NTA JEE Main 2025 April 3rd Shift 1

All five letter words are made using all the letters A, B, C, D, E and arranged as in an English dictionary with serial numbers. Let the word at serial number $$n$$ be denoted by $$W_n$$. Let the probability $$P(W_n)$$ of choosing the word $$W_n$$ satisfy $$P(W_n) = 2P(W_{n-1})$$, $$n \gt 1$$. If $$P(CDBEA) = \frac{2^{\alpha}}{2^{\beta} - 1}$$, $$\alpha, \beta \in \mathbb{N}$$, then $$\alpha + \beta$$ is equal to __________.

Let the product of the focal distances of the point $$P(4, 2\sqrt{3})$$ on the hyperbola $$H : \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$$ be 32. Let the length of the conjugate axis of H be p and the length of its latus rectum be q. Then $$p^2 + q^2$$ is equal to __________.

Let $$\vec{a} = \hat{i} + \hat{j} + \hat{k}$$, $$\vec{b} = 3\hat{i} + 2\hat{j} - \hat{k}$$, $$\vec{c} = \lambda\hat{j} + \mu\hat{k}$$ and $$\hat{d}$$ be a unit vector such that $$\vec{a} \times \hat{d} = \vec{b} \times \hat{d}$$ and $$\vec{c} \cdot \hat{d} = 1$$. If $$\vec{c}$$ is perpendicular to $$\vec{a}$$, then $$|3\lambda\hat{d} + \mu\vec{c}|^2$$ is equal to __________.

If the number of seven-digit numbers, such that the sum of their digits is even, is $$m \cdot n \cdot 10^n$$; $$m, n \in \{1, 2, 3, \ldots, 9\}$$, then $$m + n$$ is equal to __________.

The area of the region bounded by the curve $$y = \max\{|x|, x|x-2|\}$$, then x-axis and the lines $$x = -2$$ and $$x = 4$$ is equal to __________.

During the melting of a slab of ice at 273 K at atmospheric pressure :

Consider a completely full cylindrical water tank of height 1.6 m and cross-sectional area 0.5 m$$^2$$. It has a small hole in its side at a height 90 cm from the bottom. Assume, the cross-sectional area of the hole to be negligibly small as compared to that of the water tank. If a load 50 kg is applied at the top surface of the water in the tank then the velocity of the water coming out at the instant when the hole is opened is : $$(g = 10\,\text{m/s}^2)$$

Choose the correct logic circuit for the given truth table having inputs A and B.

The radiation pressure exerted by a 450 W light source on a perfectly reflecting surface placed at 2m away from it, is :

A wire of length 25 m and cross-sectional area 5 mm$$^2$$ having resistivity of $$2 \times 10^{-6}\,\Omega$$ m is bent into a complete circle. The resistance between diametrically opposite points will be