WB JEE 2025

Let $p(x)$ be a real polynomial of least degree which has a local maximum at $x=1$ and a local minimum at $x=3$. If $p(1)=6$ and $p(3)=2$, then $p^{\prime}(0)$ is equal to :

If $x=-1$ and $x=2$ are extreme points of $f(x)=\alpha \log |x|+\beta x^2+x,(x \neq 0)$, then

Let $f$ be a function which is differentiable for all real $x$. If $f(2)=-4$ and $f^{\prime}(x) \geq 6$ for all $x \in[2,4]$, then

Let $\phi(x)=f(x)+f(2 a-x), x \in[0,2 a]$ and $f^{\prime \prime}(x)>0$ for all $x \in[0, a]$. Then $\phi(x)$ is

Let $f(x)=x^3, x \in[-1,1]$. Then which of the following are correct?

Let $f(\theta)=\left|\begin{array}{ccc}1 & \cos \theta & -1 \\ -\sin \theta & 1 & -\cos \theta \\ -1 & \sin \theta & 1\end{array}\right|$.
Suppose $A$ and $B$ are respectively maximum and minimum values of $f(\theta)$.Then $(A,B)$ is equal ...

The function $f(x)=2 x^3-3 x^2-12 x+4, x \in \mathbb{R}$ has

The maximum number of common normals of $y^2=4 a x$ and $x^2=4 b y$ is equal to :

If $\left(1+x-2 x^2\right)^6=1+a_1 x+a_2 x^2+\ldots+a_{12} x^{12}$, then the value of $a_2+a_4+a_6+\ldots+a_{12}$ is

Let $x-y=0$ and $x+y=1$ be two perpendicular diameters of a circle of radius $R$. The circle will pass through the origin if $R$ is equal to

The number of common tangents to the circles $x^2+y^2-4 x-6 y-12=0, x^2+y^2+6 x+18 y+26=0$ is

If $z_1, z_2$ are complex numbers such that $\frac{2 z_1}{3 z_2}$ is a purely imaginary number, then the value of $\left|\frac{z_1-z_2}{z_1+z_2}\right|$ is

If $\left|Z_1\right|=\left|Z_2\right|=\left|Z_3\right|=1$ and $Z_1+Z_2+Z_3=0$, then the area of the triangle whose vertices are $Z_1, Z_2, Z_3$ is

Let $\omega(\neq 1)$ be a cubic root of unity. Then the minimum value of the set $\left\{\mid a+b \omega+c \omega^2\right\}^2 ; a, b, c$ are distinct non-zero integers} equals

The value of the integral $\int\limits_3^6 \frac{\sqrt{x}}{\sqrt{9-x}+\sqrt{x}} d x$ is

$\int_\limits{-1}^1 \frac{x^3+|x|+1}{x^2+2|x|+1} d x$ is equal to

Let $f(x)=\max \{x+|x|, x-[x]\}$, where $[x]$ stands for the greatest integer not greater than $x$. Then $\int\limits_{-3}^3 f(x) d x$ has the value

$\int\limits_0^{1 \cdot 5}\left[x^2\right] d x$ is equal to

The value of the integral $\int_0^{\pi / 2} \log \left(\frac{4+3 \sin x}{4+3 \cos x}\right) d x$ is

If $f(x)=\int\limits_0^{\sin ^2 x} \sin ^{-1} \sqrt{t} d t$ and $g(x)=\int\limits_0^{\cos ^2 x} \cos ^{-1} \sqrt{t} d t$, then the value of $f(x)+g(x)$ is

The value of $\int\limits_{-100}^{100} \frac{\left(x+x^3+x^5\right)}{\left(1+x^2+x^4+x^6\right)} d x$ is

If $x=\int\limits_0^y \frac{1}{\sqrt{1+9 t^2}} d t$ and $\frac{d^2 y}{d x^2}=a y$, then $a$ is equal to

The population $p(t)$ at time $t$ of a certain mouse species follows the differential equation
\(\frac{d p(t)}{d t}=0.5 p(t)-450\)
If $p(0)=850$, then the time at which the population becomes zero is

If ' $f$ ' is the inverse function of ' $g$ ' and $g^{\prime}(x)=\frac{1}{1+x^n}$, then the value of $f^{\prime}(x)$ is

Let $f(x)$ be a second degree polynomial. If $f(1)=f(-1)$ and $p, q, r$ are in A.P., then $f^{\prime}(p), f^{\prime}(q), f^{\prime}(r)$ are

If $g(f(x))=|\sin x|$ and $f(g(x))=(\sin \sqrt{x})^2$, then

Let $u+v+w=3, u, v, w \in \mathbb{R}$ and $f(x)=u x^2+v x+w$ be such that $f(x+y)=f(x)+f(y)+x y$, $\forall x, y \in \mathbb{R}$. Then $f(1)$ is equal to

If $f(x)$ and $g(x)$ are two polynomials such that $\phi(x)=f\left(x^3\right)+x g\left(x^3\right)$ is divisible by $x^2+x+1$, then

If $f(x)=\frac{3 x-4}{2 x-3}$, then $f(f(f(x)))$ will be

The number of solutions of $\sin ^{-1} x+\sin ^{-1}(1-x)=\cos ^{-1} x$ is

If $\cos ^{-1} \alpha+\cos ^{-1} \beta+\cos ^{-1} \gamma=3 \pi$, then $\alpha(\beta+\gamma)+\beta(\gamma+\alpha)+\gamma(\alpha+\beta)$ is equal to

If $f(x)=\left\{\begin{array}{ll}x^2+3 x+a, & x \leq 1 \\ b x+2, & x>1\end{array}, x \in \mathbb{R}\right.$, is everywhere differentiable, then :

Let $f(x)=|1-2 x|$, then

Let $f(x)=|x-\alpha|+|x-\beta|$, where $\alpha, \beta$ are the roots of the equation $x^2-3 x+2=0$. Then the number of points in $[\alpha,\beta]$ at which $f$ is not differentiable is

Let $f(x)$ be continuous on $[0,5]$ and differentiable in $(0,5)$. If $f(0)=0$ and $\left|f^{\prime}(x)\right| \leq \frac{1}{5}$ for all $x$ in $(0,5)$, then $\forall x$ in $[0,5]$

$\lim\limits_{x \rightarrow 0} \frac{\tan \left(\left[-\pi^2\right] x^2\right)-x^2 \tan \left(\left[-\pi^2\right]\right)}{\sin ^2 x}$ equals

A function $f: \mathbb{R} \rightarrow \mathbb{R}$, satisfies $f\left(\frac{x+y}{3}\right)=\frac{f(x)+f(y)+f(0)}{3}$ for all $x, y \in \mathbb{R}$.
If the function ' $f$ ' is differentiable at $x=0$, then $f$ is

The set of points of discontinuity of the function $f(x)=x-[x], x \in \mathbb{R}$ is

A function $f$ is defined by $f(x)=2+(x-1)^{2 / 3}$ on $[0,2]$. Which of the following statements is incorrect?

Let $f:[0,1] \rightarrow \mathbb{R}$ and $g:[0,1] \rightarrow \mathbb{R}$ be defined as follows :
$\left.\begin{array}{rl}f(x) & =1 \text { if } x \text { is rational } \\ & =0 \text { if } x \text { is irrational }\end{array}\right]$ and
$...

Let $a_n$ denote the term independent of $x$ in the expansion of $\left[x+\frac{\sin (1 / n)}{x^2}\right]^{3 n}$, then $\lim \limits_{n \rightarrow \infty} \frac{\left(a_n\right) n!}{{ }^{3 n} P_n}$ equals

The expression $2^{4 n}-15 n-1$, where $n \in \mathbb{N}$ (the set of natural numbers) is divisible by

If $P$ is a non-singular matrix of order $5 \times 5$ and the sum of the elements of each row is 1 , then the sum of the elements of each row in $P^{-1}$ is

If for a matrix $A,|A|=6$ and adj $A=\left[\begin{array}{ccc}1 & -2 & 4 \\ 4 & 1 & 1 \\ -1 & k & 0\end{array}\right]$, then $k$ is equal to

If the matrix $\left(\begin{array}{ccc}0 & a & a \\ 2 b & b & -b \\ c & -c & c\end{array}\right)$ is orthogonal, then the values of $a, b, c$ are

Suppose $\alpha, \beta, \gamma$ are the roots of the equation $x^3+q x+r=0($ with $r \neq 0)$ and they are in A.P. Then the rank of the matrix $\left(\begin{array}{lll}\alpha & \beta & \gamma \\ \beta & \gamma & \alpha \\ \gamma & \alpha & ...

An $n \times n$ matrix is formed using 0, 1 and $-$1 as its elements. The number of such matrices which are skew symmetric is

If $a, b, c$ are positive real numbers each distinct from unity, then the value of the determinant $\left|\begin{array}{ccc}1 & \log _a b & \log _a c \\ \log _b a & 1 & \log _b c \\ \log _c a & \log _c b & 1\end{array}\right|$ is

Let $A=\left[\begin{array}{ccc}5 & 5 \alpha & \alpha \\ 0 & \alpha & 5 \alpha \\ 0 & 0 & 5\end{array}\right]$. If $|A|^2=25$, then $|\alpha|$ equals to

If $\operatorname{adj} B=A,|P|=|Q|=1$, then $\operatorname{adj}\left(Q^{-1} B P^{-1}\right)=$