JEE Main 2024 Jan 27 Shift 1 — Mathematics

Let $S = \{1, 2, 3, \ldots, 10\}$. Suppose $M$ is the set of all the subsets of $S$, then the relation $R = \{(A, B) : A \cap B \neq \phi;\ A, B \in M\}$ is :

The function $f : \mathbf{N} - \{1\} \to \mathbf{N}$; defined by $f(n) =$ the highest prime factor of $n$, is :

If $S = \{z \in \mathbf{C} : |z - i| = |z + i| = |z - 1|\}$, then, $n(S)$ is :

Consider the matrix $f(x) = \begin{bmatrix} \cos x & -\sin x & 0 \\ \sin x & \cos x & 0 \\ 0 & 0 & 1 \end{bmatrix}$.

Given below are two statements :

Statement I : $f(-x)$ is the inverse of the matrix $f(x)$.

Statement II : $f(x)\, f(y) = f(x + y)$.

In the light of the above statements, choose the correct answer from the options given below

$^{n-1}C_r = (k^2 - 8)\ ^nC_{r+1}$ if and only if :

If A denotes the sum of all the coefficients in the expansion of $(1 - 3x + 10x^2)^n$ and B denotes the sum of the coefficients in the expansion of $(1 + x^2)^n$, then :

The number of common terms in the progressions $4, 9, 14, 19, \ldots$, up to $25^{\text{th}}$ term and $3, 6, 9, 12, \ldots$, up to $37^{\text{th}}$ term is :

Consider the function.

$$f(x) = \begin{cases} \dfrac{a(7x - 12 - x^2)}{b|x^2 - 7x + 12|} & , \ x < 3 \\ 2^{\frac{\sin(x-3)}{x - [x]}} & , \ x > 3 \\ b & , \ x = 3 , \end{cases}$$

where $[x]$ denotes the greatest integer less than or equal to $x$. If S denotes the set of all ordered pairs (a, b) such that $f(x)$ is continuous at $x = 3$, then the number of elements in S is :

If $a = \lim\limits_{x \to 0} \dfrac{\sqrt{1 + \sqrt{1 + x^4}} - \sqrt{2}}{x^4}$ and $b = \lim\limits_{x \to 0} \dfrac{\sin^2 x}{\sqrt{2} - \sqrt{1 + \cos x}}$, then the value of $ab^3$ is :

If the shortest distance of the parabola $y^2 = 4x$ from the centre of the circle $x^2 + y^2 - 4x - 16y + 64 = 0$ is $d$, then $d^2$ is equal to :

If $\displaystyle\int_0^1 \frac{1}{\sqrt{3 + x} + \sqrt{1 + x}}\, dx = a + b\sqrt{2} + c\sqrt{3}$, where a, b, c are rational numbers, then $2a + 3b - 4c$ is equal to :

If (a, b) be the orthocentre of the triangle whose vertices are (1, 2), (2, 3) and (3, 1), and $I_1 = \displaystyle\int_a^b x \sin(4x - x^2)\, dx$, $I_2 = \displaystyle\int_a^b \sin(4x - x^2)\, dx$, then $36 \dfrac{I_1}{I_2}$ is equal to :

Let $x = x(t)$ and $y = y(t)$ be solutions of the differential equations $\dfrac{dx}{dt} + ax = 0$ and $\dfrac{dy}{dt} + by = 0$ respectively, $a, b \in \mathbf{R}$. Given that $x(0) = 2$; $y(0) = 1$ and $3y(1) = 2x(1)$, the value of $t$, for which $x(t) = y(t)$, is :

Four distinct points $(2k, 3k)$, $(1, 0)$, $(0, 1)$ and $(0, 0)$ lie on a circle for $k$ equal to :

The portion of the line $4x + 5y = 20$ in the first quadrant is trisected by the lines $L_1$ and $L_2$ passing through the origin. The tangent of an angle between the lines $L_1$ and $L_2$ is :

The length of the chord of the ellipse $\dfrac{x^2}{25} + \dfrac{y^2}{16} = 1$, whose mid point is $\left(1, \dfrac{2}{5}\right)$, is equal to :

The distance, of the point $(7, -2, 11)$ from the line $\dfrac{x - 6}{1} = \dfrac{y - 4}{0} = \dfrac{z - 8}{3}$ along the line $\dfrac{x - 5}{2} = \dfrac{y - 1}{-3} = \dfrac{z - 5}{6}$, is :

If the shortest distance between the lines $\dfrac{x - 4}{1} = \dfrac{y + 1}{2} = \dfrac{z}{-3}$ and $\dfrac{x - \lambda}{2} = \dfrac{y + 1}{4} = \dfrac{z - 2}{-5}$ is $\dfrac{6}{\sqrt{5}}$, then the sum of all possible values of $\lambda$ is :

Let $\vec{a} = \hat{i} + 2\hat{j} + \hat{k}$, $\vec{b} = 3(\hat{i} - \hat{j} + \hat{k})$. Let $\vec{c}$ be the vector such that $\vec{a} \times \vec{c} = \vec{b}$ and $\vec{a} \cdot \vec{c} = 3$. Then $\vec{a} \cdot \left( (\vec{c} \times \vec{b}) - \vec{b} - \vec{c} \right)$ is equal to :

Let $a_1, a_2, \ldots, a_{10}$ be 10 observations such that $\sum\limits_{k=1}^{10} a_k = 50$ and $\sum\limits_{\forall\, k < j} a_k \cdot a_j = 1100$. Then the standard deviation of $a_1, a_2, \ldots, a_{10}$ is equal to :

If $\alpha$ satisfies the equation $x^2 + x + 1 = 0$ and $(1 + \alpha)^7 = A + B\alpha + C\alpha^2$, $A, B, C \geq 0$, then $5(3A - 2B - C)$ is equal to ________.

Let $A = \begin{bmatrix} 2 & 0 & 1 \\ 1 & 1 & 0 \\ 1 & 0 & 1 \end{bmatrix}$, $B = [B_1, B_2, B_3]$, where $B_1, B_2, B_3$ are column matrics, and

$$AB_1 = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix},\ AB_2 = \begin{bmatrix} 2 \\ 3 \\ 0 \end{bmatrix},\ AB_3 = \begin{bmatrix} 3 \\ 2 \\ 1 \end{bmatrix}$$

If $\alpha = |B|$ and $\beta$ is the sum of all the diagonal elements of $B$, then $\alpha^3 + \beta^3$ is equal to ________.

Let for a differentiable function $f : (0, \infty) \to \mathbf{R}$, $f(x) - f(y) \geq \log_e\left(\dfrac{x}{y}\right) + x - y,\ \forall\ x, y \in (0, \infty)$.

Then $\sum\limits_{n=1}^{20} f'\left(\dfrac{1}{n^2}\right)$ is equal to ________.

If $8 = 3 + \dfrac{1}{4}(3 + p) + \dfrac{1}{4^2}(3 + 2p) + \dfrac{1}{4^3}(3 + 3p) + \cdots \infty$, then the value of $p$ is ________.

Let $f(x) = x^3 + x^2 f'(1) + x f''(2) + f'''(3)$, $x \in \mathbf{R}$. Then $f'(10)$ is equal to ________.

Let the area of the region $\{(x, y) : x - 2y + 4 \geq 0,\ x + 2y^2 \geq 0,\ x + 4y^2 \leq 8,\ y \geq 0\}$ be $\dfrac{m}{n}$, where $m$ and $n$ are coprime numbers. Then $m + n$ is equal to ________.

If the solution of the differential equation $(2x + 3y - 2)\, dx + (4x + 6y - 7)\, dy = 0$, $y(0) = 3$, is $\alpha x + \beta y + 3 \log_e |2x + 3y - \gamma| = 6$, then $\alpha + 2\beta + 3\gamma$ is equal to ________.

The least positive integral value of $\alpha$, for which the angle between the vectors $\alpha\hat{i} - 2\hat{j} + 2\hat{k}$ and $\alpha\hat{i} + 2\alpha\hat{j} - 2\hat{k}$ is acute, is ________.

A fair die is tossed repeatedly until a six is obtained. Let X denote the number of tosses required and let $a = P(X = 3)$, $b = P(X \geq 3)$ and $c = P(X \geq 6 \mid X > 3)$. Then $\dfrac{b + c}{a}$ is equal to ________.

Let the set of all $a \in \mathbf{R}$ such that the equation $\cos 2x + a \sin x = 2a - 7$ has a solution be $[p, q]$ and $r = \tan 9° - \tan 27° - \dfrac{1}{\cot 63°} + \tan 81°$, then $pqr$ is equal to ________.