NTA JEE Mains 28th Jan 2026 Shift 2

If $$\sum_{r=1}^{25}\left( \frac{r}{r^{4}+r^{2}+1} \right)=\frac{p}{q},$$ where p and q are positive integers such that gcd(p,q)=1, then p+q is equal to ___________

Three persons enter in a lift at the ground floor. The lift will go upto $$10^{th}$$ floor. The number of ways, in which the three persons can exit the lift at three different floors, if the lift does not stop at first, second and third floors, is equal to________

Let $$A=\begin{bmatrix}3 & -4 \\1 & -1 \end{bmatrix}$$ and B be two matrices such that $$A^{100}=100B+I$$. Then the sum of all the elements of $$B^{100}$$ is_______

Let $$ f$$ be a differentiable function satisfying $$f(x)=1-2x+\int_{0}^{x} e^{(x-t)}f(t)dt, x \in \mathbb{R}$$ and let $$g(x)=\int_{0}^{x} (f(t)+2)^{15}(t-4)^{6}(t+12)^{17}dt, x \in \mathbb{R}.$$ If p and q are respectively the points of local minima and local maxima of g, then the value of $$\mid p+q \mid $$ is equal to _________

If the distance of the point $$P(43, \alpha, \beta), \beta<0,$$ from the line $$\overrightarrow{r} = 4\widehat{i}-\widehat{k}+\mu(2\widehat{i}+3\widehat{k}), \mu \in \mathbb{R}$$ along a line with direction ratios 3, -1, 0 is $$13\sqrt{10},$$ then $$ \alpha ^{2}+ \beta^{2}$$ is equal to______

Two p-n junction diodes $$D_{1}\text{ and }D_{2}$$ are connected as shown in figure. A and B are input signals and C is the output. The given circuit will function as a________.

A small block of mass m slides down from the top of a frictionless inclined surface, while the inclined plane is moving towards left with constant acceleration $$a_{0}$$. The angle between U1e inclined plane and ground is O and its base length is L. Assuming that initially the small block is at the top of the inclined plane, the time it takes to reach the lowest point of the inclined plane is ___ .

Number of photons of equal energy emitted per second by a 6 mW laser source operating at 663 nm is ____ . (Given: $$h=6.63\times 10^{-34}J.s\text{ and }c=3\times 10^{8} m/s$$)

As shown in the figure, a spring is kept in a stretched position with some extension by holding the masses 1 kg and 0.2 kg with a separation more than spring natural length and are released. Assuming the horizontal serface to be frictionless, the angular frequency (in SI unit) of the system is:

A Wheatstone bridge is initially at room temperature and all arms of the bridge have same value of resistances $$(R_{1}=R_{2}=R_{3}=R_{4})$$. When $$R_{3}$$ resistance is heated to some temperature, its resistance value has gone up by 10%. The potential difference $$(V_{a}-V_{b})$$ (after $$R_{3}$$ is heated) is ____ V.