JEE Main 2024 Apr 9 Shift 1 — Mathematics

If the domain of the function $f(x)=\sin^{-1}\left(\frac{x-1}{2x+3}\right)$ is $\mathbf{R}-(\alpha, \beta)$, then $12\alpha\beta$ is equal to :

Let $\alpha, \beta$ be the roots of the equation $x^2+2\sqrt{2}\,x-1=0$. The quadratic equation, whose roots are $\alpha^4+\beta^4$ and $\frac{1}{10}\left(\alpha^6+\beta^6\right)$, is :

Let $\lambda, \mu \in \mathbf{R}$. If the system of equations

$$3x+5y+\lambda z=3$$ $$7x+11y-9z=2$$ $$97x+155y-189z=\mu$$

has infinitely many solutions, then $\mu+2\lambda$ is equal to :

The coefficient of $x^{70}$ in $x^2(1+x)^{98}+x^3(1+x)^{97}+x^4(1+x)^{96}+\ldots+x^{54}(1+x)^{46}$ is $^{99}C_p-{}^{46}C_q$. Then a possible value of $p+q$ is :

If the sum of the series $\frac{1}{1\cdot(1+\mathrm{d})}+\frac{1}{(1+\mathrm{d})(1+2\mathrm{d})}+\ldots+\frac{1}{(1+9\mathrm{d})(1+10\mathrm{d})}$ is equal to 5, then $50\mathrm{d}$ is equal to :

Let $f(x)=ax^3+bx^2+cx+41$ be such that $f(1)=40$, $f'(1)=2$ and $f''(1)=4$. Then $a^2+b^2+c^2$ is equal to :

A variable line L passes through the point $(3, 5)$ and intersects the positive coordinate axes at the points A and B. The minimum area of the triangle OAB, where O is the origin, is :

Let $\int \frac{2-\tan x}{3+\tan x}\,\mathrm{d}x = \frac{1}{2}\left(\alpha x+\log_{\mathrm{e}}|\beta \sin x+\gamma \cos x|\right)+\mathrm{C}$, where C is the constant of integration. Then $\alpha+\frac{\gamma}{\beta}$ is equal to :

The parabola $y^2=4x$ divides the area of the circle $x^2+y^2=5$ in two parts. The area of the smaller part is equal to :

The solution of the differential equation $(x^2+y^2)\mathrm{d}x-5xy\,\mathrm{d}y=0$, $y(1)=0$, is :

The solution curve, of the differential equation $2y\,\frac{\mathrm{d}y}{\mathrm{d}x}+3=5\,\frac{\mathrm{d}y}{\mathrm{d}x}$, passing through the point $(0, 1)$ is a conic, whose vertex lies on the line :

Let $f(x)=x^2+9$, $g(x)=\frac{x}{x-9}$ and $\mathrm{a}=f{\circ}g(10)$, $\mathrm{b}=g{\circ}f(3)$. If e and $l$ denote the eccentricity and the length of the latus rectum of the ellipse $\frac{x^2}{\mathrm{a}}+\frac{y^2}{\mathrm{b}}=1$, then $8\mathrm{e}^2+l^2$ is equal to.

Let a circle passing through $(2, 0)$ have its centre at the point $(\mathrm{h}, \mathrm{k})$. Let $(x_{\mathrm{c}}, y_{\mathrm{c}})$ be the point of intersection of the lines $3x+5y=1$ and $(2+\mathrm{c})x+5\mathrm{c}^2y=1$. If $\mathrm{h}=\lim\limits_{\mathrm{c}\to 1} x_{\mathrm{c}}$ and $\mathrm{k}=\lim\limits_{\mathrm{c}\to 1} y_{\mathrm{c}}$, then the equation of the circle is :

A ray of light coming from the point $\mathrm{P}(1, 2)$ gets reflected from the point Q on the $x$-axis and then passes through the point $\mathrm{R}(4, 3)$. If the point $\mathrm{S}(\mathrm{h}, \mathrm{k})$ is such that PQRS is a parallelogram, then $\mathrm{hk}^2$ is equal to :

Let the line L intersect the lines $x-2=-y=z-1$, $2(x+1)=2(y-1)=z+1$ and be parallel to the line $\frac{x-2}{3}=\frac{y-1}{1}=\frac{z-2}{2}$. Then which of the following points lies on L?

The shortest distance between the lines $\frac{x-3}{4}=\frac{y+7}{-11}=\frac{z-1}{5}$ and $\frac{x-5}{3}=\frac{y-9}{-6}=\frac{z+2}{1}$ is :

Let $\vec{\mathrm{OA}}=2\vec{a}$, $\vec{\mathrm{OB}}=6\vec{a}+5\vec{b}$ and $\vec{\mathrm{OC}}=3\vec{b}$, where O is the origin. If the area of the parallelogram with adjacent sides $\vec{\mathrm{OA}}$ and $\vec{\mathrm{OC}}$ is 15 sq. units, then the area (in sq. units) of the quadrilateral OABC is equal to :

Let three vectors $\vec{a}=\alpha\hat{i}+4\hat{j}+2\hat{k}$, $\vec{b}=5\hat{i}+3\hat{j}+4\hat{k}$, $\vec{c}=x\hat{i}+y\hat{j}+z\hat{k}$ form a triangle such that $\vec{c}=\vec{a}-\vec{b}$ and the area of the triangle is $5\sqrt{6}$. If $\alpha$ is a positive real number, then $|\vec{c}|^2$ is equal to :

The frequency distribution of the age of students in a class of 40 students is given below.

Age	15	16	17	18	19	20
No of Students	5	8	5	12	$x$	$y$

If the mean deviation about the median is 1.25, then $4x+5y$ is equal to :

Let $|\cos\theta\,\cos(60-\theta)\,\cos(60+\theta)|\le\frac{1}{8}$, $\theta\in[0, 2\pi]$. Then, the sum of all $\theta\in[0, 2\pi]$, where $\cos 3\theta$ attains its maximum value, is :

Let $\mathrm{A}=\{2, 3, 6, 7\}$ and $\mathrm{B}=\{4, 5, 6, 8\}$. Let R be a relation defined on $\mathrm{A}\times\mathrm{B}$ by $(\mathrm{a}_1, \mathrm{b}_1)\,\mathrm{R}\,(\mathrm{a}_2, \mathrm{b}_2)$ if and only if $\mathrm{a}_1+\mathrm{a}_2=\mathrm{b}_1+\mathrm{b}_2$. Then the number of elements in R is ________.

The sum of the square of the modulus of the elements in the set $\{z=\mathrm{a}+\mathrm{ib} : \mathrm{a}, \mathrm{b}\in\mathbf{Z},\ z\in\mathbf{C},\ |z-1|\le 1,\ |z-5|\le|z-5\mathrm{i}|\}$ is ________.

Let A be a non-singular matrix of order 3. If $\det(3\,\mathrm{adj}(2\,\mathrm{adj}((\det \mathrm{A})\mathrm{A})))=3^{-13}\cdot 2^{-10}$ and $\det(3\,\mathrm{adj}(2\mathrm{A}))=2^{\mathrm{m}}\cdot 3^{\mathrm{n}}$, then $|3\mathrm{m}+2\mathrm{n}|$ is equal to ________.

Let $f:(0, \pi)\to\mathbf{R}$ be a function given by $f(x)=\begin{cases}\left(\frac{8}{7}\right)^{\frac{\tan 8x}{\tan 7x}}, & 0<x<\frac{\pi}{2}\\ \mathrm{a}-8, & x=\frac{\pi}{2}\\ \left(1+|\cot x|\right)^{\frac{\mathrm{b}}{\mathrm{a}}|\tan x|}, & \frac{\pi}{2}<x<\pi\end{cases}$

where $\mathrm{a}, \mathrm{b}\in\mathbf{Z}$. If $f$ is continuous at $x=\frac{\pi}{2}$, then $\mathrm{a}^2+\mathrm{b}^2$ is equal to ________.

The remainder when $428^{2024}$ is divided by 21 is ________.

If a function $f$ satisfies $f(\mathrm{m}+\mathrm{n})=f(\mathrm{m})+f(\mathrm{n})$ for all $\mathrm{m}, \mathrm{n}\in\mathbf{N}$ and $f(1)=1$, then the largest natural number $\lambda$ such that $\sum\limits_{\mathrm{k}=1}^{2022} f(\lambda+\mathrm{k})\le(2022)^2$ is equal to ________.

Let the set of all positive values of $\lambda$, for which the point of local minimum of the function $(1+x(\lambda^2-x^2))$ satisfies $\frac{x^2+x+2}{x^2+5x+6}<0$, be $(\alpha, \beta)$. Then $\alpha^2+\beta^2$ is equal to ________.

Let $\lim\limits_{\mathrm{n}\to\infty}\left(\frac{\mathrm{n}}{\sqrt{\mathrm{n}^4+1}}-\frac{2\mathrm{n}}{(\mathrm{n}^2+1)\sqrt{\mathrm{n}^4+1}}+\frac{\mathrm{n}}{\sqrt{\mathrm{n}^4+16}}-\frac{8\mathrm{n}}{(\mathrm{n}^2+4)\sqrt{\mathrm{n}^4+16}}+\ldots+\frac{\mathrm{n}}{\sqrt{\mathrm{n}^4+\mathrm{n}^4}}-\frac{2\mathrm{n}\cdot\mathrm{n}^2}{(\mathrm{n}^2+\mathrm{n}^2)\sqrt{\mathrm{n}^4+\mathrm{n}^4}}\right)$ be $\frac{\pi}{\mathrm{k}}$, using only the principal values of the inverse trigonometric functions. Then $\mathrm{k}^2$ is equal to ________.

Let the centre of a circle, passing through the points $(0, 0)$, $(1, 0)$ and touching the circle $x^2+y^2=9$, be $(\mathrm{h}, \mathrm{k})$. Then for all possible values of the coordinates of the centre $(\mathrm{h}, \mathrm{k})$, $4(\mathrm{h}^2+\mathrm{k}^2)$ is equal to ________.

Let a, b and c denote the outcome of three independent rolls of a fair tetrahedral die, whose four faces are marked 1, 2, 3, 4. If the probability that $ax^2+bx+c=0$ has all real roots is $\frac{\mathrm{m}}{\mathrm{n}}$, $\gcd(\mathrm{m}, \mathrm{n})=1$, then $\mathrm{m}+\mathrm{n}$ is equal to ________.