JEE Main 2024 (Online) 31st January Morning Shift

Let \(\mathrm{Q}\) and \(\mathrm{R}\) be the feet of perpendiculars from the point \(\mathrm{P}(a, a, a)\) on the lines \(x=y, z=1\) and \(x=-y, z=-1\) respectively. If \(\angle \mathrm{QPR}\) is a right angle, then \(12 a^2\) is equal to _...

$$\text { If } f(x)=\left|\begin{array}{ccc} x^3 & 2 x^2+1 & 1+3 x \\ 3 x^2+2 & 2 x & x^3+6 \\ x^3-x & 4 & x^2-2 \end{array}\right| \text { for all } x \in \mathbb{R} \text {, then } 2 f(0)+f^{\prime}(0) \text { is equal to }$$

The area of the region \(\left\{(x, y): y^2 \leq 4 x, x<4, \frac{x y(x-1)(x-2)}{(x-3)(x-4)}>0, x \neq 3\right\}\) is

Let \(a\) be the sum of all coefficients in the expansion of \(\left(1-2 x+2 x^2\right)^{2023}\left(3-4 x^2+2 x^3\right)^{2024}\) and \(b=\lim _\limits{x \rightarrow 0}\left(\frac{\int_0^x \frac{\log (1+t)}{t^{2024}+1} d t}{x^2}\right)\). I...

In the expansion of \((1+x)\left(1-x^2\right)\left(1+\frac{3}{x}+\frac{3}{x^2}+\frac{1}{x^3}\right)^5, x \neq 0\), the sum of the coefficients of $x^3$ and \(x^{-13}\) is equal to __________.

If one of the diameters of the circle \(x^2+y^2-10 x+4 y+13=0\) is a chord of another circle \(\mathrm{C}\), whose center is the point of intersection of the lines \(2 x+3 y=12\) and \(3 x-2 y=5\), then the radius of the circle $$\mathrm{C}...

If \(\alpha\) denotes the number of solutions of \(|1-i|^x=2^x\) and \(\beta=\left(\frac{|z|}{\arg (z)}\right)\), where $$z=\frac{\pi}{4}(1+i)^4\left[\frac{1-\sqrt{\pi} i}{\sqrt{\pi}+i}+\frac{\sqrt{\pi}-i}{1+\sqrt{\pi} i}\right], i=\sqrt{-1...

If the integral \(525 \int_\limits0^{\frac{\pi}{2}} \sin 2 x \cos ^{\frac{11}{2}} x\left(1+\operatorname{Cos}^{\frac{5}{2}} x\right)^{\frac{1}{2}} d x\) is equal to \((n \sqrt{2}-64)\), then \(n\) is equal to _________.

Let \(S=(-1, \infty)\) and \(f: S \rightarrow \mathbb{R}\) be defined as
\(f(x)=\int_\limits{-1}^x\left(e^t-1\right)^{11}(2 t-1)^5(t-2)^7(t-3)^{12}(2 t-10)^{61} d t \text {, }\)
Let \(\mathrm{p}=\) Sum of squares of the values of \(x\), whe...

Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be a function defined by \(f(x)=\frac{4^x}{4^x+2}\) and \(M=\int_\limits{f(a)}^{f(1-a)} x \sin ^4(x(1-x)) d x, N=\int_\limits{f(a)}^{f(1-a)} \sin ^4(x(1-x)) d x ; a \neq \frac{1}{2}\). If $$\alph...

Let \(y=y(x)\) be the solution of the differential equation \(\frac{d y}{d x}=\frac{(\tan x)+y}{\sin x(\sec x-\sin x \tan x)}, x \in\left(0, \frac{\pi}{2}\right)\) satisfying the condition \(y\left(\frac{\pi}{4}\right)=2\). Then, $$y\left(\...

The solution curve of the differential equation
\(y \frac{d x}{d y}=x\left(\log _e x-\log _e y+1\right), x>0, y>0\) passing through the point \((e, 1)\) is

If \(f(x)=\frac{4 x+3}{6 x-4}, x \neq \frac{2}{3}\) and \((f \circ f)(x)=g(x)\), where \(g: \mathbb{R}-\left\{\frac{2}{3}\right\} \rightarrow \mathbb{R}-\left\{\frac{2}{3}\right\}\), then \((g ogog)(4)\) is equal to

If the foci of a hyperbola are same as that of the ellipse \(\frac{x^2}{9}+\frac{y^2}{25}=1\) and the eccentricity of the hyperbola is \(\frac{15}{8}\) times the eccentricity of the ellipse, then the smaller focal distance of the point $$\l...

Let the foci and length of the latus rectum of an ellipse \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1, a>b b e( \pm 5,0)\) and \(\sqrt{50}\), respectively. Then, the square of the eccentricity of the hyperbola $$\frac{x^2}{b^2}-\frac{y^2}{a^2 b^2}=...

For \(\alpha, \beta, \gamma \neq 0\), if \(\sin ^{-1} \alpha+\sin ^{-1} \beta+\sin ^{-1} \gamma=\pi\) and \((\alpha+\beta+\gamma)(\alpha-\gamma+\beta)=3 \alpha \beta\), then \(\gamma\) equals

\(\lim _\limits{x \rightarrow 0} \frac{e^{2|\sin x|}-2|\sin x|-1}{x^2}\)

Let \(g(x)\) be a linear function and $$f(x)=\left\{\begin{array}{cl}g(x) & , x \leq 0 \\ \left(\frac{1+x}{2+x}\right)^{\frac{1}{x}} & , x>0\end{array}\right.$$, is continuous at \(x=0\). If \(f^{\prime}(1)=f(-1)\), then the value \(g(3)\) ...

If the system of linear equations
$$\begin{aligned} & x-2 y+z=-4 \\ & 2 x+\alpha y+3 z=5 \\ & 3 x-y+\beta z=3 \end{aligned}$$
has infinitely many solutions, then \(12 \alpha+13 \beta\) is equal to

The total number of words (with or without meaning) that can be formed out of the letters of the word 'DISTRIBUTION' taken four at a time, is equal to __________.

Three rotten apples are accidently mixed with fifteen good apples. Assuming the random variable \(x\) to be the number of rotten apples in a draw of two apples, the variance of \(x\) is

Two marbles are drawn in succession from a box containing 10 red, 30 white, 20 blue and 15 orange marbles, with replacement being made after each drawing. Then the probability, that first drawn marble is red and second drawn marble is white...

Let \(\mathrm{S}\) be the set of positive integral values of \(a\) for which \(\frac{a x^2+2(a+1) x+9 a+4}{x^2-8 x+32} < 0, \forall x \in \mathbb{R}\). Then, the number of elements in \(\mathrm{S}\) is :

For \(0 < c < b < a\), let \((a+b-2 c) x^2+(b+c-2 a) x+(c+a-2 b)=0\) and \(\alpha \neq 1\) be one of its root. Then, among the two statements
(I) If \(\alpha \in(-1,0)\), then \(b\) cannot be the geometric mean of $a$ and \(c\)
(II) If $$\a...

The sum of the series \(\frac{1}{1-3 \cdot 1^2+1^4}+\frac{2}{1-3 \cdot 2^2+2^4}+\frac{3}{1-3 \cdot 3^2+3^4}+\ldots\) up to 10 -terms is

Let \(A=\{1,2,3,4\}\) and \(R=\{(1,2),(2,3),(1,4)\}\) be a relation on \(\mathrm{A}\). Let \(\mathrm{S}\) be the equivalence relation on \(\mathrm{A}\) such that \(R \subset S\) and the number of elements in \(\mathrm{S}\) is \(\mathrm{n}\)...

Let \(\alpha, \beta, \gamma, \delta \in \mathbb{Z}\) and let \(A(\alpha, \beta), B(1,0), C(\gamma, \delta)\) and \(D(1,2)\) be the vertices of a parallelogram \(\mathrm{ABCD}\). If \(A B=\sqrt{10}\) and the points \(\mathrm{A}\) and $$\math...

Let \(\vec{a}=3 \hat{i}+\hat{j}-2 \hat{k}, \vec{b}=4 \hat{i}+\hat{j}+7 \hat{k}\) and \(\vec{c}=\hat{i}-3 \hat{j}+4 \hat{k}\) be three vectors. If a vectors \(\vec{p}\) satisfies \(\vec{p} \times \vec{b}=\vec{c} \times \vec{b}\) and $$\vec{p...

The distance of the point \(Q(0,2,-2)\) form the line passing through the point \(P(5,-4, 3)\) and perpendicular to the lines \(\vec{r}=(-3 \hat{i}+2 \hat{k})+\lambda(2 \hat{i}+3 \hat{j}+5 \hat{k}), \lambda \in \mathbb{R}\) and $$\vec{r}=(\...

Let \(\vec{a}\) and \(\vec{b}\) be two vectors such that \(|\vec{a}|=1,|\vec{b}|=4\), and \(\vec{a} \cdot \vec{b}=2\). If \(\vec{c}=(2 \vec{a} \times \vec{b})-3 \vec{b}\) and the angle between \(\vec{b}\) and \(\vec{c}\) is \(\alpha\), then...