The order of the differential equation \(\left[1+\left(\frac{d y}{d x}\right)^{5}\right]^{\frac{2}{3}}=\frac{d^{3} y}{d x^{3}}\)

If \(a, b, c\) are in AP, \(b-a, c-b\) and \(a\) are in GP, then \(a: b: c\) is

The radius and centre of circle which equation is \(3 x^2+3 y^2+6 x+12 y+5=0\) is

If \(f(x) = x^3 + \dfrac{3}{2}x^2 + 3x + 3\), then \(f(x)\) is

If \(\int_{\log 2}^{x} \frac{d y}{\sqrt{e^{y}-1}}=\frac{\pi}{6}\), then \(x\) is equal to

If \(f(x)\) satisfies the relation \(2 f(x)+f(1-x)=x^{2}\) for all real \(x\), then \(f(x)\) is

On each working day of a school there are six periods. The number of ways in which five subjects are arranged if each subject is allotted at least one period and no period remains vacant is

Match the coordination compounds in Column I having the given type of hybridisation of \(\mathrm{M}^{\mathrm{n}+}\) ion and magnetic moment as given in Column II.
\(\begin{array}{|l|l|l|l|} \hline & \text { Coordination compounds. } & & \text { Hybridisation \& Magnetic nature. } \\ \hline \text { A } & \mathrm{Ni}(\mathrm{CO})_4 & \text { P } & \mathrm{sp}^3, \quad \mu=5.92 \mathrm{BM} \\ \hline \text { B } & {\left[\mathrm{Ni}(\mathrm{CN})_4\right]^{2-}} & \text { Q } & \mathrm{sp}^3, \quad \mu=2.84 \mathrm{BM} \\ \hline \text { C } & {\left[\mathrm{Ni}(\mathrm{Cl})_4\right]^{2-}} & \text { R } & \mathrm{sp}^3, \quad \mu=0 \\ \hline \text { D } & {[\mathrm{MnBr}]^{2-}} & \text { S } & \mathrm{dsp}^2, \quad \mu=0 \\ \hline \end{array}\)

When 0.1 mole of \(\mathrm{MnO}_4{ }^{2-}\) is oxidised, the quantity of electricity required to completely oxidise \(\mathrm{MnO}_4{ }^{2-}\) to \(\mathrm{MnO}_4^{-}\) is

The height at which the acceleration due to gravity becomes \(\dfrac{g}{16}\) (where, \(g=\) acceleration due to gravity on the surface of the earth) in terms of \(R\) is, if \(R\) is the radius of earth.

The value of \(\cos \left(\frac{\pi}{4}-x\right) \cos \left(\frac{\pi}{4}-y\right)\)
\(-\sin \left(\frac{\pi}{4}-x\right) \sin \left(\frac{\pi}{4}-y\right)\) is equal to

If \(n(A)=p\) and \(n(B)=q\), then the numbers of relations from the set \(A\) to the set \(B\) is

The decomposition of \(\mathrm{PH}_3\) follows first order kinetics. The time required for \(3 / 4^{\text {th }}\) of \(\mathrm{PH}_3\) to decompose is 75.76 s . Calculate the fraction of original amount of \(\mathrm{PH}_3\) which will remain after 2.0 minutes.