NTA JEE Mains 24th Jan 2026 Shift 1

Let $$S=\frac{1}{25!}+\frac{1}{3!23!}+\frac{1}{5!21!}+...$$ up to 13 terms. If $$13S=\frac{2^{k}}{n!},k\epsilon N$$, then n+k ia equal to

LetA(l, 0), B(2, -1) and $$C\left(\frac{7}{3}, \frac{4}{3}\right)$$ be three points. If the equation of the bisector of the angle ABC is $$\alpha x+\beta y=5$$, then the value of $$\alpha^{2} +\beta^{2}$$ is

Let the lines $$L_{1}:\overrightarrow{r}=\widehat{i}+2\widehat{j}+3\widehat{k}+\lambda(2\widehat{i}+3\widehat{j}+4\widehat{k}),\lambda \in \mathbb{R}$$ and $$L_{2}:\overrightarrow{r}=(4\widehat{i}+\widehat{j})+\mu (5\widehat{i}+2\widehat{j}+\widehat{k}),\mu \in \mathbb{R}$$, interest at the point R. Let P and Q be the points lying on lines $$L_{1}$$ and $$L_{2}$$ respectively, such that $$|\overrightarrow{PR}|=\sqrt{29}$$ and $$|\overrightarrow{PQ}|=\sqrt{\frac{47}{3}}$$. If the point P lies in the first octant, then $$27(QR)^{2}$$ is equal to

Let each of the two ellipses $$E_{1}:\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1,(a > b)$$ and $$E_{2}:\frac{x^{2}}{A^{2}}+\frac{y^{2}}{B^{2}}=1,(A > B)$$ have eccentricity $$\frac{4}{5}$$. Let the lengths of the latus recta of $$E_{1}\text{ and }E_{2}$$ be $$l_{1}\text{ and }l_{2}$$ respectively, such that $$2\ l_{1}^{2}=9\ l_{2}$$. If the distance between the foci of $$E_{1}$$ is 8, then the distance between the foci of $$E_{2}$$ is

Consider an $$A.P:a_{1},a_{2},...a_{n};a_{1} > 0$$. If $$a_{2}-a_{1}=\frac{-3}{4},a_{n}=\frac{1}{4}a_{1}$$, and $$\sum_{i=1}^{n}a_{i}=\frac{525}{2}$$, then $$\sum_{i=1}^{17}a_{i}$$ is equal to

Let $$\overrightarrow{r}=2\widehat{i}+\widehat{j}-2\widehat{k}, \overrightarrow{b}=\widehat{i}+\widehat{j}\text{ and }\overrightarrow{c}=\overrightarrow{a}\times \overrightarrow{b}$$. Let $$\overrightarrow{d}$$ be a vector such that $$|\overrightarrow{d}-\overrightarrow{a}|=\sqrt{11},|\overrightarrow{c}\times \overrightarrow{d}|=3$$ and the angle between $$\overrightarrow{c}\text{ and }\overrightarrow{d}$$ is $$\frac{\pi}{4}$$. Then $$\overrightarrow{a}. \overrightarrow{d}$$ is equal to

Let $$S= z \left\{\in \mathbb{C}:|\frac{z-6i}{z-2i}|=1\text{ and }|\frac{z-8+2i}{z+2i}|=\frac{3}{5} \right\}$$. Then $$\sum_{z\in s}^{}|z|^{2}$$ is equal to

Let a circle of radius 4 pass through the origin O , the points $$A(-\sqrt{3}a,0)$$ and $$B(0,-\sqrt{2}b)$$. where a and b are real parameters and $$ab\neq 0$$. Then the locus of the centroid of $$\triangle OAB$$ is a circle of radius.

The mean and variance of a data of 10 observations are 1O and 2, respectively. If an observations $$\alpha$$ in this data is replaced by $$\beta$$, then the mean and variance become 10.1 and 1.99, respectively. Then $$\alpha+\beta$$ equals

If $$\cos x=\frac{5}{12}$$ for some $$x\in \left(\pi,\frac{3\pi}{2}\right)$$, then $$\sin 7x \left(\cos \frac{13x}{2}+\sin \frac{13x}{2}\right)+\cos 7x\left(\cos \frac{13x}{2}-\sin \frac{13x}{2}\right)$$ is equal to