NTA JEE Mains 21st Jan 2026 Shift 2

If $$\left(\frac{1}{{}^{15}C_0} + \frac{1}{{}^{15}C_1}\right) \left(\frac{1}{{}^{15}C_1} + \frac{1}{{}^{15}C_2}\right) \cdots \left(\frac{1}{{}^{15}C_{12}} + \frac{1}{{}^{15}C_{13}}\right) = \frac{\alpha^{13}}{{}^{14}C_0 {}^{14}C_1 \cdots {}^{14}C_{12}}$$ then $$30\alpha$$ is equal to __________

If $$\displaystyle \int_{0}^{1} 4\cot^{-1}(1 - 2x + 4x^2)\,dx = a\tan^{-1}(2) - b\log_e(5)$$, where $$a,b\epsilon N$$ then (2a + b} is equal to _________

Let [·] denote the greatest integer function and $$f(x) = \lim_{n \to \infty} \frac{1}{n^3} \sum_{k=1}^{n} \left[\frac{k^2}{3^x}\right]$$. Then $$12 \sum_{j=1}^{\infty} f(j)$$ is equal to _______

If P is a point on the circle $$x^{2}+y^{2}=4$$, Q is a point on the straight line 5x + y + 2 = 0 and x- y + 1 = 0 is the perpendicular bisector of PQ, then 13 times the sum of abscissa of all such points P is __________

Let the maximum value of $$\left(\sin^{-1}x\right)^2+\left(\cos^{-1}x\right)^2$$ for $$x\epsilon \left[-\frac{\sqrt{13}}{2},\frac{1}{\sqrt{2}}\right]$$ be $$\frac{m}{n}\pi^{2}$$, where gcd
(m, n) = l. Then m + n is equal to ____________

The kinetic energy of a simple harmonic oscillator is oscillating with angular frequency of 176 rad/ s. The frequency of this simple harmonic oscillator is _____Hz. [take $$\pi = \frac{22}{7}$$]

Consider two identical metallic spheres of radius R each having charge Q and mass m. Their centers have an initial separation of 4R. Both the spheres are given an initial speed of u towards each other. The minimum value of u, so that they can just touch each other is :
(Take $$k= \frac{1}{4\pi \epsilon_{0}}$$ and assume $$kQ^{2}$$ > $$Gm^{2}$$ where G is the Gravitational constant)

Given below are two statements :
Statement I : In a Young's double slit experiment, the angular separation of fringes will increase as the screen is moved away from the plane of the slits
Statement II: In a Young's double slit experiment, the angular separation of fringes will increase when monochromatic source is replaced by another monochromatic source of higher wavelength
In the light of the above statements, choose the correct answer from the options given below :

A battery with EMF E and internal resistance r is connected across a resistance R. The power consumption in R will be maximum when :

The charge stored by the capacitor C in the given circuit in the steady state is ______ $$\mu C$$.