If \(\tan \left(\frac{\pi}{4}+\frac{y}{2}\right)=\tan ^3\left(\frac{\pi}{4}+\frac{x}{2}\right)\) then \(\frac{3 \sin x+\sin ^3 x}{1+3 \sin ^2 x}=\)

The number of five digit numbers divisible by 5 that can be formed using the numbers \(0,1,2,3,4,5\) without repetition is

Suppose two tangents PA and PB are drawn to the circle centered at \(\mathrm{C}(1,2)\) from the point \(\mathrm{P}(16,7)\). If the area of the quadrilateral PACB is 75 square units, then the radius of the circle is

Find the value(s) of \(k\) such that the distance between the two parallel lines represented by \((x-2 y)^2+k(x-2 y)=0\) is 3 units

Two balls are drawn from an urn containing 7 white, 6 red and 8 black balls one after the other without replacement. Then the probability that atleast one of them is white, is

The matrix \(A=\left[\begin{array}{cc}\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\ \frac{-1}{\sqrt{2}} & \frac{-1}{\sqrt{2}}\end{array}\right]\) is

\(\int e^{4 x^2+8 x-4}(x+1) \cos \left(3 x^2+6 x-4\right) d x=\)

Drops of liquid of density \(d\) are floating half immersed in a liquid of density \(\rho\). If the surface tension of the liquid is \(T\), then the radius of the drop is

If \(\mathbf{a}, \mathbf{b}, \mathbf{c}\) are three non-coplanar vectors, then match the items of List-I with those of List-II.


The correct answer is
A \(\quad\) B \(\quad\) B \(\quad\) D

An earthquake generates both transverse \(S\) and longitudinal \(P\) waves in the earth with speeds \(4.5 \mathrm{~km} \mathrm{~s}^{-1}\) and \(8.0 \mathrm{~km} \mathrm{~s}^{-1}\), respectively. A seismograph records that the first \(P\)-wave arrives 3.5 minutes earlier than the first S-wave. From the seismograph, the epicentre of the earthquake is located at a distance.

The length and area of cross-section of a copper wire are respectively 30 m and \(6 \times 10^{-7} \mathrm{~m}^2\). If the resistivity of copper is \(1.7 \times 10^{-8} \Omega \mathrm{~m}\), then the resistance of the wire is

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:

The elastic potential energy stored in a copper rod of length one metre and area of cross-section \(1 \mathrm{~mm}^2\) when stretched by 1 mm is
(Young's modulus of copper \(=1.2 \times 10^{11} \mathrm{Nm}^{-2}\) )