NTA JEE Mains 22nd Jan 2026 Shift 2 - Mathematics

Let the locus of the mid-point of the chord through the origin O of the parabola $$y^{2}= 4x$$ be the curve S. Let P be any point on S. Then the locus of the point, which internally divides OP in the ratio 3 :1, is:

If $$\lim_{x \rightarrow 0} \frac{e^{(a-1)x}+2\cos bx+(c-2)e^{-x}}{x \cos x-\log_{e}{(1+x)}} =2$$, then $$a^{2}+b^{2}+c^{2}$$ is equal to :

Let f and g be functions satisfying f(x+ y) =f(x)f(y), f (1) =7 and g(x+ y) = g(xy), g(1) =1, for all $$x,y \epsilon N$$. If $$\sum_{x=1}^n \left(\frac{f(x)}{g(x)}\right) = 19607$$, then n is equal to:

Let n be the number obtained on rolling a fair die. If the probability that the system
x - ny + z = 6
x + (n - 2)y + (n + 1)z = 8
(n - 1)y + z = 1
has a unique solution is $$\frac{k}{6}$$, then the sum of k and all possible values of n is:

Let $$P(10, 2\sqrt{15})$$ be a point on the hyperbola $$\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$$, whose foci are S and S'. if the length of its latus rectum is 8, then the square of the area of $$\Delta PSS'$$ is equal to:

Let $$S= \left\{z \in \mathbb{C}: 4z^{2}+ \overline{z}=0 \right\}$$. Then $$\sum_{z\in S} |z|^{2}$$ is equal to:

Among the statements
(S1) : If A(5, -1) and B(-2, 3) are two vertices of a triangle, whose orthocentre is (0, 0), then its third vertex is (- 4,- 7) and
(S2) : If positive numbers 2a, b, c are three consecutive terms of an A.P., then the lines ax + by + c = 0 are concurrent at (2,-2),

lf the mean deviation about the median of the numbers k, 2k, 3k, ..... , 1000k is 500, then $$k^{2}$$ is equal to :

let $$\alpha, \beta$$ be the roots of the quadratic equation $$12x^{2}-20x+3\lambda=0, \lambda\in \mathbb{Z}$$. If $$\frac{1}{2}\leq |\beta-\alpha|\leq\frac{3}{2}$$, then the sum of all possible values of $$\lambda$$ is :

Let the domain of the function f(x) = $$\log_{3}\log_{5}(7-\log_{2}(x^{2}-10x+85))+\sin^{-1}\left(|\frac{3x-7}{17-x}|\right)$$ be $$(\alpha, \beta)$$. Then $$\alpha + \beta$$ is equal to :

Let S and S' be the foci of the ellipse $$\frac{x^{2}}{25}+\frac{y^{2}}{9}=1$$ and $$P(\alpha , \beta)$$ be a point on the ellipse in the first quadrant. If $$(SP)^{2}+(S'P)^{2}-SP\cdot S'P=37$$, then $$\alpha^{2}+\beta^{2}$$ is equal to :

Let $$f(x)= [x]^{2}-[x+3]-3, x\in \mathbb R$$, where $$[\cdot]$$ is the greatest integer funtion. Then

The area of the region $$A= \left\{(x,y): 4x^{2}+y^{2}\leq 8 \text{and } y^{2} \leq 4x \right\}$$ is :

If y=y(x) satisfies the differential equation
$$16(\sqrt{x+9\sqrt{x}})(4+\sqrt{9+\sqrt{x}}) \cos{y}dy=(1+2 \sin y)dx, x>0 \text{and} y(256) = \frac{\pi}{2}, y(49)=\alpha$$, then $$2\sin \alpha$$ is equal to :

Let $$\left[\cdot\right]$$ denote the greatest integer function, and let f (x) = $$\min \left\{\sqrt{2x},x^{2}\right\}$$. Let S = $$\left\{x \in (-2,2): \text{the function,} g(x)= |x|\left[x^{2}\right]\text{is discontinuous at x} \right\}.$$ Then $$\sum_{x\in S}f(x)$$ equals

The number of elements in the relation $$R= \left\{(x,y): 4x^{2}+y^{2}<52,x,y\in Z\right\}$$ is

If $$X=\begin{bmatrix}x \\y \\z \end{bmatrix}$$ is a solution of the system of equations AX= B, where adj $$A= \begin{bmatrix}4 & 2 & 2 \\-5 & 0 & 5 \\1 & -2 & 3 \end{bmatrix}$$ and $$B=\begin{bmatrix}4 \\0 \\2 \end{bmatrix}$$, then |x+y+z| is equal to :

Let L be the line $$\frac{x+1}{2}=\frac{y+1}{3}=\frac{z+3}{6}$$ and let S be the set of all points (a, b, c) on L, whose distance from the line $$\frac{x+1}{2}=\frac{y+1}{3}=\frac{z-9}{0}$$a long the line L is 7. Then $$\sum_{(a,b,c)\in S} (a+b+c) $$ is equal to :

Let $$C_{r}$$ denote the coefficient of $$x^{r}$$ in the binomial expansion of $$(1+x)^{n}, n\in N, 0\leq r\leq n$$. If $$P_{n}= C_{0}-C_{1}+\frac{2^{2}}{3}C_{2}-\frac{2^{3}}{4}C_{3}+.....+\frac{(-2)^{n}}{n+1}C_{n}, \text{then the value of} \sum_{n=1}^{25} \frac{1}{P_{2n}} $$ equals.

Let $$\overrightarrow{a}= 2\widehat{i}-\widehat{j}+\widehat{k}$$ and $$\overrightarrow{b}= \lambda \widehat{j}+2\widehat{k}, \lambda\in Z$$ be two vectors. Let $$\overrightarrow{c}= \overrightarrow{a} \times \overrightarrow{b} \text{and } \overrightarrow{d}$$ be a vector of magnitude 2 in yz-plane. If $$|\overrightarrow{c}|=\sqrt{53}$$, then the maximum possible value of $$\left(\overrightarrow{c}\cdot\overrightarrow{d}\right)^{2}$$ is equal to :

Let S be the set of the first 11 natural numbers. Then the number of elements in $$ A= \ B \subseteq S:n(B)\geq 2 $$, and the product of all elements of B is even} 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 $$a, b, c$$ are 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 ____ .