JEE Main 2025 Apr 2 Shift 1 — Mathematics

Let the focal chord PQ of the parabola $y^2 = 4x$ make an angle of $60°$ with the positive $x$-axis, where P lies in the first quadrant. If the circle, whose one diameter is PS, S being the focus of the parabola, touches the $y$-axis at the point $(0, \alpha)$, then $5\alpha^2$ is equal to :

Let $z$ be a complex number such that $|z| = 1$. If $\frac{2 + k^2 z}{k + \bar{z}} = kz$, $k \in \mathbf{R}$, then the maximum distance of $k + ik^2$ from the circle $|z - (1 + 2i)| = 1$ is :

If $\theta \in [-2\pi, 2\pi]$, then the number of solutions of $2\sqrt{2}\cos^2\theta + \left(2 - \sqrt{6}\right)\cos\theta - \sqrt{3} = 0$, is equal to :

If S and S' are the foci of the ellipse $\frac{x^2}{18} + \frac{y^2}{9} = 1$ and P be a point on the ellipse, then $\min(\mathrm{SP} \cdot \mathrm{S'P}) + \max(\mathrm{SP} \cdot \mathrm{S'P})$ is equal to :

Let ABCD be a tetrahedron such that the edges AB, AC and AD are mutually perpendicular. Let the areas of the triangles ABC, ACD and ADB be 5, 6 and 7 square units respectively. Then the area (in square units) of the $\Delta$BCD is equal to :

The largest $n \in \mathbf{N}$ such that $3^n$ divides $50!$ is :

For $\alpha, \beta, \gamma \in \mathbf{R}$, if $\lim_{x \to 0} \frac{x^2 \sin\alpha x + (\gamma - 1)\mathrm{e}^{x^2}}{\sin 2x - \beta x} = 3$, then $\beta + \gamma - \alpha$ is equal to :

If the function $f(x) = 2x^3 - 9ax^2 + 12a^2x + 1$, where $a > 0$, attains its local maximum and local minimum values at $p$ and $q$, respectively, such that $p^2 = q$, then $f(3)$ is equal to :

The number of sequences of ten terms, whose terms are either 0 or 1 or 2, that contain exactly five 1s and exactly three 2s, is equal to :

Let one focus of the hyperbola $\mathrm{H}: \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ be at $\left(\sqrt{10}, 0\right)$ and the corresponding directrix be $x = \frac{9}{\sqrt{10}}$. If $e$ and $l$ respectively are the eccentricity and the length of the latus rectum of H, then $9(e^2 + l)$ is equal to :

If $\vec{a}$ is a nonzero vector such that its projections on the vectors $2\hat{i} - \hat{j} + 2\hat{k}$, $\hat{i} + 2\hat{j} - 2\hat{k}$ and $\hat{k}$ are equal, then a unit vector along $\vec{a}$ is :

Let A be the set of all functions $f: \mathbf{Z} \to \mathbf{Z}$ and R be a relation on A such that $\mathrm{R} = \{(\mathrm{f}, \mathrm{g}) : f(0) = g(1) \text{ and } f(1) = g(0)\}$. Then R is :

Let the vertices Q and R of the triangle PQR lie on the line $\frac{x+3}{5} = \frac{y-1}{2} = \frac{z+4}{3}$, $\mathrm{QR} = 5$ and the coordinates of the point P be $(0, 2, 3)$. If the area of the triangle PQR is $\frac{\mathrm{m}}{\mathrm{n}}$ then :

Let $f: \mathbf{R} \to \mathbf{R}$ be a twice differentiable function such that $(\sin x \cos y)(f(2x + 2y) - f(2x - 2y)) = (\cos x \sin y)(f(2x + 2y) + f(2x - 2y))$, for all $x, y \in \mathbf{R}$. If $f'(0) = \frac{1}{2}$, then the value of $24 f''\left(\frac{5\pi}{3}\right)$ is :

Let $\mathrm{a} \in \mathbf{R}$ and A be a matrix of order $3 \times 3$ such that $\det(\mathrm{A}) = -4$ and $\mathrm{A} + \mathrm{I} = \begin{bmatrix} 1 & \mathrm{a} & 1 \\ 2 & 1 & 0 \\ \mathrm{a} & 1 & 2 \end{bmatrix}$, where I is the identity matrix of order $3 \times 3$. If $\det((\mathrm{a}+1)\operatorname{adj}((\mathrm{a}-1)\mathrm{A}))$ is $2^\mathrm{m} 3^\mathrm{n}$, $\mathrm{m}, \mathrm{n} \in \{0, 1, 2, \ldots, 20\}$, then $\mathrm{m} + \mathrm{n}$ is equal to :

The term independent of $x$ in the expansion of $\left(\frac{(x+1)}{\left(x^{2/3} + 1 - x^{1/3}\right)} - \frac{(x-1)}{\left(x - x^{1/2}\right)}\right)^{10}$, $x > 1$, is :

Let $\mathrm{P_n} = \alpha^\mathrm{n} + \beta^\mathrm{n}$, $\mathrm{n} \in \mathbf{N}$. If $\mathrm{P_{10}} = 123$, $\mathrm{P_9} = 76$, $\mathrm{P_8} = 47$ and $\mathrm{P_1} = 1$, then the quadratic equation having roots $\frac{1}{\alpha}$ and $\frac{1}{\beta}$ is :

Let $\mathrm{A} = \begin{bmatrix} \alpha & -1 \\ 6 & \beta \end{bmatrix}$, $\alpha > 0$, such that $\det(\mathrm{A}) = 0$ and $\alpha + \beta = 1$. If I denotes $2 \times 2$ identity matrix, then the matrix $(\mathrm{I} + \mathrm{A})^8$ is :

Let $\mathrm{a_1}, \mathrm{a_2}, \mathrm{a_3}, \ldots$ be in an A.P. such that $\sum_{\mathrm{k}=1}^{12} \mathrm{a_{2k-1}} = -\frac{72}{5}\mathrm{a_1}$, $\mathrm{a_1} \neq 0$. If $\sum_{\mathrm{k}=1}^{\mathrm{n}} \mathrm{a_k} = 0$, then n is :

If the system of linear equations

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

has infinitely many solutions, then the value of $22\beta - 9\alpha$ is :

If the area of the region $\left\{(x, y) : \left|4 - x^2\right| \leq y \leq x^2, \ y \leq 4, \ x \geq 0\right\}$ is $\left(\frac{80\sqrt{2}}{\alpha} - \beta\right)$, $\alpha, \beta \in \mathbf{N}$, then $\alpha + \beta$ is equal to __________.

The absolute difference between the squares of the radii of the two circles passing through the point $(-9, 4)$ and touching the lines $x + y = 3$ and $x - y = 3$, is equal to __________.

Let $[\cdot]$ denote the greatest integer function. If $\int_0^{\mathrm{e}^3} \left[\frac{1}{\mathrm{e}^{x-1}}\right] \mathrm{d}x = \alpha - \log_\mathrm{e} 2$, then $\alpha^3$ is equal to __________.

Let $f : \mathbf{R} \to \mathbf{R}$ be a thrice differentiable odd function satisfying $f'(x) \geq 0$, $f''(x) = f(x)$, $f(0) = 0$, $f'(0) = 3$. Then $9f(\log_\mathrm{e} 3)$ is equal to __________.

Three distinct numbers are selected randomly from the set $\{1, 2, 3, \ldots, 40\}$. If the probability, that the selected numbers are in an increasing G.P., is $\frac{\mathrm{m}}{\mathrm{n}}$, $\gcd(\mathrm{m}, \mathrm{n}) = 1$, then $\mathrm{m} + \mathrm{n}$ is equal to __________.