Let \(S\) be the focus of the parabola \(y^2=4 a x\) and \(\mathrm{PQ}\) be a focal chord such that \(\mathrm{SP}=\alpha\) and \(\mathrm{SQ}=\alpha^{\prime}\). Then \(\frac{1}{\alpha}+\frac{1}{\alpha^{\prime}}=\)

If \(y=\log \left(\frac{1+x}{1-x}\right)^{1 / 4}-\frac{1}{2} \tan ^{-1}(x)\), then \(\frac{d y}{d x}\) at \(x=\frac{1}{\sqrt{2}}\) equals

If \(D, E\) and \(F\) are respectively the mid-points of \(A B, A C\) and \(B C\) in \(\triangle A B C\), then \(\overrightarrow{\mathbf{B E}}+\overrightarrow{\mathbf{A F}}\) is equal to :

\(\alpha, \beta, \gamma\) are the roots of the equation \(x^3-10 x^2+7 x+8=0\). Match the following and choose the correct answer.

A. \(\alpha+\beta+\gamma\)	(1) \(-\frac{43}{4}\)
B. \(\alpha^2+\beta^2+\gamma^2\)	(2) \(-\frac{7}{8}\)
C. \(\frac{1}{\alpha}+\frac{1}{\beta}+\frac{1}{\gamma}\)	(3) 86
D. \(\frac{\alpha}{\beta \gamma}+\frac{\beta}{\gamma \alpha}+\frac{\gamma}{\alpha \beta}\)	(4) 0
	(5) 10

If \(\lim _{x \rightarrow 0}\left(\frac{\cos 4 x+a \cos 2 x+b}{x^4}\right)\) is finite, then the values of \(a, b\) are respectively :

The point \(P(a, b)\) lies on the straight line \(3 x+2 y=13\) and the point \(Q(b, a)\) lies on the straight line \(4 x-y=5\). Then the equation of the line \(P Q\) is

If \(x=\frac{2.5}{(2 !) 3}+\frac{2.5 .7}{(3 !) 3^2}+\frac{2.5 .7 .9}{(4 !) 3^3}+\ldots\) then \(x^2+8 x+8=\)

If the binding energy of $N^{14}$ is $7.5 \mathrm{MeV}$ per nucleon and that of $N^{15}$ is$7.7 \mathrm{MeV}$ per nucleon. then the energy required to remove a neutron from $N^{15}$ is

A particle of mass 0.5 g and charge \(10 \mu \mathrm{C}\) is subjected to a uniform electric field of \(8 \mathrm{NC}^{-1}\). If the particle is initially at rest, the velocity of the particle after a time of 5 seconds is

Identify the reaction which does not liberate \(\mathrm{N}_2\)

(i) \(\mathrm{H}_3 \mathrm{PO}_4(a q) \rightleftharpoons \mathrm{H}^{+}(a q)+\mathrm{H}_2 \mathrm{PO}_4^{-}(a q)\)
(ii) \(\mathrm{H}_2 \mathrm{PO}_4^{-}(a q) \rightleftharpoons \mathrm{H}^{+}(a q)+\mathrm{HPO}_4^{2-}(a q)\)
(iii) \(\mathrm{HPO}_4^{2-}(a q) \rightleftharpoons \mathrm{H}^{+}(a q)+\mathrm{PO}_4^{3-}(a q)\)
The equilibrium constants for the above reactions at a certain temperature are \(K_1, K_2\) and \(K_3\) respectively. The equilibrium constant for the reaction \(\mathrm{H}_3 \mathrm{PO}_4(a q) \rightleftharpoons 3 \mathrm{H}^{+}(a q)+\mathrm{PO}_4^{3-}(a q)\) in terms of \(K_1, K_2\) and \(K_3\) is

Two bodies of masses \(12 \mathrm{~kg}\) and \(6 \mathrm{~kg}\) are projected simultaneously with velocities \(15 \mathrm{~ms}^{-1}\) and \(20 \mathrm{~ms}^{-1}\) respectively from the top of a tower of height \(25 \mathrm{~m}\).
The body of mass \(12 \mathrm{~kg}\) is projected vertically upwards and the body of mass \(6 \mathrm{~kg}\) is projected horizontally. The maximum height reached by the centre of mass of the system of two bodies from the ground.
\[
\left(\mathrm{g}=10 \mathrm{~ms}^{-2}\right)
\]

The displacement \(y(\) in \(\mathrm{cm})\) in case of a simple harmonic wave is given by \(y=\frac{10}{\pi} \sin \left(2000 \pi t-\frac{\pi x}{17}\right)\). The period and maximum velocity of the particles in the medium will respectively be