NTA JEE Mains 22nd Jan 2025 Shift 2

If $$\sum_{r=1}^{30}\f\frac{r^{2}({}^{30}C_{r})^{2}}{{}^{30}C_{r-1}}=\alpha \t\times 2^{29}$$, then $$\alpha$$ is equal to______.

Let A = {1, 2, 3}. The number of relations on A, containing (1,2) and (2,3), which are reflexive and transitive but not symmetric, is ______ -

Let A(6,8),$$B(10\cos \alpha, -10\sin \alpha)$$ and $$C(-10\sin \alpha, 10\cos \alpha)$$. be the vertices of a triangle. If L(a, 9) and G(h, k) be its orthocenter and centroid respectively, then $$(5a − 3h + 6k + 100 \sin 2\alpha)$$ is equals to_______.

Let y = f(x) be the solution of the differential equation $$\frac{dy}{dx}+\frac{xy}{x^{2}-1}=\frac{x^{6}+4x}{\sqrt{1-x^{2}}},-1 < x < 1$$ such that f(0)=0.If $$6\int_{-\frac{1}{2}}^{\frac{1}{2}}f(x)dx=2\pi - \alpha$$ then $$\alpha^{2}$$ is equal to ______.

Let the distance between two parallel lines be 5 units and a point P lie between the lines at a unit distance from one of them. An equilateral triangle PQR is formed such that Q lies on one of the parallel lines, while R lies on the other. Then $$(QR)^{2}$$ is equal to______.

To obtain the given truth table, following logic gate should be placed at G:

A small rigid spherical ball of mass M is dropped in a long vertical tube containing glycerine. The velocity of the ball becomes constant after some time. If the density of glycerine is half of the density of the ball, then the viscous force acting on the ball will be (consider g as acceleration due to gravity)

The torque due to the force $$(2\widehat{i}+\widehat{j}+2\widehat{k})$$ about the origin, acting on a particle whose position vector is $$(\widehat{i}+\widehat{j}+\widehat{k})$$, would be

A symmetric thin biconvex lens is cut into four equal parts by two planes AB and CD as shown in figure. If the power of original lens is 4 D then the power of a part of the divided lens is

For a short dipole placed at origin O, the dipole moment is along x-axis, as shown in the figure. If the electric potential and electric field at A are $$V_{\circ}$$ and $$E_{\circ}$$, respectively, then the correct combination of the electric potential and electric field, respectively, at point B on the -axis is given by