NTA JEE Main 2025 April 3rd Shift 1

If $$y(x) = \begin{vmatrix} \sin x & \cos x & \sin x + \cos x + 1 \\ 27 & 28 & 27 \\ 1 & 1 & 1 \end{vmatrix}$$, $$x \in \mathbb{R}$$, then $$\frac{d^2y}{dx^2} + y$$ is equal to

Let g be a differentiable function such that $$\int_0^x g(t)\,dt = x - \int_0^x tg(t)\,dt$$, $$x \geq 0$$ and let $$y = y(x)$$ satisfy the differential equation $$\frac{dy}{dx} - y\tan x = 2(x+1)\sec x \cdot g(x)$$, $$x \in \left[0, \frac{\pi}{2}\right)$$. If $$y(0) = 0$$, then $$y\left(\frac{\pi}{3}\right)$$ is equal to

A line passes through the origin and makes equal angles with the positive coordinate axes. It intersects the lines $$L_1 : 2x + y + 6 = 0$$ and $$L_2 : 4x + 2y - p = 0$$, $$p \gt 0$$, at the points A and B, respectively. If $$AB = \frac{9}{\sqrt{2}}$$ and the foot of the perpendicular from the point A on the line $$L_2$$ is M, then $$\frac{AM}{BM}$$ is equal to

Let $$z \in \mathbb{C}$$ be such that $$\frac{z^2 + 3i}{z - 2 + i} = 2 + 3i$$. Then the sum of all possible values of $$z^2$$ is

Let $$f(x) = \int x^3\sqrt{3 - x^2}\,dx$$. If $$5f(\sqrt{2}) = -4$$, then $$f(1)$$ is equal to

Let $$a_1, a_2, a_3, \ldots$$ be a G.P. of increasing positive numbers. If $$a_3 a_5 = 729$$ and $$a_2 + a_4 = \frac{111}{4}$$, then $$24(a_1 + a_2 + a_3)$$ is equal to

Let the domain of the function $$f(x) = \log_2 \log_4 \log_6 (3 + 4x - x^2)$$ be $$(a, b)$$. If $$\int_0^{b-a} [x^2]\,dx = p - \sqrt{q} - \sqrt{r}$$, $$p, q, r \in \mathbb{N}$$, $$\gcd(p, q, r) = 1$$, then $$p + q + r$$ is equal to

The radius of the smallest circle which touches the parabolas $$y = x^2 + 2$$ and $$x = y^2 + 2$$ is

Let $$f(x) = \begin{cases} (1 + ax)^{1/x}, & x \lt 0 \\ 1 + b, & x = 0 \\ \frac{(x+4)^{1/2} - 2}{(x+c)^{1/3} - 2}, & x \gt 0 \end{cases}$$ be continuous at $$x = 0$$. Then $$e^a bc$$ is equal to

Line $$L_1$$ passes through the point $$(1, 2, 3)$$ and is parallel to z-axis. Line $$L_2$$ passes through the point $$(\lambda, 5, 6)$$ and is parallel to y-axis. Let for $$\lambda = \lambda_1, \lambda_2$$, $$\lambda_2 \lt \lambda_1$$, the shortest distance between the two lines be 3. Then the square of the distance of the point $$(\lambda_1, \lambda_2, 7)$$ from the line $$L_1$$ is