JEE Main 2024 (Online) 8th April Evening Shift

If the shortest distance between the lines \(\frac{x-\lambda}{2}=\frac{y-4}{3}=\frac{z-3}{4}\) and \(\frac{x-2}{4}=\frac{y-4}{6}=\frac{z-7}{8}\) is \(\frac{13}{\sqrt{29}}\), then a value of \(\lambda\) is :

Let \(\mathrm{P}(\alpha, \beta, \gamma)\) be the image of the point \(\mathrm{Q}(1,6,4)\) in the line \(\frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3}\). Then \(2 \alpha+\beta+\gamma\) is equal to ________

If the function \(f(x)=2 x^3-9 \mathrm{ax}^2+12 \mathrm{a}^2 x+1, \mathrm{a}> 0\) has a local maximum at \(x=\alpha\) and a local minimum at \(x=\alpha^2\), then \(\alpha\) and \(\alpha^2\) are the roots of the equation :

Let \(\mathrm{A}\) be the region enclosed by the parabola \(y^2=2 x\) and the line \(x=24\). Then the maximum area of the rectangle inscribed in the region \(\mathrm{A}\) is ________.

The area of the region in the first quadrant inside the circle \(x^2+y^2=8\) and outside the parabola \(y^2=2 x\) is equal to :

If the term independent of \(x\) in the expansion of \(\left(\sqrt{\mathrm{a}} x^2+\frac{1}{2 x^3}\right)^{10}\) is 105 , then \(\mathrm{a}^2\) is equal to :

If the image of the point \((-4,5)\) in the line \(x+2 y=2\) lies on the circle \((x+4)^2+(y-3)^2=r^2\), then \(r\) is equal to:

The sum of all possible values of \(\theta \in[-\pi, 2 \pi]\), for which \(\frac{1+i \cos \theta}{1-2 i \cos \theta}\) is purely imaginary, is equal to :

Let \(\int_\limits\alpha^{\log _e 4} \frac{\mathrm{d} x}{\sqrt{\mathrm{e}^x-1}}=\frac{\pi}{6}\). Then \(\mathrm{e}^\alpha\) and \(\mathrm{e}^{-\alpha}\) are the roots of the equation :

Let \(y=y(x)\) be the solution curve of the differential equation \(\sec y \frac{\mathrm{d} y}{\mathrm{~d} x}+2 x \sin y=x^3 \cos y, y(1)=0\). Then \(y(\sqrt{3})\) is equal to:

Let \(\alpha|x|=|y| \mathrm{e}^{x y-\beta}, \alpha, \beta \in \mathbf{N}\) be the solution of the differential equation \(x \mathrm{~d} y-y \mathrm{~d} x+x y(x \mathrm{~d} y+y \mathrm{~d} x)=0,y(1)=2\). Then \(\alpha+\beta\) is equal to ___...

Let $$f(x)=\left\{\begin{array}{ccc}-\mathrm{a} & \text { if } & -\mathrm{a} \leq x \leq 0 \\ x+\mathrm{a} & \text { if } & 0< x \leq \mathrm{a}\end{array}\right.$$ where \(\mathrm{a}> 0\) and \(\mathrm{g}(x)=(f(|x|)-|f(x)|) / 2\). Then the...

Let \(\mathrm{S}\) be the focus of the hyperbola \(\frac{x^2}{3}-\frac{y^2}{5}=1\), on the positive \(x\)-axis. Let \(\mathrm{C}\) be the circle with its centre at \(\mathrm{A}(\sqrt{6}, \sqrt{5})\) and passing through the point $$\mathrm{S...

If \(\int \frac{1}{\sqrt[5]{(x-1)^4(x+3)^6}} \mathrm{~d} x=\mathrm{A}\left(\frac{\alpha x-1}{\beta x+3}\right)^B+\mathrm{C}\), where \(\mathrm{C}\) is the constant of integration, then the value of \(\alpha+\beta+20 \mathrm{AB}\) is _______...

For \(\mathrm{a}, \mathrm{b}>0\), let $$f(x)= \begin{cases}\frac{\tan ((\mathrm{a}+1) x)+\mathrm{b} \tan x}{x}, & x< 0 \\ 3, & x=0 \\ \frac{\sqrt{\mathrm{a} x+\mathrm{b}^2 x^2}-\sqrt{\mathrm{a} x}}{\mathrm{~b} \sqrt{\mathrm{a}} x \sqrt{x}},...

If \(\alpha=\lim _\limits{x \rightarrow 0^{+}}\left(\frac{\mathrm{e}^{\sqrt{\tan x}}-\mathrm{e}^{\sqrt{x}}}{\sqrt{\tan x}-\sqrt{x}}\right)\) and \(\beta=\lim _\limits{x \rightarrow 0}(1+\sin x)^{\frac{1}{2} \cot x}\) are the roots of the qu...

If the system of equations \(x+4 y-z=\lambda, 7 x+9 y+\mu z=-3,5 x+y+2 z=-1\) has infinitely many solutions, then \((2 \mu+3 \lambda)\) is equal to :

If \(\alpha \neq \mathrm{a}, \beta \neq \mathrm{b}, \gamma \neq \mathrm{c}\) and $$\left|\begin{array}{lll}\alpha & \mathrm{b} & \mathrm{c} \\ \mathrm{a} & \beta & \mathrm{c} \\ \mathrm{a} & \mathrm{b} & \gamma\end{array}\right|=0$$, then $...

The number of ways five alphabets can be chosen from the alphabets of the word MATHEMATICS, where the chosen alphabets are not necessarily distinct, is equal to:

There are three bags \(X, Y\) and \(Z\). Bag \(X\) contains 5 one-rupee coins and 4 five-rupee coins; Bag \(Y\) contains 4 one-rupee coins and 5 five-rupee coins and Bag \(Z\) contains 3 one-rupee coins and 6 five-rupee coins. A bag is sele...

The number of distinct real roots of the equation \(|x+1||x+3|-4|x+2|+5=0\), is _______

In an increasing geometric progression of positive terms, the sum of the second and sixth terms is \(\frac{70}{3}\) and the product of the third and fifth terms is 49. Then the sum of the \(4^{\text {th }}, 6^{\text {th }}\) and $$8^{\text ...

An arithmetic progression is written in the following way

The sum of all the terms of the 10th row is _________.

Let \(A=\{2,3,6,8,9,11\}\) and \(B=\{1,4,5,10,15\}\). Let \(R\) be a relation on \(A \times B\) defined by
\((a, b) R(c, d)\) if and only if \(3 a d-7 b c\) is an even integer. Then the relation \(R\) is

Let \(\mathrm{a}, \mathrm{b}, \mathrm{c} \in \mathbf{N}\) and \(\mathrm{a}< \mathrm{b}< \mathrm{c}\). Let the mean, the mean deviation about the mean and the variance of the 5 observations \(9,25, a, b, c\) be 18, 4 and \(\frac{136}{5}\), r...

If the line segment joining the points \((5,2)\) and \((2, a)\) subtends an angle \(\frac{\pi}{4}\) at the origin, then the absolute value of the product of all possible values of $a$ is :

Let a ray of light passing through the point \((3,10)\) reflects on the line \(2 x+y=6\) and the reflected ray passes through the point \((7,2)\). If the equation of the incident ray is \(a x+b y+1=0\), then \(a^2+b^2+3 a b\) is equal to __...

If the value of \(\frac{3 \cos 36^{\circ}+5 \sin 18^{\circ}}{5 \cos 36^{\circ}-3 \sin 18^{\circ}}\) is \(\frac{a \sqrt{5}-b}{c}\), where \(a, b, c\) are natural numbers and \(\operatorname{gcd}(a, c)=1\), then \(a+b+c\) is equal to :

Let \(\overrightarrow{\mathrm{a}}=\hat{i}+2 \hat{j}+3 \hat{k}, \overrightarrow{\mathrm{b}}=2 \hat{i}+3 \hat{j}-5 \hat{k}\) and \(\overrightarrow{\mathrm{c}}=3 \hat{i}-\hat{j}+\lambda \hat{k}\) be three vectors. Let $$\overrightarrow{\mathrm...

Let \(\overrightarrow{\mathrm{a}}=4 \hat{i}-\hat{j}+\hat{k}, \overrightarrow{\mathrm{b}}=11 \hat{i}-\hat{j}+\hat{k}\) and \(\overrightarrow{\mathrm{c}}\) be a vector such that $$(\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}) \tim...