NTA JEE Main 2025 April 4th Shift 2

If $$\alpha$$ is a root of the equation $$x^2 + x + 1 = 0$$ and $$\sum_{k=1}^{n} \left(\alpha^k + \frac{1}{\alpha^k}\right)^2 = 20$$, then n is equal to ______.

If $$\int \frac{(\sqrt{1+x^2}+x)^{10}}{(\sqrt{1+x^2}-x)^9} dx = \frac{1}{m}\left((\sqrt{1+x^2}+x)^n\left(n\sqrt{1+x^2}-x\right)\right) + C$$ where C is the constant of integration and $$m, n \in \mathbb{N}$$, then $$m + n$$ is equal to

A card from a pack of 52 cards is lost. From the remaining 51 cards, n cards are drawn and are found to be spades. If the probability of the lost card to be a spade is $$\frac{11}{50}$$, the n is equal to

Let m and n, (m < n) be two 2-digit numbers. Then the total numbers of pairs (m, n), such that gcd(m, n) = 6, is ______.

Let the three sides of a triangle ABC be given by the vectors $$2\hat{i} - \hat{j} + \hat{k}$$, $$\hat{i} - 3\hat{j} - 5\hat{k}$$ and $$3\hat{i} - 4\hat{j} - 4\hat{k}$$. Let G be the centroid of the triangle ABC. Then $$6\left(|\vec{AG}|^2 + |\vec{BG}|^2 + |\vec{CG}|^2\right)$$ is equal to ______.

A radioactive material P first decays into Q and then Q decays to non-radioactive material R. Which of the following figure represents time dependent mass of P, Q and R?

There are 'n' number of identical electric bulbs, each is designed to draw a power p independently from the mains supply. They are now joined in series across the main supply. The total power drawn by the combination is :

Consider a rectangular sheet of solid material of length $$l = 9$$ cm and width $$d = 4$$ cm. The coefficient of linear expansion is $$\alpha = 3.1 \times 10^{-5}$$ K$$^{-1}$$ at room temperature and one atmospheric pressure. The mass of sheet $$m = 0.1$$ kg and the specific heat capacity $$C_v = 900$$ J kg$$^{-1}$$K$$^{-1}$$. If the amount of heat supplied to the material is $$8.1 \times 10^2$$ J then change in area of the rectangular sheet is :

Given below are two statements :
Statement (I) : The dimensions of Planck's constant and angular momentum are same.
Statement (II) : In Bohr's model electron revolve around the nucleus only in those orbits for which angular momentum is integral multiple of Planck's constant.
In the light of the above statements, choose the most appropriate answer from the options given below :

A cylindrical rod of length 1 m and radius 4 cm is mounted vertically. It is subjected to a shear force of $$10^5$$ N at the top. Considering infinitesimally small displacement in the upper edge, the angular displacement $$\theta$$ of the rod axis from its original position would be : (shear moduli, $$G = 10^{10}$$ N/m$$^2$$)