WB JEE 2026

Four natural numbers selected at random are multiplied together, then the probability that the digit in the unit's place in the product be $1,3,7$ or 9 is

A mapping is selected at random from all mappings $f: A \rightarrow A$, where set $A=\{1,2,3 \ldots, n\}$. If the probability that the mapping is injective is $\frac{3}{32}$, then the value of $n$ is

Consider the sequence of numbers $(1,2,3, \ldots \ldots, 13)$. A person choose three numbers at random from the sequence. The probability that the chosen three number form an A.P. is

If $A_1, A_2, A_3, \ldots, A_{1006}$ be independent events such that $P\left(A_l\right)=\frac{1}{2 i},(i=1,2, \ldots, 1006)$ and the probability that none of the events occurs be $\frac{\alpha!}{2^a(\beta!)^2}$ ，then

If ' $a$ ' is an integer lying in $[-5,30]$, then the probability that the graph of $y=x^2+2(a+4) x-5 a+64$ lies above the $x-$ axis is

Let 1 lies between the roots of the equation $y^2-m y+1=0$ and $[x]$ denotes the greatest integer function. Then the value of $\left[\left(\frac{4|x|}{x^2+16}\right)^m\right]$ is

The equation $x^3+5 x^2+p x+q=0$ and $x^3+7 x^2+p x+r=0$ have two roots in common. If the third root of each equation is represented by $x_1$ and $x_2$ respectively, the GCD of $x_1, x_2$ will be

If $\left(4^{\sec ^2 \alpha}\right) x^2+2 x+\left(\beta^2-\beta+\frac{1}{2}\right)=0$ has real roots，then the value／values of $\left(\cos \alpha+\cos ^{-1} \beta\right)$ is／are

If $0<\alpha<\beta<\gamma<\frac{\pi}{2}$, then the equation $\frac{1}{x-\sin \alpha}+\frac{1}{x-\sin \beta}+\frac{1}{x-\sin \gamma}=0$ has

If $\alpha, \beta$ are the roots of the equation $x^2-p x+q=0$ and $\alpha>0, \beta>0$, then $\alpha^{\frac{1}{4}}+\beta^{\frac{1}{4}}=\left(p+6 \sqrt{p}+4 q^{\frac{1}{4}} \sqrt{p+2 \sqrt{q}}\right)^k$, where $K$ is

Let $a_1, a_2, a_3 \ldots$ are in G.P. such that $n>m, a_n>a_m$ and $a_1+a_n=66, a_2 \cdot a_{n-1}=128$. If $\sum_{r=1}^n a_r=126$, then $n$ is

If $t_n$ denotes the $n^{\text {th }}$ term of an A.P. and $t_p=\frac{1}{q}, t_q=\frac{1}{p}$, then which one of the following options is a root of the equation $(p+2 q-3 r) x^2+(q+2 x-3 p) x+(r+2 p-3 q)=0 ?$

On the set $\mathbb{R}$ of real numbers the relation $\rho$, defined by $\mathrm{x} \rho \mathrm{y}(\mathrm{x}, \mathrm{y} \in \mathbb{R})$ iff

Let $A_1, A_2, \ldots, A_6$ are six sets, each with four elements and $B_1, B_2, \ldots ., B_n$ are $n$ sets, each with two elements. Let $S=A_1 \cup A_2 \cup \ldots \cup A_6=B_1 \cup B_2 \cup \ldots \cup B_n$.
Given that each element of $S...

The ends $A$, $B$ of a straight line segment of constant length $c$ slide upon the fixed rectangular axes $O X, O Y$ respectively. If the rectangle $O A P B$ completed, then the locus of the foot of perpendicular drawn from $P$ to $A B$ is

Intercepts of the plane $\vec{r} \cdot \vec{n}=d(\neq 0)$ on the coordinate axes respectively are

Consider a square $A B C D$ of diagonal length 2a. The square is folded along the diagonal $A C$ so that the plane of $\triangle A B C$ is perpendicular to the plane of $\triangle A D C$. In this case the shortest distance between $A B$ and...

The point of intersection of $\vec{r} \times \vec{a}=\vec{b} \times \vec{a}$ and $\vec{r} \times \vec{b}=\vec{a} \times \vec{b}$, where $\vec{a}=\hat{i}+\hat{j}$ and $\vec{b}=2 \hat{i}-\hat{k}$ is

$\theta$ elimination from the equation $x^2+y^2=\frac{x \cos 3 \theta+y \sin 3 \theta}{\cos ^3 \theta}=\frac{y \cos 3 \theta-x \sin 3 \theta}{\sin ^3 \theta}$ will be

The general solution of the equation $\sin ^{100} \mathrm{x}-\cos ^{100} \mathrm{x}=1$ is

A vector given by $\vec{P}=f(t) \hat{i}+g(t)+\hat{k}$ moves in such a way that it is always parallel to the vector $\vec{Q}=-f^{\prime \prime}(t) \hat{i}+f^{\prime}(t) \hat{j}+\hat{k}$.

Let $\vec{a}=(x, y, z)$ be the vector with $|\vec{a}|=2 \sqrt{3}$, which makes equal angles with the vector $\vec{b}=(y,-2 z, 3 x)$ and $\vec{c}=(2 z, 3 x,-y)$ and is perpendicular to the vector $\vec{d}=(1,-1,2)$. If the angle between $\ve...

Let $\vec{r}=\sin x(\vec{a} \times \vec{b})+\cos y(\vec{b} \times \vec{c})+2(\vec{c} \times \vec{a})$ ，where $\vec{a}, \vec{b}$ and $\vec{c}$ are three non－coplanar vectors． It is given that $\vec{r}$ is perpendicular to $(\vec{a}+\vec{b}+\...

If $\vec{a}=\hat{i}+\hat{j}+\hat{k}, \vec{b}=\hat{i}-\hat{j}+\hat{k}, \vec{c}=\hat{i}+2 \hat{j}-\hat{k}$, then the value of $\left|\begin{array}{lll}\vec{a} \cdot \vec{a} & \vec{a} \cdot \vec{b} & \vec{a} \cdot \vec{c} \\ \vec{b} \cdot \vec...

The position vectors of two adjacent sides $\overrightarrow{O A}$ and $\overrightarrow{O B}$ of a rectangle $O A C B$ are $\vec{a}$ and $\vec{b}$ respectively, where $O$ is the origin. If $16|\vec{a} \times \vec{b}|=3(|\vec{a}|+|\vec{b}|)^2...