JEE Main 2024 (Online) 4th April Morning Shift

Let the point, on the line passing through the points \(P(1,-2,3)\) and \(Q(5,-4,7)\), farther from the origin and at a distance of 9 units from the point \(P\), be \((\alpha, \beta, \gamma)\). Then \(\alpha^2+\beta^2+\gamma^2\) is equal to...

Let the sum of the maximum and the minimum values of the function \(f(x)=\frac{2 x^2-3 x+8}{2 x^2+3 x+8}\) be \(\frac{m}{n}\), where \(\operatorname{gcd}(\mathrm{m}, \mathrm{n})=1\). Then \(\mathrm{m}+\mathrm{n}\) is equal to :

One of the points of intersection of the curves \(y=1+3 x-2 x^2\) and \(y=\frac{1}{x}\) is \(\left(\frac{1}{2}, 2\right)\). Let the area of the region enclosed by these curves be $$\frac{1}{24}(l \sqrt{5}+\mathrm{m})-\mathrm{n} \log _{\math...

The sum of all rational terms in the expansion of \(\left(2^{\frac{1}{5}}+5^{\frac{1}{3}}\right)^{15}\) is equal to :

Let $$a=1+\frac{{ }^2 \mathrm{C}_2}{3 !}+\frac{{ }^3 \mathrm{C}_2}{4 !}+\frac{{ }^4 \mathrm{C}_2}{5 !}+...., \mathrm{b}=1+\frac{{ }^1 \mathrm{C}_0+{ }^1 \mathrm{C}_1}{1 !}+\frac{{ }^2 \mathrm{C}_0+{ }^2 \mathrm{C}_1+{ }^2 \mathrm{C}_2}{2 !}...

A square is inscribed in the circle \(x^2+y^2-10 x-6 y+30=0\). One side of this square is parallel to \(y=x+3\). If \(\left(x_i, y_i\right)\) are the vertices of the square, then \(\Sigma\left(x_i^2+y_i^2\right)\) is equal to:

Let \(\alpha\) and \(\beta\) be the sum and the product of all the non-zero solutions of the equation \((\bar{z})^2+|z|=0, z \in C\). Then \(4(\alpha^2+\beta^2)\) is equal to :

$$\text { Let } f(x)=\left\{\begin{array}{lr} -2, & -2 \leq x \leq 0 \\ x-2, & 0< x \leq 2 \end{array} \text { and } \mathrm{h}(x)=f(|x|)+|f(x)| \text {. Then } \int_\limits{-2}^2 \mathrm{~h}(x) \mathrm{d} x\right. \text { is equal to: }$$

If the shortest distance between the lines \(\frac{x+2}{2}=\frac{y+3}{3}=\frac{z-5}{4}\) and \(\frac{x-3}{1}=\frac{y-2}{-3}=\frac{z+4}{2}\) is \(\frac{38}{3 \sqrt{5}} \mathrm{k}\), and $$\int_\limits 0^{\mathrm{k}}\left[x^2\right] \mathrm{d...

If \(\int_0^{\frac{\pi}{4}} \frac{\sin ^2 x}{1+\sin x \cos x} \mathrm{~d} x=\frac{1}{\mathrm{a}} \log _{\mathrm{e}}\left(\frac{\mathrm{a}}{3}\right)+\frac{\pi}{\mathrm{b} \sqrt{3}}\), where \(\mathrm{a}, \mathrm{b} \in \mathrm{N}\), then $$...

If the solution \(y=y(x)\) of the differential equation \((x^4+2 x^3+3 x^2+2 x+2) \mathrm{d} y-(2 x^2+2 x+3) \mathrm{d} x=0\) satisfies \(y(-1)=-\frac{\pi}{4}\), then \(y(0)\) is equal to :

Let the solution \(y=y(x)\) of the differential equation \(\frac{\mathrm{d} y}{\mathrm{~d} x}-y=1+4 \sin x\) satisfy \(y(\pi)=1\). Then \(y\left(\frac{\pi}{2}\right)+10\) is equal to __________.

Let \(f(x)=x^5+2 \mathrm{e}^{x / 4}\) for all \(x \in \mathbf{R}\). Consider a function \(g(x)\) such that \((g \circ f)(x)=x\) for all \(x \in \mathbf{R}\). Then the value of \(8 g^{\prime}(2)\) is :

If the domain of the function \(\sin ^{-1}\left(\frac{3 x-22}{2 x-19}\right)+\log _{\mathrm{e}}\left(\frac{3 x^2-8 x+5}{x^2-3 x-10}\right)\) is \((\alpha, \beta]\), then \(3 \alpha+10 \beta\) is equal to:

Let \(f: \mathbf{R} \rightarrow \mathbf{R}\) be a function given by
$$f(x)= \begin{cases}\frac{1-\cos 2 x}{x^2}, & x < 0 \\ \alpha, & x=0, \\ \frac{\beta \sqrt{1-\cos x}}{x}, & x>0\end{cases}$$
where \(\alpha, \beta \in \mathbf{R}\). If $$f...

If \(\lim _\limits{x \rightarrow 1} \frac{(5 x+1)^{1 / 3}-(x+5)^{1 / 3}}{(2 x+3)^{1 / 2}-(x+4)^{1 / 2}}=\frac{\mathrm{m} \sqrt{5}}{\mathrm{n}(2 \mathrm{n})^{2 / 3}}\), where \(\operatorname{gcd}(\mathrm{m}, \mathrm{n})=1\), then $$8 \mathrm...

Let \(\alpha \in(0, \infty)\) and $$A=\left[\begin{array}{lll}1 & 2 & \alpha \\ 1 & 0 & 1 \\ 0 & 1 & 2\end{array}\right]$$. If $$\operatorname{det}\left(\operatorname{adj}\left(2 A-A^T\right) \cdot \operatorname{adj}\left(A-2 A^T\right)\rig...

If the system of equations
$$\begin{aligned} & x+(\sqrt{2} \sin \alpha) y+(\sqrt{2} \cos \alpha) z=0 \\ & x+(\cos \alpha) y+(\sin \alpha) z=0 \\ & x+(\sin \alpha) y-(\cos \alpha) z=0 \end{aligned}$$
has a non-trivial solution, then $$\alpha...

Let \(A\) be a square matrix of order 2 such that \(|A|=2\) and the sum of its diagonal elements is \(-\)3 . If the points \((x, y)\) satisfying \(\mathrm{A}^2+x \mathrm{~A}+y \mathrm{I}=\mathrm{O}\) lie on a hyperbola, whose transverse axi...

Let \(A\) be a \(3 \times 3\) matrix of non-negative real elements such that $$A\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right]=3\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right]$$. Then the maximum value of $$\operatorname{det}(\math...

Let the length of the focal chord PQ of the parabola \(y^2=12 x\) be 15 units. If the distance of \(\mathrm{PQ}\) from the origin is \(\mathrm{p}\), then \(10 \mathrm{p}^2\) is equal to __________.

There are 5 points \(P_1, P_2, P_3, P_4, P_5\) on the side \(A B\), excluding \(A\) and \(B\), of a triangle \(A B C\). Similarly there are 6 points \(\mathrm{P}_6, \mathrm{P}_7, \ldots, \mathrm{P}_{11}\) on the side \(\mathrm{BC}\) and 7 p...

Three urns A, B and C contain 7 red, 5 black; 5 red, 7 black and 6 red, 6 black balls, respectively. One of the urn is selected at random and a ball is drawn from it. If the ball drawn is black, then the probability that it is drawn from ur...

If 2 and 6 are the roots of the equation \(a x^2+b x+1=0\), then the quadratic equation, whose roots are \(\frac{1}{2 a+b}\) and \(\frac{1}{6 a+b}\), is :

Let the first three terms 2, p and q, with \(q \neq 2\), of a G.P. be respectively the \(7^{\text {th }}, 8^{\text {th }}\) and \(13^{\text {th }}\) terms of an A.P. If the \(5^{\text {th }}\) term of the G.P. is the \(n^{\text {th }}\) ter...

In a survey of 220 students of a higher secondary school, it was found that at least 125 and at most 130 students studied Mathematics; at least 85 and at most 95 studied Physics; at least 75 and at most 90 studied Chemistry; 30 studied both...

Let \(\alpha, \beta \in \mathbf{R}\). Let the mean and the variance of 6 observations \(-3,4,7,-6, \alpha, \beta\) be 2 and 23, respectively. The mean deviation about the mean of these 6 observations is :

The vertices of a triangle are \(\mathrm{A}(-1,3), \mathrm{B}(-2,2)\) and \(\mathrm{C}(3,-1)\). A new triangle is formed by shifting the sides of the triangle by one unit inwards. Then the equation of the side of the new triangle nearest to...

Let a unit vector which makes an angle of \(60^{\circ}\) with \(2 \hat{i}+2 \hat{j}-\hat{k}\) and an angle of \(45^{\circ}\) with \(\hat{i}-\hat{k}\) be \(\vec{C}\). Then $$\vec{C}+\left(-\frac{1}{2} \hat{i}+\frac{1}{3 \sqrt{2}} \hat{j}-\fr...

Let \(\mathrm{ABC}\) be a triangle of area \(15 \sqrt{2}\) and the vectors \(\overrightarrow{\mathrm{AB}}=\hat{i}+2 \hat{j}-7 \hat{k}, \overrightarrow{\mathrm{BC}}=\mathrm{a} \hat{i}+\mathrm{b} \hat{j}+\mathrm{c} \hat{k}\) and $$\overrighta...