NTA JEE Mains 24th Jan 2026 Shift 1 - Mathematics

The value of $$\dfrac{\sqrt{3}  \operatorname{cosec} 20^{\circ}-\sec20^{\circ}}{\cos20^{\circ}\cos40^{\circ}\cos60^{\circ}\cos80^{\circ}}$$ is equal to:

From a lot containing 10 defective and 90 non-defective bulbs, 8 bulbs are selected one by one with replacement. Then the probability of getting at least 7 defective bulbs is

Let 729, 81 , 9, 1, ... be a sequence and $$P_{n}$$ denote the product of the first n terms of this sequence.
If $$2\sum_{n=1}^{40}(P_{n})^{\frac{1}{n}}=\frac{3^{\alpha}-1}{3^{\beta}}$$ and $$gcd(\alpha\beta)=1$$ then $$\alpha+\beta$$ is equal to

Let $$A_{1}$$ be the bounded area enclosed by the curves $$y=x^{2}+2,x+Y=8$$ and y-axis that lies in the first quadrant. Let $$A_{2}$$ be the bounded area enclosed by the curves $$y=x^{2}+2,y^{2}=x,x=2$$ and y-axis that lies in the first quadrant. Then $$A_{1}-A_{2}$$ is equal to

Let $$f(t)=\int_{}^{}\left(\frac{1-\sin(\log_{e}{t})}{1-\cos(\log_{e}{t})}\right)dt,t > 1$$.
If $$f(e^{\pi/2})=-e^{\pi/2}\text{ and }f(e^{\pi/4})=\alpha e^{\pi/4}$$, then $$\alpha$$ equals

The number of the real solutions of the equation:
$$x|x+3|+|x-1|-2=0$$ is

If the function $$f(x)=\frac{e^{x}(e^{\tan x-x}-1)+\log_{e}{(\sec  x+\tan x)}-x}{\tan x-x}$$ is continuous at x = 0, then the value of f(O) is equal to

Let R be a relation defined on the set {1 , 2, 3, 4} x { l, 2, 3, 4} by R = {((a, b), (c, d)): 2a + 3b = 3c + 4d}.
Then the number of elements in R is

If the domain of the function $$f(x)=\log_{(10x^{2}-17x+7)}{(18x^{2}-11x+1)}$$ is $$(-\infty ,a)\cup (b,c)\cup (d,\infty)-{e}$$ and 90(a + b + c + d + e) equals:

Let $$\alpha, \beta \epsilon \mathbb R$$ be such that the fonction $$f(x) =\left\{\begin{array}{||}2\alpha(x^{2}-2)+2\beta x & \quad,{x<1}\\(\alpha +3)x+(\alpha -\beta) & \quad ,{x\geq 1}\\\end{array}\right.$$ be differentiable at all $$x\epsilon \beta$$. Then $$34(\alpha +\mathbb R)$$ is equal to

Let $$S=\dfrac{1}{25!}+\dfrac{1}{3!23!}+\dfrac{1}{5!21!}+...$$ up to 13 terms. If $$13S=\dfrac{2^k}{n!},\ \ k\in\mathbf{N}$$, then $$n+k$$ is 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 $$\cot x=\frac{5}{12}$$ 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

Let a differentiable function f satisfy the equation $$\int_{0}^{36}f(\frac{tx}{36})dt=4\alpha f(x)$$. If y=f(x) is a standard parabola passing through the points (2, 1) and $$(-4,\beta)$$, then $$/beta^{\alpha}$$ is equal to______.

The number of $$3\times 2$$ matrices A, which can be formed using the elements of the set {-2, -1 , 0, 1, 2} such that the sum of all the diagonal elements of $$A^{T}A$$ is 5, is_____

Let a line L passing through the point P (1, 1, 1) be perpendicular to the lines $$\frac{x-4}{4}=\frac{y-1}{1}=\frac{z-1}{1}$$ and $$\frac{x-17}{1}=\frac{y-71}{1}=\frac{z}{0}$$. Let the line L intersect the yz-plane at the point Q. Another line parallel to L and passing through the point S (1, 0, - 1) intersects the yz-plane at the point R. Then the square of the area of the parallelogram PQRS is equal to ___.

The number of numbers greater than 5000, less than 9000 and divisible by 3, that can be formed using the digits 0, 1, 2, 5, 9, if the repetition of the digits is allowed, is______.

Let $$(2a, a)$$ be the largest interval in which the function $$f(t)=\frac{|t+1|}{t^{2}},t < 0$$, is strictly decreasing. Then the local maximum value of the function $$g(x)=2\log_{e}(x-2)+a x^{2}+4x-a,x > 2$$, is______.