The letters of the word "COCHIN" are permuted and all the permutations are arranged in alphabetical order as in an English dictionary. The number of words that appear before the word "COCHIN" is

The equation \(\dfrac{x^2}{2-\lambda}+\dfrac{y^2}{\lambda-5}+1=0\) represents an ellipse, if

\(\text { If } y=\sqrt{\sin x+y} \text { then find } \dfrac{d y}{d x} \text { at } x=0, \quad y=1\)

\(\text { If } A=\left(\begin{array}{ll}
1 & 2 \\
0 & 1
\end{array}\right) \quad P=\left(\begin{array}{cc}
\cos \theta & \sin \theta \\
-\sin \theta & \cos \theta
\end{array}\right) \quad Q=P^T A P, \quad \text { then } P Q^{2014} P^T \text { is equal to }\)

The distance between the foci of a hyperbola is 16 and its eccentricity is \(\sqrt{2}\). Then its equation is

\(\int \frac{\sin x \cos x}{\sqrt{1-\sin ^{4} x}} d x=\)

A variable line \(\frac{x}{a}+\frac{y}{b}=1\) is such that \(a+b=4\). The locus of the mid-point of the portion of the line intercepted between the axes is

Two subatomic particles 1 and 2, with the same kinetic energies have their de- Broglie wavelengths as \(\lambda_1 \& \lambda_2\) and masses as 3 m and 6 m respectively.
Determine the ratio \(\lambda_1: \lambda_2\).

A first order reaction proceeds to \(90 \%\) completion. What will be the approximate time taken for \(90 \%\) completion in relation to \(t_{1 / 2}\) of the reaction?

The following results were obtained during study of the reaction \(2 \mathrm{NO}(\mathrm{g})+\mathrm{Cl}_2(\mathrm{~g}) \rightarrow 2 \mathrm{NOCl}(\mathrm{g})\). Determine the value of \([\mathrm{X}]\) in \(\mathrm{mol} / \mathrm{L}\)
\(\begin{array}{|c|c|c|c|} \hline \text{Experiment} & \text{[NO] mol/L} & \left[\mathrm{Cl}_2\right] \text{ mol/L} & \text{Initial rate of formation. [NOCl] mol/L/min} \\ \hline \text{I} & 0.2 & 0.2 & 6.0 \times 10^{-3} \\ \hline \text{II} & 0.2 & 0.4 & 2.4 \times 10^{-2} \\ \hline \text{III} & 0.4 & 0.2 & 1.2 \times 10^{-2} \\ \hline \text{IV} & X & 0.6 & 1.35 \times 10^{-1} \\ \hline \end{array}\)

Highest molar conductivity is given by

If \(x\) occurs is the expansion of \(\left(x+\frac{1}{x}\right)^{n}\), then its coefficient is

The maximum value of \(Z=10 x+16 y\), subject to constraints \(x \geq 0, y \geq 0, x+y \leq 12,2 x+y \leq 20\) is