JEE Main 2024 (Online) 4th April Evening Shift

Let \(\mathrm{P}\) be the point of intersection of the lines \(\frac{x-2}{1}=\frac{y-4}{5}=\frac{z-2}{1}\) and \(\frac{x-3}{2}=\frac{y-2}{3}=\frac{z-3}{2}\). Then, the shortest distance of \(\mathrm{P}\) from the line \(4 x=2 y=z\) is

Consider a line \(\mathrm{L}\) passing through the points \(\mathrm{P}(1,2,1)\) and \(\mathrm{Q}(2,1,-1)\). If the mirror image of the point \(\mathrm{A}(2,2,2)\) in the line \(\mathrm{L}\) is \((\alpha, \beta, \gamma)\), then $$\alpha+\bet...

Let \(f(x)=3 \sqrt{x-2}+\sqrt{4-x}\) be a real valued function. If \(\alpha\) and \(\beta\) are respectively the minimum and the maximum values of \(f\), then \(\alpha^2+2 \beta^2\) is equal to

The area (in sq. units) of the region described by \(\left\{(x, y): y^2 \leq 2 x \text {, and } y \geq 4 x-1\right\}\) is

If the coefficients of \(x^4, x^5\) and \(x^6\) in the expansion of \((1+x)^n\) are in the arithmetic progression, then the maximum value of \(n\) is:

Let \(\mathrm{C}\) be a circle with radius \(\sqrt{10}\) units and centre at the origin. Let the line \(x+y=2\) intersects the circle \(\mathrm{C}\) at the points \(\mathrm{P}\) and \(\mathrm{Q}\). Let \(\mathrm{MN}\) be a chord of $$\mathr...

The area (in sq. units) of the region \(S=\{z \in \mathbb{C}:|z-1| \leq 2 ;(z+\bar{z})+i(z-\bar{z}) \leq 2, \operatorname{lm}(z) \geq 0\}\) is

Let \(f(x)=\int_0^x\left(t+\sin \left(1-e^t\right)\right) d t, x \in \mathbb{R}\). Then, \(\lim _\limits{x \rightarrow 0} \frac{f(x)}{x^3}\) is equal to

If the value of the integral \(\int\limits_{-1}^1 \frac{\cos \alpha x}{1+3^x} d x\) is \(\frac{2}{\pi}\).Then, a value of \(\alpha\) is

Let \(y=y(x)\) be the solution of the differential equation \((x+y+2)^2 d x=d y, y(0)=-2\). Let the maximum and minimum values of the function \(y=y(x)\) in \(\left[0, \frac{\pi}{3}\right]\) be \(\alpha\) and \(\beta\), respectively. If $$(...

Let \(y=y(x)\) be the solution of the differential equation \((x^2+4)^2 d y+(2 x^3 y+8 x y-2) d x=0\). If \(y(0)=0\), then \(y(2)\) is equal to

Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be a thrice differentiable function such that \(f(0)=0, f(1)=1, f(2)=-1, f(3)=2\) and \(f(4)=-2\). Then, the minimum number of zeros of $$\left(3 f^{\prime} f^{\prime \prime}+f f^{\prime \prime \...

Consider the function \(f: \mathbb{R} \rightarrow \mathbb{R}\) defined by \(f(x)=\frac{2 x}{\sqrt{1+9 x^2}}\). If the composition of $$f, \underbrace{(f \circ f \circ f \circ \cdots \circ f)}_{10 \text { times }}(x)=\frac{2^{10} x}{\sqrt{1+...

Consider a hyperbola \(\mathrm{H}\) having centre at the origin and foci on the \(\mathrm{x}\)-axis. Let \(\mathrm{C}_1\) be the circle touching the hyperbola \(\mathrm{H}\) and having the centre at the origin. Let \(\mathrm{C}_2\) be the c...

If \(\int \operatorname{cosec}^5 x d x=\alpha \cot x \operatorname{cosec} x\left(\operatorname{cosec}^2 x+\frac{3}{2}\right)+\beta \log _x\left|\tan \frac{x}{2}\right|+\mathrm{C}\)
where \(\alpha, \beta \in \mathbb{R}\) and \(\mathrm{C}\) i...

Given that the inverse trigonometric function assumes principal values only. Let \(x, y\) be any two real numbers in \([-1,1]\) such that \(\cos ^{-1} x-\sin ^{-1} y=\alpha, \frac{-\pi}{2} \leq \alpha \leq \pi\).
Then, the minimum value of ...

If the function
$$f(x)= \begin{cases}\frac{72^x-9^x-8^x+1}{\sqrt{2}-\sqrt{1+\cos x}}, & x \neq 0 \\ a \log _e 2 \log _e 3 & , x=0\end{cases}$$
is continuous at \(x=0\), then the value of \(a^2\) is equal to

Let \(A\) be a \(2 \times 2\) symmetric matrix such that $$A\left[\begin{array}{l}1 \\ 1\end{array}\right]=\left[\begin{array}{l}3 \\ 7\end{array}\right]$$ and the determinant of \(A\) be 1 . If \(A^{-1}=\alpha A+\beta I\), where \(I\) is a...

Let $$A=\left[\begin{array}{ll}1 & 2 \\ 0 & 1\end{array}\right]$$ and \(B=I+\operatorname{adj}(A)+(\operatorname{adj} A)^2+\ldots+(\operatorname{adj} A)^{10}\).
Then, the sum of all the elements of the matrix \(B\) is:

Let \(P Q\) be a chord of the parabola \(y^2=12 x\) and the midpoint of \(P Q\) be at \((4,1)\). Then, which of the following point lies on the line passing through the points \(\mathrm{P}\) and \(\mathrm{Q}\) ?

There are 4 men and 5 women in Group A, and 5 men and 4 women in Group B. If 4 persons are selected from each group, then the number of ways of selecting 4 men and 4 women is ________.

In a tournament, a team plays 10 matches with probabilities of winning and losing each match as \(\frac{1}{3}\) and \(\frac{2}{3}\) respectively. Let \(x\) be the number of matches that the team wins, and $y$ be the number of matches that t...

If the mean of the following probability distribution of a radam variable \(\mathrm{X}\) :

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Consider a triangle \(\mathrm{ABC}\) having the vertices \(\mathrm{A}(1,2), \mathrm{B}(\alpha, \beta)\) and \(\mathrm{C}(\gamma, \delta)\) and angles \(\angle A B C=\frac{\pi}{6}\) and \(\angle B A C=\frac{2 \pi}{3}\). If the points $$\math...

The value of \(\frac{1 \times 2^2+2 \times 3^2+\ldots+100 \times(101)^2}{1^2 \times 2+2^2 \times 3+\ldots .+100^2 \times 101}\) is

Let three real numbers \(a, b, c\) be in arithmetic progression and \(a+1, b, c+3\) be in geometric progression. If \(a>10\) and the arithmetic mean of \(a, b\) and \(c\) is 8, then the cube of the geometric mean of \(a, b\) and \(c\) is

Let a relation \(\mathrm{R}\) on \(\mathrm{N} \times \mathbb{N}\) be defined as: \(\left(x_1, y_1\right) \mathrm{R}\left(x_2, y_2\right)\) if and only if \(x_1 \leq x_2\) or \(y_1 \leq y_2\).
Consider the two statements:
(I) \(\mathrm{R}\) ...

Let \(S=\left\{\sin ^2 2 \theta:\left(\sin ^4 \theta+\cos ^4 \theta\right) x^2+(\sin 2 \theta) x+\left(\sin ^6 \theta+\cos ^6 \theta\right)=0\right.\) has real roots \(\}\). If \(\alpha\) and \(\beta\) be the smallest and largest elements o...

Let \(\vec{a}=\hat{i}+\hat{j}+\hat{k}, \vec{b}=2 \hat{i}+4 \hat{j}-5 \hat{k}\) and \(\vec{c}=x \hat{i}+2 \hat{j}+3 \hat{k}, x \in \mathbb{R}\).
If \(\vec{d}\) is the unit vector in the direction of \(\vec{b}+\vec{c}\) such that $$\vec{a} \c...

For \(\lambda>0\), let \(\theta\) be the angle between the vectors \(\vec{a}=\hat{i}+\lambda \hat{j}-3 \hat{k}\) and \(\vec{b}=3 \hat{i}-\hat{j}+2 \hat{k}\). If the vectors \(\vec{a}+\vec{b}\) and \(\vec{a}-\vec{b}\) are mutually perpendicu...