Let $\mathrm{S}_{\mathrm{n}}=\sum_{\mathrm{k}=1}^{\mathrm{n}} \frac{\mathrm{n}}{\mathrm{n}^2+\mathrm{kn}+\mathrm{k}^2}$ and $\mathrm{T}_{\mathrm{n}}=\sum_{\mathrm{k}=0}^{\mathrm{n}-1} \frac{\mathrm{n}}{\mathrm{n}^2+\mathrm{kn}+\mathrm{k}^2}$ for $\mathrm{n}=1,2,3, \ldots$. Then,

A satellite is moving with a constant speed ' $V$ ' in a circular orbit about the earth. An object...

A light ray travelling in glass medium is incident on glass-air interface at an angle of incidence $\theta$. The reflected...

Let $\mathrm{P}\left(\mathrm{x}_1, \mathrm{y}_1\right)$ and $\mathrm{Q}\left(\mathrm{x}_2, \mathrm{y}_2\right), \mathrm{y}_1<0, \mathrm{y}_2<0$, be the end points of the latus rectum of the ellipse $\mathrm{x}^2+4 \mathrm{y}^2=4$. The equations of parabolas with latus rectum PQ are

A straight line through the vertex P of a triangle PQR intersects the side QR at the point S and...

A wooden block performs SHM on a frictionless surface with frequency, $\mathrm{v}_0$. The block carries a charge +Q on its...

The density of a solid ball is to be determined in an experiment. The diameter of the ball is measured...

Let $\mathrm{f}(\mathrm{x})$ be a non-constant twice differentiable function defined on $(-\infty, \infty)$ such that $\mathrm{f}(\mathrm{x})=\mathrm{f}(1-\mathrm{x})$ and $\mathrm{f}^{\prime}\left(\frac{1}{4}\right)=0$. Then (A) (B)...

The edges of a parallelopiped are of unit length and are parallel to non-coplanar unit vectors $\hat{a}, \hat{b}, \hat{c}$ such...

A ball of mass 0.2 kg rests on a vertical post of height 5 m . A bullet of mass...

Which of the field patterns given below is valid for electric field as well as for magnetic field?

Consider the two curves $C_1: y^2=4 x, C_2: x^2+y^2-6 x+1=0$. Then,

Match the reactions in column I with appropriate types of steps/reactive intermediate involved in these reactions as given in column...

Match the transformations in column I with appropriate options in column II

The number of hexagonal faces that are present in a truncated octahedron is

In 1 L saturated solution of $\mathrm{AgCl}\left[\mathrm{K}_{\mathrm{sp}}(\mathrm{AgCl})=1.6 \times 10^{-10}\right], 0.1 \mathrm{~mol}$ of $\mathrm{CuCl}\left[\mathrm{K}_{\mathrm{sp}}(\mathrm{CuCl})=1.0 \times 10^{-6}\right]$ is added. The resultant concentration...

Let $\mathrm{g}(\mathrm{x})=\frac{(\mathrm{x}-1)^{\mathrm{n}}}{\log \cos ^{\mathrm{m}}(\mathrm{x}-1)} ; 0<\mathrm{x}<2, \mathrm{~m}$ and n are integers, $\mathrm{m} \neq 0, \mathrm{n}>0$, and let p be the left hand derivative of $|\mathrm{x}-1|$ at $x=1$. If $\lim _{x \rightarrow 1^{+}} g(x)=p$, then

The volume (in mL ) of $0.1 \mathrm{M} \mathrm{AgNO}_3$ required for complete precipitation of chloride ions present in 30 mL...