NTA JEE Main 2025 April 02 Shift 1

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

Let $$f : \mathbb{R} \to \mathbb{R}$$ be a thrice differentiable odd function satisfying $$f'(x) \geq 0$$, $$f''(x) = f(x)$$, $$f(0) = 0$$, $$f'(0) = 3$$. Then $$9f(\log_e 3)$$ is equal to _____.

If the area of the region $$\{(x,y) : |4 - x^2| \leq y \leq x^2, y \leq 4, x \geq 0\}$$ is $$\left(\frac{80\sqrt{2}}{\alpha} - \beta\right)$$, $$\alpha, \beta \in \mathbb{N}$$, then $$\alpha + \beta$$ 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 $$\frac{m}{n}$$, $$\gcd(m, n) = 1$$, then $$m + n$$ 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 _____.

A light wave is propagating with plane wave fronts of the type $$x + y + z$$ = constant. The angle made by the direction of wave propagation with the x-axis is:

The equation for real gas is given by $$\left(P + \frac{a}{V^2}\right)(V - b) = RT$$, where P, V, T and R are the pressure, volume, temperature and gas constant, respectively. The dimension of $$ab^{-2}$$ is equivalent to that of:

A cord of negligible mass is wound around the rim of a wheel supported by spokes with negligible mass. The mass of wheel is 10 kg and radius is 10 cm and it can freely rotate without any friction. Initially the wheel is at rest. If a steady pull of 20 N is applied on the cord, the angular velocity of the wheel, after the cord is unwound by 1 m, would be:

A slanted object AB is placed on one side of convex lens as shown in the diagram. The image is formed on the opposite side. Angle made by the image with principal axis is:

Consider two infinitely large plane parallel conducting plates as shown below. The plates are uniformly charged with a surface charge density $$+\sigma$$ and $$-2\sigma$$. The force experienced by a point charge $$+q$$ placed at the mid point between two plates will be: