NTA JEE Main 2025 April 02 Shift 2

Let $$y = y(x)$$ be the solution of the differential equation $$\frac{dy}{dx} + 2y\sec^2 x = 2\sec^2 x + 3\tan x \cdot \sec^2 x$$ such that $$y(0) = \frac{5}{4}$$. Then $$12\left(y\left(\frac{\pi}{4}\right) - e^{-2}\right)$$ is equal to __________.

If the sum of the first 10 terms of the series $$\frac{4 \cdot 1}{1 + 4 \cdot 1^4} + \frac{4 \cdot 2}{1 + 4 \cdot 2^4} + \frac{4 \cdot 3}{1 + 4 \cdot 3^4} + \ldots$$ is $$\frac{m}{n}$$, where $$\gcd(m, n) = 1$$, then $$m + n$$ is equal to __________.

If $$y = \cos\left(\frac{\pi}{3} + \cos^{-1}\frac{x}{2}\right)$$, then $$(x - y)^2 + 3y^2$$ is equal to __________.

Let $$A(4, -2)$$, $$B(1, 1)$$ and $$C(9, -3)$$ be the vertices of a triangle ABC. Then the maximum area of the parallelogram AFDE, formed with vertices D, E and F on the sides BC, CA and AB of the triangle ABC respectively, is __________.

If the set of all $$a \in \mathbb{R} - \{1\}$$, for which the roots of the equation $$(1 - a)x^2 + 2(a - 3)x + 9 = 0$$ are positive is $$(-\infty, -\alpha] \cup [\beta, \gamma)$$, then $$2\alpha + \beta + \gamma$$ is equal to __________.

Given below are two statements : one is labelled as Assertion (A) and the other is labelled as Reason (R).

Assertion (A) : Net dipole moment of a polar linear isotropic dielectric substance is not zero even in the absence of an external electric field.

Reason (R) : In absence of an external electric field, the different permanent dipoles of a polar dielectric substance are oriented in random directions.

In the light of the above statements, choose the most appropriate answer from the options given below :

In a moving coil galvanometer, two moving coils $$M_1$$ and $$M_2$$ have the following particulars :
$$R_1 = 5\,\Omega$$, $$N_1 = 15$$, $$A_1 = 3.6 \times 10^{-3}\,\text{m}^2$$, $$B_1 = 0.25\,\text{T}$$
$$R_2 = 7\,\Omega$$, $$N_2 = 21$$, $$A_2 = 1.8 \times 10^{-3}\,\text{m}^2$$, $$B_2 = 0.50\,\text{T}$$
Assuming that torsional constant of the springs are same for both coils, what will be the ratio of voltage sensitivity of $$M_1$$ and $$M_2$$ ?

The moment of inertia of a circular ring of mass M and diameter r about a tangential axis lying in the plane of the ring is :

Two water drops each of radius 'r' coalesce to form a bigger drop. If 'T' is the surface tension, the surface energy released in this process is :

An electron with mass 'm' with an initial velocity $$(t = 0)$$ $$\vec{v} = v_0\hat{i}$$ $$(v_0 \gt 0)$$ enters a magnetic field $$\vec{B} = B_0\hat{j}$$. If the initial de-Broglie wavelength at $$t = 0$$ is $$\lambda_0$$ then its value after time 't' would be :