JEE Main 2024 Apr 9 Shift 2 — Mathematics

Let the range of the function $f(x)=\frac{1}{2+\sin 3x+\cos 3x}, x \in \mathbb{R}$ be $[a, b]$. If $\alpha$ and $\beta$ are respectively the A.M. and the G.M. of $a$ and $b$, then $\frac{\alpha}{\beta}$ is equal to

Let $z$ be a complex number such that the real part of $\frac{z-2i}{z+2i}$ is zero. Then, the maximum value of $|z-(6+8i)|$ is equal to

Let $\alpha, \beta;\ \alpha>\beta$, be the roots of the equation $x^2-\sqrt{2}x-\sqrt{3}=0$. Let $\mathrm{P}_n=\alpha^n-\beta^n, n \in \mathbb{N}$. Then $\left(11\sqrt{3}-10\sqrt{2}\right)\mathrm{P}_{10}+\left(11\sqrt{2}+10\right)\mathrm{P}_{11}-11\mathrm{P}_{12}$ is equal to

Let $B=\begin{bmatrix} 1 & 3 \\ 1 & 5 \end{bmatrix}$ and $A$ be a $2\times 2$ matrix such that $AB^{-1}=A^{-1}$. If $BCB^{-1}=A$ and $C^4+\alpha C^2+\beta I=O$, then $2\beta-\alpha$ is equal to

$\lim\limits_{x \to 0} \dfrac{e-(1+2x)^{\frac{1}{2x}}}{x}$ is equal to

The sum of the coefficient of $x^{2/3}$ and $x^{-2/5}$ in the binomial expansion of $\left(x^{2/3}+\frac{1}{2}x^{-2/5}\right)^{9}$ is

Let $a, ar, ar^2, \ldots\ldots$ be an infinite G.P. If $\sum\limits_{n=0}^{\infty} ar^n = 57$ and $\sum\limits_{n=0}^{\infty} a^3r^{3n} = 9747$, then $a+18r$ is equal to

If $\log_e y = 3\sin^{-1}x$, then $(1-x^2)y''-xy'$ at $x=\frac{1}{2}$ is equal to

$\lim\limits_{x \to \frac{\pi}{2}}\left(\dfrac{\int_{x^3}^{(\pi/2)^3}\left(\sin\left(2t^{1/3}\right)+\cos\left(t^{1/3}\right)\right)dt}{\left(x-\frac{\pi}{2}\right)^2}\right)$ is equal to

The area (in square units) of the region enclosed by the ellipse $x^2+3y^2=18$ in the first quadrant below the line $y=x$ is

The integral $\int_{1/4}^{3/4} \cos\left(2\cot^{-1}\sqrt{\dfrac{1-x}{1+x}}\right)dx$ is equal to

The value of the integral $\int_{-1}^{2} \log_e\left(x+\sqrt{x^2+1}\right)dx$ is

Let $\int_{0}^{x}\sqrt{1-\left(y'(t)\right)^2}\,dt=\int_{0}^{x} y(t)\,dt$, $0 \le x \le 3$, $y \ge 0$, $y(0)=0$. Then at $x=2$, $y''+y+1$ is equal to

Two vertices of a triangle ABC are $A(3, -1)$ and $B(-2, 3)$, and its orthocentre is $P(1,1)$. If the coordinates of the point $C$ are $(\alpha,\beta)$ and the centre of the of the circle circumscribing the triangle PAB is $(h,k)$, then the value of $(\alpha+\beta)+2(h+k)$ equals

Let the foci of a hyperbola $H$ coincide with the foci of the ellipse $E: \dfrac{(x-1)^2}{100}+\dfrac{(y-1)^2}{75}=1$ and the eccentricity of the hyperbola $H$ be the reciprocal of the eccentricity of the ellipse $E$. If the length of the transverse axis of $H$ is $\alpha$ and the length of its conjugate axis is $\beta$, then $3\alpha^2+2\beta^2$ is equal to

Consider the line L passing through the points $(1, 2, 3)$ and $(2, 3, 5)$. The distance of the point $\left(\dfrac{11}{3}, \dfrac{11}{3}, \dfrac{19}{3}\right)$ from the line L along the line $\dfrac{3x-11}{2}=\dfrac{3y-11}{1}=\dfrac{3z-19}{2}$ is equal to

Between the following two statements:

Statement I: Let $\vec{a}=\hat{i}+2\hat{j}-3\hat{k}$ and $\vec{b}=2\hat{i}+\hat{j}-\hat{k}$. Then the vector $\vec{r}$ satisfying $\vec{a}\times\vec{r}=\vec{a}\times\vec{b}$ and $\vec{a}\cdot\vec{r}=0$ is of magnitude $\sqrt{10}$.

Statement II: In a triangle $ABC$, $\cos 2A+\cos 2B+\cos 2C \ge -\dfrac{3}{2}$.

Let $\vec{a}=2\hat{i}+\alpha\hat{j}+\hat{k}$, $\vec{b}=-\hat{i}+\hat{k}$, $\vec{c}=\beta\hat{j}-\hat{k}$, where $\alpha$ and $\beta$ are integers and $\alpha\beta=-6$. Let the values of the ordered pair $(\alpha, \beta)$, for which the area of the parallelogram of diagonals $\vec{a}+\vec{b}$ and $\vec{b}+\vec{c}$ is $\dfrac{\sqrt{21}}{2}$, be $(\alpha_1, \beta_1)$ and $(\alpha_2, \beta_2)$. Then $\alpha_1^2+\beta_1^2-\alpha_2\beta_2$ is equal to

If the variance of the frequency distribution

$x$	$c$	$2c$	$3c$	$4c$	$5c$	$6c$
$f$	$2$	$1$	$1$	$1$	$1$	$1$

is 160, then the value of $c \in \mathbb{N}$ is

If an unbiased dice is rolled thrice, then the probability of getting a greater number in the $i^{\text{th}}$ roll than the number obtained in the $(i-1)^{\text{th}}$ roll, $i=2, 3$, is equal to

Let $A=\{(x, y): 2x+3y=23, x, y \in \mathbb{N}\}$ and $B=\{x: (x, y) \in A\}$. Then the number of one-one functions from $A$ to $B$ is equal to __________.

Consider the matrices : $A=\begin{bmatrix} 2 & -5 \\ 3 & m \end{bmatrix}$, $B=\begin{bmatrix} 20 \\ m \end{bmatrix}$ and $X=\begin{bmatrix} x \\ y \end{bmatrix}$. Let the set of all $m$, for which the system of equations $AX=B$ has a negative solution (i.e., $x<0$ and $y<0$), be the interval $(a, b)$. Then $8\int_{a}^{b}|A|\,dm$ is equal to ________.

The number of integers, between 100 and 1000 having the sum of their digits equals to 14, is ____________.

If $\left(\dfrac{1}{\alpha+1}+\dfrac{1}{\alpha+2}+\ldots\ldots+\dfrac{1}{\alpha+1012}\right)-\left(\dfrac{1}{2\cdot 1}+\dfrac{1}{4\cdot 3}+\dfrac{1}{6\cdot 5}+\ldots\ldots+\dfrac{1}{2024\cdot 2023}\right)=\dfrac{1}{2024}$, then $\alpha$ is equal to __________.

Let the set of all values of $p$, for which $f(x)=\left(p^2-6p+8\right)\left(\sin^2 2x-\cos^2 2x\right)+2(2-p)x+7$ does not have any critical point, be the interval $(a, b)$. Then $16ab$ is equal to ________.

For a differentiable function $f: \mathbb{R} \to \mathbb{R}$, suppose $f'(x)=3f(x)+\alpha$, where $\alpha \in \mathbb{R}$, $f(0)=1$ and $\lim\limits_{x \to -\infty} f(x)=7$. Then $9f\left(-\log_e 3\right)$ is equal to ________.

Consider the circle $C: x^2+y^2=4$ and the parabola $P: y^2=8x$. If the set of all values of $\alpha$, for which three chords of the circle $C$ on three distinct lines passing through the point $(\alpha, 0)$ are bisected by the parabola $P$ is the interval $(p, q)$, then $(2q-p)^2$ is equal to ________.

Let $A$, $B$ and $C$ be three points on the parabola $y^2=6x$ and let the line segment $AB$ meet the line $L$ through $C$ parallel to the $x$-axis at the point $D$. Let $M$ and $N$ respectively be the feet of the perpendiculars from $A$ and $B$ on $L$. Then $\left(\dfrac{AM \cdot BN}{CD}\right)^2$ is equal to ________.

The square of the distance of the image of the point $(6, 1, 5)$ in the line $\dfrac{x-1}{3}=\dfrac{y}{2}=\dfrac{z-2}{4}$, from the origin is ________.

Let the inverse trigonometric functions take principal values. The number of real solutions of the equation $2\sin^{-1}x+3\cos^{-1}x=\dfrac{2\pi}{5}$, is ________.