NTA JEE Main 2025 April 4th Shift 1

If the area of the region $$\{(x, y) : |x - 5| \le y \le 4\sqrt{x}\}$$ is A, then 3A is equal to ________.

Let $$A = \begin{bmatrix} \cos\theta & 0 & -\sin\theta \\ 0 & 1 & 0 \\ \sin\theta & 0 & \cos\theta \end{bmatrix}$$. If for some $$\theta \in (0, \pi)$$, $$A^2 = A^T$$, then the sum of the diagonal elements of the matrix $$(A + I)^3 + (A - I)^3 - 6A$$ is equal to ________.

Let $$A = \{z \in \mathbb{C} : |z - 2 - i| = 3\}$$, $$B = \{z \in \mathbb{C} : \text{Re}(z - iz) = 2\}$$ and $$S = A \cap B$$. Then $$\displaystyle\sum_{z \in S} |z|^2$$ is equal to ________.

Let C be the circle $$x^2 + (y - 1)^2 = 2$$, $$E_1$$ and $$E_2$$ be two ellipses whose centres lie at the origin and major axes lie on x-axis and y-axis respectively. Let the straight line $$x + y = 3$$ touch the curves C, $$E_1$$ and $$E_2$$ at $$P(x_1, y_1)$$, $$Q(x_2, y_2)$$ and $$R(x_3, y_3)$$ respectively. Given that P is the mid-point of the line segment QR and $$PQ = \dfrac{2\sqrt{2}}{3}$$, the value of $$9(x_1 y_1 + x_2 y_2 + x_3 y_3)$$ is equal to ________.

Let m and n be the number of points at which the function $$f(x) = \max\{x, x^3, x^5, \ldots, x^{21}\}$$, $$x \in \mathbb{R}$$, is not differentiable and not continuous, respectively. Then $$m + n$$ is equal to ________.

The mean free path and the average speed of oxygen molecules at 300 K and 1 atm are $$3 \times 10^{-7}$$ m and 600 m/s, respectively. Find the frequency of its collisions.

A small mirror of mass m is suspended by a massless thread of length $$l$$. Then the small angle through which the thread will be deflected when a short pulse of laser of energy E falls normal on the mirror (c = speed of light in vacuum and g = acceleration due to gravity)

Two liquids A and B have $$\theta_A$$ and $$\theta_B$$ as contact angles in a capillary tube. If $$K = \cos\theta_A / \cos\theta_B$$, then identify the correct statement:

Which of the following are correct expressions for torque acting on a body?
A. $$\vec{\tau} = \vec{r} \times \vec{L}$$
B. $$\vec{\tau} = \dfrac{d}{dt}(\vec{r} \times \vec{p})$$
C. $$\vec{\tau} = \vec{r} \times \dfrac{d\vec{p}}{dt}$$
D. $$\vec{\tau} = I\vec{\alpha}$$
E. $$\vec{\tau} = \vec{r} \times \vec{F}$$
($$\vec{r}= position vector;$$ $$\vec{p}=linear momentum;$$ $$\vec{L}= angular momentum;$$ $$\vec{\alpha} = angular acceleration;$$ $$I = moment of inertia;$$ $$\vec{F} = force; t = time$$)
Choose the correct answer from the options given below :

In a Young's double slit experiment, the slits are separated by 0.2 mm. If the slits separation is increased to 0.4 mm, the percentage change of the fringe width is: