The number of ways of dividing 200 dissimilar things into 10 groups each containing 20 elements is

The line \(x-2=0\) cuts the circle \(x^2+y^2-8 x-2 y+8=0\) at \(A\) and \(B\). The equation of the circle passing through the points \(A\) and \(B\) and having least radius is

\(\sum_{k=0}^{12} \frac{1}{\sin \left((k+1) \frac{\pi}{6}+\frac{\pi}{4}\right) \sin \left(\frac{k \pi}{6}+\frac{\pi}{4}\right)}=\)

The normal at a point on the parabola $y^2=4x$ passes through $(5,0)$. If two more normals to this parabola also pass through$(5,0),$  then the centroid of the triangle formed by the feet of these three normals is

If the equation whose roots are \(p\) times the roots of \(x^4-2 a x^3+4 b x^2+8 a x+16=0\) is a reciprocal equation, then \(|p|=\)

The domain of \(f(x)=\cos ^{-1}\left(\frac{x-3}{2}\right)-\log _{10}(4-x)\) is

If \(z=x+i y\) and \(z^2=(i \bar{z})^2\), then

The area of the parallelogram, whose diagonals are $2\hat{i}-\hat{j}+\hat{k}$ and $\hat{i}+3\hat{j}-\hat{k},$ is equal to

\[
(-1+i \sqrt{3})^{60}=
\]

The horizontal distance between a tower and a builidng is \(10 \sqrt{3}\) units. If the angle of depression of the foot of the building from the top of the tower is \(60^{\circ}\) and the angle of elevation of the top of the building from the foot of the tower is \(30^{\circ}\), then the sum of the heights of the tower and the building is

The standard reduction potentials of \(\mathrm{Zn}^{2+}\left|\mathrm{Zn}, \mathrm{Cu}^{2+}\right| \mathrm{Cu}\) and \(\mathrm{Ag}^{+} \mid \mathrm{Cu}\) and \(\mathrm{Ag}^{+} \mid \mathrm{Ag}\) are respectively \(-0.76,0.34\) and \(0.8 \mathrm{~V}\). The following cells were constructed :
(1) \(\mathrm{Zn}\left|\mathrm{Zn}^{2+}\right|\left|\mathrm{Cu}^{2+}\right| \mathrm{Cu}\)
(2) \(\mathrm{Zn}\left|\mathrm{Zn}^{2+}\right|\left|\mathrm{Ag}^{+}\right| \mathrm{Ag}\)
(3) \(\mathrm{Cu}\left|\mathrm{Cu}^{2+} \| \mathrm{Ag}^{+}\right| \mathrm{Ag}\)
What is the correct order of \(E_{\text {cell }}^{\circ}\) of these cells?

The initial rates of decrease of \(\mathrm{I}_2\) in acetone - iodine reaction catalysed by \(\mathrm{H}^{+}\) are given in the table.
\(\begin{array}{|c|c|c|c|c|}
\hline \begin{array}{c}
\text { Experiment } \\
\end{array} & \begin{array}{c}
\text { Initial } \\
{\left[\mathrm{I}_2\right]} \\
\left(\mathrm{mol} \mathrm{L}^{-1}\right)
\end{array} & \begin{array}{c}
\text { Initial } \\
{\left[\mathrm{H}^{+}\right]} \\
\left(\mathrm{mol} \mathrm{L}^{-1}\right)
\end{array} & \begin{array}{c}
\text { Initial } \\
{\left[\mathrm{CH}_3 \mathrm{COCH}_3\right]} \\
\left(\mathrm{mol} \mathrm{L}^{-1}\right)
\end{array} & \begin{array}{c}
\text { Initial rate of decrease of } \mathrm{I}_2 \\
\left(\mathrm{mol} \mathrm{L}^{-1} \mathrm{~s}^{-1}\right)
\end{array} \\
\hline 1 & 0.01 & 0.1 & 0.1 & 0.096 \\
2 & 0.01 & 0.2 & 0.1 & 0.192 \\
3 & 0.02 & 0.2 & 0.1 & 0.192 \\
4 & 0.01 & 0.2 & 0.2 & 0.384 \\
\hline
\end{array}\)
The order with respect to \(\mathrm{I}_2, \mathrm{H}^{+}\), acetone and total order of the reaction respectively are:

If the radical axis of the circles \(x^2+y^2+2 g x+2 f y+c=0\) and \(2 x^2+2 y^2+3 x+8 y+2 c=0\) touches the circle \(x^2+y^2+2 x+2 y+1=0\), then