WB JEE 2024

If \(n\) is a positive integer, the value of \((2 n+1){ }^n C_0+(2 n-1){ }^n C_1+(2 n-3){ }^n C_2 +\ldots .+1 \cdot{ }^n C_n\) is

Chords \(\mathrm{AB}\) & \(\mathrm{CD}\) of a circle intersect at right angle at the point \(\mathrm{P}\). If the length of AP, PB, CP, PD are 2, 6, 3, 4 units respectively, then the radius of the circle is

If two circles which pass through the points \((0, a)\) and \((0,-a)\) and touch the line \(\mathrm{y}=\mathrm{m} x+\mathrm{c}\), cut orthogonally then

If \(z_1\) and \(z_2\) be two roots of the equation \(z^2+a z+b=0, a^2<4 b\), then the origin, \(\mathrm{z}_1\) and \(\mathrm{z}_2\) form an equilateral triangle if

If \(\cos \theta+i \sin \theta, \theta \in \mathbb{R}\), is a root of the equation
\(a_0 x^n+a_1 x^{n-1}+\ldots .+a_{n-1} x+a_n=0, a_0, a_1, \ldots . a_n \in \mathbb{R}, a_0 \neq 0,\)
then the value of $$a_1 \sin \theta+a_2 \sin 2 \theta+\l...

All values of a for which the inequality \(\frac{1}{\sqrt{a}} \int_\limits1^a\left(\frac{3}{2} \sqrt{x}+1-\frac{1}{\sqrt{x}}\right) \mathrm{d} x<4\) is satisfied, lie in the interval

For any integer \(\mathrm{n}, \int_\limits0^\pi \mathrm{e}^{\cos ^2 x} \cdot \cos ^3(2 n+1) x \mathrm{~d} x\) has the value :

If \(\mathrm{f}(x)=\frac{\mathrm{e}^x}{1+\mathrm{e}^x}, \mathrm{I}_1=\int_\limits{\mathrm{f}(-\mathrm{a})}^{\mathrm{f}(\mathrm{a})} x \mathrm{~g}(x(1-x)) \mathrm{d} x\) and $$\mathrm{I}_2=\int_\limits{\mathrm{f}(-\mathrm{a})}^{\mathrm{f}(\m...

Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be a differentiable function and \(f(1)=4\). Then the value of \(\lim _\limits{x \rightarrow 1} \int_\limits4^{f(x)} \frac{2 t}{x-1} d t\), if \(f^{\prime}(1)=2\) is

Let \(\mathrm{I}(\mathrm{R})=\int_\limits0^{\mathrm{R}} \mathrm{e}^{-\mathrm{R} \sin x} \mathrm{~d} x, \mathrm{R}>0\). then,

\(\lim _\limits{n \rightarrow \infty} \frac{1}{n^{k+1}}[2^k+4^k+6^k+\ldots .+(2 n)^k]=\)

\(\text { The points of extremum of } \int_\limits0^{x^2} \frac{t^2-5 t+4}{2+e^t} d t \text { are }\)

Let \(\mathrm{f}\) be a differential function with \(\lim _\limits{x \rightarrow \infty} \mathrm{f}(x)=0\). If \(\mathrm{y}^{\prime}+\mathrm{yf}^{\prime}(x)-\mathrm{f}(x) \mathrm{f}^{\prime}(x)=0\), $$\lim _\limits{x \rightarrow \infty} y(x...

If \(x y^{\prime}+y-e^x=0, y(a)=b\), then \(\lim _\limits{x \rightarrow 1} y(x)\) is

If \(\mathrm{U}_{\mathrm{n}}(\mathrm{n}=1,2)\) denotes the \(\mathrm{n}^{\text {th }}\) derivative \((\mathrm{n}=1,2)\) of \(\mathrm{U}(x)=\frac{\mathrm{L} x+\mathrm{M}}{x^2-2 \mathrm{~B} x+\mathrm{C}}\) (L, M, B, C are constants), then $$\...

\(\text { If } y=\tan ^{-1}\left[\frac{\log _e\left(\frac{e}{x^2}\right)}{\log _e\left(e x^2\right)}\right]+\tan ^{-1}\left[\frac{3+2 \log _e x}{1-6 \cdot \log _e x}\right] \text {, then } \frac{d^2 y}{d x^2}=\)

The equation \(\mathrm{r} \cos \theta=2 \mathrm{a} \sin ^2 \theta\) represents the curve

A line of fixed length \(\mathrm{a}+\mathrm{b} . \mathrm{a} \neq \mathrm{b}\) moves so that its ends are always on two fixed perpendicular straight lines. The locus of a point which divides the line into two parts of length a and b is

With origin as a focus and \(x=4\) as corresponding directrix, a family of ellipse are drawn. Then the locus of an end of minor axis is

Let \(\mathrm{f}: \mathbb{R} \rightarrow \mathbb{R}\) be a function defined by \(\mathrm{f}(x)=\frac{\mathrm{e}^{|x|}-\mathrm{e}^{-x}}{\mathrm{e}^x+\mathrm{e}^{-x}}\), then

For every real number \(x \neq-1\), let \(\mathrm{f}(x)=\frac{x}{x+1}\).
Write \(\mathrm{f}_1(x)=\mathrm{f}(x)\) & for \(\mathrm{n} \geq 2, \mathrm{f}_{\mathrm{n}}(x)=\mathrm{f}\left(\mathrm{f}_{\mathrm{n}-1}(x)\right)\). Then $$\mathrm{f}_...

The equation \(2^x+5^x=3^x+4^x\) has

The function \(\mathrm{f}: \mathbb{R} \rightarrow \mathbb{R}\) defined by \(\mathrm{f}(x)=\mathrm{e}^x+\mathrm{e}^{-x}\) is :

Choose the correct statement :

In a plane \(\vec{a}\) and \(\vec{b}\) are the position vectors of two points A and B respectively. A point $P$ with position vector \(\overrightarrow{\mathrm{r}}\) moves on that plane in such a way that $$|\overrightarrow{\vec{r}}-\vec{a}|...

The locus of the midpoint of the system of parallel chords parallel to the line \(y=2 x\) to the hyperbola \(9 x^2-4 y^2=36\) is

\(\text { If } \int \frac{\log _e\left(x+\sqrt{1+x^2}\right)}{\sqrt{1+x^2}} \mathrm{~d} x=\mathrm{f}(\mathrm{g}(x))+\mathrm{c} \text { then }\)

$$ \text { Let } f(x)=\left|\begin{array}{ccc} \cos x & x & 1 \\ 2 \sin x & x^3 & 2 x \\ \tan x & x & 1 \end{array}\right| \text {, then } \lim _\limits{x \rightarrow 0} \frac{f(x)}{x^2}= $$

If \(\alpha, \beta\) are the roots of the equation \(a x^2+b x+c=0\) then \(\lim _\limits{x \rightarrow \beta} \frac{1-\cos \left(a x^2+b x+c\right)}{(x-\beta)^2}\) is

If \(\left(x^2 \log _x 27\right) \cdot \log _9 x=x+4\) then the value of \(x\) is

If $$A=\left(\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta\end{array}\right)$$ and \(\theta=\frac{2 \pi}{7}\), then \(A^{100}=A \times A \times \ldots .(100\) times) is equal to

$$ \text { If }\left|\begin{array}{lll} x^k & x^{k+2} & x^{k+3} \\ y^k & y^{k+2} & y^{k+3} \\ z^k & z^{k+2} & z^{k+3} \end{array}\right|=(x-y)(y-z)(z-x)\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right) \text {, then } $$

If $$\left[\begin{array}{ll}2 & 1 \\ 3 & 2\end{array}\right] \cdot A \cdot\left[\begin{array}{cc}-3 & 2 \\ 5 & -3\end{array}\right]=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]$$, then \(A=\)

Let $$A=\left(\begin{array}{ccc}1 & -1 & 0 \\ 0 & 1 & -1 \\ 1 & 1 & 1\end{array}\right), B=\left(\begin{array}{l}2 \\ 1 \\ 7\end{array}\right)$$
Then for the validity of the result \(\mathrm{AX}=\mathrm{B}, \mathrm{X}\) is

Let $$A=\left[\begin{array}{ccc}0 & 0 & -1 \\ 0 & -1 & 0 \\ -1 & 0 & 0\end{array}\right]$$, then

If \(\mathrm{a}_{\mathrm{i}}, \mathrm{b}_{\mathrm{i}}, \mathrm{c}_{\mathrm{i}} \in \mathbb{R}(\mathrm{i}=1,2,3)\) and \(x \in \mathbb{R}\) and $$\left|\begin{array}{lll}\mathrm{a}_1+b_1 x & a_1 x+b_1 & c_1 \\ \mathrm{a}_2+b_2 x & a_2 x+b_2 ...

\(\triangle \mathrm{OAB}\) is an equilateral triangle inscribed in the parabola \(\mathrm{y}^2=4 \mathrm{a} x, \mathrm{a}>0\) with O as the vertex, then the length of the side of \(\triangle \mathrm{O A B}\) is

The numbers \(1,2,3, \ldots \ldots, \mathrm{m}\) are arranged in random order. The number of ways this can be done, so that the numbers \(1,2, \ldots \ldots ., \mathrm{r}(\mathrm{r}<\mathrm{m})\) appears as neighbours is

Five balls of different colours are to be placed in three boxes of different sizes. The number of ways in which we can place the balls in the boxes so that no box remains empty is

\(\text { If } 1000!=3^n \times m \text { where } m \text { is an integer not divisible by } 3 \text {, then } n=\)

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-\...