NTA JEE Main 2025 April 3rd Shift 2

Let I be the identity matrix of order $$3 \times 3$$ and for the matrix $$A = \begin{bmatrix} \lambda & 2 & 3 \\ 4 & 5 & 6 \\ 7 & -1 & 2 \end{bmatrix}$$, $$|A| = -1$$. Let B be the inverse of the matrix $$\text{adj}(A \cdot \text{adj}(A^2))$$. Then $$|(\lambda B + I)|$$ is equal to ________.

Let $$(1 + x + x^2)^{10} = a_0 + a_1 x + a_2 x^2 + \ldots + a_{20} x^{20}$$. If $$(a_1 + a_3 + a_5 + \ldots + a_{19}) - 11a_2 = 121k$$, then k is equal to ________.

If $$\displaystyle\lim_{x \to 0} \left(\dfrac{\tan x}{x}\right)^{1/x^2} = p$$, then $$96 \log_e p$$ is equal to ________.

Let $$\vec{a} = \hat{i} + 2\hat{j} + \hat{k}$$, $$\vec{b} = 3\hat{i} - 3\hat{j} + 3\hat{k}$$, $$\vec{c} = 2\hat{i} - \hat{j} + 2\hat{k}$$ and $$\vec{d}$$ be a vector such that $$\vec{b} \times \vec{d} = \vec{c} \times \vec{d}$$ and $$\vec{a} \cdot \vec{d} = 4$$. Then $$|\vec{a} \times \vec{d}|^2$$ is equal to ________.

If the equation of the hyperbola with foci $$(4, 2)$$ and $$(8, 2)$$ is $$3x^2 - y^2 - \alpha x + \beta y + \gamma = 0$$, then $$\alpha + \beta + \gamma$$ is equal to ________.

A magnetic dipole experiences a torque of $$80\sqrt{3}$$ N m when placed in uniform magnetic field in such a way that dipole moment makes angle of 60° with magnetic field. The potential energy of the dipole is:

In the resonance experiment, two air columns (closed at one end) of 100 cm and 120 cm long, give 15 beats per second when each one is sounding in the respective fundamental modes. The velocity of sound in the air column is:

Two cylindrical vessels of equal cross sectional area of $$2 \text{ m}^2$$ contain water upto height 10 m and 6 m, respectively. If the vessels are connected at their bottom, then the work done by the force of gravity is: (Density of water is $$10^3 \text{ kg/m}^3$$ and $$g = 10 \text{ m/s}^2$$)

Width of one of the two slits in a Young's double slit interference experiment is half of the other slit. The ratio of the maximum to the minimum intensity in the interference pattern is:

An ideal gas exists in a state with pressure $$P_0$$, volume $$V_0$$. It is isothermally expanded to 4 times of its initial volume $$(V_0)$$, then isobarically compressed to its original volume. Finally the system is heated isochorically to bring it to its initial state. The amount of heat exchanged in this process is: