JEE Main 2024 (Online) 31st January Evening Shift

Let \((\alpha, \beta, \gamma)\) be the mirror image of the point \((2,3,5)\) in the line \(\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}\). Then, \(2 \alpha+3 \beta+4 \gamma\) is equal to

The shortest distance, between lines \(L_1\) and \(L_2\), where \(L_1: \frac{x-1}{2}=\frac{y+1}{-3}=\frac{z+4}{2}\) and \(L_2\) is the line, passing through the points \(\mathrm{A}(-4,4,3), \mathrm{B}(-1,6,3)\) and perpendicular to the line...

A line passes through \(A(4,-6,-2)\) and \(B(16,-2,4)\). The point \(P(a, b, c)\), where \(a, b, c\) are non-negative integers, on the line \(A B\) lies at a distance of 21 units, from the point \(A\). The distance between the points $$P(a,...

Let \(f: \rightarrow \mathbb{R} \rightarrow(0, \infty)\) be strictly increasing function such that \(\lim _\limits{x \rightarrow \infty} \frac{f(7 x)}{f(x)}=1\). Then, the value of $$\lim _\limits{x \rightarrow \infty}\left[\frac{f(5 x)}{f(...

If the function \(f:(-\infty,-1] \rightarrow(a, b]\) defined by \(f(x)=e^{x^3-3 x+1}\) is one - one and onto, then the distance of the point \(P(2 b+4, a+2)\) from the line \(x+e^{-3} y=4\) is :

The area of the region enclosed by the parabolas \(y=4 x-x^2\) and \(3 y=(x-4)^2\) is equal to :

Let the coefficient of \(x^r\) in the expansion of \((x+3)^{n-1}+(x+3)^{n-2}(x+2)+(x+3)^{n-3}(x+2)^2+\ldots \ldots \ldots .+(x+2)^{n-1}\) be \(\alpha_r\). If \(\sum_\limits{r=0}^n \alpha_r=\beta^n-\gamma^n, \beta, \gamma \in \mathbb{N}\), t...

Let a variable line passing through the centre of the circle \(x^2+y^2-16 x-4 y=0\), meet the positive co-ordinate axes at the points \(A\) and \(B\). Then the minimum value of \(O A+O B\), where \(O\) is the origin, is equal to

Let \(z_1\) and \(z_2\) be two complex numbers such that \(z_1+z_2=5\) and \(z_1^3+z_2^3=20+15 i\) Then, \(\left|z_1^4+z_2^4\right|\) equals -

Let \(f, g:(0, \infty) \rightarrow \mathbb{R}\) be two functions defined by \(f(x)=\int\limits_{-x}^x\left(|t|-t^2\right) e^{-t^2} d t\) and \(g(x)=\int\limits_0^{x^2} t^{1 / 2} e^{-t} d t\). Then, the value of $$9\left(f\left(\sqrt{\log _e...

\(\left|\frac{120}{\pi^3} \int_\limits0^\pi \frac{x^2 \sin x \cos x}{\sin ^4 x+\cos ^4 x} d x\right| \text { is equal to }\) ________.

The temperature \(T(t)\) of a body at time \(t=0\) is \(160^{\circ} \mathrm{F}\) and it decreases continuously as per the differential equation \(\frac{d T}{d t}=-K(T-80)\), where \(K\) is a positive constant. If $$T(15)=120^{\circ} \mathrm...

Let \(y=y(x)\) be the solution of the differential equation
\(\sec ^2 x d x+\left(e^{2 y} \tan ^2 x+\tan x\right) d y=0,0< x<\frac{\pi}{2}, y(\pi / 4)=0\).
If \(y(\pi / 6)=\alpha\), then \(e^{8 \alpha}\) is equal to ____________.

Let \(P\) be a parabola with vertex \((2,3)\) and directrix \(2 x+y=6\). Let an ellipse \(E: \frac{x^2}{a^2}+\frac{y^2}{b^2}=1, a>b\), of eccentricity \(\frac{1}{\sqrt{2}}\) pass through the focus of the parabola \(P\). Then, the square of ...

If \(a=\sin ^{-1}(\sin (5))\) and \(b=\cos ^{-1}(\cos (5))\), then \(a^2+b^2\) is equal to

Consider the function \(f:(0, \infty) \rightarrow \mathbb{R}\) defined by \(f(x)=e^{-\left|\log _e x\right|}\). If \(m\) and \(n\) be respectively the number of points at which \(f\) is not continuous and \(f\) is not differentiable, then $...

If \(\lim _\limits{x \rightarrow 0} \frac{a x^2 e^x-b \log _e(1+x)+c x e^{-x}}{x^2 \sin x}=1\), then \(16\left(a^2+b^2+c^2\right)\) is equal to ________.

Let \(A\) be a \(3 \times 3\) real matrix such that
$$A\left(\begin{array}{l}
1 \\
0 \\
1
\end{array}\right)=2\left(\begin{array}{l}
1 \\
0 \\
1
\end{array}\right), A\left(\begin{array}{l}
-1 \\
0 \\
1
\end{array}\right)=4\left(\begin{array...

Let A be a \(3 \times 3\) matrix and \(\operatorname{det}(A)=2\). If \(n=\operatorname{det}(\underbrace{\operatorname{adj}(\operatorname{adj}(\ldots . .(\operatorname{adj} A))}_{2024-\text { times }}))\), then the remainder when \(n\) is di...

The number of ways in which 21 identical apples can be distributed among three children such that each child gets at least 2 apples, is

If for some \(m, n ;{ }^6 C_m+2\left({ }^6 C_{m+1}\right)+{ }^6 C_{m+2}>{ }^8 C_3\) and \({ }^{n-1} P_3:{ }^n P_4=1: 8\), then \({ }^n P_{m+1}+{ }^{\mathrm{n}+1} C_m\) is equal to

A coin is biased so that a head is twice as likely to occur as a tail. If the coin is tossed 3 times, then the probability of getting two tails and one head is

Let \(a, b, c\) be the lengths of three sides of a triangle satistying the condition \(\left(a^2+b^2\right) x^2-2 b(a+c) x+\left(b^2+c^2\right)=0\). If the set of all possible values of \(x\) is the interval \((\alpha, \beta)\), then $$12\l...

Let \(2^{\text {nd }}, 8^{\text {th }}\) and \(44^{\text {th }}\) terms of a non-constant A. P. be respectively the \(1^{\text {st }}, 2^{\text {nd }}\) and \(3^{\text {rd }}\) terms of a G. P. If the first term of the A. P. is 1, then the ...

Let \(A=\{1,2,3, \ldots \ldots \ldots \ldots, 100\}\). Let \(R\) be a relation on \(\mathrm{A}\) defined by \((x, y) \in R\) if and only if \(2 x=3 y\). Let \(R_1\) be a symmetric relation on \(A\) such that \(R \subset R_1\) and the number...

Let the mean and the variance of 6 observations \(a, b, 68,44,48,60\) be \(55\) and \(194\), respectively. If \(a>b\), then \(a+3 b\) is

Let \(A(a, b), B(3,4)\) and \(C(-6,-8)\) respectively denote the centroid, circumcentre and orthocentre of a triangle. Then, the distance of the point \(P(2 a+3,7 b+5)\) from the line \(2 x+3 y-4=0\) measured parallel to the line $$x-2 y-1=...

Let \(A(-2,-1), B(1,0), C(\alpha, \beta)\) and \(D(\gamma, \delta)\) be the vertices of a parallelogram \(A B C D\). If the point \(C\) lies on \(2 x-y=5\) and the point \(D\) lies on \(3 x-2 y=6\), then the value of $$|\alpha+\beta+\gamma+...

The number of solutions, of the equation \(e^{\sin x}-2 e^{-\sin x}=2\), is :

Let \(\vec{a}=3 \hat{i}+2 \hat{j}+\hat{k}, \vec{b}=2 \hat{i}-\hat{j}+3 \hat{k}\) and \(\vec{c}\) be a vector such that \((\vec{a}+\vec{b}) \times \vec{c}=2(\vec{a} \times \vec{b})+24 \hat{j}-6 \hat{k}\) and $$(\vec{a}-\vec{b}+\hat{i}) \cdot...