JEE Main 2024 (Online) 5th April Morning Shift

Let \(\mathrm{d}\) be the distance of the point of intersection of the lines \(\frac{x+6}{3}=\frac{y}{2}=\frac{z+1}{1}\) and \(\frac{x-7}{4}=\frac{y-9}{3}=\frac{z-4}{2}\) from the point \((7,8,9)\). Then \(\mathrm{d}^2+6\) is equal to :

If the line \(\frac{2-x}{3}=\frac{3 y-2}{4 \lambda+1}=4-z\) makes a right angle with the line \(\frac{x+3}{3 \mu}=\frac{1-2 y}{6}=\frac{5-z}{7}\), then \(4 \lambda+9 \mu\) is equal to :

For the function
\(f(x)=\sin x+3 x-\frac{2}{\pi}\left(x^2+x\right), \text { where } x \in\left[0, \frac{\pi}{2}\right],\)
consider the following two statements :
(I) \(f\) is increasing in \(\left(0, \frac{\pi}{2}\right)\).
(II) $$f^{\prime...

Let a rectangle ABCD of sides 2 and 4 be inscribed in another rectangle PQRS such that the vertices of the rectangle ABCD lie on the sides of the rectangle PQRS. Let a and b be the sides of the rectangle PQRS when its area is maximum. Then ...

Let \(f(x)=x^5+2 x^3+3 x+1, x \in \mathbf{R}\), and \(g(x)\) be a function such that \(g(f(x))=x\) for all \(x \in \mathbf{R}\). Then \(\frac{g(7)}{g^{\prime}(7)}\) is equal to :

The area of the region enclosed by the parabolas \(y=x^2-5 x\) and \(y=7 x-x^2\) is ________.

If the constant term in the expansion of \(\left(1+2 x-3 x^3\right)\left(\frac{3}{2} x^2-\frac{1}{3 x}\right)^9\) is \(\mathrm{p}\), then \(108 \mathrm{p}\) is equal to ________.

Let a circle C of radius 1 and closer to the origin be such that the lines passing through the point \((3,2)\) and parallel to the coordinate axes touch it. Then the shortest distance of the circle C from the point \((5,5)\) is :

Consider the following two statements :
Statement I: For any two non-zero complex numbers $$z_1, z_2,(|z_1|+|z_2|)\left|\frac{z_1}{\left|z_1\right|}+\frac{z_2}{\left|z_2\right|}\right| \leq 2\left(\left|z_1\right|+\left|z_2\right|\right) \t...

The value of \(\int_\limits{-\pi}^\pi \frac{2 y(1+\sin y)}{1+\cos ^2 y} d y\) is :

The integral \(\int_\limits0^{\pi / 4} \frac{136 \sin x}{3 \sin x+5 \cos x} \mathrm{~d} x\) is equal to :

If \(y=y(x)\) is the solution of the differential equation \(\frac{\mathrm{d} y}{\mathrm{~d} x}+2 y=\sin (2 x), y(0)=\frac{3}{4}\), then \(y\left(\frac{\pi}{8}\right)\) is equal to :

Let the line \(2 x+3 y-\mathrm{k}=0, \mathrm{k}>0\), intersect the \(x\)-axis and \(y\)-axis at the points \(\mathrm{A}\) and \(\mathrm{B}\), respectively. If the equation of the circle having the line segment \(A B\) as a diameter is $$x^2...

If \(S=\{a \in \mathbf{R}:|2 a-1|=3[a]+2\{a \}\}\), where \([t]\) denotes the greatest integer less than or equal to \(t\) and \(\{t\}\) represents the fractional part of \(t\), then \(72 \sum_\limits{a \in S} a\) is equal to _________.

Let \(A=\{1,3,7,9,11\}\) and \(B=\{2,4,5,7,8,10,12\}\). Then the total number of one-one maps \(f: A \rightarrow B\), such that \(f(1)+f(3)=14\), is :

If the function \(f(x)=\frac{\sin 3 x+\alpha \sin x-\beta \cos 3 x}{x^3}, x \in \mathbf{R}\), is continuous at \(x=0\), then \(f(0)\) is equal to :

Let \(f\) be a differentiable function in the interval \((0, \infty)\) such that \(f(1)=1\) and \(\lim _\limits{t \rightarrow x} \frac{t^2 f(x)-x^2 f(t)}{t-x}=1\) for each \(x>0\). Then \(2 f(2)+3 f(3)\) is equal to _________.

Let A and B be two square matrices of order 3 such that \(\mathrm{|A|=3}\) and \(\mathrm{|B|=2}\). Then $$|\mathrm{A}^{\mathrm{T}} \mathrm{A}(\operatorname{adj}(2 \mathrm{~A}))^{-1}(\operatorname{adj}(4 \mathrm{~B}))(\operatorname{adj}(\mat...

If the system of equations
$$\begin{array}{r} 11 x+y+\lambda z=-5 \\ 2 x+3 y+5 z=3 \\ 8 x-19 y-39 z=\mu \end{array}$$
has infinitely many solutions, then \(\lambda^4-\mu\) is equal to :

Suppose \(\mathrm{AB}\) is a focal chord of the parabola \(y^2=12 x\) of length \(l\) and slope \(\mathrm{m}<\sqrt{3}\). If the distance of the chord \(\mathrm{AB}\) from the origin is \(\mathrm{d}\), then \(l \mathrm{~d}^2\) is equal to __...

The number of ways of getting a sum 16 on throwing a dice four times is ________.

From a lot of 10 items, which include 3 defective items, a sample of 5 items is drawn at random. Let the random variable \(X\) denote the number of defective items in the sample. If the variance of \(X\) is \(\sigma^2\), then $$96 \sigma^2$...

The coefficients \(a, b, c\) in the quadratic equation \(a x^2+b x+c=0\) are chosen from the set \(\{1,2,3,4,5,6,7,8\}\). The probability of this equation having repeated roots is :

The number of distinct real roots of the equation \(|x||x+2|-5|x+1|-1=0\) is __________.

If \(\frac{1}{\sqrt{1}+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+\ldots+\frac{1}{\sqrt{99}+\sqrt{100}}=m\) and \(\frac{1}{1 \cdot 2}+\frac{1}{2 \cdot 3}+\ldots+\frac{1}{99 \cdot 100}=\mathrm{n}\), then the point \((\mathrm{m}, \mathrm{n})\) lie...

Let \(a_1, a_2, a_3, \ldots\) be in an arithmetic progression of positive terms.
Let \(A_k=a_1^2-a_2^2+a_3^2-a_4^2+\ldots+a_{2 k-1}^2-a_{2 k}^2\).
If \(\mathrm{A}_3=-153, \mathrm{~A}_5=-435\) and $$\mathrm{a}_1^2+\mathrm{a}_2^2+\mathrm{a}_3...

Let two straight lines drawn from the origin \(\mathrm{O}\) intersect the line \(3 x+4 y=12\) at the points \(\mathrm{P}\) and \(\mathrm{Q}\) such that \(\triangle \mathrm{OPQ}\) is an isosceles triangle and $$\angle \mathrm{POQ}=90^{\circ}...

Suppose \(\theta \in\left[0, \frac{\pi}{4}\right]\) is a solution of \(4 \cos \theta-3 \sin \theta=1\). Then \(\cos \theta\) is equal to :

Let \(\overrightarrow{\mathrm{a}}=\hat{i}-3 \hat{j}+7 \hat{k}, \overrightarrow{\mathrm{b}}=2 \hat{i}-\hat{j}+\hat{k}\) and \(\overrightarrow{\mathrm{c}}\) be a vector such that $$(\overrightarrow{\mathrm{a}}+2 \overrightarrow{\mathrm{b}}) \...

If \(\mathrm{A}(1,-1,2), \mathrm{B}(5,7,-6), \mathrm{C}(3,4,-10)\) and \(\mathrm{D}(-1,-4,-2)\) are the vertices of a quadrilateral ABCD, then its area is :