NTA JEE Mains 29th Jan 2025 Shift 2

If $$24 \int_{0}^{\frac{\pi}{4}} \left(\sin\left|4x-\frac{\pi}{12}\right| + \left[2\sin x\right]\right) dx = 2\pi + \alpha,$$ , where $$[\cdot]$$ denotes the greatest integer function, then $$\alpha$$ is equal to _______.

Let $$a_1, a_2, \ldots, a_{2024}$$ be an Arithmetic Progression such that $$a_1 + (a_5 + a_{10} + a_{15} + \cdots + a_{2020}) + a_{2024} = 2233$$. Then $$a_1 + a_2 + a_3 + \cdots + a_{2024}$$ is equal to _______

If $$\lim_{t \to 0} \left(\int_{0}^{1} (3x+5)^t \, dx\right)^{\frac{1}{t}} = \frac{\alpha}{5e}\left(\frac{8}{5}\right)^{\frac{2}{3}}$$, then $$\alpha$$ is equal to ________

Let $$y^{2} = 12x$$ be the parabola and S be its focus. Let PQ be a focal chord of the parabola such that (SP)(SQ) = $$\frac{147}{4}$$. Let C be the circle described taking PQ as a diameter. If the equation of a circle C is $$64x^2 + 64y^2 - \alpha x - 64\sqrt{3}\,y = \beta$$, then $$\beta - \alpha$$ is equal to ________.

Let integers $$a,b \in [-3,3]$$ be such that $$a+b \neq 0$$. Then the number of all possible ordered pairs (a, b), for which $$\left|\frac{z-a}{z+b}\right| = 1$$ and $$\begin{vmatrix} z+1 & \omega & \omega^2 \\ \omega & z+\omega^2 & 1 \\ \omega^2 & 1 & z+\omega \end{vmatrix} = 1,\quad z \in \mathbb{C}$$, where $$\omega$$ and $$\omega^{2}$$ are the roots of $$x^{2} + x + 1 = 0$$, is equal to_________.

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

Assertion (A) :
Three identical spheres of same mass undergo one dimensional motion as shown in figure with initial velocities $$v_A = 5\,\text{m/s},\; v_B = 2\,\text{m/s},\; v_C = 4\,\text{m/s}$$. If we wait sufficiently long for elastic collision to happen, then $$v_A = 4\,$$$$\text{m/s}$$$$,\; v_B = 2\,$$$$\text{m/s}$$$$,\; v_C = 5\,$$$$\text{m/s}$$$$, will be the final velocities.
Reason (R): In an elastic collision between identical masses, two objects exchange their velocities. In the light of the above statements, choose the correct answer from the options given below :

Two identical symmetric double convex lenses of focal length f are cut into two equal parts $$L_{1}, L_{2}$$ by AB plane and $$L_{3}, L_{4}$$ by XY plane as shown in figure respectively. The ratio of focal lengths of lenses $$L_{1}$$ and $$L_{3}$$ i

Two bodies A and B of equal mass are suspended from two massless springs of spring constant $$k_{1}$$ and $$k_{2}$$, respectively. If the bodies oscillate vertically such that their amplitudes are equal, the ratio of the maximum velocity of A to the maximum velocity of B is

Given below are two statements. One is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A) : With the increase in the pressure of an ideal gas, the volume falls off more rapidly in an isothermal process in comparison to the adiabatic process.
Reason (R) : In isothermal process, PV = constant, while in adiabatic process $$PV^{\gamma}$$ = constant. Here $$\gamma$$ is the ratio of specific heats, P is the pressure and V is the volume of the ideal gas. In the light of the above statements, choose the correct answer from the options given below :

The truth table for the circuit given below is :