.

The variance $$\sigma^2$$ of the data

$$x_i$$	0	1	5	6	10	12	17
$$f_i$$	3	2	3	2	6	3	3

is _________.

The area of the region enclosed by the parabola $$(y-2)^2=x-1$$, the line $$x-2 y+4=0$$ and the positive coordinate axes is _________.

The number of symmetric relations defined on the set $$\{1,2,3,4\}$$ which are not reflexive is _________.

Let $$\alpha=\sum_\limits{k=0}^n\left(\frac{\left({ }^n C_k\right)^2}{k+1}\right)$$ and $$\beta=\sum_\limits{k=0}^{n-1}\left(\frac{{ }^n C_k{ }^n C_{k+1}}{k+2}\right)$$ If $$5 \alpha=6 \beta$$, then $$n$$ equals _______.

Let $$S_n$$ be the sum to $$n$$-terms of an arithmetic progression $$3,7,11$$, If $$40<\left(\frac{6}{n(n+1)} \sum_\limits{k=1}^n S_k\right)<42$$, then $$n$$ equals ________.