NTA JEE Mains 24th Jan 2026 Shift 2

$$\left(\frac{1}{3}+\frac{4}{7}\right)+\left( \frac{1}{3^{2}}+\frac{1}{3}\times\frac{4}{7}+\frac{4^{2}}{7^{2}} \right)+\left(\frac{1}{3^{3}}+\frac{1}{3^{2}}\times\frac{4}{7}+\frac{1}{3}\times\frac{4^{2}}{7^{2}}+\frac{4^{3}}{7^{3}} \right)+......$$ upto infinite term, is equal to

Let [t] denote the greatest integer less than or equal to t. If the function $$f(x) = \begin{cases} b^2 \sin\!\left(\dfrac{\pi}{2}\left[\dfrac{\pi}{2}(\cos x + \sin x)\cos x\right]\right), & x < 0 \\[10pt] \dfrac{\sin x - \dfrac{1}{2}\sin 2x}{x^3}, & x > 0 \\[10pt] a, & x = 0 \end{cases}$$ is continuous at x = 0,then $$a^{2} + b^{2}$$ is equal to

The smallest positive integral value of a, for which all the roots of $$x^{4} - ax^{2} + 9 = 0$$ are real and distinct, is equal to

Let $$f(x)=\int_{}^{} \frac{7x^{10}+9x^{8}}{(1+x^{2}+2x^{9})^{2}}dx, x>0, \lim_{x \rightarrow 0}f(x)=0$$ and $$f(1)=\frac{1}{4.}$$ If $$A= \begin{bmatrix}0 & 0 & 1 \\ \frac{1}{4} & f'(1) & 1 \\ \alpha^{2} & 4 & 1 \end{bmatrix}$$ and B = adj(adj A) be such that |B| = 81 , then $$\alpha^{2}$$ is equal to

If the domain of the function f(x) = $$\sin^{-1}\frac{1}{x^{2}-2x-2}$$, is $$\left[-\infty, \alpha\right] \cup \left[\beta,\gamma\right]\cup \left[\delta,\infty\right],$$ then $$\alpha+\beta+\gamma+\delta$$ is equal to

Consider the following three statements for the function $$f: (0, \infty ) \rightarrow \mathbb R$$ defined by
$$f(x)= |\log_{e}{x}|-|x-1|:$$
(I)f is differentiable at all x &gt; 0.
(II)f is increasing in (0, 1).
(III)f is decreasing in (1, $$\infty$$).
Then.

Let the angles made with the positive x-axis by two straight lines drawn from the point P(2, 3) and meeting the line x + y = 6 at a distance $$\sqrt{\frac{2}{3}}$$ from the point P be $$\theta_{1}$$ and $$\theta_{2}$$. Then the value of $$(\theta_{1}+\theta_{2})$$ is :

Let $$P[P_{ij}]$$ and $$Q=[q_{ij}]$$ be two square matrices of order 3 such that $$q_{ij}= 2^{(i+j-1)}p_{ij}$$ and $$\det (Q)=2^{10}.$$ Then the value of det(adj(adj P)) is:

Let $$X= \left\{x\in N:1\leq x\leq19 \right\}$$ and for some $$a,b \in \mathbb R, Y = \left\{ax+b:x\in X\right\}.$$ If the mean and variance of the elements of Y are 30 and 750, respectively, then the sum of all possible values of b is

The largest value of n, for which $$40^{n}$$ divides 60! , is