Let $\int x^3 \sin x \mathrm{~d} x=g(x)+C$, where $C$ is the constant of integration. If $8\left(g\left(\frac{\pi}{2}\right)+g^{\prime}\left(\frac{\pi}{2}\right)\right)=\alpha \pi^3+\beta \pi^2+\gamma, \alpha, \beta, \gamma \in Z$, then $\alpha+\beta-\gamma$ equals :

Let $f$ be a differentiable function on $\mathbb{R}$ such that $f(2)=1,\ f'(2)=4$. Let $\displaystyle \lim_{x\to 0}\big(f(2+x)\big)^{\frac{3}{x}}=e^{\alpha}$. Then the number of times the curve $y=4x^3-4x^2-4(\alpha-7)x-\alpha$ meets the $x$-axis is:

A spherical chocolate ball has a layer of ice-cream of uniform thickness around it. When the thickness of the ice-cream layer is 1 cm , the ice-cream melts at the rate of $81 \mathrm{~cm}^3 / \mathrm{min}$ and the thickness of the ice-cream layer decreases at the rate of $\frac{1}{4 \pi} \mathrm{~cm} / \mathrm{min}$. The surface area (in $\mathrm{cm}^2$ ) of the chocolate ball (without the ice-cream layer) is :

From a group of 7 batsmen and 6 bowlers, 10 players are to be chosen for a team, which should include atleast 4 batsmen and atleast 4 bowlers. One batsmen and one bowler who are captain and vice-captain respectively of the team should be included. Then the total number of ways such a selection can be made, is

Let the shortest distance from $(a,0)$, $a>0$, to the parabola $y^{2}=4x$ be $4$. Then the equation of the circle passing through the point $(a,0)$ and the focus of the parabola, with centre on the axis of the parabola, is:

Let $C_1$ be the circle in the third quadrant of radius 3 , that touches both coordinate axes. Let $C_2$ be the circle with centre $(1,3)$ that touches $\mathrm{C}_1$ externally at the point $(\alpha, \beta)$. If $(\beta-\alpha)^2=\frac{m}{n}$ , $\operatorname{gcd}(m, n)=1$, then $m+n$ is equal to

Let $x=x(y)$ be the solution of the differential equation $y=\left(x-y \frac{\mathrm{~d} x}{\mathrm{~d} y}\right) \sin \left(\frac{x}{y}\right), y>0$ and $x(1)=\frac{\pi}{2}$. Then $\cos (x(2))$ is equal to :

Let P be the parabola, whose focus is $(-2,1)$ and directrix is $2 x+y+2=0$. Then the sum of the ordinates of the points on P, whose abscissa is $-$2, is

The mean and standard deviation of $100$ observations are $40$ and $5.1$, respectively. By mistake one observation is taken as $50$ instead of $40$. If the correct mean and the correct standard deviation are $\mu$ and $\sigma$ respectively, then $10(\mu+\sigma)$ is equal to

Let $\triangle ABC$ be the triangle such that the equations of lines $AB$ and $AC$ are $3y-x=2$ and $x+y=2$, respectively, and the points $B$ and $C$ lie on the $x$-axis. If $P$ is the orthocentre of $\triangle ABC$, then the area of $\triangle PBC$ is equal to

$\displaystyle \lim_{x\to 0^{+}}\frac{\tan\big(5x^{1/5}\big),\ln(1+3x^{2})}{\big(\tan^{-1}(3\sqrt{2})\big),\big(e^{x\sqrt{3}}-1\big)}$ is equal to

Let $y=y(x)$ be the solution curve of the differential equation $x(x^{2}+e^{x})^{2}dy+\big(e^{x}(x-2)y-x^{3}\big)dx=0, x>0,$ passing through the point $(1,0)$. Then $y(2)$ is equal to:

Let $A$ be a $3 \times 3$ matrix such that $|\operatorname{adj}(\operatorname{adj}(\operatorname{adj} \mathrm{A}))|=81$.

If $S=\left\{n \in \mathbb{Z}:(|\operatorname{adj}(\operatorname{adj} A)|)^{\frac{(n-1)^2}{2}}=|A|^{\left(3 n^2-5 n-4\right)}\right\}$, then $\sum_\limits{n \in S}\left|A^{\left(n^2+n\right)}\right|$ is equal to :

If $\displaystyle \lim_{x \to 0} \csc x \left( \sqrt{2\cos^2 x + 3\cos x} - \sqrt{\cos^2 x + \sin x + 4} \right)$ is :

The remainder when $\big((64)^{(64)}\big)^{(64)}$ is divided by $7$ is:

Let $f:\mathbb{R}-{0}\to\mathbb{R}$ be a function such that $f(x) - 6f\left(\frac{1}{x}\right) = \frac{35}{3x} - \frac{5}{2}.$ If $\displaystyle \lim_{x\to 0}\left(\frac{1}{x} + f(x)\right) = \beta,\ \alpha, \beta \in \mathbb{R},$ then $\alpha + 2\beta$ is equal to :

If for $\theta\in\left[-\dfrac{\pi}{3},0\right]$, the points $(x,y)=\big(3\tan(\theta+\tfrac{\pi}{3}),,2\tan(\theta+\tfrac{\pi}{6})\big)$ lie on $xy+\alpha x+\beta y+\gamma=0$, then $\alpha^{2}+\beta^{2}+\gamma^{2}$ is:

The product of all the rational roots of the equation $ (x^2 - 9x + 11)^2 - (x - 4)(x - 5) = 3 $ is equal to

Let the system of equations :$ \begin{aligned} & 2 x+3 y+5 z=9 \\ & 7 x+3 y-2 z=8 \\ & 12 x+3 y-(4+\lambda) z=16-\mu \end{aligned}$$

have infinitely many solutions. Then the radius of the circle centred at $(\lambda, \mu)$ and touching the line $4 x=3 y$ is :

For some $ n \ne 10 $, let the coefficients of the 5th, 6th and 7th terms in the binomial expansion of $ (1 + x)^{n+4} $ be in A.P. Then the largest coefficient in the expansion of $ (1 + x)^{n+4} $ is:

Let the set of all values of $p\in\mathbb{R}$, for which both the roots of the equation $x^{2}-(p+2)x+(2p+9)=0$ are negative real numbers, be the interval $(\alpha,\beta)$. Then $\beta-2\alpha$ is:

Let $f(x) = \dfrac{9x^2 + 16}{2^{2x+1} + 2^{x+4} + 32}$. Then the value of $8 \big( f\left(\dfrac{1}{15}\right) + f\left(\dfrac{2}{15}\right) + \dots + f\left(\dfrac{50}{15}\right) \big)$ is equal to

Among the statements (S1): The set ${z\in\mathbb{C}\setminus{-i}:\ |z|=1\ \text{ and }\ \dfrac{z-i}{z+i}\ \text{is purely real}}$ contains exactly two elements and (S2): The set ${z\in\mathbb{C}\setminus{-1}:\ |z|=1\ \text{ and }\ \dfrac{z-1}{z+1}\ \text{is purely imaginary}}$ contains infinitely many elements.

If $I(m,n) = \displaystyle \int_0^1 x^{m-1}(1-x)^{n-1} dx, ; m,n > 0$, then $I(9,14) + I(10,13)$ is

The integral $\displaystyle \int_{0}^{\pi}\frac{(x+3)\sin x}{1+3\cos^{2}x}dx$ is equal to

If $\alpha$ and $\beta$ are the roots of the equation $2x^2 - 3x - 2i = 0$, where $i = \sqrt{-1}$, then $16 \cdot \text{Re}\left( \dfrac{\alpha^{19} + \beta^{19} + \alpha^{11} + \beta^{11}}{\alpha^5 + \beta^5} \right) \cdot \text{Im}\left( \dfrac{\alpha^{19} + \beta^{19} + \alpha^{11} + \beta^{11}}{\alpha^5 + \beta^5} \right)$ is equal to

Let the line $L$ pass through $(1,1,1)$ and intersect the lines $\dfrac{x-1}{2} = \dfrac{y+1}{3} = \dfrac{z-1}{4}$ and $\dfrac{x-3}{1} = \dfrac{y-4}{2} = \dfrac{z}{1}$. Then, which of the following points lies on the line $L$?

Let the product of the focal distances of the point $\left( \sqrt{3}, \dfrac{1}{2} \right)$ on the ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$, $(a > b)$, be $\dfrac{7}{4}$. Then the absolute difference of the eccentricities of two such ellipses is

If the area of the region bounded by the curves $y = 4 - \dfrac{x^2}{4}$ and $y = \dfrac{x-4}{2}$ is equal to $\alpha$, then $6\alpha$ equals

Let $y = y(x)$ be the solution of the differential equation $\left(xy - 5x^2\sqrt{1 + x^2}\right)dx + (1 + x^2)dy = 0$, $y(0) = 0$. Then $y(\sqrt{3})$ is equal to

Let the angle $\theta,;0<\theta<\tfrac{\pi}{2}$ between two unit vectors $\hat a$ and $\hat b$ be $\sin^{-1}\left(\tfrac{\sqrt{65}}{9}\right)$. If the vector $\vec c=3\hat a+6\hat b+9(\hat a\times\hat b)$, then the value of $9(\vec c\cdot\hat a)-3(\vec c\cdot\hat b)$ is

Let in a $\triangle ABC$, the length of the side $AC$ be $6$, the vertex $B$ be $(1, 2, 3)$ and the vertices $A, C$ lie on the line $\dfrac{x - 6}{3} = \dfrac{y - 7}{2} = \dfrac{z - 7}{-2}$. Then the area (in sq. units) of $\triangle ABC$ is:

Let $x_1,x_2,x_3,x_4$ be in a geometric progression. If $2,7,9,5$ are subtracted respectively from $x_1,x_2,x_3,x_4$, then the resulting numbers are in an arithmetic progression. Then the value of $\dfrac1{24}(x_1x_2x_3x_4)$ is

Let the line passing through the points $(-1, 2, 1)$ and parallel to the line $\dfrac{x - 1}{2} = \dfrac{y + 1}{3} = \dfrac{z}{4}$ intersect the line $\dfrac{x + 2}{3} = \dfrac{y - 3}{2} = \dfrac{z - 4}{1}$ at the point $P$. Then the distance of $P$ from the point $Q(4, -5, 1)$ is

If the shortest distance between the lines $\dfrac{x-1}{2}=\dfrac{y-2}{3}=\dfrac{z-3}{4}$ and $\dfrac{x}{1}=\dfrac{y}{\alpha}=\dfrac{z-5}{1}$ is $\dfrac{5}{\sqrt6}$, then the sum of all possible values of $\alpha$ is

$A$ and $B$ alternately throw a pair of dice. $A$ wins if he throws a sum of $5$ before $B$ throws a sum of $8$, and $B$ wins if he throws a sum of $8$ before $A$ throws a sum of $5$. The probability that $A$ wins if $A$ makes the first throw, is

Let $x=-1$ and $x=2$ be the critical points of the function $f(x)=x^{3}+ax^{2}+b\log_{e}|x|+1,;x\neq0$. Let $m$ and $M$ respectively be the absolute minimum and the absolute maximum values of $f$ in the interval $\left[-2,-\dfrac{1}{2}\right]$. Then $|M+m|$ is equal to (take $\log_{e}2=0.7$):

Let $\vec{a} = \hat{i} + 2\hat{j} + 3\hat{k}$, $\vec{b} = 3\hat{i} + \hat{j} - \hat{k}$ and $\vec{c}$ be three vectors such that $\vec{c}$ is coplanar with $\vec{a}$ and $\vec{b}$. If the vector $\vec{c}$ is perpendicular to $\vec{b}$ and $\vec{a} \cdot \vec{c} = 5$, then $|\vec{c}|$ is equal to

Let $f:\mathbb{R}\to\mathbb{R}$ be a polynomial of degree four having extreme values at $x=4$ and $x=5$. If $\displaystyle \lim_{x\to 0}\frac{f(x)}{x^{2}}=5$, then $f(2)$ is equal to:

Consider the region $R = {(x, y) : x \le y \le 9 - \dfrac{11}{3}x^2, , x \ge 0}$. The area of the largest rectangle of sides parallel to the coordinate axes and inscribed in $R$ is

Let $y=y(x)$ be the solution of the differential equation $(x^{2}+1),y'-2xy=(x^{4}+2x^{2}+1)\cos x$, with $y(0)=1$. Then $\displaystyle \int_{-3}^{3} y(x),dx$ is:

The area of the region ${(x, y) : x^2 + 4x + 2 \le y \le |x + 2|}$ is equal to

If the locus of $z\in\mathbb{C}$, such that $\operatorname{Re}!\left(\dfrac{z-1}{2z+i}\right)+\operatorname{Re}!\left(\dfrac{z-1}{2z-i}\right)=2$, is a circle of radius $r$ and center $(a,b)$, then $\dfrac{15ab}{r^2}$ is equal to:

Let $S_n = \dfrac{1}{2} + \dfrac{1}{6} + \dfrac{1}{12} + \dfrac{1}{20} + \dots$ up to $n$ terms. If the sum of the first six terms of an A.P. with first term $-p$ and common difference $p$ is $\sqrt{2026}, S_{2025}$, then the absolute difference between $20^{\text{th}}$ and $15^{\text{th}}$ terms of the A.P. is

Let $a_n$ be the $n^{\text{th}}$ term of an A.P. If $S_n=a_1+a_2+\cdots+a_n=700$, $a_6=7$ and $S_7=7$, then $a_n$ is equal to:

For a statistical data $x_1, x_2, \ldots, x_{10}$ of $10$ values, a student obtained the mean as $5.5$ and $\sum_{i=1}^{10} x_i^2 = 371$. He later found that he had noted two values in the data incorrectly as $4$ and $5$, instead of the correct values $6$ and $8$, respectively. The variance of the corrected data is

A bag contains $19$ unbiased coins and one coin with head on both sides. One coin drawn at random is tossed and a head turns up. If the probability that the drawn coin was unbiased is $\dfrac{m}{n}$ with $\gcd(m,n)=1$, then $n^2-m^2$ is equal to:

Let the lines $3x - 4y - \alpha = 0$, $8x - 11y - 33 = 0$, and $2x - 3y + \lambda = 0$ be concurrent. If the image of the point $(1,2)$ in the line $2x - 3y + \lambda = 0$ is $\left(\dfrac{57}{13}, -\dfrac{40}{13}\right)$, then $|\alpha \lambda|$ is equal to

If the range of the function $f(x)=\dfrac{5-x}{x^2-3x+2}$, $x\ne1,2$, is $(-\infty,\alpha]\cup[\beta,\infty)$, then $\alpha^2+\beta^2$ is equal to:f

If the system of equations $2x - y + z = 4$, $5x + \lambda y + 3z = 12$, $100x - 47y + \mu z = 212$ has infinitely many solutions, then $\mu - 2\lambda$ is equal to

The number of real roots of the equation $x|x-2|+3|x-3|+1=0$ is:

Let circle $C$ be the image of $x^2 + y^2 - 2x + 4y - 4 = 0$ in the line $2x - 3y + 5 = 0$ and $A$ be the point on $C$ such that $OA$ is parallel to $x$-axis and $A$ lies on the right hand side of the centre $O$ of $C$. If $B(\alpha,\beta)$, with $\beta < 4$, lies on $C$ such that the length of the arc $AB$ is $(1/6)^{\text{th}}$ of the perimeter of $C$, then $\beta - \sqrt{3},\alpha$ is equal to

Let the length of a latus rectum of an ellipse $\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1$ be $10$. If its eccentricity is the minimum value of $f(t)=t^{2}+t+\dfrac{11}{12}$, $t\in\mathbb{R}$, then $a^{2}+b^{2}$ is equal to:

In an arithmetic progression, if $S_{40} = 1030$ and $S_{12} = 57$, then $S_{30} - S_{10}$ is equal to

Let $e_1$ and $e_2$ be the eccentricities of the ellipse $\dfrac{x^2}{b^2}+\dfrac{y^2}{25}=1$ and the hyperbola $\dfrac{x^2}{16}-\dfrac{y^2}{b^2}=1$, respectively. If $b<5$ and $e_1e_2=1$, then the eccentricity of the ellipse having its axes along the coordinate axes and passing through all four foci (two of the ellipse and two of the hyperbola) is:

Let $A = [a_{ij}]$ be a square matrix of order $2$ with entries either $0$ or $1$. Let $E$ be the event that $A$ is an invertible matrix. Then the probability $\mathrm{P}(E)$ is

If the equation of the line passing through the point $ \left( 0, -\frac{1}{2}, 0 \right) $ and perpendicular to the lines $ \vec{r} = \lambda \left( \hat{i} + a\hat{j} + b\hat{k} \right) $ and $ \vec{r} = \left( \hat{i} - \hat{j} - 6\hat{k} \right) + \mu \left( -b \hat{i} + a\hat{j} + 5\hat{k} \right) $ is $ \frac{x-1}{-2} = \frac{y+4}{d} = \frac{z-c}{-4} $, then $ a+b+c+d $ is equal to :

Let $(2,3)$ be the largest open interval in which the function $f(x)=2\log_e(x-2)-x^2+ax+1$ is strictly increasing and $(b,c)$ be the largest open interval in which the function $g(x)=(x-1)^3(x+2-a)^2$ is strictly decreasing. Then $100(a+b-c)$ is equal to

If the sum of the second, fourth and sixth terms of a G.P. of positive terms is $21$ and the sum of its eighth, tenth and twelfth terms is $15309$, then the sum of its first nine terms is:

Let $A={,x\in(0,\pi)-{\tfrac{\pi}{2}}: \log_{(2/\pi)}|\sin x|+\log_{(2/\pi)}|\cos x|=2,}$ and $B={,x\ge 0:\sqrt{x}(\sqrt{x}-4)-3|\sqrt{x}-2|+6=0,}$. Then $n(A\cup B)$ is equal to

If the area of the region ${(x,y):, 1+x^2 \le y \le \min{x+7,; 11-3x}}$ is $A$, then $3A$ is equal to:

For some $a, b,$ let $f(x)=\left|\begin{array}{ccc}\mathrm{a}+\frac{\sin x}{x} & 1 & \mathrm{~b} \\ \mathrm{a} & 1+\frac{\sin x}{x} & \mathrm{~b} \\ \mathrm{a} & 1 & \mathrm{~b}+\frac{\sin x}{x}\end{array}\right|, x \neq 0, \lim \limits_{x \rightarrow 0} f(x)=\lambda+\mu \mathrm{a}+\nu \mathrm{b}.$ Then $(\lambda+\mu+v)^2$ is equal to :

The number of solutions of the equation

$ \cos 2\theta \cos \frac{\theta}{2} + \cos \frac{5\theta}{2} = 2\cos^3 \frac{5\theta}{2} $ in $ \left[ -\frac{\pi}{2}, \frac{\pi}{2} \right] $ is :

The area of the region enclosed by the curves $y=\mathrm{e}^x, y=\left|\mathrm{e}^x-1\right|$ and $y$-axis is :

$\text{Consider the lines } L_{1}: , x-1=y-2=z \quad \text{and} \quad L_{2}: , x-2=y=z-1.$ $\text{Let the feet of the perpendiculars from the point } P(5,1,-3) \text{ on } L_{1} \text{ and } L_{2} \text{ be } Q \text{ and } R \text{ respectively.}$ $\text{If the area of the triangle } PQR \text{ is } A, \text{ then } 4A^{2}\text{ is equal to:}$

If $\alpha>\beta>\gamma>0$, then the expression $\cot ^{-1}\left\{\beta+\frac{\left(1+\beta^2\right)}{(\alpha-\beta)}\right\}+\cot ^{-1}\left\{\gamma+\frac{\left(1+\gamma^2\right)}{(\beta-\gamma)}\right\}+\cot ^{-1}\left\{\alpha+\frac{\left(1+\alpha^2\right)}{(\gamma-\alpha)}\right\}$ is equal to :

$A={(\alpha,\beta)\in\mathbb{R}\times\mathbb{R}:\ |\alpha-1|\le 4\ \text{and}\ |\beta-5|\le 6}$ $B={(\alpha,\beta)\in\mathbb{R}\times\mathbb{R}:\ 16(\alpha-2)^2+9(\beta-6)^2\le 144}.$ Then

If the system of equations $x+2y-3z=2$, $2x+\lambda y+5z=5$, $14x+3y+\mu z=33$ has infinitely many solutions, then $\lambda+\mu$ is equal to:

Let $\vec a$ and $\vec b$ be vectors of the same magnitude such that $\displaystyle \frac{\lvert\vec a+\vec b\rvert+\lvert\vec a-\vec b\rvert}{\lvert\vec a+\vec b\rvert-\lvert\vec a-\vec b\rvert}=\sqrt2+1.$ Then $\displaystyle \frac{\lvert\vec a+\vec b\rvert^{2}}{\lvert\vec a\rvert^{2}}$ is:

Suppose $A$ and $B$ are the coefficients of $30^{\text{th}}$ and $12^{\text{th}}$ terms respectively in the binomial expansion of $(1+x)^{2n-1}$. If $2A=5B$, then $n$ is equal to:

Let the system of equations

x + 5y - z = 1

4x + 3y - 3z = 7

24x + y + λz = μ

λ, μ ∈ ℝ, have infinitely many solutions. Then the number of the solutions of this system,

if x, y, z are integers and satisfy 7 ≤ x + y + z ≤ 77, is :

Let the position vectors of three vertices of a triangle be $4\vec p+\vec q-3\vec r$, $-5\vec p+\vec q+2\vec r$ and $2\vec p-\vec q+2\vec r$. If the position vectors of the orthocenter and the circumcenter of the triangle are $\dfrac{\vec p+\vec q+\vec r}{4}$ and $\alpha \vec p+\beta \vec q+\gamma \vec r$ respectively, then $\alpha+2\beta+5\gamma$ is equal to:

Let $p$ be the number of all triangles that can be formed by joining the vertices of a regular $n$-gon $P$, and $q$ be the number of all quadrilaterals that can be formed by joining the vertices of $P$. If $p+q=126$, then the eccentricity of the ellipse $\dfrac{x^2}{16}+\dfrac{y^2}{n}=1$ is:

If $7 = 5 + \frac{1}{7}(5+\alpha) + \frac{1}{7^2}(5+2\alpha) + \frac{1}{7^3}(5+3\alpha) + \cdots$, then the value of $\alpha$ is:

Let a random variable $X$ take values $0,1,2,3$ with $P(X=0)=P(X=1)=p$, $P(X=2)=P(X=3)$ and $E(X^2)=2E(X)$. Then the value of $8p-1$ is:

If the equation of the parabola with vertex $V!\left(\frac{3}{2},,3\right)$ and directrix $x+2y=0$ is $\alpha x^2+\beta y^2-\gamma xy-30x-60y+225=0$, then $\alpha+\beta+\gamma$ is equal to:

If the orthocenter of the triangle formed by the lines y = x + 1, y = 4x - 8 and y = mx + c is at (3, -1), then m - c is :

Let $\vec a = 3\hat{i}-\hat{j}+2\hat{k}$, $\vec b=\vec a \times (\hat{i}-2\hat{k})$ and $\vec c=\vec b \times \hat{k}$. Then the projection of $\vec c-2\hat{j}$ on $\vec a$ is:

The equation of the chord of the ellipse $\dfrac{x^2}{25}+\dfrac{y^2}{16}=1$, whose mid-point is $(3,1)$, is:

Let the points $\left(\dfrac{11}{2},,\alpha\right)$ lie on or inside the triangle with sides $x+y=11$, $x+2y=16$ and $2x+3y=29$. Then the product of the smallest and the largest values of $\alpha$ is equal to:

Group $A$ consists of $7$ boys and $3$ girls, while group $B$ consists of $6$ boys and $5$ girls. The number of ways $4$ boys and $4$ girls can be invited for a picnic if $5$ of them must be from group $A$ and the remaining $3$ from group $B$, is equal to:

Let $[x]$ denote the greatest integer function, and let $m$ and $n$ respectively be the numbers of the points where the function $f(x) = [x] + |x - 2|$, $-2 < x < 3$, is not continuous and not differentiable. Then $m + n$ is equal to:

The number of real solution(s) of the equation $x^2 + 3x + 2 = \min{|x - 3|,; |x + 2|}$ is:

Let $f:(0,\infty)\to\mathbb{R}$ be a function which is differentiable at all points of its domain and satisfies the condition $x^2 f'(x)=2x f(x)+3$, with $f(1)=4$. Then $2f(2)$ is equal to:

The function $f:(-\infty,\infty)\to(-\infty,1)$, defined by $f(x)=\dfrac{2^x-2^{-x}}{2^x+2^{-x}}$, is:

et $A(x,y,z)$ be a point in $xy$-plane, which is equidistant from three points $(0,3,2)$, $(2,0,3)$ and $(0,0,1)$. Let $B=(1,4,-1)$ and $C=(2,0,-2)$. Then among the statements (S1): $\triangle ABC$ is an isosceles right angled triangle, and (S2): the area of $\triangle ABC$ is $\dfrac{9\sqrt{2}}{2}$,

If $f(x)=\dfrac{2^x}{,2^x+\sqrt{2},},; x\in\mathbb{R}$, then $\displaystyle \sum_{k=1}^{81} f!\left(\dfrac{k}{82}\right)$ is equal to

The sum of the squares of all the roots of the equation $x^2 + |2x - 3| - 4 = 0$ is

The relation $R={(x,y): x,y\in\mathbb{Z}\ \text{and}\ x+y\ \text{is even}}$ is:

Let $(a_n)$ be a sequence such that $a_0=0$, $a_1=\dfrac{1}{2}$ and $2a_{n+2}=5a_{n+1}-3a_n,; n=0,1,2,\ldots$. Then $\displaystyle \sum_{k=1}^{100} a_k$ is equal to

$\displaystyle \cos\left(\sin^{-1}\frac{3}{5}+\sin^{-1}\frac{5}{13}+\sin^{-1}\frac{33}{65}\right)$ is equal to:

If $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{96 x^2 \cos ^2 x}{\left(1+e^x\right)} \mathrm{d} x=\pi\left(\alpha \pi^2+\beta\right), \alpha, \beta \in \mathbb{Z}$, then $(\alpha+\beta)^2$ equals

Let $\mathrm{T}_{\mathrm{r}}$ be the $\mathrm{r}^{\text {th }}$ term of an A.P. If for some $\mathrm{m}, \mathrm{T}_{\mathrm{m}}=\frac{1}{25}, \mathrm{~T}_{25}=\frac{1}{20}$, and $20 \sum\limits_{\mathrm{r}=1}^{25} \mathrm{~T}_{\mathrm{r}}=13$, then $5 \mathrm{~m} \sum\limits_{\mathrm{r}=\mathrm{m}}^{2 \mathrm{~m}} \mathrm{~T}_{\mathrm{r}}$ is equal to

Let $\binom{n}{r-1}=28$, $\binom{n}{r}=56$ and $\binom{n}{r+1}=70$. Let $A(4\cos t,,4\sin t)$, $B(2\sin t,,-2\cos t)$ and $C(3r-n,,r^{2}-n-1)$ be the vertices of a triangle $ABC$, where $t$ is a parameter. If $(3x-1)^{2}+(3y)^{2}=\alpha$ is the locus of the centroid of triangle $ABC$, then $\alpha$ equals

Two numbers $k_{1}$ and $k_{2}$ are randomly chosen from the set of natural numbers. Then, the probability that the value of $i^{k_{1}}+i^{k_{2}}$ $(i=\sqrt{-1})$ is non-zero equals

Three defective oranges are accidentally mixed with seven good ones and, on looking at them, it is not possible to differentiate between them. Two oranges are drawn at random from the lot. If $x$ denotes the number of defective oranges, then the variance of $x$ is

Let $ABCD$ be a trapezium whose vertices lie on the parabola $y^{2}=4x$. Let the sides $AD$ and $BC$ of the trapezium be parallel to the $y$-axis. If the diagonal $AC$ is of length $\dfrac{25}{4}$ and it passes through the point $(1,0)$, then the area of $ABCD$ is

If the image of the point $(4,4,3)$ in the line $\dfrac{x-1}{2}=\dfrac{y-2}{1}=\dfrac{z-1}{3}$ is $(\alpha,\beta,\gamma)$, then $\alpha+\beta+\gamma$ is equal to

Let, for some function $y=f(x)$, $\displaystyle \int_{0}^{x} t,f(t),dt = x^{2}f(x)$ for $x>0$ and $f(2)=3$. Then $f(6)$ is equal to

The sum of all local minimum values of the function

$\mathrm{f}(x)=\left\{\begin{array}{lr} 1-2 x, & x<-1 \\ \frac{1}{3}(7+2|x|), & -1 \leq x \leq 2 \\ \frac{11}{18}(x-4)(x-5), & x>2 \end{array}\right.$

The number of different $5$-digit numbers greater than $50000$ that can be formed using the digits $0,1,2,3,4,5,6,7$, such that the sum of their first and last digits is not more than $8$, is

Let the equation of the circle, which touches $x$-axis at the point $(a,0)$, $a>0$, and cuts off an intercept of length $b$ on $y$-axis be $x^{2}+y^{2}-\alpha x+\beta y+\gamma=0$. If the circle lies below $x$-axis, then the ordered pair $(2a,,b^{2})$ is equal to

Let $f:\mathbb{R}\to\mathbb{R}$ be a function defined by $f(x)=(2+3a)x^{2}+\dfrac{a+2}{a-1}x+b$, $a\ne1$. If $f(x+y)=f(x)+f(y)+1-\dfrac{2}{7}xy$, then the value of $28\displaystyle\sum_{i=1}^{5}\lvert f(i)\rvert$ is

Let $O$ be the origin, the point $A$ be $z_1=\sqrt{3}+2\sqrt{2},i$, the point $B$ $(z_2)$ be such that $\sqrt{3},|z_2|=|z_1|$ and $\arg(z_2)=\arg(z_1)+\dfrac{\pi}{6}$. Then

The area (in sq. units) of the region ${(x,y): 0\le y\le 2|x|+1,; 0\le y\le x^{2}+1,; |x|\le 3}$ is

The area of the region bounded by the curves $x(1+y^{2})=1$ and $y^{2}=2x$ is:

Bag $B_1$ contains 6 white and 4 blue balls, Bag $B_2$ contains 4 white and 6 blue balls, and Bag $B_3$ contains 5 white and 5 blue balls. One of the bags is selected at random and a ball is drawn from it. If the ball is white, then the probability that the ball is drawn from Bag $B_2$ is:

Let $f:\mathbb{R}\to\mathbb{R}$ be a twice differentiable function such that $f(2)=1$. If $F(x)=x f(x)$ for all $x\in\mathbb{R}$, $\displaystyle\int_{0}^{2} x F''(x),dx=6$ and $\displaystyle\int_{0}^{2} x^{2} F''(x),dx=40$, then $F'(2)+\displaystyle\int_{0}^{2} F(x),dx$ is equal to:

If the midpoint of a chord of the ellipse $\dfrac{x^{2}}{9}+\dfrac{y^{2}}{4}=1$ is $\left(\sqrt{2},,\dfrac{4}{3}\right)$, and the length of the chord is $\dfrac{2\sqrt{\alpha}}{3}$, then $\alpha$ is:

The square of the distance of the point $\left(\dfrac{15}{7},,\dfrac{32}{7},,7\right)$ from the line $\dfrac{x+1}{3}=\dfrac{y+3}{5}=\dfrac{z+5}{7}$ in the direction of the vector $\hat{i}+4\hat{j}+7\hat{k}$ is:

Two equal sides of an isosceles triangle are along $-x+2y=4$ and $x+y=4$. If $m$ is the slope of its third side, then the sum of all possible distinct values of $m$ is:

Let the coefficients of three consecutive terms $T_r, T_{r+1}$ and $T_{r+2}$ in the binomial expansion of $(a+b)^{12}$ be in a G.P. Let $p$ be the number of all possible values of $r$. Let $q$ be the sum of all rational terms in the binomial expansion of $(\sqrt{3}+\sqrt[3]{4})^{12}$. Then $p+q$ is equal to

Let $f:\mathbb{R}-{0}\to(-\infty,1)$ be a polynomial of degree $2$, satisfying $f(x)f\left(\dfrac{1}{x}\right)=f(x)+f\left(\dfrac{1}{x}\right)$. If $f(K)=-2K$, then the sum of squares of all possible values of $K$ is:

Let [x] denote the greatest integer less than or equal to x. Then the domain of $ f(x) = \sec^{-1}(2[x] + 1) $ is:

Let S be the set of all the words that can be formed by arranging all the letters of the word GARDEN. From the set S, one word is selected at random. The probability that the selected word will NOT have vowels in alphabetical order is:

If A and B are the points of intersection of the circle $x^2 + y^2 - 8x = 0$ and the hyperbola $\frac{x^2}{9} - \frac{y^2}{4} = 1$ and a point P moves on the line $2x - 3y + 4 = 0$, then the centroid of $\Delta PAB$ lies on the line :

If $\sum\limits_{r=1}^{13}\left\{\frac{1}{\sin \left(\frac{\pi}{4}+(r-1) \frac{\pi}{6}\right) \sin \left(\frac{\pi}{4}+\frac{r \pi}{6}\right)}\right\}=a \sqrt{3}+b, a, b \in Z$, then $a^2+b^2$ is equal to :

If $\alpha + i\beta$ and $\gamma + i\delta$ are the roots of $x^2 - (3 - 2i)x - (2i - 2) = 0$, $i = \sqrt{-1}$, then $\alpha \gamma + \beta \delta$ is equal to:

>Let $A, B, C$ be three points in xy-plane, whose position vector are given by $\sqrt{3} \hat{i}+\hat{j}, \hat{i}+\sqrt{3} \hat{j}$ and $a \hat{i}+(1-a) \hat{j}$ respectively with respect to the origin O . If the distance of the point C from the line bisecting the angle between the vectors $\overrightarrow{\mathrm{OA}}$ and $\overrightarrow{\mathrm{OB}}$ is $\frac{9}{\sqrt{2}}$, then the sum of all the possible values of $a$ is :

If $f(x)=\displaystyle\int \frac{1}{x^{1/4}\left(1+x^{1/4}\right)},dx,; f(0)=-6$, then $f(1)$ is equal to:

Let $\mathrm{A}=\left[\begin{array}{cc}\frac{1}{\sqrt{2}} & -2 \\ 0 & 1\end{array}\right]$ and $\mathrm{P}=\left[\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta\end{array}\right], \theta>0$. If $\mathrm{B}=\mathrm{PAP}{ }^{\top}, \mathrm{C}=\mathrm{P}^{\top} \mathrm{B}^{10} \mathrm{P}$ and the sum of the diagonal elements of $C$ is $\frac{m}{n}$, where $\operatorname{gcd}(m, n)=1$, then $m+n$ is :

Let $f$ be a real valued continuous function defined on the positive real axis such that $g(x)=\int\limits_0^x t f(t) d t$. If $g\left(x^3\right)=x^6+x^7$, then value of $\sum\limits_{r=1}^{15} f\left(r^3\right)$ is :

Let the values of $\lambda$ for which the shortest distance between the lines $\frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{4}$
and $\frac{x-\lambda}{3} = \frac{y-4}{4} = \frac{z-5}{5}$ is $\frac{1}{\sqrt{6}}$ be $\lambda_1$ and $\lambda_2$. Then the radius of the circle passing through the
points $(0, 0), (\lambda_1, \lambda_2)$ and $(\lambda_2, \lambda_1)$ is

et $f:[0,3]\to A$ be defined by $,f(x)=2x^3-15x^2+36x+7,$ and $g:[0,\infty)\to B$ be defined by $,g(x)=\dfrac{x^{2025}}{x^{2025}+1}.$ If both the functions are onto and $S={x\in\mathbb{Z},:,x\in A\ \text{or}\ x\in B},$ then $n(S)$ is equal to:

Let $f(x)=x-1$ and $g(x)=e^x$ for $x\in\mathbb{R}$. If $\dfrac{dy}{dx}=e^{-2\sqrt{x}}g\big(f(f(x))\big)-\dfrac{y}{\sqrt{x}}, y(0)=0$, then $y(1)$ is:

For positive integers $n$, if $4a_n=(n^2+5n+6)$ and $S_n=\displaystyle\sum_{k=1}^{n}\frac{1}{a_k},$ then the value of $50,S_{2025}$ is:

If $ \frac{1}{1^4} + \frac{1}{2^4} + \frac{1}{3^4} + \ldots \infty= \frac{\pi^4}{90} $,

$\frac{1}{1^4} + \frac{1}{3^4} + \frac{1}{5^4} + \ldots \infty= \alpha $,

$ \frac{1}{2^4} + \frac{1}{4^4} + \frac{1}{6^4} + \ldots \infty= \beta $,

then $ \frac{\alpha}{\beta} $ is equal to :

Let a be the length of a side of a square OABC with O being the origin. Its side OA makes an acute angle $\alpha $ with the positive x-axis and the equations of its diagonals are $(\sqrt{3}+1)x+(\sqrt{3}-1)y=0$ and $(\sqrt{3}-1)x-(\sqrt{3}+1)y+8\sqrt{3}=0$. Then $a$² is equal to :

A line passing through the point $P(a, 0)$ makes an acute angle $\alpha$ with the positive x-axis. Let this line be rotated about the point $P$ through an angle $\dfrac{\alpha}{2}$ in the clockwise direction. If in the new position, the slope of the line is $2 - \sqrt{3}$ and its distance from the origin is $\dfrac{1}{\sqrt{2}}$, then the value of $3a^2 \tan^2 \alpha - 2\sqrt{3}$ is:

Let the ellipse $3x^2 + py^2 = 4$ pass through the centre $C$ of the circle $x^2 + y^2 - 2x - 4y - 11 = 0$ of radius $r$. Let $f_1, f_2$ be the focal distances of the point $C$ on the ellipse. Then $6f_1f_2 - r$ is equal to:

The sum of the squares of the roots of $|x - 2|^2 + |x - 2| - 2 = 0$ and the squares of the roots of $x^2 - 2|x - 3| - 5 = 0$, is:

The value of $ \cot^{-1} \left( \frac{\sqrt{1 + \tan^2(2)} - 1}{\tan(2)} \right) - \cot^{-1} \left( \frac{\sqrt{1 + \tan^2\left(\frac{1}{2}\right)} + 1}{\tan\left(\frac{1}{2}\right)} \right) $ is equal to

Let $A = {\theta \in [0,2\pi] : 1 + 10,\mathrm{Re}\left(\dfrac{2\cos\theta + i\sin\theta}{\cos\theta - 3i\sin\theta}\right) = 0}$. Then $\displaystyle \sum_{\theta \in A} \theta^2$ is equal to

Given below are two statements: Statement I: $\displaystyle \lim_{x \to 0} \left( \tan^{-1}x + \log_e \dfrac{\sqrt{1+x}}{1-x} - 2x \right) = \dfrac{2}{5}$ Statement II: $\displaystyle \lim_{x \to 1} \left( x^{\frac{1}{x-1}} \right) = \dfrac{1}{e^2}$ In the light of the above statements, choose the correct answer from the options given below:

The integral $\displaystyle \int_{-1}^{\tfrac{3}{2}} \left( |\pi^2 x \sin(\pi x)| \right) dx$ is equal to

Let $ A = \begin{bmatrix} 2 & 2+p & 2+p+q \\ 4 & 6+2p & 8+3p+2q \\ 6 & 12+3p & 20+6p+3q \end{bmatrix} $.

If $ \det(\text{adj}(\text{adj}(3A))) = 2^m \cdot 3^n $, $ m, n \in \mathbb{N} $, then $ m + n $ is equal to

Let $A={0,1,2,3,4,5}$. Let $R$ be a relation on $A$ defined by $(x,y)\in R$ iff $\max{x,y}\in{3,4}$. Then among the statements $(S_1):$ The number of elements in $R$ is $18$, $(S_2):$ The relation $R$ is symmetric but neither reflexive nor transitive, choose the correct option:

Suppose$ \displaystyle f(x)=\frac{(x^{2}+2-x),\tan x;\sqrt{\tan^{-1}!\left(\frac{x^{2}-x+1}{x}\right)}}{(7x^{2}+3x+1)^{3}}. $ Then the value of $f'(0)$ is equal to:

Let $f(x)$ be a positive function and $I_{1}=\int_{-\tfrac{1}{2}}^{1} 2x,f\left(2x(1-2x)\right),dx$ and $I_{2}=\int_{-1}^{2} f\left(x(1-x)\right),dx$. Then the value of $\dfrac{I_{2}}{I_{1}}$ is equal to

Let $\vec a,\vec b,\vec c$ be three non-zero vectors such that $\vec b$ and $\vec c$ are non-collinear. If $\ \vec a+5\vec b\ $ is collinear with $\vec c$, and $\ \vec b+6\vec c\ $ is collinear with $\vec a$, and $\ \vec a+\alpha,\vec b+\beta,\vec c=\vec 0$, then $\alpha+\beta$ is equal to:

There are 12 points in a plane, no three of which are in the same straight line, except 5 points which are collinear. Then the total number of triangles that can be formed with the vertices at any three of these 12 points is

Let $(5, \tfrac{a}{4})$ be the circumcenter of a triangle with vertices $A(a, -2)$, $B(a, 6)$ and $C\left(\tfrac{a}{4}, -2\right)$. Let $\alpha$ denote the circumradius, $\beta$ denote the area and $\gamma$ denote the perimeter of the triangle. Then $\alpha + \beta + \gamma$ is:

If $A$ and $B$ are two events such that $P(A)=0.7,\ P(B)=0.4$ and $P(A\cap \overline{B})=0.5$, where $\overline{B}$ denotes the complement of $B$, then $P!\left(B,\middle|,(A\cup \overline{B})\right)$ is equal to

If $\alpha,\ -\tfrac{\pi}{2} < \alpha < \tfrac{\pi}{2}$ is the solution of $4\cos\theta + 5\sin\theta = 1$, then the value of $\tan\alpha$ is:

Let $\vec a=\hat i+2\hat j+\hat k$ and $\vec b=2\hat i+\hat j-\hat k$. Let $\vec c$ be a unit vector in the plane of the vectors $\vec a$ and $\vec b$ and be perpendicular to $\vec a$. Then such a vector $\vec c$ is:

If in a G.P. of $64$ terms, the sum of all the terms is $7$ times the sum of the odd terms of the G.P., then the common ratio of the G.P. is equal to:

Let the function $f(x)=\dfrac{x}{3}+\dfrac{3}{x}+3,\ x\ne0$ be strictly increasing in $(-\infty,\alpha_1)\cup(\alpha_2,\infty)$ and strictly decreasing in $(\alpha_3,\alpha_4)\cup(\alpha_4,\alpha_5)$. Then $\displaystyle \sum_{i=1}^{5}\alpha_i^{2}$ is equal to

A fair die is thrown until $2$ appears. Then, the probability that $2$ appears in an even number of throws is:

The number of integral terms in the expansion of $\left(5^{\tfrac{1}{2}}+7^{\tfrac{1}{8}}\right)^{1016}$ is:

In an A.P., the sixth term $a_6 = 2$. If the product $a_1 a_4 a_9$ is the greatest, then the common difference of the A.P. is equal to:

Let α be a solution of $x^2 + x + 1 = 0$, and for some a and b in

$R, \begin{bmatrix} 4 & a & b \end{bmatrix} \begin{bmatrix} 1 & 16 & 13 \\ -1 & -1 & 2 \\ -2 & -14 & -8 \end{bmatrix} = \begin{bmatrix} 0 & 0 & 0 \end{bmatrix}$. If $\frac{4}{\alpha^4} + \frac{m}{\alpha^a} + \frac{n}{\alpha^b} = 3$, then m + n is equal to _______

$\text { Let } A=\left[\begin{array}{lll} 1 & 0 & 0 \\ 0 & \alpha & \beta \\ 0 & \beta & \alpha \end{array}\right] \text { and }|2 \mathrm{~A}|^3=2^{21} \text { where } \alpha, \beta \in Z \text {, Then a value of } \alpha \text { is }$

Let $R$ be a relation on $\mathbb{Z} \times \mathbb{Z}$ defined by $(a,b) R (c,d)$ if and only if $ad - bc$ is divisible by $5$. Then $R$ is:

If $f(x)=\left\{\begin{array}{cc}2+2 x, & -1 \leq x < 0 \\ 1-\frac{x}{3}, & 0 \leq x \leq 3\end{array} ; g(x)=\left\{\begin{array}{cc}-x, & -3 \leq x \leq 0 \\ x, & 0 < x \leq 1\end{array}\right.\right.$, then range of $(f o g)(x)$

Let $O$ be the origin and the position vectors of $A$ and $B$ be $\vec{A} = 2\hat{i} + 2\hat{j} + \hat{k}$ and $\vec{B} = 2\hat{i} + 4\hat{j} + 4\hat{k}$ respectively. If the internal bisector of $\angle AOB$ meets the line $AB$ at $C$, then the length of $OC$ is:

In a $\triangle ABC$, suppose $y = x$ is the equation of the bisector of the angle $B$ and the equation of the side $AC$ is $2x - y = 2$. If $2AB = BC$ and the points $A$ and $B$ are respectively $(4,6)$ and $(\alpha, \beta)$, then $\alpha + 2\beta$ is equal to:

For $x \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$, if $y(x)=\int \frac{\operatorname{cosec} x+\sin x}{\operatorname{cosec} x \sec x+\tan x \sin ^2 x} d x$, and $\lim _\limits{x \rightarrow\left(\frac{\pi}{2}\right)^{-}} y(x)=0$ then $y\left(\frac{\pi}{4}\right)$ is equal to

Let $\mathrm{A}$ be a square matrix such that $\mathrm{AA}^{\mathrm{T}}=\mathrm{I}$. Then $\frac{1}{2} A\left[\left(A+A^T\right)^2+\left(A-A^T\right)^2\right]$ is equal to

If $z = \dfrac{1}{2} - 2i$ is such that $|z + 1| = \alpha z + \beta (1 + i)$, $i = \sqrt{-1}$ and $\alpha, \beta \in \mathbb{R}$, then $\alpha + \beta$ is equal to:

Consider the function $f:\left[\frac{1}{2}, 1\right] \rightarrow \mathbb{R}$ defined by $f(x)=4 \sqrt{2} x^3-3 \sqrt{2} x-1$. Consider the statements

(I) The curve $y=f(x)$ intersects the $x$-axis exactly at one point.

(II) The curve $y=f(x)$ intersects the $x$-axis at $x=\cos \frac{\pi}{12}$

Let $P Q R$ be a triangle with $R(-1,4,2)$. Suppose $M(2,1,2)$ is the mid point of $\mathrm{PQ}$. The distance of the centroid of $\triangle \mathrm{PQR}$ from the point of intersection of the lines $\frac{x-2}{0}=\frac{y}{2}=\frac{z+3}{-1}$ and $\frac{x-1}{1}=\frac{y+3}{-3}=\frac{z+1}{1}$

$\mathop {\lim }\limits_{x \to {\pi \over 2}} \left( {{1 \over {{{\left( {x - {\pi \over 2}} \right)}^2}}}\int\limits_{{x^3}}^{{{\left( {{\pi \over 2}} \right)}^3}} {\cos \left( {{t^{{1 \over 3}}}} \right)dt} } \right)$ is equal to

A function $y=f(x)$ satisfies $f(x) \sin 2 x+\sin x-\left(1+\cos ^2 x\right) f^{\prime}(x)=0$ with condition $f(0)=0$. Then, $f\left(\frac{\pi}{2}\right)$ is equal to

If $\displaystyle \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\left(\dfrac{x^2\cos x}{1+x^2}+\dfrac{1+\sin^2 x}{1+e^{\sin 2x}}\right)dx = \dfrac{\pi}{4}(\pi+a)-2$, then the value of $a$ is:

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

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

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

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

Let a unit vector $\hat{\mathbf{u}}=x\mathbf{i}+y\mathbf{j}+z\mathbf{k}$ make angles $\dfrac{\pi}{2},\ \dfrac{\pi}{3}$ and $\dfrac{2\pi}{3}$ with the vectors $\dfrac{1}{\sqrt{2}}\mathbf{i}+\dfrac{1}{\sqrt{2}}\mathbf{k},\ \dfrac{1}{\sqrt{2}}\mathbf{j}+\dfrac{1}{\sqrt{2}}\mathbf{k},\ \dfrac{1}{\sqrt{2}}\mathbf{i}+\dfrac{1}{\sqrt{2}}\mathbf{j}$ respectively. If $\vec{\mathbf{v}}=\dfrac{1}{\sqrt{2}}\mathbf{i}+\dfrac{1}{\sqrt{2}}\mathbf{j}+\dfrac{1}{\sqrt{2}}\mathbf{k}$, then $|\hat{\mathbf{u}}-\vec{\mathbf{v}}|^{2}$ 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 ${(1,2),(1,3)}\subset R$, then the number of elements in $R$ is:

The mean and variance of five observations are $\dfrac{24}{5}$ and $\dfrac{194}{25}$ respectively. If the mean of the first four observations is $\dfrac{7}{2}$, then the variance of the first four observations is:

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

Let $y=\log_e!\left(\dfrac{1-x^2}{1+x^2}\right)$, with $-1
1 732
2 736
3 742
4 746
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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (29 January Evening Shift) PYQ

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