Let $f(x)=2x+\tan^{-1}x$ and $g(x)=\log_{e}\!\big(\sqrt{1+x^{2}}+x\big),\ x\in[0,3]$. Then

Let $f(x)=\begin{vmatrix} 1+\sin^{2}x & \cos^{2}x & \sin 2x\\ \sin^{2}x & 1+\cos^{2}x & \sin 2x\\ \sin^{2}x & \cos^{2}x & 1+\sin 2x \end{vmatrix},\ x\in\left[\dfrac{\pi}{6},\dfrac{\pi}{3}\right].$ If $\alpha$ and $\beta$ respectively are the maximum and the minimum values of $f$, then

The area enclosed by the closed curve $\mathcal{C}$ given by the differential equation $\dfrac{dy}{dx}+\dfrac{x+a}{\,y-2\,}=0,\quad y(1)=0$ is $4\pi$. Let $P$ and $Q$ be the points of intersection of the curve $\mathcal{C}$ with the $y$-axis. If the normals at $P$ and $Q$ on $\mathcal{C}$ intersect the $x$-axis at points $R$ and $S$ respectively, then the length of the line segment $RS$ is:

If the orthocentre of the triangle whose vertices are $(1,2)$, $(2,3)$ and $(3,1)$ is $(\alpha,\beta)$, then the quadratic equation whose roots are $\alpha+4\beta$ and $4\alpha+\beta$ is:

If $y=y(x)$ is the solution curve of the differential equation $\dfrac{dy}{dx}+y\tan x=x\sec x,\ 0\le x\le \dfrac{\pi}{3},\ y(0)=1$, then $y\!\left(\dfrac{\pi}{6}\right)$ is equal to:

Let $9=x_{1} < x_{2} < \ldots < x_{7}$ be in an A.P. with common difference d. If the standard deviation of $x_{1}, x_{2}..., x_{7}$ is 4 and the mean is $\bar{x}$, then $\bar{x}+x_{6}$ is equal to :

Let $P(S)$ denote the power set of $S=\{1,2,3,\ldots,10\}$. Define the relations $R_{1}$ and $R_{2}$ on $P(S)$ as $A\,R_{1}\,B \iff (A\cap B^{c})\cup(B\cap A^{c})=\varnothing$ and $A\,R_{2}\,B \iff A\cup B^{c}=B\cup A^{c}$, for all $A,B\in P(S)$. Then:

The sum of the absolute maximum and minimum values of the function $f(x)=\lvert x^{2}-5x+6\rvert-3x+2$ in the interval $[-1,3]$ is equal to:

Let $\vec a=5\hat{\imath}-\hat{\jmath}-3\hat{k}$ and $\vec b=\hat{\imath}+3\hat{\jmath}+5\hat{k}$ be two vectors. Then which one of the following statements is TRUE?

For the system of linear equations $\alpha x+y+z=1,\quad x+\alpha y+z=1,\quad x+y+\alpha z=\beta$, which one of the following statements is **NOT** correct?

The value of the integral $\displaystyle \int_{-\pi/4}^{\pi/4}\frac{x+\pi/4}{\,2-\cos 2x\,}\,dx$ is:

If $y(x)=x^{x},\ x>0$, then $y''(2)-2y'(2)$ is equal to:

The area of the region given by $\{(x,y):\, xy\le 8,\ 1\le y\le x^{2}\}$ is:

The number of integral values of $k$ for which one root of the equation $2x^{2}-8x+k=0$ lies in the interval $(1,2)$ and its other root lies in the interval $(2,3)$ is:

Let $a,b$ be two real numbers such that $ab<0$. If the complex number $\dfrac{1+ai}{\,b+i\,}$ is of unit modulus and $a+ib$ lies on the circle $|z-1|=|2z|$, then a possible value of $\dfrac{1+[a]}{4b}$, where $[\,\cdot\,]$ is the greatest integer function, is:

If $A=\dfrac12\begin{bmatrix}1 & \sqrt{3}\\ -\sqrt{3} & 1\end{bmatrix}$, then:

Two dice are thrown independently. Let \(A\) be the event that the number on the \(1^{\text{st}}\) die is less than the number on the \(2^{\text{nd}}\) die; \(B\) be the event that the number on the \(1^{\text{st}}\) die is even and that on the \(2^{\text{nd}}\) die is odd; and \(C\) be the event that the number on the \(1^{\text{st}}\) die is odd and that on the \(2^{\text{nd}}\) die is even. Then:

Let \(\vec a = 2\hat i - 7\hat j + 5\hat k\), \(\vec b = \hat i + \hat k\) and \(\vec c = \hat i + 2\hat j - 3\hat k\) be three given vectors. If \(\vec r\) is a vector such that \(\vec r \times \vec a = \vec c \times \vec a\) and \(\vec r \cdot \vec b = 0\), then \(|\vec r|\) is equal to:

Let $\alpha x = \exp(x^{\beta} y^{\gamma})$ be the solution of the differential equation $2x^{2}y\,dy - (1 - xy^{2})\,dx = 0,\ x>0,\ y(2)=\sqrt{\log_{e}2}.$ Then $\alpha + \beta - \gamma$ equals:

Let $f:\mathbb{R}-\{0,1\}\to\mathbb{R}$ be a function such that $f(x)+f\!\left(\frac{1}{1-x}\right)=1+x.$ Then $f(2)$ is equal to:

Let $S = \left\{ {x \in R:0 < x < 1\,\mathrm{and}\,2{{\tan }^{ - 1}}\left( {{{1 - x} \over {1 + x}}} \right) = {{\cos }^{ - 1}}\left( {{{1 - {x^2}} \over {1 + {x^2}}}} \right)} \right\}$.

If $\mathrm{n(S)}$ denotes the number of elements in $\mathrm{S}$ then :

Let $A = \{x \in \mathbb{R} : [x + 3] + [x + 4] \le 3\}$, and $B = \left\{ x \in \mathbb{R} : 3^x \left( \sum_{r=1}^{\infty} \frac{3}{10^r} \right)^{x - 3} < 3^{-3x} \right\}$, where $[\,]$ denotes the greatest integer function. Then,

If the system of equations $x + y + a z = b$ $2x + 5y + 2z = 6$ $x + 2y + 3z = 3$ has infinitely many solutions, then $2a + 3b$ is equal to :

The straight lines $l_1$ and $l_2$ pass through the origin and trisect the line segment of the line $L : 9x + 5y = 45$ between the axes. If $m_1$ and $m_2$ are the slopes of the lines $l_1$ and $l_2$, then the point of intersection of the line $y = (m_1 + m_2)x$ with $L$ lies on :

One vertex of a rectangular parallelepiped is at the origin $O$ and the lengths of its edges along the $x$, $y$ and $z$ axes are $3,\,4$ and $5$ units respectively. Let $P$ be the vertex $(3,4,5)$. Then the shortest distance between the diagonal $OP$ and an edge parallel to the $z$–axis, not passing through $O$ or $P$, is:

The mean and variance of a set of $15$ numbers are $12$ and $14$ respectively. The mean and variance of another set of $15$ numbers are $14$ and $\sigma^{2}$ respectively. If the variance of all the $30$ numbers in the two sets is $13$, then $\sigma^{2}$ is equal to:

Let $A = [a_{ij}]_{2\times 2}$, where $a_{ij}\ne 0$ for all $i,j$ and $A^{2}=I$. Let $a$ be the sum of all diagonal elements of $A$ and $b=\lvert A\rvert$ (i.e., $b=\det A$). Then $3a^{2}+4b^{2}$ is equal to:

Let $5f(x)+4f\!\left(\dfrac{1}{x}\right)=\dfrac{1}{x}+3,\; x>0.$ Then $18\displaystyle\int_{1}^{2} f(x)\,dx$ is equal to:

Let $I(x)=\displaystyle \int \frac{x^{2}\big(x\sec^{2}x+\tan x\big)}{(x\tan x+1)^{2}}\,dx.$ If $I(0)=0$, then $I\!\left(\frac{\pi}{4}\right)$ is equal to:

Let $\vec{a}=2\hat{i}+3\hat{j}+4\hat{k}$, $\vec{b}=\hat{i}-2\hat{j}-2\hat{k}$ and $\vec{c}=-\hat{i}+4\hat{j}+3\hat{k}$. If $\vec{d}$ is a vector perpendicular to both $\vec{b}$ and $\vec{c}$, and $\vec{a}\cdot\vec{d}=18$, then $\lvert \vec{a}\times \vec{d}\rvert^{2}$ is equal to:

Let $a_{1},a_{2},a_{3},\ldots,a_{n}$ be $n$ positive consecutive terms of an arithmetic progression. If $d>0$ is its common difference, then \[ \lim_{n\to\infty}\sqrt{\frac{d}{n}} \left(\frac{1}{\sqrt{a_{1}}+\sqrt{a_{2}}} +\frac{1}{\sqrt{a_{2}}+\sqrt{a_{3}}} +\cdots +\frac{1}{\sqrt{a_{n-1}}+\sqrt{a_{n}}}\right) \] is:

The sum of all the roots of the equation $\lvert x^{2}-8x+15\rvert-2x+7=0$ is:

If the ratio of the fifth term from the beginning to the fifth term from the end in the expansion of $\left(\sqrt[4]{2}+\dfrac{1}{\sqrt[4]{3}}\right)^{n}$ is $\sqrt{6}:1$, then the third term from the beginning is:

If $2x^{y}+3y^{x}=20$, then $\dfrac{dy}{dx}$ at $(2,2)$ is equal to:

If the coefficient of $x^{7}$ in $\left(a x^{2}+\dfrac{1}{2 b x}\right)^{11}$ and $x^{-7}$ in $\left(a x-\dfrac{1}{3 b x^{2}}\right)^{11}$ are equal, then:

The area bounded by the curves $y=\lvert x-1\rvert+\lvert x-2\rvert$ and $y=3$ is equal to:

Let $a\ne b$ be two non-zero real numbers. Then the number of elements in the set $X=\{\, z\in\mathbb{C} : \operatorname{Re}(a z^{2}+bz)=a \text{ and } \operatorname{Re}(b z^{2}+a z)=b \,\}$ is equal to:

Let $P$ be a square matrix such that $P^{2}=I-P$. For $\alpha,\beta,\gamma,\delta\in\mathbb{N}$, if $P^{\alpha}+P^{\beta}=\gamma I-29P$ and $P^{\alpha}-P^{\beta}=\delta I-13P$, then $\alpha+\beta+\gamma-\delta$ is equal to:

Among the statements: (S1): $2023^{2022}-1999^{2022}$ is divisible by $8$. (S2): $13(13)^n-12n-13$ is divisible by $144$ for infinitely many $n\in\mathbb{N}$.

If the solution curve $f(x,y)=0$ of the differential equation $(1+\log_e x)\frac{dx}{dy}-x\log_e x=e^y,\; x>0,$ passes through the points $(1,0)$ and $(\alpha,2)$, then $\alpha^\alpha$ is equal to:

Let \(f(x)\) be a function satisfying \(f(x)+f(\pi-x)=\pi^{2}\), \(\forall x\in\mathbb{R}\). Then \(\displaystyle \int_{0}^{\pi} f(x)\sin x\,dx\) is equal to:

\[ \lim_{n\to\infty} \left\{ \left(2^{\tfrac12}-2^{\tfrac13}\right)\left(2^{\tfrac12}-2^{\tfrac15}\right)\cdots\left(2^{\tfrac12}-2^{\tfrac{1}{2n+1}}\right) \right\} \] is equal to:

In a group of 100 persons, 75 speak English and 40 speak Hindi. Each person speaks at least one of the two languages. If the number of persons who speak only English is $\alpha$ and the number of persons who speak only Hindi is $\beta$, then the eccentricity of the ellipse \[ 25\big(\beta^2 x^2 + \alpha^2 y^2\big)=\alpha^2\beta^2 \] is:

Three dice are rolled. If the probability of getting different numbers on the three dice is $\frac{p}{q}$, where $p$ and $q$ are co-prime, then $q-p$ is equal to :

For the system of equations \[ \begin{cases} x+y+z=6,\\ x+2y+\alpha z=10,\\ x+3y+5z=\beta, \end{cases} \] which one of the following is **NOT** true?

Let the sets A and B denote the domain and range respectively of the function $f(x)=\frac{1}{\sqrt{\lceil x\rceil-x}}$, where $\lceil x\rceil$ denotes the smallest integer greater than or equal to $x$. Then among the statements

(S1) : $A \cap B=(1, \infty)-\mathbb{N}$ and

(S2) : $A \cup B=(1, \infty)$

$ \text{Let } A = \begin{bmatrix} 2 & 1 & 0 \ 1 & 2 & -1 \ 0 & -1 & 2 \end{bmatrix}. \text{ If } \left| \operatorname{adj} \big( \operatorname{adj} (\operatorname{adj}(2A)) \big) \right| = (16)^n, \text{ then } n \text{ is equal to:} $

$ \displaystyle \lim_{x\to 0} \left( \frac{1-\cos^2(3x)}{\cos^3(4x)} \right)\left( \frac{\sin^3(4x)}{(\log_e(2x+1))^5} \right) \text{ is equal to } \underline{\ \ \ \ \ \ \ \ \ \ }. $

Let $f(x)=\frac{\sin x+\cos x-\sqrt{2}}{\sin x-\cos x}, x \in[0, \pi]-\left\{\frac{\pi}{4}\right\}$. Then $f\left(\frac{7 \pi}{12}\right) f^{\prime \prime}\left(\frac{7 \pi}{12}\right)$ is equal to

In a bolt factory, machines $A, B$ and $C$ manufacture respectively $20 \%, 30 \%$ and $50 \%$ of the total bolts. Of their output 3, 4 and 2 percent are respectively defective bolts. A bolt is drawn at random from the product. If the bolt drawn is found the defective, then the probability that it is manufactured by the machine $C$ is :

$ \text{The number of ways in which 5 girls and 7 boys can be seated at a round table so that no two girls sit together, is:} $

$ \text{The shortest distance between the lines } \dfrac{x-4}{4}=\dfrac{y+2}{5}=\dfrac{z+3}{3} \text{ and } \dfrac{x-1}{3}=\dfrac{y-3}{4}=\dfrac{z-4}{2} \text{ is:} $

$ \text{Let } R \text{ be the focus of the parabola } y^{2}=20x \text{ and the line } y=mx+c \text{ intersect the parabola at two points } P \text{ and } Q. $ $ \text{Let the point } G(10,10) \text{ be the centroid of the triangle } PQR. \text{ If } c-m=6, \text{ then } (PQ)^{2} \text{ is:} $

$ \text{The area of the region } {(x,y): x^{2}\le y \le 8-x^{2},; y\le 7} \text{ is:} $

$ \text{Let } C(\alpha,\beta) \text{ be the circumcenter of the triangle formed by the lines } 4x+3y=69,; 4y-3x=17,; x+7y=61. $ $ \text{Then } (\alpha-\beta)^2+\alpha+\beta \text{ is equal to:} $

$ \text{Let } I(x)=\int \frac{x+1}{x,(1+x e^{x})^{2}},dx,; x>0.\ \text{If } \lim_{x\to\infty} I(x)=0,\ \text{then } I(1) \text{ is equal to:} $

$ \text{Let } \alpha,\beta,\gamma \text{ be the three roots of } x^{3}+bx+c=0. \text{ If } \beta\gamma=1=-\alpha,\ \text{then } b^{3}+2c^{3}-3\alpha^{3}-6\beta^{3}-8\gamma^{3} \text{ is equal to:} $

$ \text{Let } S_K=\dfrac{1+2+\cdots+K}{K} \text{ and } \displaystyle\sum_{j=1}^{n} S_j^{2}=\dfrac{n}{A}\big(Bn^{2}+Cn+D\big),\ \text{where } A,B,C,D\in\mathbb{N} \text{ and } A \text{ has least value. Then:} $

Let $P=\left[\begin{array}{cc}\frac{\sqrt{3}}{2} & \frac{1}{2} \\ -\frac{1}{2} & \frac{\sqrt{3}}{2}\end{array}\right], A=\left[\begin{array}{ll}1 & 1 \\ 0 & 1\end{array}\right]$ and $Q=P A P^{T}$. If $P^{T} Q^{2007} P=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]$, then $2 a+b-3 c-4 d$ equal to :

If for $z=\alpha+i \beta,|z+2|=z+4(1+i)$, then $\alpha+\beta$ and $\alpha \beta$ are the roots of the equation :

$ \text{If the points with position vectors } \alpha\hat{i}+10\hat{j}+13\hat{k},; 6\hat{i}+11\hat{j}+11\hat{k},; \dfrac{9}{2}\hat{i}+\beta\hat{j}-8\hat{k} \text{ are collinear, then } (19\alpha-6\beta)^2 \text{ is equal to:} $

Let $A=\left\{\theta \in(0,2 \pi): \frac{1+2 i \sin \theta}{1-i \sin \theta}\right.$ is purely imaginary $\}$. Then the sum of the elements in $\mathrm{A}$ is :

$25^{190}-19^{190}-8^{190}+2^{190}$ is divisible by:

$ \text{The integral } \displaystyle \int \left[ \left(\frac{x}{2}\right)^{x} + \left(\frac{2}{x}\right)^{x} \right] \ln!\left(\frac{e x}{2}\right), dx \text{ is equal to:} $

The probability that the random variable $X$ takes value $x$ is given by $P(X = x) = k(x + 1)3^{-x}, \; x = 0, 1, 2, 3, \ldots$ where $k$ is a constant. Then $P(X \ge 2)$ is equal to:

The set $A={1,2,3,4,5,6,7}$. The relation $R={(x,y)\in A\times A:\ x+y=7}$ is:

The mean and variance of $12$ observations are $\dfrac{9}{2}$ and $4$ respectively. Later, it was observed that two observations were considered as $9$ and $10$ instead of $7$ and $14$ respectively. If the correct variance is $\dfrac{m}{n}$, where $m$ and $n$ are coprime, then $m+n$ is equal to:

The absolute difference of the coefficients of $x^{10}$ and $x^{7}$ in the expansion of $\left(2x^{2}+\dfrac{1}{2x}\right)^{11}$ is equal to:

The area of the quadrilateral $ABCD$ with vertices $A(2,1,1)$, $B(1,2,5)$, $C(-2,-3,5)$ and $D(1,-6,-7)$ is equal to:

If the number of words (with or without meaning) that can be formed using all the letters of the word MATHEMATICS — in which C and S do not come together — is $(6!)k$, then $k$ is equal to:

The numbers $\alpha>\beta>0$ are the roots of the equation $a x^{2}+b x+1=0$, and $\displaystyle \lim_{x\to \frac{1}{\alpha}} \left( \frac{1-\cos!\big(x^{2}+bx+a\big)}{2(1-a x)^{2}} \right)^{\tfrac{1}{2}} = \frac{1}{k}!\left(\frac{1}{\beta}-\frac{1}{\alpha}\right).$ Then $k$ is equal to:

The value of $36,(4\cos^{2}9^\circ-1)(4\cos^{2}27^\circ-1)(4\cos^{2}81^\circ-1)(4\cos^{2}243^\circ-1)$ is:

$A=\left[\begin{array}{cc}1 & 5 \\ \lambda & 10\end{array}\right], \mathrm{A}^{-1}=\alpha \mathrm{A}+\beta \mathrm{I}$ and $\alpha+\beta=-2$, then $4 \alpha^{2}+\beta^{2}+\lambda^{2}$ is equal to :

The set $S$ is all values of $\theta\in[-\pi,\pi]$ for which the system $x+y+\sqrt{3},z=0,\quad -x+(\tan\theta),y+\sqrt{7},z=0,\quad x+y+(\tan\theta),z=0$ has a non-trivial solution. Then $\dfrac{120}{\pi}\displaystyle\sum_{\theta\in S}\theta$ is equal to:

The function $I(x)=\int e^{\sin^{2}x},(\cos x\sin 2x-\sin x),dx$ with $I(0)=1$. Then $I!\left(\dfrac{\pi}{3}\right)$ is equal to:

The shortest distance between the lines $\dfrac{x+2}{1}=\dfrac{y}{-2}=\dfrac{z-5}{2}$ and $\dfrac{x-4}{1}=\dfrac{y-1}{2}=\dfrac{z+3}{0}$ is:

Let the ellipse $E:{x^2} + 9{y^2} = 9$ intersect the positive x and y-axes at the points A and B respectively. Let the major axis of E be a diameter of the circle C. Let the line passing through A and B meet the circle C at the point P. If the area of the triangle with vertices A, P and the origin O is ${m \over n}$, where m and n are coprime, then $m - n$ is equal to :

Let O be the origin and the position vector of the point P be $ - \widehat i - 2\widehat j + 3\widehat k$. If the position vectors of the points A, B and C are $ - 2\widehat i + \widehat j - 3\widehat k,2\widehat i + 4\widehat j - 2\widehat k$ and $ - 4\widehat i + 2\widehat j - \widehat k$ respectively, then the projection of the vector $\overrightarrow {OP} $ on a vector perpendicular to the vectors $\overrightarrow {AB} $ and $\overrightarrow {AC} $ is :

The coefficient of $x^{7}$ in $\left(ax-\dfrac{1}{bx^{2}}\right)^{13}$ and the coefficient of $x^{-5}$ in $\left(ax+\dfrac{1}{bx^{2}}\right)^{13}$ are equal. Then $a^{4}b^{4}$ is equal to:

The arc $PQ$ of a circle subtends a right angle at its centre $O$. The midpoint of the arc $PQ$ is $R$. If $\overrightarrow{OP}=\vec{u}$, $\overrightarrow{OR}=\vec{v}$ and $\overrightarrow{OQ}=\alpha\vec{u}+\beta\vec{v}$, then $\alpha,\ \beta^{2}$ are the roots of the equation:

Let $N$ denote the sum of the numbers obtained when two dice are rolled. If the probability that $2^{N} < N!$ is $\dfrac{m}{n}$, where $m$ and $n$ are coprime, then $4m-3n$ is equal to:

The square tin of side $30\ \text{cm}$ is made into an open-top box by cutting a square of side $x$ from each corner and folding up the flaps. If the volume of the box is maximum, then its surface area (in $\text{cm}^2$) is:

$96\cos\frac{\pi}{33},\cos\frac{2\pi}{33},\cos\frac{4\pi}{33},\cos\frac{8\pi}{33},\cos\frac{16\pi}{33}$ is equal to:

The system of linear equations $2x - y + 3z = 5$ $3x + 2y - z = 7$ $4x + 5y + \alpha z = \beta,$ which of the following is NOT correct?

A line segment $AB$ of length $\lambda$ moves such that the points $A$ and $B$ remain on the periphery of a circle of radius $\lambda$. Then the locus of the point that divides the line segment $AB$ in the ratio $2:3$ is a circle of radius:

The function $f$ is differentiable and satisfies $x^{2}f(x)-x=4\displaystyle\int_{0}^{x} t f(t),dt$, with $f(1)=\dfrac{2}{3}$. Then $18f(3)$ is equal to:

The first term $\alpha$ and common ratio $r$ of a geometric progression are positive integers. If the sum of squares of its first three terms is $33033$, then the sum of these three terms is equal to:

The complex number $z=x+iy$ is such that $\dfrac{2z-3i}{2z+i}$ is purely imaginary. If $x+y^{2}=0$, then $y^{4}+y^{2}-y$ is equal to:

The set $S = \{ z = x + i y : \dfrac{2z - 3i}{4z + 2i} \text{ is a real number} \}$ is given. Then which of the following is **NOT correct**?

Let the number $(22)^{2022} + (2022)^{22}$ leave the remainder $\alpha$ when divided by $3$ and $\beta$ when divided by $7$. Then $(\alpha^2 + \beta^2)$ is equal to:

Let $g(x) = f(x) + f(1 - x)$ and $f''(x) > 0, \; x \in (0, 1)$. If $g$ is decreasing in the interval $(0, \alpha)$ and increasing in the interval $(\alpha, 1)$, then $\tan^{-1}(2\alpha) + \tan^{-1}\!\left(\dfrac{1}{\alpha}\right) + \tan^{-1}\!\left(\dfrac{\alpha + 1}{\alpha}\right)$ is equal to:

If the coefficients of $x$ and $x^2$ in $(1 + x)^p (1 - x)^q$ are $4$ and $-5$ respectively, then $2p + 3q$ is equal to:

Let $f$ be a continuous function satisfying $\displaystyle \int_{0}^{t^2} \big(f(x) + x^2\big)\,dx = \dfrac{4}{3}t^3, \; \forall t > 0.$ Then $f\!\left(\dfrac{\pi^2}{4}\right)$ is equal to:

For $\alpha, \beta, \gamma, \delta \in \mathbb{N}$, if $\displaystyle \int \left( \left(\dfrac{x}{e}\right)^{2x} + \left(\dfrac{e}{x}\right)^{2x} \right) \log_e x \, dx = \dfrac{1}{\alpha} \left(\dfrac{x}{e}\right)^{\beta x} - \dfrac{1}{\gamma} \left(\dfrac{e}{x}\right)^{\delta x} + C$, where $e = \displaystyle \sum_{n=0}^{\infty} \dfrac{1}{n!}$ and $C$ is the constant of integration, then $\alpha + 2\beta + 3\gamma - 4\delta$ is equal to:

Let $\mu$ be the mean and $\sigma$ be the standard deviation of the distribution where $\displaystyle \sum f_i = 62$. If $[x]$ denotes the greatest integer $\le x$, then $[\mu^2 + \sigma^2]$ is equal to:

Let $\vec{a} = 2\hat{i} + 7\hat{j} - \hat{k}$, $\vec{b} = 3\hat{i} + 5\hat{k}$ and $\vec{c} = \hat{i} - \hat{j} + 2\hat{k}$. Let $\vec{d}$ be a vector which is perpendicular to both $\vec{a}$ and $\vec{b}$, and $\vec{c} \cdot \vec{d} = 12$. Then $(-\hat{i} + \hat{j} - \hat{k}) \cdot (\vec{c} \times \vec{d})$ is equal to:

If the points $\mathbf{P}$ and $\mathbf{Q}$ are respectively the circumcenter and the orthocentre of a $\triangle ABC$, then $\overrightarrow{PA} + \overrightarrow{PB} + \overrightarrow{PC}$ is equal to:

Let $A$ be the point $(1, 2)$ and $B$ be any point on the curve $x^2 + y^2 = 16$. If the centre of the locus of the point $P$, which divides the line segment $AB$ in the ratio $3 : 2$, is the point $C(\alpha, \beta)$, then the length of the line segment $AC$ is:

Let $A = \{2, 3, 4\}$ and $B = \{8, 9, 12\}$. Then the number of elements in the relation $R = \{ ((a_1, b_1), (a_2, b_2)) \in (A \times B, A \times B) : a_1 \text{ divides } b_2 \text{ and } a_2 \text{ divides } b_1 \}$ is:

Let $p, q \in \mathbb{R}$ and $(1 - \sqrt{3}i)^{200} = 2^{199}(p + iq),\ i = \sqrt{-1}$ Then $p + q + q^2$ and $p - q + q^2$ are roots of the equation.

Let $R$ be a rectangle given by the lines $x = 0$, $x = 2$, $y = 0$ and $y = 5$. Let $A(\alpha, 0)$ and $B(0, \beta)$, $\alpha \in [0, 2]$ and $\beta \in [0, 5]$, be such that the line segment $AB$ divides the area of the rectangle $R$ in the ratio $4 : 1$. Then, the mid-point of $AB$ lies on a:

Let $N$ denote the number that turns up when a fair die is rolled. If the probability that the system of equations $x + y + z = 1$ , $2x + Ny + 2z = 2$, $3x + 3y + Nz = 3$ . has a unique solution is $\dfrac{k}{6}$, then the sum of the value of $k$ and all possible values of $N$ is:

Let $S = \{ M = [a_{ij}], \; a_{ij} \in \{0, 1, 2\}, \; 1 \le i, j \le 2 \}$ be a sample space and $A = \{ M \in S : M \text{ is invertible} \}$ be an event. Then $P(A)$ is equal to:

The relation $\mathsf{R} = \{(a,b) : \gcd(a,b)=1,\ 2a \ne b,\ a,b \in \mathbb{Z}\}$ is:

Let $x_1, x_2, \ldots, x_{100}$ be in an arithmetic progression, with $x_1 = 2$ and their mean equal to $200$. If $y_i = i(x_i - i), \; 1 \le i \le 100$, then the mean of $y_1, y_2, \ldots, y_{100}$ is:

$\displaystyle \lim_{t\to 0}\left(1^{\frac{1}{\sin^2 t}}+2^{\frac{1}{\sin^2 t}}+\cdots+n^{\frac{1}{\sin^2 t}}\right)^{\sin^2 t}$ is equal to

Let $w_1$ be the point obtained by the rotation of $z_1 = 5 + 4i$ about the origin through a right angle in the anticlockwise direction, and $w_2$ be the point obtained by the rotation of $z_2 = 3 + 5i$ about the origin through a right angle in the clockwise direction. Then the principal argument of $w_1 - w_2$ is equal to:

$\tan^{-1}\!\left(\dfrac{1+\sqrt{3}}{3+\sqrt{3}}\right) + \sec^{-1}\!\left(\sqrt{\dfrac{8+4\sqrt{3}}{6+3\sqrt{3}}}\right)$ is equal to:

Let $y = y(x)$ be a solution curve of the differential equation \[ (1 - x^2 y^2)\,dx = y\,dx + x\,dy. \] If the line $x = 1$ intersects the curve $y = y(x)$ at $y = 2$ and the line $x = 2$ intersects the curve $y = y(x)$ at $y = \alpha$, then a value of $\alpha$ is:

Let $y=y(x)$ be the solution of the differential equation $x^{3}\,dy+(xy-1)\,dx=0,\quad x>0,$ with $y\!\left(\dfrac{1}{2}\right)=3-e.$ Then $y(1)$ is equal to:

The value of the integral \[ \int_{-\log_e 2}^{\log_e 2} e^x \left( \log_e\!\left( e^x + \sqrt{1 + e^{2x}} \right) \right) dx \] is equal to:

Let $\Omega$ be the sample space and $A \subseteq \Omega$ be an event. Given below are two statements: (S1): If $P(A)=0$, then $A=\varnothing$ (S2): If $P(A)=1$, then $A=\Omega$ Then:

The area enclosed by the curves $y^2 + 4x = 4$ and $y - 2x = 2$ is:

For any vector $\vec{a} = a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k}$, with $10|a_i| < 1, \; i = 1, 2, 3$, consider the following statements: (A): $\max \{|a_1|, |a_2|, |a_3|\} \le |\vec{a}|$ (B): $|\vec{a}| \le 3 \max \{|a_1|, |a_2|, |a_3|\}$

The equation $x^{2}-4x+[x]+3 = x[x]$, where $[x]$ denotes the greatest integer function, has:

 no solution  

Let $\mathbf{A}$ be a $2 \times 2$ matrix with real entries such that $\mathbf{A}' = \alpha \mathbf{A} + \mathbf{I}$, where $\alpha \in \mathbb{R} - \{-1, 1\}$. If $\det(\mathbf{A}^2 - \mathbf{A}) = 4$, then the sum of all possible values of $\alpha$ is equal to:

If $A$ and $B$ are two non-zero $n \times n$ matrices such that $A^{2}+B=A^{2}B$, then:

$A^{2}=I \text{ or } B=I$  

Let $f(x) = \lfloor x^2 - x \rfloor + | -x + \lfloor x \rfloor |$, where $x \in \mathbb{R}$ and $\lfloor t \rfloor$ denotes the greatest integer less than or equal to $t$. Then, $f$ is:

For three positive integers $p, q, r$, $x^{p q^{2}} = y^{q r} = z^{p^{2} r}$ and $r = pq + 1$ such that $3,\ 3\log_{y}x,\ 3\log_{z}y,\ 7\log_{x}z$ are in A.P. with common difference $\dfrac{1}{2}$. Then $r - p - q$ is equal to:

Consider ellipses $\mathbf{E_k} : kx^2 + k^2y^2 = 1, \; k = 1, 2, \ldots, 20$. Let $\mathbf{C_k}$ be the circle which touches the four chords joining the end points (one on minor axis and another on major axis) of the ellipse $\mathbf{E_k}$. If $r_k$ is the radius of the circle $\mathbf{C_k}$, then the value of \[ \sum_{k=1}^{20} \dfrac{1}{r_k^2} \] is:

Let $\triangle PQR$ be a triangle. The points $A, B,$ and $C$ are on the sides $QR, RP,$ and $PQ$ respectively such that $\dfrac{QA}{AR}=\dfrac{RB}{BP}=\dfrac{PC}{CQ}=\dfrac{1}{2}$. Then $\dfrac{\operatorname{Area}(\triangle PQR)}{\operatorname{Area}(\triangle ABC)}$ is equal to:

Area of the region \[ \{(x, y) : x^2 + (y - 2)^2 \le 4, \; x^2 \ge 2y\} \] is:

Let $$ f(x)= \begin{cases} x^{2}\sin\!\left(\dfrac{1}{x}\right), & x\ne 0,\\[6pt] 0, & x=0 \end{cases} $$ Then at $x=0$:

Let $\vec{a}$ be a non-zero vector parallel to the line of intersection of the two planes described by $\hat{i} + \hat{j}, \; \hat{i} + \hat{k}$ and $\hat{i} - \hat{j}, \; \hat{j} - \hat{k}$. If $\theta$ is the angle between the vector $\vec{a}$ and the vector $\vec{b} = 2\hat{i} - 2\hat{j} + \hat{k}$ and $\vec{a} \cdot \vec{b} = 6$, then the ordered pair $(\theta, |\vec{a} \times \vec{b}|)$ is equal to:

Let $f(x)$ be a function such that $f(x+y)=f(x)\cdot f(y)$ for all $x,y\in \mathbb{N}$. If $f(1)=3$ and $\displaystyle \sum_{k=1}^{n} f(k)=3279$, then the value of $n$ is:

The number of triplets , where  are distinct non negative integers satisfying , is :

Let $f(x)$ be a function such that $f(x+y)=f(x)\cdot f(y)$ for all $x,y\in \mathbb{N}$. If $f(1)=3$ and $\displaystyle \sum_{k=1}^{n} f(k)=3279$, then the value of $n$ is:

An organization awarded 48 medals in event 'A', 25 in event 'B' and 18 in event 'C'. If these medals went to total 60 men and only five men got medals in all the three events, then, how many received medals in exactly two of three events?

The number of real solutions of the equation $3\left(x^{2}+\dfrac{1}{x^{2}}\right)-2\left(x+\dfrac{1}{x}\right)+5=0$ is:

The number of integral solutions $x$ of $\log _{\left(x+\frac{7}{2}\right)}\left(\frac{x-7}{2 x-3}\right)^{2} \geq 0$ is :

The number of integers, greater than 7000 that can be formed, using the digits 3, 5, 6, 7, 8 without repetition is :

The sum of the coefficients of three consecutive terms in the binomial expansion of $(1+x)^{\,n+2}$, which are in the ratio $1:3:5$, is equal to:

Let $\vec{\alpha}=4\hat{i}+3\hat{j}+5\hat{k}$ and $\vec{\beta}=\hat{i}+2\hat{j}-4\hat{k}$. Let $\vec{\beta}_{1}$ be parallel to $\vec{\alpha}$ and $\vec{\beta}_{2}$ be perpendicular to $\vec{\alpha}$. If $\vec{\beta}=\vec{\beta}_{1}+\vec{\beta}_{2}$, then the value of $5\,\vec{\beta}_{2}\cdot(\hat{i}+\hat{j}+\hat{k})$ is:

If the system of linear equations $7x + 11y + \alpha z = 13$ $5x + 4y + 7z = \beta$ $175x + 194y + 57z = 361$ has infinitely many solutions, then $\alpha + \beta + 2$ is equal to:

If the system of equations $x + 2y + 3z = 3$ $4x + 3y - 4z = 4$ $8x + 4y - \lambda z = 9 + \mu$ has infinitely many solutions, then the ordered pair $(\lambda,\mu)$ is equal to:

For $a \in \mathbb{C}$, let $A = \{\, z \in \mathbb{C} : \Re(a + \bar z) > \Im(\bar a + z) \,\}$ and $B = \{\, z \in \mathbb{C} : \Re(a + \bar z) < \Im(\bar a + z) \,\}$. Then among the two statements: (S1): If $\Re(a), \Im(a) > 0$, then the set $A$ contains all the real numbers. (S2): If $\Re(a), \Im(a) < 0$, then the set $B$ contains all the real numbers.

The set of all values of $a$ for which $\displaystyle \lim_{x\to a}\big([x-5]-[2x+2]\big)=0$, where $[\alpha]$ denotes the greatest integer less than or equal to $\alpha$, is equal to:

Let $a,b,c,d$ be positive real numbers such that $a+b+c+d=11$. If the maximum value of $a^5 b^3 c^2 d$ is $3750\beta$, then the value of $\beta$ is:

The locus of the mid-points of the chords of the circle $C_{1} : (x-4)^{2}+(y-5)^{2}=4$ which subtend an angle $\theta_{i}$ at the centre of the circle $C_{1}$, is a circle of radius $r_{i}$. If $\theta_{1}=\dfrac{\pi}{3}$, $\theta_{3}=\dfrac{2\pi}{3}$ and $r_{1}^{2}=r_{2}^{2}+r_{3}^{2}$, then $\theta_{2}$ is equal to:

If the letters of the word MATHS are permuted and all possible words so formed are arranged as in a dictionary with serial numbers, then the serial number of the word THAMS is :

Let $y=y(x)$ be the solution of the differential equation $(x^{2}-3y^{2})\,dx+3xy\,dy=0$, with $y(1)=1$. Then $6y^{2}(e)$ is equal to:

Let $A=\{1,3,4,6,9\}$ and $B=\{2,4,5,8,10\}$. Let $R$ be a relation defined on $A\times B$ such that $R=\{\,((a_1,b_1),(a_2,b_2)) : a_1 \le b_2 \text{ and } b_1 \le a_2 \,\}$. Then the number of elements in the set $R$ is:

If $f(x)=\dfrac{2^{2x}}{2^{2x}+2},\ x\in\mathbb{R}$, then $f\!\left(\dfrac{1}{2023}\right)+f\!\left(\dfrac{2}{2023}\right)+\cdots+f\!\left(\dfrac{2022}{2023}\right)$ is equal to:

If the $1011^{\text{th}}$ term from the end in the binomial expansion of \(\left(\dfrac{4x}{5}-\dfrac{5}{2x}\right)^{2022}\) is \(1024\) times the $1011^{\text{th}}$ term from the beginning, then \(|x|\) is equal to:

The number of square matrices of order $5$ with entries from the set $\{0,1\}$, such that the sum of all the elements in each row is $1$ and the sum of all the elements in each column is also $1$, is:

If \[ \begin{vmatrix} x+1 & x & x \\ x & x+\lambda & x \\ x & x & x+\lambda^2 \end{vmatrix} = \dfrac{9}{8}\,(103x+81), \] then $\lambda,\ \dfrac{\lambda}{3}$ are the roots of the equation:

Let the six numbers $a_{1},a_{2},a_{3},a_{4},a_{5},a_{6}$ be in A.P. and $a_{1}+a_{3}=10$. If the mean of these six numbers is $\dfrac{19}{2}$ and their variance is $\sigma^{2}$, then $8\sigma^{2}$ is equal to:

Let the mean of 6 observations $1, 2, 4, 5, x, y$ be $5$ and their variance be $10$. Then their mean deviation about the mean is equal to:

The value of $\left(\dfrac{\,1+\sin\frac{2\pi}{9}+i\cos\frac{2\pi}{9}\,}{\,1+\sin\frac{2\pi}{9}-i\cos\frac{2\pi}{9}\,}\right)^{3}$ is:

If $f:\mathbb{R}\to\mathbb{R}$ is a continuous function satisfying \[ \int_{0}^{\pi/2} f(\sin 2x)\,\sin x\,dx \;+\; \alpha \int_{0}^{\pi/4} f(\cos 2x)\,\cos x\,dx \;=\; 0, \] then the value of $\alpha$ is:

If $f(x)=x^{3}-x^{2}f'(1)+x f''(2)-f'''(3),\ x\in\mathbb{R}$, then:

Let $f$ and $g$ be two functions defined by \[ f(x)= \begin{cases} x+1, & x<0,\\[2pt] |x-1|, & x\ge 0 \end{cases} \qquad\text{and}\qquad g(x)= \begin{cases} x+1, & x<0,\\[2pt] 1, & x\ge 0. \end{cases} \] Then $(g\circ f)(x)$ is:

$\displaystyle \int_{\tfrac{3\sqrt{2}}{4}}^{\tfrac{3\sqrt{3}}{4}} \dfrac{48}{\sqrt{9-4x^{2}}}\,dx$ is equal to:

Let $y=y(x)$ be the solution of the differential equation \[ \frac{dy}{dx}+\frac{5}{x(x^5+1)}\,y=\frac{(x^5+1)^2}{x^7},\quad x>0. \] If $y(1)=2$, then $y(2)$ is equal to:

The vector $\vec{a}=-\hat{i}+2\hat{j}+\hat{k}$ is rotated through a right angle, passing through the $y$-axis in its way and the resulting vector is $\vec{b}$. Then the projection of $3\vec{a}+\sqrt{2}\,\vec{b}$ on $\vec{c}=5\hat{i}+4\hat{j}+3\hat{k}$ is:

Let the function $f:[0,2]\to\mathbb{R}$ be defined as \[ f(x)= \begin{cases} e^{\min\{x^2,\; x-[x]\}}, & x\in[0,1),\\[4pt] e^{[\,x-\log_e x\,]}, & x\in[1,2], \end{cases} \] where $[t]$ denotes the greatest integer less than or equal to $t$. Then the value of the integral $\displaystyle \int_{0}^{2} x f(x)\,dx$ is:

Let $x=2$ be a local minima of the function $f(x)=2x^{4}-18x^{2}+8x+12,\ x\in(-4,4)$. If $M$ is the local maximum value of the function $f$ in $(-4,4)$, then $M=$

The domain of the function \[ f(x)=\frac{1}{\sqrt{[x]^2-3[x]-10}} \] (where $[x]$ denotes the greatest integer less than or equal to $x$) is:

Two dice $A$ and $B$ are rolled. Let the numbers obtained on $A$ and $B$ be $\alpha$ and $\beta$ respectively. If the variance of $\alpha-\beta$ is $\dfrac{p}{q}$, where $p$ and $q$ are co-prime, then the sum of the positive divisors of $p$ is equal to:

The value of $\displaystyle \lim_{n\to\infty} \frac{1+2-3+4+5-6+\cdots+(3n-2)+(3n-1)-3n} {\sqrt{2n^{4}+4n+3}-\sqrt{n^{4}+5n+4}}$ is:

Let $\mathbf{A}=\begin{bmatrix} 1 & \tfrac{1}{51} \\[2pt] 0 & 1 \end{bmatrix}$. If $\mathbf{B}=\begin{bmatrix} 1 & 2 \\ -1 & -1 \end{bmatrix}\mathbf{A}\begin{bmatrix} -1 & -2 \\ 1 & 1 \end{bmatrix}$, then the sum of all the elements of the matrix $\displaystyle \sum_{n=1}^{50} \mathbf{B}^n$ is equal to:

Let $M$ be the maximum value of the product of two positive integers when their sum is $66$. Let the sample space $S=\{\,x\in\mathbb{Z}: x(66-x)\ge \tfrac{5}{9}M\,\}$ and the event $A=\{\,x\in S:\ x\ \text{is a multiple of }3\,\}$. Then $P(A)$ is equal to:

Let $y=y(x),\ y>0$, be a solution curve of the differential equation \[ (1+x^2)\,dy = y(x-y)\,dx. \] If $y(0)=1$ and $y(2\sqrt{2})=\beta$, then

Let $f(x)=\displaystyle \int \frac{2x}{(x^{2}+1)(x^{2}+3)}\,dx$. If $f(3)=\dfrac{1}{2}(\log_{e}5-\log_{e}6)$, then $f(4)$ is equal to:

Let the lines \[ \ell_1:\ \frac{x+5}{3}=\frac{y+4}{1}=\frac{z-\alpha}{-2} \quad\text{and}\quad \ell_2:\ 3x+2y+z-2=0\;=\;x-3y+2z-13 \] be coplanar. If the point $P(a,b,c)$ on $\ell_1$ is nearest to the point $Q(-4,-3,2)$, then $|a|+|b|+|c|$ is equal to:

The distance of the point $P(4,6,-2)$ from the line passing through the point $(-3,2,3)$ and parallel to a line with direction ratios $3,3,-1$ is equal to:

The number of five digit numbers, greater than 40000 and divisible by 5 , which can be formed using the digits 1, 3, 7 and 9 without repetition, is equal to :

Let $y(x)=(1+x)(1+x^{2})(1+x^{4})(1+x^{8})(1+x^{16})$. Then $y' - y''$ at $x=-1$ is equal to:

Let $C$ be the circle in the complex plane with centre $z_0=\tfrac{1}{2}(1+3i)$ and radius $r=1$. Let $z_1=1+i$ and the complex number $z_2$ be outside the circle $C$ such that $\lvert z_1-z_0\rvert\,\lvert z_2-z_0\rvert=1$. If $z_0,z_1$ and $z_2$ are collinear, then the smaller value of $\lvert z_2\rvert^2$ is equal to:

Consider the lines $L_{1}$ and $L_{2}$ given by $L_{1}:\ \dfrac{x-1}{2}=\dfrac{y-3}{1}=\dfrac{z-2}{2}$ $L_{2}:\ \dfrac{x-2}{1}=\dfrac{y-2}{2}=\dfrac{z-3}{3}$ A line $L_{3}$ having direction ratios $1,-1,-2$ intersects $L_{1}$ and $L_{2}$ at the points $P$ and $Q$ respectively. Then the length of line segment $PQ$ is:

If the point $(\alpha, \dfrac{7\sqrt{3}}{3})$ lies on the curve traced by the mid-points of the line segments of the lines $x\cos\theta + y\sin\theta = 7, \theta \in (0, \dfrac{\pi}{2})$ between the co-ordinates axes, then $\alpha$ is equal to:

The points of intersection of the line $ax+by=0,\ (a\ne b)$ and the circle $x^{2}+y^{2}-2x=0$ are $A(\alpha,0)$ and $B(1,\beta)$. The image of the circle with $AB$ as a diameter in the line $x+y+2=0$ is:

Let $\alpha, \beta$ be the roots of the quadratic equation $x^{2}+\sqrt{6}x+3=0$. Then $\dfrac{\alpha^{23}+\beta^{23}+\alpha^{14}+\beta^{14}}{\alpha^{15}+\beta^{15}+\alpha^{10}+\beta^{10}}$ is equal to:

Let $z_{1}=2+3i$ and $z_{2}=3+4i$. The set $S=\left\{\,z\in\mathbb{C}:\ |z-z_{1}|^{2}-|z-z_{2}|^{2}=|z_{1}-z_{2}|^{2}\,\right\}$ represents a

Let $S_{1}$ and $S_{2}$ be respectively the sets of all $a\in \mathbb{R}\setminus\{0\}$ for which the system of linear equations $ax+2ay-3az=1$ $(2a+1)x+(2a+3)y+(a+1)z=2$ $(3a+5)x+(a+5)y+(a+2)z=3$ has unique solution and infinitely many solutions. Then

Let $\mathrm{P}\left(\dfrac{2\sqrt{3}}{\sqrt{7}}, \dfrac{6}{\sqrt{7}}\right), \mathrm{Q}, \mathrm{R}$ and $\mathrm{S}$ be four points on the ellipse $9x^{2}+4y^{2}=36$. Let $\mathrm{PQ}$ and $\mathrm{RS}$ be mutually perpendicular and pass through the origin. If $\dfrac{1}{(PQ)^{2}}+\dfrac{1}{(RS)^{2}}=\dfrac{p}{q}$, where $p$ and $q$ are coprime, then $p+q$ is equal to:

Let $y=y(x)$ be the solution curve of the differential equation $\displaystyle \frac{dy}{dx}=\frac{y}{x}\bigl(1+xy^{2}(1+\log_{e}x)\bigr),\ x>0,\ y(1)=3.$ Then $\displaystyle \frac{y^{2}(x)}{9}$ is equal to:

If the local maximum value of the function $f(x)=\left(\dfrac{\sqrt{3}e}{2\sin x}\right)^{\sin^{2}x},; x\in\left(0,\dfrac{\pi}{2}\right),$ is $\dfrac{k}{e},$ then $\left(\dfrac{k}{e}\right)^{8}+\dfrac{k^{8}}{e^{5}}+k^{8}$ is equal to:

$ \text{The area of the region enclosed by the curve } y=x^{3} \text{ and its tangent at the point } (-1,-1) \text{ is: } $

The equations of two sides of a variable triangle are $x=0$ and $y=3$, and its third side is a tangent to the parabola $y^{2}=6x$. The locus of its circumcentre is:

Let $D$ be the domain of the function $f(x)=\sin^{-1}\!\left(\log_{3x}\!\left(\dfrac{6+2\log_{3}x}{-5x}\right)\right)$. If the range of the function $g: D \to \mathbb{R}$ defined by $g(x)=x-[x]$ (where $[x]$ is the greatest integer function) is $(\alpha,\beta)$, then $\alpha^{2}+\dfrac{5}{\beta}$ is equal to:

The foot of the perpendicular from the point $(2,0,5)$ on the line $\dfrac{x+1}{2}=\dfrac{y-1}{5}=\dfrac{z+1}{-1}$ is $(\alpha,\beta,\gamma)$. Then, which of the following is NOT correct?

The area of the region enclosed by the curve $f(x)=\max\{\sin x,\cos x\},\ -\pi \le x \le \pi$ and the $x$-axis is:

The number of functions $f:\{1,2,3,4\}\to \{\,a\in \mathbb{Z}\mid |a|\le 8\,\}$ satisfying $f(n)+\dfrac{1}{n}f(n+1)=1,\ \forall\, n\in\{1,2,3\}$ is:

Let $PQ$ be a focal chord of the parabola $y^{2}=36x$ of length $100$, making an acute angle with the positive $x$-axis. Let the ordinate of $P$ be positive and $M$ be the point on the line segment $PQ$ such that $PM:MQ=3:1$. Then which of the following points does NOT lie on the line passing through $M$ and perpendicular to the line $PQ$?

Let $N$ be the sum of the numbers appeared when two fair dice are rolled and let the probability that $N-2,\ \sqrt{3N},\ N+2$ are in geometric progression be $\dfrac{k}{48}$. Then the value of $k$ is:

Let $\vec{a}=\hat{i}+4\hat{j}+2\hat{k}$, $\vec{b}=3\hat{i}-2\hat{j}+7\hat{k}$ and $\vec{c}=2\hat{i}-\hat{j}+4\hat{k}$. If a vector $\vec{d}$ satisfies $\vec{d}\times\vec{b}=\vec{c}\times\vec{b}$ and $\vec{d}\cdot\vec{a}=24$, then $|\vec{d}|^{2}$ is equal to:

For the system of linear equations

$2 x+4 y+2 a z=b$

$x+2 y+3 z=4$

$2 x-5 y+2 z=8$

which of the following is NOT correct?

If the function $f(x) = \left\{ {\matrix{ {(1 + |\cos x|)^{\lambda \over {|\cos x|}}} & , & {0 < x < {\pi \over 2}} \cr \mu & , & {x = {\pi \over 2}} \cr e^{{{\cot 6x} \over {{}\cot 4x}}} & , & {{\pi \over 2} < x < \pi } \cr } } \right.$

is continuous at $x = {\pi \over 2}$, then $9\lambda + 6{\log _e}\mu + {\mu ^6} - {e^{6\lambda }}$ is equal to

Let $y=y_1(x)$ and $y=y_2(x)$ be the solution curves of the differential equation $\dfrac{dy}{dx}=y+7$ with initial conditions $y_1(0)=0$ and $y_2(0)=1$ respectively. Then the curves $y=y_1(x)$ and $y=y_2(x)$ intersect at:

two points  

infinite number of points  

one point  

The shortest distance between the lines $x+1=2y=-12z$ and $x=y+2=6z-6$ is:

A coin is biased so that the head is 3 times as likely to occur as tail. This coin is tossed until a head or three tails occur. If $X$ denotes the number of tosses of the coin, then the mean of $X$ is:

$\dfrac{37}{16}$  

$\dfrac{21}{16}$  

 $\dfrac{15}{16}$

Let $T$ and $C$ respectively be the transverse and conjugate axes of the hyperbola $16x^{2}-y^{2}+64x+4y+44=0$. Then the area of the region above the parabola $x^{2}=y+4$, below the transverse axis $T$ and on the right of the conjugate axis $C$ is:

The set of all $a\in\mathbb{R}$ for which the equation $x|x-1|+|x+2|+a=0$ has exactly one real root, is:

Let the function $f(x)=2x^{3}+(2p-7)x^{2}+3(2p-9)x-6$ have a maxima for some value of $x<0$ and a minima for some value of $x>0$. Then, the set of all values of $p$ is:

Evaluate the integral $ \displaystyle \int_{0}^{\infty}\frac{6}{e^{3x}+6e^{2x}+11e^{x}+6},dx $ is equal to:

Let $z$ be a complex number such that $\left|\dfrac{z-2i}{z+i}\right|=2,\ z\ne -i$. Then $z$ lies on the circle of radius $2$ and centre:

Let $s_1,s_2,s_3,\ldots,s_{10}$ respectively be the sum to $12$ terms of $10$ A.P.s whose first terms are $1,2,3,\ldots,10$ and the common differences are $1,3,5,\ldots,19$ respectively. Then $\sum_{i=1}^{10}s_i$ is equal to:

$ \text{The integral } 16 \int_{1}^{2} \frac{dx}{x^{3}(x^{2}+2)^{2}} \text{ is equal to:}$

Let $B=\left[\begin{array}{lll}1 & 3 & \alpha \\ 1 & 2 & 3 \\ \alpha & \alpha & 4\end{array}\right], \alpha > 2$ be the adjoint of a matrix $A$ and $|A|=2$. Then $\left[\begin{array}{ccc}\alpha & -2 \alpha & \alpha\end{array}\right] B\left[\begin{array}{c}\alpha \\ -2 \alpha \\ \alpha\end{array}\right]$$ is equal to :

Let $f:\mathbb{R}\to\mathbb{R}$ be a function defined by $f(x)=\log_{\sqrt{m}}\!\left(\sqrt{2}(\sin x-\cos x)+m-2\right)$, for some $m$, such that the range of $f$ is $[0,2]$. Then the value of $m$ is ______

The fractional part of the number $\dfrac{4^{2022}}{15}$ is equal to:

$ \text{For } x\in\mathbb{R}, \text{ two real valued functions } f(x) \text{ and } g(x) \text{ are such that } g(x)=\sqrt{x}+1 \text{ and } (f\circ g)(x)=x+3-\sqrt{x}. \text{ Then } f(0) \text{ is equal to: } $

Let $y=y(t)$ be a solution of the differential equation $\dfrac{dy}{dt}+\alpha y=\gamma e^{-\beta t}$ where $\alpha>0$, $\beta>0$ and $\gamma>0$. Then $\displaystyle \lim_{t\to\infty} y(t)$

For the differentiable function $f:\mathbb{R}\setminus{0}\to\mathbb{R}$, let $3f(x)+2f!\left(\dfrac{1}{x}\right)=\dfrac{1}{x}-10$. Then $\left|,f(3)+f'!\left(\dfrac{1}{4}\right)\right|$ is equal to:

Let $A=\begin{bmatrix}\dfrac{1}{\sqrt{10}} & \dfrac{3}{\sqrt{10}}\\[4pt]-\dfrac{3}{\sqrt{10}} & \dfrac{1}{\sqrt{10}}\end{bmatrix}$ and $B=\begin{bmatrix}1 & -i\\[2pt] 0 & 1\end{bmatrix}$, where $i=\sqrt{-1}$.

$\displaystyle \max_{0\le x\le \pi}\left\{x-2\sin x\cos x+\frac{1}{3}\sin(3x)\right\}=$

Evaluate the sum: $\displaystyle \sum_{k=0}^{6} \binom{51-k}{3}$

$\text{The number of symmetric matrices of order }3\text{, with all entries from the set }{0,1,2,3,4,5,6,7,8,9}\text{ is:}$

Let $f(x)=2x^{n}+\lambda$, $\lambda\in \mathbb{R}$, $n\in \mathbb{N}$, and $f(4)=133$, $f(5)=255$. Then the sum of all the positive integer divisors of $\bigl(f(3)-f(2)\bigr)$ is:

Let $a_1, a_2, a_3, \dots$ be a G.P. of increasing positive numbers. Let the sum of its $6^{th}$ and $8^{th}$ terms be $2$ and the product of its $3^{rd}$ and $5^{th}$ terms be $\dfrac{1}{9}$. Then $6(a_2 + a_4)(a_4 + a_6)$ is equal to

Let $\lambda\ne 0$ be a real number. Let $\alpha,\beta$ be the roots of the equation $14x^{2}-31x+3\lambda=0$ and $\alpha,\gamma$ be the roots of the equation $35x^{2}-53x+4\lambda=0$. Then $\dfrac{3\alpha}{\beta}$ and $\dfrac{4\alpha}{\gamma}$ are the roots of the equation

All words, with or without meaning, are made using all the letters of the word MONDAY. These words are written as in a dictionary with serial numbers. The serial number of the word MONDAY is :

Let $B$ and $C$ be the two points on the line $y+x=0$ such that $B$ and $C$ are symmetric with respect to the origin. Suppose $A$ is a point on $y-2x=2$ such that $\triangle ABC$ is an equilateral triangle. Then, the area of the $\triangle ABC$ is:

Three rotten apples are mixed accidentally with seven good apples and four apples are drawn one by one without replacement. Let the random variable $X$ denote the number of rotten apples. If $\mu$ and $\sigma^{2}$ represent the mean and variance of $X$, respectively, then $10(\mu^{2}+\sigma^{2})$ is equal to: