NTA JEE Mains 22nd Jan 2026 Shift 2

Let S be the set of the first 11 natural numbers. Then the number of elements in $$A= \left\{B \subseteq S:n(B)\geq 2, \text{and the product of all elements of Bis even}\right\}$$ is __________.

Let $$\cos(\alpha+\beta)= -\frac{1}{10} \text{and} \sin (\alpha -\beta)= \frac{3}{8}$$, where $$0<\alpha<\frac{\pi}{3}$$ and $$0<\beta<\frac{\pi}{4}$$. If $$\tan 2\alpha = \frac{3(1-r\sqrt{5})}{\sqrt{11}(s+\sqrt{5})}, r,s\in N$$, then r + s is equal to __________.

Let a vector $$\vec{a} = \sqrt{2}\,\hat{i} - \hat{j} + \lambda \hat{k}, \quad \lambda > 0,$$ make an obtuse angle with the vector $$\vec{b} = -\lambda^{2}\hat{i} + 4\sqrt{2}\,\hat{j} + 4\sqrt{2}\,\hat{k}$$ and an angle $$\theta, \dfrac{\pi}{6} < \theta < \dfrac{\pi}{2}$$, with the positive z-axis. If the set of all possible values of $$\lambda$$ is $$( \alpha, \beta) - \{\gamma\}$$, then $$\alpha + \beta + \gamma$$ is equal to __________.

Suppose a, b, care in A.P. and $$a^{2}, 2b^{2}$$, c^{2} are in G.P. If a < b < c and a + b + c = 1, then $$9(a^{2}+b^{2}+c^{2})$$ is equal to _____________.

Let $$\left[\cdot\right]$$ be the greatest integer function. If $$(\alpha = \int_{0}^{64} \left( x^{1/3} - [x^{1/3}] \right)\, dx $$, then $$\frac{1}{\pi} \int_{0}^{\alpha\pi } \left( \frac{\sin^{2}\theta } {\sin^{6}\theta + \cos^{6}\theta} \right) d\theta$$ is equal to ____ .

The wavelength of light, while it is passing through water is 540 nm. The refractive index of water is $$\frac{4}{3}$$. The wavelength of the same light when it is passing through a transparent medium having refractive index of $$\frac{3}{2}$$ is ____________nm.

Given below are two statements:
Statement I : An object moves from position $$r_{1}$$ to position $$r_{2}$$ under a conservative force field $$\overrightarrow{F}$$.
The work done by the force is W = $$\int_{r_{1}}^{r_{2}} \overrightarrow{F}.\overrightarrow{dr}.$$
Statement II: Any object moving from one location to another location can follow infinite number of paths. Therefore, the amount of work done by the object changes with the path it follows for a conservative force.
In the light of the above statements, choose the correct answer from the options given below :

In an open organ pipe $$v_{3}$$ and $$v_{6}$$ are $$3^{rd}$$ and $$6^{th}$$ harmonic frequencies, respectively. If $$v_{6} - v_{3}$$ = 2200 Hz then length of the pipe is ____ mm .
(Take velocity of sound in air is 330 m/s.)

When a part of a straight capillary tube is placed vertically in a liquid, the liquid raises uptocertain height h. If the inner radius of the capillary tube, density of the liquid and surface tension of the liquid decrease by 1 % each, then the height of the liquid in the tube will change by __ %.

Consider two boxes containing ideal gases A and B such that their temperatures, pressures and number densities are same. The molecular size of A is half of that of B and mass of molecule A is four times that of B. If the collision frequency in gas B is $$32 \times 10^{18}/s$$ then collision frequency in gas A is _____________/s.