JEE Main 2024 Apr 6 Shift 2 — Mathematics

Let $f(x)=\frac{1}{7-\sin 5x}$ be a function defined on $\mathbf{R}$. Then the range of the function $f(x)$ is equal to :

Let $A=\{1, 2, 3, 4, 5\}$. Let $R$ be a relation on $A$ defined by $xRy$ if and only if $4x \le 5y$. Let $m$ be the number of elements in $R$ and $n$ be the minimum number of elements from $A \times A$ that are required to be added to $R$ to make it a symmetric relation. Then $m+n$ is equal to :

If $z_1, z_2$ are two distinct complex number such that $\left|\dfrac{z_1-2z_2}{\frac{1}{2}-z_1\bar{z}_2}\right|=2$, then

If $A$ is a square matrix of order 3 such that $\det(A)=3$ and $\det(\operatorname{adj}(-4\ \operatorname{adj}(-3\ \operatorname{adj}(3\ \operatorname{adj}((2A)^{-1})))))=2^{\mathrm{m}}\, 3^{\mathrm{n}}$, then $\mathrm{m}+2\mathrm{n}$ is equal to :

If all the words with or without meaning made using all the letters of the word “NAGPUR” are arranged as in a dictionary, then the word at $315^{\text{th}}$ position in this arrangement is :

Let $0 \le \mathrm{r} \le \mathrm{n}$. If $^{\mathrm{n}+1}C_{\mathrm{r}+1} : {^{\mathrm{n}}C_{\mathrm{r}}} : {^{\mathrm{n}-1}C_{\mathrm{r}-1}} = 55:35:21$, then $2\mathrm{n}+5\mathrm{r}$ is equal to :

A software company sets up m number of computer systems to finish an assignment in 17 days. If 4 computer systems crashed on the start of the second day, 4 more computer systems crashed on the start of the third day and so on, then it took 8 more days to finish the assignment. The value of m is equal to :

If the function $f(x)=\left(\frac{1}{x}\right)^{2x};\ x>0$ attains the maximum value at $x=\frac{1}{\mathrm{e}}$ then :

$\lim\limits_{\mathrm{n}\to\infty} \dfrac{\left(1^2-1\right)(\mathrm{n}-1)+\left(2^2-2\right)(\mathrm{n}-2)+\cdots+\left((\mathrm{n}-1)^2-(\mathrm{n}-1)\right)\cdot 1}{\left(1^3+2^3+\cdots+\mathrm{n}^3\right)-\left(1^2+2^2+\cdots+\mathrm{n}^2\right)}$ is equal to :

Suppose for a differentiable function $h$, $h(0)=0$, $h(1)=1$ and $h'(0)=h'(1)=2$. If $g(x)=h(\mathrm{e}^x)\mathrm{e}^{h(x)}$, then $g'(0)$ is equal to :

If the area of the region $\left\{(x,y):\ \frac{\mathrm{a}}{x^2} \le y \le \frac{1}{x},\ 1 \le x \le 2,\ 0 < \mathrm{a} < 1\right\}$ is $(\log_{\mathrm{e}} 2)-\frac{1}{7}$ then the value of $7\mathrm{a}-3$ is equal to :

If $\int \dfrac{1}{\mathrm{a}^2\sin^2 x + \mathrm{b}^2\cos^2 x}\,\mathrm{d}x = \dfrac{1}{12}\tan^{-1}(3\tan x) + \text{constant}$, then the maximum value of $\mathrm{a}\sin x + \mathrm{b}\cos x$, is :

Suppose the solution of the differential equation $\dfrac{\mathrm{d}y}{\mathrm{d}x} = \dfrac{(2+\alpha)x-\beta y+2}{\beta x-2\alpha y-(\beta\gamma-4\alpha)}$ represents a circle passing through origin. Then the radius of this circle is :

If the locus of the point, whose distances from the point $(2, 1)$ and $(1, 3)$ are in the ratio $5:4$, is $\mathrm{a}x^2+\mathrm{b}y^2+\mathrm{c}xy+\mathrm{d}x+\mathrm{e}y+170=0$, then the value of $\mathrm{a}^2+2\mathrm{b}+3\mathrm{c}+4\mathrm{d}+\mathrm{e}$ is equal to :

If $P(6, 1)$ be the orthocentre of the triangle whose vertices are $A(5, -2)$, $B(8, 3)$ and $C(\mathrm{h}, \mathrm{k})$, then the point $C$ lies on the circle :

Let $ABC$ be an equilateral triangle. A new triangle is formed by joining the middle points of all sides of the triangle $ABC$ and the same process is repeated infinitely many times. If $P$ is the sum of perimeters and $Q$ is be the sum of areas of all the triangles formed in this process, then :

Let $P(\alpha, \beta, \gamma)$ be the image of the point $Q(3, -3, 1)$ in the line $\dfrac{x-0}{1}=\dfrac{y-3}{1}=\dfrac{z-1}{-1}$ and $R$ be the point $(2, 5, -1)$. If the area of the triangle $PQR$ is $\lambda$ and $\lambda^2=14K$, then $K$ is equal to :

Let $\vec{a}=2\hat{i}+\hat{j}-\hat{k},\ \vec{b}=\left(\left(\vec{a}\times\left(\hat{i}+\hat{j}\right)\right)\times\hat{i}\right)\times\hat{i}$. Then the square of the projection of $\vec{a}$ on $\vec{b}$ is :

Let $\vec{a}=6\hat{i}+\hat{j}-\hat{k}$ and $\vec{b}=\hat{i}+\hat{j}$. If $\vec{c}$ is a is vector such that $|\vec{c}|\ge 6$, $\vec{a}\cdot\vec{c}=6|\vec{c}|$, $|\vec{c}-\vec{a}|=2\sqrt{2}$ and the angle between $\vec{a}\times\vec{b}$ and $\vec{c}$ is $60^{\circ}$, then $\left|\left(\vec{a}\times\vec{b}\right)\times\vec{c}\right|$ is equal to :

If three letters can be posted to any one of the 5 different addresses, then the probability that the three letters are posted to exactly two addresses is :

Let $\alpha, \beta$ be roots of $x^2+\sqrt{2}x-8=0$. If $\mathrm{U_n}=\alpha^{\mathrm{n}}+\beta^{\mathrm{n}}$, then $\dfrac{\mathrm{U}_{10}+\sqrt{2}\,\mathrm{U}_9}{2\mathrm{U}_8}$ is equal to ________.

If the system of equations $$2x+7y+\lambda z=3$$ $$3x+2y+5z=4$$ $$x+\mu y+32z=-1$$ has infinitely many solutions, then $(\lambda-\mu)$ is equal to __________ :

If $S(x)=(1+x)+2(1+x)^2+3(1+x)^3+\cdots+60(1+x)^{60}$, $x \ne 0$, and $(60)^2 S(60)=\mathrm{a(b)^b+b}$, where $\mathrm{a, b} \in \mathbf{N}$, then $(\mathrm{a+b})$ equal to ________.

Let $[t]$ denote the greatest integer less than or equal to $t$. Let $f:[0, \infty) \to \mathbf{R}$ be a function defined by $f(x)=\left[\dfrac{x}{2}+3\right]-\left[\sqrt{x}\right]$. Let $S$ be the set of all points in the interval $[0, 8]$ at which $f$ is not continuous. Then $\sum\limits_{\mathrm{a}\in S} \mathrm{a}$ is equal to ________.

Let $[t]$ denote the largest integer less than or equal to $t$. If $$\int\limits_0^3\left(\left[x^2\right]+\left[\frac{x^2}{2}\right]\right)\mathrm{d}x=\mathrm{a}+\mathrm{b}\sqrt{2}-\sqrt{3}-\sqrt{5}+\mathrm{c}\sqrt{6}-\sqrt{7},$$ where $\mathrm{a, b, c} \in \mathbf{Z}$, then $\mathrm{a+b+c}$ is equal to ________.

If the solution $y(x)$ of the given differential equation $(\mathrm{e}^y+1)\cos x\,\mathrm{d}x+\mathrm{e}^y \sin x\,\mathrm{d}y=0$ passes through the point $\left(\frac{\pi}{2}, 0\right)$, then the value of $\mathrm{e}^{y\left(\frac{\pi}{6}\right)}$ is equal to ________.

The length of the latus rectum and directrices of a hyperbola with eccentricity e are 9 and $x=\pm\frac{4}{\sqrt{3}}$, respectively. Let the line $y-\sqrt{3}x+\sqrt{3}=0$ touch this hyperbola at $(x_0, y_0)$. If m is the product of the focal distances of the point $(x_0, y_0)$, then $4\mathrm{e}^2+\mathrm{m}$ is equal to ________.

If the shortest distance between the lines $\dfrac{x-\lambda}{3}=\dfrac{y-2}{-1}=\dfrac{z-1}{1}$ and $\dfrac{x+2}{-3}=\dfrac{y+5}{2}=\dfrac{z-4}{4}$ is $\dfrac{44}{\sqrt{30}}$, then the largest possible value of $|\lambda|$ is equal to ________.

From a lot of 12 items containing 3 defectives, a sample of 5 items is drawn at random. Let the random variable $X$ denote the number of defective items in the sample. Let items in the sample be drawn one by one without replacement. If variance of $X$ is $\dfrac{\mathrm{m}}{\mathrm{n}}$, where $\gcd(\mathrm{m, n})=1$, then $\mathrm{n-m}$ is equal to ________.

In a triangle $ABC$, $BC=7$, $AC=8$, $AB=\alpha \in \mathbf{N}$ and $\cos A=\dfrac{2}{3}$. If $49\cos(3C)+42=\dfrac{\mathrm{m}}{\mathrm{n}}$, where $\gcd(\mathrm{m,n})=1$, then $\mathrm{m+n}$ is equal to ________.