NTA JEE Main 2025 April 7th Shift 2

If the function $$f(x) = \frac{\tan(\tan x) - \sin(\sin x)}{\tan x - \sin x}$$ is continuous at $$x = 0$$, then $$f(0)$$ is equal to _____.

If $$\int \left(\frac{1}{x} + \frac{1}{x^3}\right)\left(\sqrt[23]{3x^{-24} + x^{-26}}\right) dx = -\frac{\alpha}{3(\alpha+1)}(3x^\beta + x^\gamma)^{\frac{\alpha+1}{\alpha}} + C$$, $$x > 0$$, $$(\alpha, \beta, \gamma \in \mathbb{Z})$$, where C is the constant of integration, then $$\alpha + \beta + \gamma$$ is equal to _____.

For $$t > -1$$, let $$\alpha_t$$ and $$\beta_t$$ be the roots of the equation $$\left((t+2)^{1/7} - 1\right)x^2 + \left((t+2)^{1/6} - 1\right)x + \left((t+2)^{1/21} - 1\right) = 0$$. If $$\lim_{t \to -1^+} \alpha_t = a$$ and $$\lim_{t \to -1^+} \beta_t = b$$, then $$72(a + b)^2$$ is equal to _____.

Let the lengths of the transverse and conjugate axes of a hyperbola in standard form be 2a and 2b, respectively, and one focus and the corresponding directrix of this hyperbola be $$(-5, 0)$$ and $$5x + 9 = 0$$, respectively. If the product of the focal distances of a point $$\left(\alpha, 2\sqrt{5}\right)$$ on the hyperbola is p, then 4p is equal to _____.

The sum of the series $$2 \times 1 \times {^{20}C_4} - 3 \times 2 \times {^{20}C_5} + 4 \times 3 \times {^{20}C_6} - 5 \times 4 \times {^{20}C_7} + \ldots + 18 \times 17 \times {^{20}C_{20}}$$, is equal to

Given below are two statements : one is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A) : The outer body of an air craft is made of metal which protects persons sitting inside from lightning-strikes.
Reason (R) : The electric field inside the cavity enclosed by a conductor is zero.
In the light of the above statements, choose the most appropriate answer from the options given below :

Given below are two statements : one is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A) : The density of the copper $$\left(^{64}_{29}Cu\right)$$ nucleus is greater than that of the carbon $$\left(^{12}_{6}C\right)$$ nucleus.
Reason (R) : The nucleus of mass number A has a radius proportional to $$A^{1/3}$$.
In the light of the above statements, choose the most appropriate answer from the options given below :

The unit of $$\sqrt{\frac{2I}{\epsilon_0 c}}$$ is : (I = intensity of an electromagnetic wave, c : speed of light)

The dimension of $$\sqrt{\frac{\mu_0}{\epsilon_0}}$$ is equal to that of : ($$\mu_0$$ = Vacuum permeability and $$\epsilon_0$$ = Vacuum permittivity)

Given below are two statements : one is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A) : The radius vector from the Sun to a planet sweeps out equal areas in equal intervals of time and thus areal velocity of planet is constant.
Reason (R) : For a central force field the angular momentum is a constant.
In the light of about statement, Choose the most appropriate answer  from the option  given below :