NTA JEE Mains 29th Jan 2025 Shift 1

Let $$S = \left\{x : \cos^{-1} x = \pi + \sin^{-1} x+\sin^{-1}(2x+1)\right\}$$. Then $$\sum_{x \in S}^{}(2x-1)^{2}$$ is equal to_________.

Let $$F : \left(0,\infty\right)\rightarrow R$$ be a twice differentiable function. If for some $$a \neq 0,\int_{0}^{1}f(\lambda x)d\lambda = af(x),f(1)=1$$ and $$f(16)=\frac{1}{8}$$, then $$16-f'\left(\frac{1}{16}\right)$$ is equal to _______.

The number of 6-letter words, with or without meaning, that can be formed using the letters of the word MATHS such that any letter that appears in the word must appear at least twice, is _______.

Let $$S = \left\{m \in Z : A^{m^{2}}+A^{m} = 3I - A^{-6}\right\}$$, where $$ A =\begin{bmatrix}2 & -1 \\1 & 0 \end{bmatrix}$$. Then n(S) is equal to ______.

Let [t] be the greatest integer less than or equal to t. Then the least value of $$p \in N$$ for which $$\lim_{x\rightarrow 0^{+}}\left(x([\frac{1}{x}]+[\frac{2}{x}]+...+[\frac{p}{x}])-x^{2}([\frac{1}{x^{2}}]+[\frac{2^{2}}{x^{2}}]+...+[\frac{9^{2}}{x^{2}}])\right) \geq 1$$ is equal to_______.

An electric dipole of mass m, charge q, and length $$l$$ is placed in a uniform electric field $$\overrightarrow{E} = E_{\circ}\hat{i}$$. When the dipole is rotated slightly from its equilibrium position and released, the time period of its oscillations will be:

A coil of area A and N turns is rotating with angular velocity $$\omega$$ in a uniform magnetic field $$\overrightarrow{B}$$ about an axis perpendicular to $$\overrightarrow{B}$$. Magnetic flux $$\varphi$$ and induced emf ε across it, at an instant when $$\overrightarrow{B}$$ is parallel to the plane of coil, are :

Given below are two statements : one is labelled as Assertion (A) and the other is labelled as Reason (R). Assertion (A) : Choke coil is simply a coil having a large inductance but a small resistance. Choke coils are used with fluorescent mercury-tube fittings. If household electric power is directly connected to a mercury tube, the tube will be damaged. Reason (R): By using the choke coil, the voltage across the tube is reduced by a factor $$\left(R/\sqrt{R^{2}+\omega^{2}L^{2}}\right)$$, where ω frequency of the supply across resistor R and inductor L. If the choke coil were not used, the voltage across the resistor would be the same as the applied voltage. In the light of the above statements, choose the most appropriate answer from the options given below :

As shown below, bob A of a pendulum having massless string of length 'R' is released from $$60^{\circ}$$ to the vertical. It hits another bob B of half the mass that is at rest on a friction less table in the center. Assuming elastic collision, the magnitude of the velocity of bob A after the collision will be (take g as acceleration due to gravity.)

Given below are two statements : one is labelled as Assertion (A) and the other is labelled as Reason (R). Assertion (A) : Electromagnetic waves carry energy but not momentum. Reason (R): Mass of a photon is zero. In the light of the above statements, choose the most appropriate answer from the options given below :