JEE Main 2024 (Online) 5th April Evening Shift

Let \((\alpha, \beta, \gamma)\) be the image of the point \((8,5,7)\) in the line \(\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-2}{5}\). Then \(\alpha+\beta+\gamma\) is equal to :

Let the point \((-1, \alpha, \beta)\) lie on the line of the shortest distance between the lines \(\frac{x+2}{-3}=\frac{y-2}{4}=\frac{z-5}{2}\) and \(\frac{x+2}{-1}=\frac{y+6}{2}=\frac{z-1}{0}\). Then \((\alpha-\beta)^2\) is equal to ______...

Let the maximum and minimum values of \(\left(\sqrt{8 x-x^2-12}-4\right)^2+(x-7)^2, x \in \mathbf{R}\) be \(\mathrm{M}\) and \(\mathrm{m}\), respectively. Then \(\mathrm{M}^2-\mathrm{m}^2\) is equal to _________.

The area enclosed between the curves \(y=x|x|\) and \(y=x-|x|\) is :

If the constant term in the expansion of \(\left(\frac{\sqrt[5]{3}}{x}+\frac{2 x}{\sqrt[3]{5}}\right)^{12}, x \neq 0\), is \(\alpha \times 2^8 \times \sqrt[5]{3}\), then \(25 \alpha\) is equal to :

Let the circle \(C_1: x^2+y^2-2(x+y)+1=0\) and \(\mathrm{C_2}\) be a circle having centre at \((-1,0)\) and radius 2 . If the line of the common chord of \(\mathrm{C}_1\) and \(\mathrm{C}_2\) intersects the \(\mathrm{y}\)-axis at the point ...

Let ABCD and AEFG be squares of side 4 and 2 units, respectively. The point E is on the line segment AB and the point F is on the diagonal AC. Then the radius r of the circle passing through the point F and touching the line segments BC and...

Let \(S_1=\{z \in \mathbf{C}:|z| \leq 5\}, S_2=\left\{z \in \mathbf{C}: \operatorname{Im}\left(\frac{z+1-\sqrt{3} i}{1-\sqrt{3} i}\right) \geq 0\right\}\) and \(S_3=\{z \in \mathbf{C}: \operatorname{Re}(z) \geq 0\}\). Then the area of the r...

Let \(\beta(\mathrm{m}, \mathrm{n})=\int_\limits0^1 x^{\mathrm{m}-1}(1-x)^{\mathrm{n}-1} \mathrm{~d} x, \mathrm{~m}, \mathrm{n}>0\). If $$\int_\limits0^1\left(1-x^{10}\right)^{20} \mathrm{~d} x=\mathrm{a} \times \beta(\mathrm{b}, \mathrm{c}...

If \(f(t)=\int_\limits0^\pi \frac{2 x \mathrm{~d} x}{1-\cos ^2 \mathrm{t} \sin ^2 x}, 0<\mathrm{t}<\pi\), then the value of \(\int_\limits0^{\frac{\pi}{2}} \frac{\pi^2 \mathrm{dt}}{f(\mathrm{t})}\) equals __________.

The differential equation of the family of circles passing through the origin and having centre at the line \(y=x\) is :

Let \(y=y(x)\) be the solution of the differential equation
\(\frac{\mathrm{d} y}{\mathrm{~d} x}+\frac{2 x}{\left(1+x^2\right)^2} y=x \mathrm{e}^{\frac{1}{\left(1+x^2\right)}} ; y(0)=0.\)
Then the area enclosed by the curve $$f(x)=y(x) \mat...

If \(y(\theta)=\frac{2 \cos \theta+\cos 2 \theta}{\cos 3 \theta+4 \cos 2 \theta+5 \cos \theta+2}\), then at \(\theta=\frac{\pi}{2}, y^{\prime \prime}+y^{\prime}+y\) is equal to :

Let \(f, g: \mathbf{R} \rightarrow \mathbf{R}\) be defined as :
$$f(x)=|x-1| \text { and } g(x)= \begin{cases}\mathrm{e}^x, & x \geq 0 \\ x+1, & x \leq 0 .\end{cases}$$
Then the function \(f(g(x))\) is

Let ,\(f:[-1,2] \rightarrow \mathbf{R}\) be given by \(f(x)=2 x^2+x+\left[x^2\right]-[x]\), where \([t]\) denotes the greatest integer less than or equal to \(t\). The number of points, where \(f\) is not continuous, is :

Let \(\mathrm{a}>0\) be a root of the equation \(2 x^2+x-2=0\). If \(\lim _\limits{x \rightarrow \frac{1}{a}} \frac{16\left(1-\cos \left(2+x-2 x^2\right)\right)}{(1-a x)^2}=\alpha+\beta \sqrt{17}\), where \(\alpha, \beta \in Z\), then $$\al...

The values of \(m, n\), for which the system of equations
$$\begin{aligned} & x+y+z=4, \\ & 2 x+5 y+5 z=17, \\ & x+2 y+\mathrm{m} z=\mathrm{n} \end{aligned}$$
has infinitely many solutions, satisfy the equation :

Let \(\alpha \beta \neq 0\) and $$A=\left[\begin{array}{rrr}\beta & \alpha & 3 \\ \alpha & \alpha & \beta \\ -\beta & \alpha & 2 \alpha\end{array}\right]$$. If $$B=\left[\begin{array}{rrr}3 \alpha & -9 & 3 \alpha \\ -\alpha & 7 & -2 \alpha ...

Let a line perpendicular to the line \(2 x-y=10\) touch the parabola \(y^2=4(x-9)\) at the point P. The distance of the point P from the centre of the circle \(x^2+y^2-14 x-8 y+56=0\) is __________.

Let the set \(S=\{2,4,8,16, \ldots, 512\}\) be partitioned into 3 sets \(A, B, C\) with equal number of elements such that \(\mathrm{A} \cup \mathrm{B} \cup \mathrm{C}=\mathrm{S}\) and $$\mathrm{A} \cap \mathrm{B}=\mathrm{B} \cap \mathrm{C}...

60 words can be made using all the letters of the word \(\mathrm{BHBJO}\), with or without meaning. If these words are written as in a dictionary, then the \(50^{\text {th }}\) word is:

The coefficients \(\mathrm{a}, \mathrm{b}, \mathrm{c}\) in the quadratic equation \(\mathrm{a} x^2+\mathrm{bx}+\mathrm{c}=0\) are from the set \(\{1,2,3,4,5,6\}\). If the probability of this equation having one real root bigger than the oth...

The number of real solutions of the equation \(x|x+5|+2|x+7|-2=0\) is __________.

For \(x \geqslant 0\), the least value of \(\mathrm{K}\), for which \(4^{1+x}+4^{1-x}, \frac{\mathrm{K}}{2}, 16^x+16^{-x}\) are three consecutive terms of an A.P., is equal to :

If \(1+\frac{\sqrt{3}-\sqrt{2}}{2 \sqrt{3}}+\frac{5-2 \sqrt{6}}{18}+\frac{9 \sqrt{3}-11 \sqrt{2}}{36 \sqrt{3}}+\frac{49-20 \sqrt{6}}{180}+\ldots\) upto \(\infty=2+\left(\sqrt{\frac{b}{a}}+1\right) \log _e\left(\frac{a}{b}\right)\), where a ...

Let the mean and the standard deviation of the probability distribution

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Let \(\mathrm{A}(-1,1)\) and \(\mathrm{B}(2,3)\) be two points and \(\mathrm{P}\) be a variable point above the line \(\mathrm{AB}\) such that the area of \(\triangle \mathrm{PAB}\) is 10. If the locus of \(\mathrm{P}\) is $$\mathrm{a} x+\m...

The number of solutions of \(\sin ^2 x+\left(2+2 x-x^2\right) \sin x-3(x-1)^2=0\), where \(-\pi \leq x \leq \pi\), is ________.

Let \(\vec{a}=2 \hat{i}+5 \hat{j}-\hat{k}, \vec{b}=2 \hat{i}-2 \hat{j}+2 \hat{k}\) and \(\vec{c}\) be three vectors such that \((\vec{c}+\hat{i}) \times(\vec{a}+\vec{b}+\hat{i})=\vec{a} \times(\vec{c}+\hat{i})\). If $$\vec{a} \cdot \vec{c}=...

Consider three vectors \(\vec{a}, \vec{b}, \vec{c}\). Let \(|\vec{a}|=2,|\vec{b}|=3\) and \(\vec{a}=\vec{b} \times \vec{c}\). If \(\alpha \in\left[0, \frac{\pi}{3}\right]\) is the angle between the vectors \(\vec{b}\) and \(\vec{c}\), then ...