JEE Main 2024 Feb 1 Shift 1 — Mathematics

Let $S=\{z \in \mathbb{C} : |z-1|=1 \text{ and } (\sqrt{2}-1)(z+\bar{z}) - i(z-\bar{z}) = 2\sqrt{2}\}$. Let $z_1, z_2 \in S$ be such that $|z_1| = \max\limits_{z \in S} |z|$ and $|z_2| = \min\limits_{z \in S} |z|$. Then $\left|\sqrt{2}\, z_1 - z_2\right|^2$ equals :

Let $S=\{x \in \mathbb{R} : (\sqrt{3}+\sqrt{2})^x + (\sqrt{3}-\sqrt{2})^x = 10\}$. Then the number of elements in $S$ is :

If $A = \begin{bmatrix} \sqrt{2} & 1 \\ -1 & \sqrt{2} \end{bmatrix}$, $B = \begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix}$, $C = ABA^{T}$ and $X = A^{T}C^{2}A$, then $\det X$ is equal to :

If n is the number of ways five different employees can sit into four indistinguishable offices where any office may have any number of persons including zero, then n is equal to :

If the system of equations

$$2x + 3y - z = 5$$ $$x + \alpha y + 3z = -4$$ $$3x - y + \beta z = 7$$

has infinitely many solutions, then $13\,\alpha\beta$ is equal to ________.

Let $3, a, b, c$ be in A.P. and $3, a-1, b+1, c+9$ be in G.P. Then, the arithmetic mean of $a, b$ and $c$ is :

If $5f(x) + 4f\left(\dfrac{1}{x}\right) = x^2 - 2, \forall\, x \neq 0$ and $y = 9x^2 f(x)$, then $y$ is strictly increasing in :

Let $f: \mathbb{R} \to \mathbb{R}$ be defined as :

$$f(x) = \begin{cases} \dfrac{a - b\cos 2x}{x^2} & ;\ x < 0 \\ x^2 + cx + 2 & ;\ 0 \le x \le 1 \\ 2x + 1 & ;\ x > 1 \end{cases}$$

If $f$ is continuous everywhere in $\mathbb{R}$ and m is the number of points where $f$ is NOT differential then $m + a + b + c$ equals :

The area enclosed by the curves $xy + 4y = 16$ and $x + y = 6$ is equal to :

The value of the intergral $\displaystyle\int_0^{\pi/4} \dfrac{x\, dx}{\sin^4(2x) + \cos^4(2x)}$ equals :

Let $y = y(x)$ be the solution of the differential equation $\dfrac{dy}{dx} = 2x(x+y)^3 - x(x+y) - 1$, $y(0) = 1$.

Then, $\left(\dfrac{1}{\sqrt{2}} + y\left(\dfrac{1}{\sqrt{2}}\right)\right)^2$ equals :

For $0 < \theta < \pi/2$, if the eccentricity of the hyperbola $x^2 - y^2\operatorname{cosec}^2\theta = 5$ is $\sqrt{7}$ times eccentricity of the ellipse $x^2\operatorname{cosec}^2\theta + y^2 = 5$, then the value of $\theta$ is :

Let $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$, $a > b$ be an ellipse, whose eccentricity is $\dfrac{1}{\sqrt{2}}$ and the length of the latusrectum is $\sqrt{14}$. Then the square of the eccentricity of $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$ is :

Let $f: \mathbb{R} \to \mathbb{R}$ and $g: \mathbb{R} \to \mathbb{R}$ be defined as $f(x) = \begin{cases} \log_e x, & x > 0 \\ e^{-x}, & x \le 0 \end{cases}$ and $g(x) = \begin{cases} x, & x \ge 0 \\ e^{x}, & x < 0 \end{cases}$. Then, $gof: \mathbb{R} \to \mathbb{R}$ is :

Let $C: x^2 + y^2 = 4$ and $C': x^2 + y^2 - 4\lambda x + 9 = 0$ be two circles. If the set of all values of $\lambda$ so that the circles $C$ and $C'$ intersect at two distinct points, is $\mathbb{R} - [a, b]$, then the point $(8a + 12, 16b - 20)$ lies on the curve :

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

A bag contains 8 balls, whose colours are either white or black. 4 balls are drawn at random without replacement and it was found that 2 balls are white and other 2 balls are black. The probability that the bag contains equal number of white and black balls is :

Let the median and the mean deviation about the median of 7 observation 170, 125, 230, 190, 210, a, b be 170 and $\dfrac{205}{7}$ respectively. Then the mean deviation about the mean of these 7 observations is :

Let $\vec{a} = -5\hat{i} + \hat{j} - 3\hat{k}$, $\vec{b} = \hat{i} + 2\hat{j} - 4\hat{k}$ and $\vec{c} = \left(\left(\left(\vec{a} \times \vec{b}\right) \times \hat{i}\right) \times \hat{i}\right) \times \hat{i}$. Then $\vec{c} \cdot \left(-\hat{i} + \hat{j} + \hat{k}\right)$ is equal to :

If $\tan A = \dfrac{1}{\sqrt{x(x^2 + x + 1)}}$, $\tan B = \dfrac{\sqrt{x}}{\sqrt{x^2 + x + 1}}$ and

$\tan C = \left(x^{-3} + x^{-2} + x^{-1}\right)^{1/2}$, $0 < A, B, C < \dfrac{\pi}{2}$, then $A + B$ is equal to :

Let $A = \{1, 2, 3, \ldots, 20\}$. Let $R_1$ and $R_2$ two relation on $A$ such that $R_1 = \{(a, b) : b \text{ is divisible by } a\}$ $R_2 = \{(a, b) : a \text{ is an integral multiple of } b\}$. Then, number of elements in $R_1 - R_2$ is equal to ________.

Let $P = \{z \in \mathbb{C} : |z + 2 - 3i| \le 1\}$ and $Q = \{z \in \mathbb{C} : z(1 + i) + \bar{z}(1 - i) \le -8\}$. Let in $P \cap Q$, $|z - 3 + 2i|$ be maximum and minimum at $z_1$ and $z_2$ respectively. If $|z_1|^2 + 2|z_2|^2 = \alpha + \beta\sqrt{2}$, where $\alpha, \beta$ are integers, then $\alpha + \beta$ equals ________.

If the Coefficient of $x^{30}$ in the expansion of $\left(1 + \dfrac{1}{x}\right)^6 \left(1 + x^2\right)^7 \left(1 - x^3\right)^8$; $x \neq 0$ is $\alpha$, then $|\alpha|$ equals ________.

The number of elements in the set $S = \{(x, y, z) : x, y, z \in \mathbb{Z},\ x + 2y + 3z = 42,\ x, y, z \ge 0\}$ equals ________.

Let $3, 7, 11, 15, \ldots, 403$ and $2, 5, 8, 11, \ldots, 404$ be two arithmetic progressions. Then the sum, of the common terms in them, is equal to ________.

Let $\{x\}$ denote the fractional part of $x$ and $f(x) = \dfrac{\cos^{-1}\left(1 - \{x\}^2\right)\sin^{-1}(1 - \{x\})}{\{x\} - \{x\}^3}$, $x \neq 0$. If L and R respectively denotes the left hand limit and the right hand limit of $f(x)$ at $x = 0$, then $\dfrac{32}{\pi^2}\left(\mathrm{L}^2 + \mathrm{R}^2\right)$ is equal to ________.

If $\displaystyle\int_{-\pi/2}^{\pi/2} \dfrac{8\sqrt{2}\,\cos x\, dx}{\left(1 + e^{\sin x}\right)\left(1 + \sin^4 x\right)} = \alpha\pi + \beta \log_e\left(3 + 2\sqrt{2}\right)$, where $\alpha, \beta$ are integers, then $\alpha^2 + \beta^2$ equals ________.

If $x = x(t)$ is the solution of the differential equation $(t + 1)dx = \left(2x + (t + 1)^4\right)dt$, $x(0) = 2$, then, $x(1)$ equals ________.

Let the line $L: \sqrt{2}x + y = \alpha$ pass through the point of the intersection P (in the first quadrant) of the circle $x^2 + y^2 = 3$ and the parabola $x^2 = 2y$. Let the line L touch two circles $C_1$ and $C_2$ of equal radius $2\sqrt{3}$. If the centres $Q_1$ and $Q_2$ of the circles $C_1$ and $C_2$ lie on the $y$-axis, then the square of the area of the triangle $PQ_1Q_2$ is equal to ________.

Let the line of the shortest distance between the lines

$$L_1: \vec{r} = \left(\hat{i} + 2\hat{j} + 3\hat{k}\right) + \lambda\left(\hat{i} - \hat{j} + \hat{k}\right)$$ and

$$L_2: \vec{r} = \left(4\hat{i} + 5\hat{j} + 6\hat{k}\right) + \mu\left(\hat{i} + \hat{j} - \hat{k}\right)$$

intersect $L_1$ and $L_2$ at P and Q respectively. If $(\alpha, \beta, \gamma)$ is the mid point of the line segment PQ, then $2(\alpha + \beta + \gamma)$ is equal to ________.