WB JEE 2024

Two smallest squares are chosen one by one on a chess board. The probability that they have a side in common is

Two integers \(\mathrm{r}\) and \(\mathrm{s}\) are drawn one at a time without replacement from the set \(\{1,2, \ldots, \mathrm{n}\}\). Then \(\mathrm{P}(\mathrm{r} \leq \mathrm{k} / \mathrm{s} \leq \mathrm{k})=\)
(k is an integer < n)

A biased coin with probability \(\mathrm{p}(0<\mathrm{p}<1)\) of getting head is tossed until a head appears for the first time. If the probability that the number of tosses required is even is \(\frac{2}{5}\), then \(\mathrm{p}=\)

If \(\mathrm{P}(x)=\mathrm{a} x^2+\mathrm{b} x+\mathrm{c}\) and \(\mathrm{Q}(x)=-\mathrm{a} x^2+\mathrm{d} x+\mathrm{c}\) where \(\mathrm{ac} \neq 0\), then \(\mathrm{P}(x) \cdot \mathrm{Q}(x)=0\) has (a, b, c, d are real)

Let \(\mathrm{N}\) be the number of quadratic equations with coefficients from \(\{0,1,2, \ldots, 9\}\) such that 0 is a solution of each equation. Then the value of \(\mathrm{N}\) is

If \(a, b, c\) are distinct odd natural numbers, then the number of rational roots of the equation \(a x^2+b x+c=0\)

If the quadratic equation \(a x^2+b x+c=0(a>0)\) has two roots \(\alpha\) and \(\beta\) such that \(\alpha<-2\) and \(\beta>2\), then

Given an A.P. and a G.P. with positive terms, with the first and second terms of the progressions being equal. If \(a_n\) and \(b_n\) be the \(n^{\text {th }}\) term of A.P. and G.P. respectively then

If for the series \(a_1, a_2, a_3\), ...... etc, \(\mathrm{a}_{\mathrm{r}}-\mathrm{a}_{\mathrm{r}+\mathrm{i}}\) bears a constant ratio with \(\mathrm{a}_{\mathrm{r}} \cdot \mathrm{a}_{\mathrm{r}+1}\); then $$\mathrm{a}_1, \mathrm{a}_2, \mat...

If \(\alpha_1, \alpha_2, \ldots, \alpha_n\) are in A.P. with common difference \(\theta\), then the sum of the series
$$
\sec \alpha_1 \sec \alpha_2+\sec \alpha_2 \sec \alpha_3+\ldots .+\sec \alpha_{n-1} \sec \alpha_n=k\left(\tan \alpha_n-\...

In R, a relation p is defined as follows:
\(\forall a, b \in \mathbb{R}, a p\) holds iff \(a^2-4 a b+3 b^2=0\). Then

Let A be the set of even natural numbers that are < 8 & B be the set of prime integers that are \(<7\) The number of relations from A to B are

For the real numbers \(x\) & \(y\), we write \(x\) p y iff \(x-y+\sqrt{2}\) is an irrational number. Then relation p is

If \((1,5)\) be the midpoint of the segment of a line between the line \(5 x-y-4=0\) and \(3 x+4 y-4=0\), then the equation of the line will be

In \(\triangle \mathrm{ABC}\), co-ordinates of \(\mathrm{A}\) are \((1,2)\) and the equation of the medians through \(\mathrm{B}\) and C are \(x+\mathrm{y}=5\) and \(x=4\) respectively. Then the midpoint of \(\mathrm{BC}\) is

Let \(\Gamma\) be the curve \(\mathrm{y}=\mathrm{be}^{-x / a}\) & \(\mathrm{L}\) be the straight line \(\frac{x}{\mathrm{a}}+\frac{\mathrm{y}}{\mathrm{b}}=1\) where \(\mathrm{a}, \mathrm{b} \in \mathbb{R}\).
Then

A square with each side equal to '\(a\)' above the \(x\)-axis and has one vertex at the origin. One of the sides passing through the origin makes an angle \(\alpha\) \(\left(0<\alpha< \frac{\pi}{4}\right)\) with the positive direction of th...

If \(\mathrm{ABC}\) is an isosceles triangle and the coordinates of the base points are \(B(1,3)\) and \(C(-2,7)\). The coordinates of \(A\) can be

The plane \(2 x-y+3 z+5=0\) is rotated through \(90^{\circ}\) about its line of intersection with the plane \(x+y+z=1\). The equation of the plane in new position is

If the relation between the direction ratios of two lines in \(\mathbb{R}^3\) are given by
\(l+\mathrm{m}+\mathrm{n}=0,2 l \mathrm{~m}+2 \mathrm{mn}-l \mathrm{n}=0\)
then the angle between the lines is
(\(l, \mathrm{~m}, \mathrm{n}\) have t...

Angle between two diagonals of a cube will be

The expression \(\cos ^2 \phi+\cos ^2(\theta+\phi)-2 \cos \theta \cos \phi \cos (\theta+\phi)\) is

If \(0< \theta<\frac{\pi}{2}\) and \(\tan 3 \theta \neq 0\), then \(\tan \theta+\tan 2 \theta+\tan 3 \theta=0\) if \(\tan \theta \cdot \tan 2 \theta=\mathrm{k}\) where \(\mathrm{k}=\)

If \(A\) and \(B\) are acute angles such that \(\sin A=\sin ^2 B\) and \(2 \cos ^2 A=3 \cos ^2 B\), then \((A, B)=\)

A unit vector in XY-plane making an angle \(45^{\circ}\) with \(\hat{i}+\hat{j}\) and an angle \(60^{\circ}\) with \(3 \hat{i}-4 \hat{j}\) is