NTA JEE Main 2025 April 02 Shift 1

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

Let one focus of the hyperbola H: $$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$$ be at $$(\sqrt{10}, 0)$$ 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:

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 $$f : \mathbb{R} \to \mathbb{R}$$ 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 $$x, y \in \mathbb{R}$$. If $$f'(0) = \frac{1}{2}$$, then the value of $$24 f''\left(\frac{5\pi}{3}\right)$$ is:

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

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:

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

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

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:

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