NTA JEE Mains 22nd Jan 2026 Shift 1

Let the solution curve of the differential equation $$xdy-ydx=\sqrt{x^{2}+y^{2}}dx,x>0,$$

If the line $$ax+ 2y = 1$$, where $$\alpha \in R $$, does not meet the hyperbola $$x^{2}-9y^{2}=9$$, then a possible value of $$\alpha$$ is:

The coefficient of $$x^{48}$$ in $$ (1+x) + 2(1+x)^{2}+3(1+x)^{3}+....+100(1+x)^{100} $$ is equal to

Let $$ f: [1 , \infty ) \rightarrow R$$ be a differentiable function. If $$6 \int_{1}^{x} f(t)dt=3x f(x)+ x^{3}-4$$ for all $$x\geq 1$$ then the value of $$f(2)-f(3)$$ is

If $$A=\begin{bmatrix}2 & 3 \\3 & 5 \end{bmatrix}$$, then the determinant of the matrix $$ (A^{2025}-3A^{2024}+ A^{2023})$$ is

If a random variable x has the probability distribution

then $$ P(3< x\leq 6)$$ is equal to

Let the relation R on the set $$ M=\left\{ 1,2,3,...,16 \right\}$$ be given by $$ R=\left\{ (x, y): 4y= 5x-3,x,y \text{ }\epsilon \text{ }M\right\}$$.
Then the minimum number of elements required to be added in R, in order to make the relation symmetric, is equal to

Let the set of all values of r, for which the circles $$ (x+1)^{2}+(y+4)^{2}=r^{2}$$ and $$ x^{2}+y^{2}-4x-2y-4=0$$ intersect at two distinct points be the interval $$( \alpha,\beta )$$. Then $$ \alpha\beta $$ is equal to

The value of $$ \int_{\frac{\pi}{2}}^{\frac{\pi}{2}} \left( \frac{1}{[x]+4}\right)dx $$ where [.]denotes the greatest integer function, is

The number of solutions of $$ \tan^{-1}4x + \tan^{-1}6x = \frac{\pi}{6} $$, where $$ -\frac{1}{2\sqrt{6}}<x<\frac{1}{2\sqrt{6}}, $$ is equal to