WB JEE 2026

Let domain and range of $f(x)$ and $g(x)$ is $[0, \infty)$. If $f(x)$ is an increasing function, $g(x)$ is a decreasing function, $h(x)= f\{g(x)\}, h(0)=0$ and $p(x)=h\left(x^3-2 x^2+2 x\right)-h(4)$, then for all $x \in(0,2)$

Consider the curve $x=1-3 t^2, y=t-3 t^3$. The tangent to the curve at the point $t$ is inclined at an angle $\phi$ to OX and the tangent at $\mathrm{P}(-2,2)$ meets the curve again at Q . Then

Which of the following statements is always true?

Let $f(x)$ be a twice differentiable function in $[1,3]$ and $f(1)=f(3)$. Further if $\left|f^{\prime \prime}(x)\right| \leq 2$, then for all $x$ in $[1,3]$

Tangent at a point $P_1$ (other than $(0,0)$ ) on the curve $y=x^3$ meets the curve again at $P_2$. The tangent at $P_2$ meets the curve at $\mathrm{P}_3$ and so on. Then the abscissae of $\mathrm{P}_1, \mathrm{P}_2, \mathrm{P}_3, \ldots, \...

A figure is bounded by the curves $y=x^2+1, y=0, x=0$ and $x=1$. The point at which a tangent should be drawn to the curve $y=x^2+1$ for it to cut off trapezium of the greatest area from the figure is

Given $P(x)=x^4+a x^3+b x^2+c x+d$ such that $x=0$ is the only real root of $P^{\prime}(x)=0$. If $P(-1) < P(1)$, then in the interval $[-1,1]$.

The quantities $a_1, a_2, a_3, \ldots$ form an infinite decreasing G.P. If $a_1=1$, then the common ratio of the progression for which the expression $6 a_5-16 a_4-3 a_3+12 a_2$ is at a maximum is

Consider the function $y=f(x)$ defined implicitly by the equation $y^3-3 y+x=0$ on the interval $(-\infty,-2) \cup(2, \infty)$. The area of the region bounded by the curve $y=f(x)$, the $x$-axis and the lines $x=a, x=b$, where $-\infty< a< ...

If $\sum_{r=0}^{2 n} a_r(x-2)^r=\sum_{r=0}^{2 n} b_r(x-3)^r$ and $a_k=1 \forall k \geq 1$, then the value of $\frac{b_n}{{ }^{2 n+1} C_{n+1}}$ is

The term independent of $x$ in the expansion of $\left(\frac{x+1}{x^{\frac{2}{3}}-x^{\frac{1}{3}}+1}-\frac{x-1}{x-x^{\frac{1}{2}}}\right)^{15}$ is equal to

Let all the points on the curve $x^2+y^2-10 x=0$ are reflected about the line $y=x+3$. If the locus of the reflected points is in the form $x^2+y^2+g x+f y+c=0$, then the value of $(g+f+c)$ is

The expression $\sum_{k=1}^{32}(3 K+2)\left\{\sum_{r=1}^{10}\left(\sin \frac{2 r \pi}{11}-i \cos \frac{2 r \pi}{11}\right)\right\}^k$ represents

The total number of polynomials of the form $x^3+a x^2+b x+c$ which is divisible by $x^2+1$, where $a, b, c \in\{1,2,3, \ldots ., 10\}$ is

Let $Z_1, Z_2$ be the roots of the equation $Z^2+p Z+q=0$, where the coefficients $p$ and $q$ may be complex numbers and also let $A, B$ represent $Z_1, Z_2$ respectively in the complex plane. If $\angle A O B=\alpha \neq 0$ and $O A=O B$, ...

Let $f(x)$ be a real valued $f$ unction which is monotonic and differentiable. Then for any reals a and $b, \int_{f(a)}^{f(b)} 2 x\left\{b-f^{-1}(x)\right\} d x=$

If $\int_0^1\left(\sum_{r=1}^{2013} \frac{x}{x^2+r^2}\right)\left(\prod_{r=1}^{2013}\left(x^2+r^2\right)\right) d x=\frac{1}{2}\left[\left(\prod_{r=1}^{2013}\left(1+r^2\right)-K^2\right]\right.$, then $K$ is

Let $a, b, c$ be non-zero real numbers, such that $\int_0^r\left(1+\cos ^8 x\right)\left(a x^2+b x+c\right) d x=\int_0^{2^{\prime}}\left(1+\cos ^8 x\right)\left(a x^2+b x+c\right) d x$, then $a x^2+b x+c=0$ has

Let $f(x)>0$ for all $x \in \mathbb{R}$ and $f(x)$ is bounded. If $\mathop {\lim }\limits_{n \to \infty } \sum_{r-1}^n a^{r-1} \int_{(r-1) a}^{r a} \frac{f(x) d x}{f(x)+f(2 r a-a-x)}=\frac{3}{5}$ where $0< a< 1$, then the value(s) of a is a...

The least positive value of ' $a$ ' for which the equation $\int_0^x\left(t^2-8 t+13\right) d t=x \sin \frac{a}{x}$ has a solution is

The solution of the differential equation $2 x^2 y \frac{d y}{d x}=\tan \left(x^2 y^2\right)-2 x y^2$, given $y(1)=\sqrt{\frac{\pi}{2}}$ is

Let $f:(0,1) \rightarrow(0,1)$ be a differentiable function such that $f^{\prime}(x) \neq 0 \forall x \in(0,1)$ and $f\left(\frac{1}{2}\right)=\frac{\sqrt{3}}{2}$. Suppose for all $x$, $\mathop {\lim }\limits_{t \to x} \frac{\int_0^t \sqrt...

Consider the following ellipse :
$\frac{x^2}{f\left(K^2+2 K+5\right)}+\frac{y^2}{f(K+11)}=1$, where $f(x)$ is a positive decreasing function. Then the value (values) of $K$ for which the major axis coincides with $x$-axis is

The minimum length of intercept on any tangent to the ellipse $\frac{x^2}{4}+\frac{y^2}{9}=1$ cut by the circle $x^2+y^2=25$ is

Number of elements in the range set of $f(x)=\left[\frac{x}{15}\right]\left[-\frac{15}{x}\right]$, for all $x \in(0,90$ ); (where [.] denotes the greatest integer function) is

If $f$ be a real valued function defined for all real numbers $x$ such that for some fixed $a>0$, it satisfies $f(x+a)=\frac{1}{2}+\sqrt{f(x)-(f(x))^2} \forall x$, then $f(x)$ is periodic with period

Let $A=[a, \infty)$ denotes the domain, then $f:(a, \infty) \rightarrow B$, which is defined by $f(x)=2 x^3-3 x^2+6$ will have an inverse for the smallest real value of ' $a$ ' if

If the domain of $f(x)$ is $(0,1)$, then the domain of $y=f\left(e^x\right)+f(\ln |x|)$ is

If a differentiable function satisfies $(x-y) f(x+y)-(x+y) f(x-y)=2\left(x^2 y-y^3\right) \forall x, y \in \mathbb{R}$ and $f(I)=2$, then

If $\int \frac{\left(1-x^2\right)}{\sqrt{x} \sqrt{\left(1+x^2\right)^3}}=\alpha \frac{x^\beta}{\left(1+x^2\right)^\gamma}+C ; \alpha, \beta, \gamma \in \mathbb{R}$ and $C$ is constant of integration, then $\alpha: \beta: \gamma$ will be

\(\int \frac{\left(\sqrt[3]{x+\sqrt{2-x^2}}\right)\left(\sqrt[6]{1-x \sqrt{2-x^2}}\right)}{\sqrt[3]{1-x^2}} d x ;(x \in(0,1))=\)

If $\int \frac{\operatorname{cosec}^2 x-2010}{\cos ^{2010} x} d x=-\frac{f(x)}{(g(x))^{2010}}+c$, where $f\left(\frac{\pi}{4}\right)=1$; then the number of solutions of the equation $\frac{f(x)}{g(x)}=\{x\}$ in $[0,2 \pi]$ is/are (where $\{...

If $\sum\limits_{r=1}^{\infty} \tan ^{-1}\left(\frac{1}{2 r^2}\right)=a$, then $\tan a$ is equal to

Let $g(x)=a x+b$, where $a<0$ and $g$ is defined from $[1,3]$ onto $[0,2]$. Then the value of $\cot \left(\cos ^{-1}(|\sin x|+|\cos x|)+\right. \left.\sin ^{-1}(-|\cos x|-|\sin x|)\right)$ is equal to

If for two real numbers $\mathrm{a}, \mathrm{b}$ with $|\mathrm{a}| \leq 1$ and $|\mathrm{b}| \leq 1$,
$\frac{1}{3}+\frac{\sin ^{-1} a+\sin ^{-1} b}{4}+\frac{\left(\sin ^{-1} a+\sin ^{-1} b\right)^2}{16}+\frac{\left(\sin ^{-1} a+\sin ^{-1} ...

If $f(x)=x\left(1331 x^2-3630 x+3300\right)$, then for $a=\cos ^2\left(\tan ^{-1}\left(\sin \left(\cot ^{-1} 3\right)\right)\right)$

The true set of values of ' $K$ ' for which $\sin ^{-1}\left(\frac{1}{1+\sin ^2 x}\right)=\frac{K \pi}{6}$ may have a solution is

Consider a function $f(x)$ which has exactly two roots at $x=a$. If $\mathop {\lim }\limits_{x \to a}\left(\frac{\lambda f^{\prime}(x)}{f(x)}-\frac{1}{x-a}\right)=m(\neq 0)$, then the value of $\lambda$ ix

If $f(x)$ is differentiable for all $x \in \mathbb{R}$ and satisfies the relation
$x=\mathop {\lim }\limits_{n \to \infty }\frac{\left[1^2(f(x))^x\right]+\left[2^2(f(x))^x\right]+\ldots+\left[n^2(f(x))^x\right]}{n^3}$ where [.] denotes the ...

For a real number $y$, consider $(y)$ denotes the greatest integer less than or equal to $y$. If $f(x)=\frac{\tan (\pi[x-\pi])}{1+[x]^2}$, then

If $a=\mathop {\lim }\limits_{n \to \infty } \cos ^{2 n} x,(x=n \pi)$ and $b=\mathop {\lim }\limits_{n \to \infty } \cos ^{2 n} x,(x \neq n \pi)$, then numerical value of the area of the triangle whose vertices are (a, b), (-2, 1) and (2, ...

The equation $|x+1|^{\log _{(x+1)}\left(3+2 x-x^2\right)}=(x-3)|x|$ has

If $f(x)=\frac{1+x}{1-x}$ and $A$ is a matrix such that $A^3=0$, then $f(A)=$

Let us define the power of a matrix $A$ as the maximum $m \in Z^{+}$such that $A^m=I$. For two matrices $A$ and $B$ if $A^5=I$ and $A B A^{-1}=B^2$, then the power of the matrix $B$ is between

Let $\operatorname{det} A=\left|\begin{array}{ccc}\mathrm{l} & \mathrm{m} & \mathrm{n} \\ \mathrm{p} & \mathrm{q} & \mathrm{r} \\ \mathrm{l} & \mathrm{l} & \mathrm{l}\end{array}\right|$ If $(I-m)^2+(p-q)^2=9,(m-n)^2+(q-r)^2=16,(n-I)^2+(r-p)...

If the locus of mid point of any normal chord of the parabola $y^2=4 x$ is $x-\lambda=\frac{\mu}{y^2}+\frac{y^2}{v}$, where $\lambda, \mu, v \in N$, then ( $\lambda+\mu+v$ ) equals to

The parabola $y=4-x^2$ has vertex P. It intersects $x$-axis at A and B. If the parabola is translated from its initial position to a new position by moving its vertex along the line $y=x+4$, so that it intersects $x$-axis at B and C , then ...

The number of 3-digit numbers we of the form $x y z$ with $x< y, z< y$ and $x \neq 0$ is

Let 10 Bags $B_1, B_2, \ldots, B_{10}$ which contains $21,22, \ldots, 30$ different articles respectively. Then the total number of ways to bring out 10 articles from a Bag is

Suppose $A$ is denoted the set of all numbers between 1 and 700 which are divisible by 3 and let $B$ is denoted the set of all numbers between 1 and 300 which are divisible by 7 . If $C=\{(a, b) \mid a \in A, b \in B, a \neq b$ and $a+b=$ e...