JEE Main 2024 Apr 5 Shift 1 — Mathematics

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

Consider the following two statements :

Statement I : For any two non-zero complex numbers $z_1, z_2$, $$(|z_1| + |z_2|)\left|\frac{z_1}{|z_1|} + \frac{z_2}{|z_2|}\right| \le 2\,(|z_1| + |z_2|),$$ and

Statement II : If $x, y, z$ are three distinct complex numbers and $a, b, c$ are three positive real numbers such that $\frac{a}{|y-z|} = \frac{b}{|z-x|} = \frac{c}{|x-y|}$, then $$\frac{a^2}{y-z} + \frac{b^2}{z-x} + \frac{c^2}{x-y} = 1.$$

Between the above two statements,

Let A and B be two square matrices of order 3 such that $|A| = 3$ and $|B| = 2$. Then $\left|A^T A (\operatorname{adj}(2A))^{-1} (\operatorname{adj}(4B)) (\operatorname{adj}(AB))^{-1} A A^T\right|$ is equal to :

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

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

(I) $f$ is increasing in $\left(0, \frac{\pi}{2}\right)$.

(II) $f'$ is decreasing in $\left(0, \frac{\pi}{2}\right)$.

Between the above two statements,

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

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

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

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 $(a+b)^2$ is equal to :

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

The integral $\displaystyle\int_{0}^{\pi/4} \frac{136 \sin x}{3 \sin x + 5 \cos x}\, \mathrm{d}x$ is equal to :

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

Let two straight lines drawn from the origin O intersect the line $3x + 4y = 12$ at the points P and Q such that $\triangle OPQ$ is an isosceles triangle and $\angle POQ = 90°$. If $l = OP^2 + PQ^2 + QO^2$, then the greatest integer less than or equal to $l$ is :

Let the line $2x + 3y - k = 0$, $k > 0$, intersect the $x$-axis and $y$-axis at the points A and B, respectively. If the equation of the circle having the line segment AB as a diameter is $x^2 + y^2 - 3x - 2y = 0$ and the length of the latus rectum of the ellipse $x^2 + 9y^2 = k^2$ is $\dfrac{\mathrm{m}}{\mathrm{n}}$, where m and n are coprime, then $2\mathrm{m} + \mathrm{n}$ is equal to

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

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

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

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

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

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

The number of distinct real roots of the equation $|x|\,|x+2| - 5|x+1| - 1 = 0$ is ________.

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

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

Let $a_1, a_2, a_3, \ldots$ be in an arithmetic progression of positive terms.

Let $A_k = a_1^2 - a_2^2 + a_3^2 - a_4^2 + \ldots + a_{2k-1}^2 - a_{2k}^2$.

If $A_3 = -153$, $A_5 = -435$ and $a_1^2 + a_2^2 + a_3^2 = 66$, then $a_{17} - A_7$ is equal to ________.

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

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

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

Let $\vec{a} = \hat{i} - 3\hat{j} + 7\hat{k}$, $\vec{b} = 2\hat{i} - \hat{j} + \hat{k}$ and $\vec{c}$ be a vector such that $(\vec{a} + 2\vec{b}) \times \vec{c} = 3(\vec{c} \times \vec{a})$. If $\vec{a} \cdot \vec{c} = 130$, then $\vec{b} \cdot \vec{c}$ is equal to ________.

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 $96\sigma^2$ is equal to ________.