NTA JEE Mains 24th Jan 2026 Shift 2 - Mathematics

Let f be a function such that $$3f(x)+2f \left(\frac{m}{19x}\right) = 5x, x\neq 0$$, where $$m= \sum_{i-1}^9(i)^{2}$$. Then f(5) - f(2) is equal to

Lety = y (x) be a differentiable function in the interval $$(0, \infty)$$ such that y(l) = 2, and $$\lim_{t \rightarrow x} \left( \frac{t^{2}y(x)-x^{2}y(t)}{x-t} \right) = 3$$ for each x > 0. Then 2){2) is equal to

Let f(a) denote the area of the region in the first quadrant bounded by x = 0, x = 1, $$y^{2}=x$$ and y = |ax - 5| - |1 - ax| + $$ax^{2}$$. Then (f(O) + f(1)) is equal

Let the image of parabola $$x^{2}=4y$$, in the line x - y = 1 be $$(y+a)^{2}$$ = b(x-c), $$a,b,c \in N.$$ Then a + b + c is equal to

Let $$\overrightarrow{a}= 2\widehat{i}-\widehat{j}-\widehat{k}, \overrightarrow{b}=\widehat{i}+ 3\widehat{j}-\widehat{k}$$ and $$\overrightarrow{c} = 2\widehat{i}+\widehat{j}+3\widehat{k}.$$ Let $$\overrightarrow{\nu}$$ be the vector in the plane of the vectors $$\overrightarrow{a}$$ and $$\overrightarrow{b}$$, such that the length of its projection on the vector $$\overrightarrow{C}$$ is $$\frac{1}{\sqrt{14}}$$. Then $$\mid \overrightarrow{\nu} \mid$$ is euqal to

Let $$\overrightarrow{a}= 2\widehat{i}-5\widehat{j}+5\widehat{k}$$ and $$\overrightarrow{b}= \widehat{i}-\widehat{j}+3\widehat{k}$$. If $$\overrightarrow{C}$$ is a vector such that $$2(\overrightarrow{a}\times\overrightarrow{c})+3(\overrightarrow{b}\times\overrightarrow{c})= \overrightarrow{0}$$ and $$(\overrightarrow{a}-\overrightarrow{b})\cdot\overrightarrow{c}=-97,$$ then $$\mid \overrightarrow{c}\times\widehat{k} \mid^{2}$$ is equal to

The sum of all values of $$\alpha$$, for which the sho1test distance between the lines
$$\frac{x+1}{\alpha}=\frac{y-2}{-1}=\frac{z-4}{-\alpha}$$ and $$\frac{x}{\alpha}=\frac{y-1}{2}=\frac{z-1}{2\alpha}$$ is $$\sqrt{2}$$, is

Let $$a_{1},a_{2},a_{3},a_{4}$$ be an A.P. of four terms such that each term of the A.P. and its common difference $$l$$ are integers. If $$a_{1} +a_{2}+a_{3}+a_{4}= 48$$ and $$a_{1} a_{2}a_{3}a_{4} + l^{4} = 361,$$ then the largest term of the A.P. is equal to

The letters of the word "UDAYPUR" are written in all possible ways with or without meaning and these words are arranged as in a dictionary. The rank of the word "UDAYPUR" is

Let the length of the latus rectum of an ellipse $$\f\frac{x^{2}}{a^{2}}+\f\frac{y^{2}}{b^{2}}=1,(a\gt b)$$ be 30. If its eccentricity is the maximum value of the function $$f(t)=-\f\frac{3}{4}+2t-t^{2}$$ then $$(a^{2}+b^{2})$$ is equal to

$$\left(\dfrac{1}{3}+\dfrac{4}{7}\right)+\left( \dfrac{1}{3^{2}}+\dfrac{1}{3}\times\dfrac{4}{7}+\dfrac{4^{2}}{7^{2}} \right)+\left(\dfrac{1}{3^{3}}+\dfrac{1}{3^{2}}\times\dfrac{4}{7}+\dfrac{1}{3}\times\dfrac{4^{2}}{7^{2}}+\dfrac{4^{3}}{7^{3}} \right)+......$$ 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 > 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

If f(x) satisfies the relation $$f(x)=e^{x}+\int_{0}^{1}\left(y+xe^{x}\right)f(y)dy,$$ then e + f(0) is equal to ______.

Let (h, k) lie on the circle $$C: x^{2}+y^{2}=4$$ and the point (2h + l , 3k + 2) lie on an ellipse with eccentricity e. Then the value of $$\frac{5}{e^{2}}$$ is equal to __________.

The number of elements in the set $$\left\{x \in [0,180^{\circ}]:\tan (x+100^{\circ}) = \tan (x+50^{\circ}) \tan x \tan(x-50^{\circ})\right\}$$ is ___________.

Let S be a set of 5 elements and P(S) denote the power set of S. Let E be an event of choosing an ordered pair (A, B) from the set P(S) x P(S) such that $$A\cap B=\phi.$$ If
the probability of the event E is $$\frac{3^{p}}{2^{q}}$$, where p,q $$\in$$ N, then p + q is equal to __________

Let z = (1 + i) (1 + 2i) (1 + 3i) .... (l + ni), where i = $$\sqrt{-1}$$. If $$|z|^{2}$$ = 44200, then n is equal to __