JEE Main 2024 (Online) 8th April Morning Shift

Let \(P(x, y, z)\) be a point in the first octant, whose projection in the \(x y\)-plane is the point \(Q\). Let \(O P=\gamma\); the angle between \(O Q\) and the positive \(x\)-axis be \(\theta\); and the angle between \(O P\) and the posi...

If the shortest distance between the lines
$$\begin{array}{ll}
L_1: \vec{r}=(2+\lambda) \hat{i}+(1-3 \lambda) \hat{j}+(3+4 \lambda) \hat{k}, & \lambda \in \mathbb{R} \\
L_2: \vec{r}=2(1+\mu) \hat{i}+3(1+\mu) \hat{j}+(5+\mu) \hat{k}, & \mu \...

The number of critical points of the function \(f(x)=(x-2)^{2 / 3}(2 x+1)\) is

Let \(f(x)=4 \cos ^3 x+3 \sqrt{3} \cos ^2 x-10\). The number of points of local maxima of \(f\) in interval \((0,2 \pi)\) is

For the function \(f(x)=(\cos x)-x+1, x \in \mathbb{R}\), between the following two statements
(S1) \(f(x)=0\) for only one value of \(x\) in \([0, \pi]\).
(S2) \(f(x)\) is decreasing in \(\left[0, \frac{\pi}{2}\right]\) and increasing in $...

Let the area of the region enclosed by the curve \(y=\min \{\sin x, \cos x\}\) and the \(x\) axis between \(x=-\pi\) to \(x=\pi\) be \(A\). Then \(A^2\) is equal to __________.

Let the circles \(C_1:(x-\alpha)^2+(y-\beta)^2=r_1^2\) and \(C_2:(x-8)^2+\left(y-\frac{15}{2}\right)^2=r_2^2\) touch each other externally at the point \((6,6)\). If the point \((6,6)\) divides the line segment joining the centres of the ci...

Let \(z\) be a complex number such that \(|z+2|=1\) and \(\operatorname{lm}\left(\frac{z+1}{z+2}\right)=\frac{1}{5}\). Then the value of \(|\operatorname{Re}(\overline{z+2})|\) is

If the set \(R=\{(a, b): a+5 b=42, a, b \in \mathbb{N}\}\) has \(m\) elements and \(\sum_\limits{n=1}^m\left(1-i^{n !}\right)=x+i y\), where \(i=\sqrt{-1}\), then the value of \(m+x+y\) is

The value of \(k \in \mathbb{N}\) for which the integral \(I_n=\int_0^1\left(1-x^k\right)^n d x, n \in \mathbb{N}\), satisfies \(147 I_{20}=148 I_{21}\) is

Let \(f(x)\) be a positive function such that the area bounded by \(y=f(x), y=0\) from \(x=0\) to \(x=a>0\) is \(e^{-a}+4 a^2+a-1\). Then the differential equation, whose general solution is \(y=c_1 f(x)+c_2\), where \(c_1\) and \(c_2\) are...

Let \(y=y(x)\) be the solution of the differential equation \((1+y^2) e^{\tan x} d x+\cos ^2 x(1+e^{2 \tan x}) d y=0, y(0)=1\). Then \(y\left(\frac{\pi}{4}\right)\) is equal to

If the range of \(f(\theta)=\frac{\sin ^4 \theta+3 \cos ^2 \theta}{\sin ^4 \theta+\cos ^2 \theta}, \theta \in \mathbb{R}\) is \([\alpha, \beta]\), then the sum of the infinite G.P., whose first term is 64 and the common ratio is $$\frac{\al...

Let \(H: \frac{-x^2}{a^2}+\frac{y^2}{b^2}=1\) be the hyperbola, whose eccentricity is \(\sqrt{3}\) and the length of the latus rectum is \(4 \sqrt{3}\). Suppose the point \((\alpha, 6), \alpha>0\) lies on \(H\). If \(\beta\) is the product ...

Let \(I(x)=\int \frac{6}{\sin ^2 x(1-\cot x)^2} d x\). If \(I(0)=3\), then \(I\left(\frac{\pi}{12}\right)\) is equal to

The value of \(\lim _\limits{x \rightarrow 0} 2\left(\frac{1-\cos x \sqrt{\cos 2 x} \sqrt[3]{\cos 3 x} \ldots \ldots . \sqrt[10]{\cos 10 x}}{x^2}\right)\) is __________.

Let $$A=\left[\begin{array}{lll}2 & a & 0 \\ 1 & 3 & 1 \\ 0 & 5 & b\end{array}\right]$$. If \(A^3=4 A^2-A-21 I\), where \(I\) is the identity matrix of order \(3 \times 3\), then \(2 a+3 b\) is equal to

Let $$A=\left[\begin{array}{cc}2 & -1 \\ 1 & 1\end{array}\right]$$. If the sum of the diagonal elements of \(A^{13}\) is \(3^n\), then \(n\) is equal to ________.

Let \([t]\) be the greatest integer less than or equal to \(t\). Let \(A\) be the set of all prime factors of 2310 and \(f: A \rightarrow \mathbb{Z}\) be the function \(f(x)=\left[\log _2\left(x^2+\left[\frac{x^3}{5}\right]\right)\right]\)....

The number of 3-digit numbers, formed using the digits 2, 3, 4, 5 and 7, when the repetition of digits is not allowed, and which are not divisible by 3 , is equal to ________.

Let the sum of two positive integers be 24 . If the probability, that their product is not less than \(\frac{3}{4}\) times their greatest possible product, is \(\frac{m}{n}\), where \(\operatorname{gcd}(m, n)=1\), then \(n\)-\(m\) equals

Three balls are drawn at random from a bag containing 5 blue and 4 yellow balls. Let the random variables \(X\) and \(Y\) respectively denote the number of blue and yellow balls. If \(\bar{X}\) and \(\bar{Y}\) are the means of \(X\) and $$Y...

The sum of all the solutions of the equation \((8)^{2 x}-16 \cdot(8)^x+48=0\) is :

Let \(\alpha=\sum_\limits{r=0}^n\left(4 r^2+2 r+1\right){ }^n C_r\) and \(\beta=\left(\sum_\limits{r=0}^n \frac{{ }^n C_r}{r+1}\right)+\frac{1}{n+1}\). If \(140<\frac{2 \alpha}{\beta}<281\), then the value of \(n\) is _________.

Let the positive integers be written in the form :

If the \(k^{\text {th }}\) row contains exactly \(k\) numbers for every natural number \(k\), then the row in which the number 5310 will be, is __________.

If the orthocentre of the triangle formed by the lines \(2 x+3 y-1=0, x+2 y-1=0\) and \(a x+b y-1=0\), is the centroid of another triangle, whose circumcentre and orthocentre respectively are \((3,4)\) and \((-6,-8)\), then the value of $$|...

The equations of two sides \(\mathrm{AB}\) and \(\mathrm{AC}\) of a triangle \(\mathrm{ABC}\) are \(4 x+y=14\) and \(3 x-2 y=5\), respectively. The point \(\left(2,-\frac{4}{3}\right)\) divides the third side \(\mathrm{BC}\) internally in t...

If \(\sin x=-\frac{3}{5}\), where \(\pi< x <\frac{3 \pi}{2}\), then \(80\left(\tan ^2 x-\cos x\right)\) is equal to

The set of all \(\alpha\), for which the vectors \(\vec{a}=\alpha t \hat{i}+6 \hat{j}-3 \hat{k}\) and \(\vec{b}=t \hat{i}-2 \hat{j}-2 \alpha t \hat{k}\) are inclined at an obtuse angle for all \(t \in \mathbb{R}\), is

Let \(\vec{a}=9 \hat{i}-13 \hat{j}+25 \hat{k}, \vec{b}=3 \hat{i}+7 \hat{j}-13 \hat{k}\) and \(\vec{c}=17 \hat{i}-2 \hat{j}+\hat{k}\) be three given vectors. If \(\vec{r}\) is a vector such that $$\vec{r} \times \vec{a}=(\vec{b}+\vec{c}) \ti...