Let \(S_n=\sum_{k=0}^n \frac{n}{n^2+k n+k^2}\) and \(T_n=\sum_{k=0}^{n-1} \frac{n}{n^2+k n+k^2}\), for \(n=1,2,3, \ldots\), then

The common tangents to the circle ${\mathrm{x}}^2+{\mathrm{y}}^2=2$ and the parabola ${\mathrm{y}}^2=8\mathrm{x}$ touch the circle at the points P, and Q the parabola at the points R, S. Then the area of the quadrilateral PQRS is

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Consider the function \(f:(-\infty, \infty) \rightarrow(-\infty, \infty)\) defined by \(f(x)=\frac{x^2-a x+1}{x^2+a x+1} ; 0 < a < 2\)Question:
Which of the following is true?

If a chord, which is not a tangent, of the parabola $y^2=16x$ has the equation $2x+y=p$, and midpoint (h, k), then which of the following is (are) possible value(s) of $p,h$ and $k$?
 

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A circle \(C\) of radius 1 is inscribed in an equilateral \(\triangle P Q R\). The points of contact of \(C\) with the sides \(P Q, Q R, R P\) are \(D, E, F\) respectively. The line \(P Q\) is given by the equation
\(\sqrt{3} x+y-6=0\) and the point \(D\) is \(\left(\frac{3 \sqrt{3}}{2}, \frac{3}{2}\right)\). Further, it is given that the origin and the centre of \(C\) are on the same side of the line \(P Q\).Question:
The equation of circle \(C\) is

Two fair dice, each with faces numbered $1,2,3,4,5$ and $6$, are rolled together and the sum of the numbers on the faces is observed. This process is repeated till the sum is either a prime number or a perfect square. Suppose the sum turns out to be a perfect square before it turns out to be a prime number. If $p$ is the probability that this perfect square is an odd number, then the value of $14p$ is_____

Consider the set of eight vectors $\text{V}=\left\{\text{a}\widehat{i}+\text{b}\widehat{\text{j}}+\text{c}\widehat{\text{k}}:\; a,\; b,\; c \in \left\{-1,\; 1\right\}\right\}$. Three non-coplanar vectors can be chosen from $\text{V}$ in $2^{\text{p}}$ ways. Then $\text{p}$ is

A pendulum consists of a bob of mass \(m=0.1 \mathrm{~kg}\) and a massless inextensible string of length \(L=1.0 \mathrm{~m}\). It is suspended from a fixed point at height \(H=0.9 \mathrm{~m}\) above a frictionless horizontal floor. Initially, the bob of the pendulum is lying on the floor at rest vertically below the point of suspension. A horizontal impulse \(P=0.2 \mathrm{~kg}-\mathrm{m} / \mathrm{s}\) is imparted to the bob at some instant. After the bob slides for some distance, the string becomes taut and the bob lifts off the floor. The magnitude of the angular momentum of the pendulum about the point of suspension just before the bob lifts off is \(J \mathrm{~kg}-\mathrm{m}^{2} / \mathrm{s}\). The kinetic energy of the pendulum just after the liftoff is \(K\) Joules.
The value of $K$ is ____.

If \(\lim _{x \rightarrow \infty}\left(\frac{x^{2}+x+1}{x+1}-a x-b\right)=4\), then

Match the statements/expressions given in Column I with the values given in Column II.

In plotting stress versus strain curves for two materials \(P\) and \(Q\), a student by mistake puts strain on the \(y\)-axis and stress on the \(x\)-axis as shown in the figure. Then the correct statement(s) is(are)

Two particles of mass \(m\) each are tied at the ends of a light string of length \(2 a\). The whole system is kept on a frictionless horizontal surface with the string held tight so that each mass is at a distance \(a\) from the centre \(P\) (as shown in the figure). Now, the mid-point of the string is pulled vertically upwards with a small but constant force \(F\). As a result, the particles move towards each other on the surface.
The magnitude of acceleration, when the separation between them becomes \(2 x\), is

The number of optically active products obtained from the complete ozonolysis of the given compound is :