In a potentiometer of $10$ wires, the balance point is obtained on the $6^{\text{th }}$ wire. To shift the balance point to $8^{\text{th }}$ wire, we should

A sound source is moving towards a stationary listener with \(1 / 10\) th of the speed of sound. The ratio of apparent to real frequency is

A body is projected vertically upwards from the surface of the earth with a velocity sufficient to carry it to infinity. The time taken by it to reach a height of three times the radius of the earth is (acceleration due to gravity \(=9.8 \mathrm{~ms}^{-2}\) and radius of the earth \(=6400 \mathrm{~km}\) )

A car moving with uniform acceleration covers the distance of \(200 \mathrm{~m}\) in first 2 seconds and the distance of \(220 \mathrm{~m}\) in next 4 seconds. The velocity of the car after 7 seconds is

The total random kinetic energy of \(1 \mathrm{~g}\) of helium at \(100 \mathrm{~K}\) will be \(\left(\mathrm{R}=8.3 \mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}\right)\)

A particle is released from a height $H$. At a certain height its kinetic energy is half of its potential energy with reference to the surface of the earth. Height and speed of the particle at that instant are respectively

Match the following List I with List II.
\(\begin{array}{|c|c|c|}
\hline & \text { List I } & \text { List II } \\
\hline \text { (A) } & \text { Transverse wave (i) } & \begin{array}{l}
\text {Vibrations parallel to the } \\
\text {direction of propagation }
\end{array} \\
\hline \text { (B) } & \begin{array}{l}
\text {Longitudinal wave }
\end{array} & \begin{array}{l}
\text {Vibrations perpendicular to } \\
\text {the direction of propagation }
\end{array} \\
\hline \text { (C) } & \text { Beats } & \begin{array}{l}
\text {Superposition of waves } \\
\text {travelling in the opposite directions }
\end{array} \\
\hline \text { (D) } & \text { Stationary waves (iv) } & \begin{array}{l}
\text {Superposition of waves } \\
\text {travelling in same direction }
\end{array} \\
\hline
\end{array}\)
The correct answer is

If \(\mathbf{a}=\hat{i}+\hat{j}+\hat{k}, \mathbf{c}=\hat{j}-\hat{k}, \mathbf{a} \times \mathbf{b}=\mathbf{c}, \mathbf{a} \cdot \mathbf{b}=3\), then \(\mathbf{b}=\)

\(\int_0^{\frac{\pi}{2}} \frac{\sum_{n=0}^4\left(\frac{n \pi}{4}+x\right)}{\cos x+\sin x} d x=\)

A coil of inductance $0.1 \mathrm{H}$ and resistance $110 \mathrm{\Omega }$ is connected to a source of $110 \mathrm{V}$ and $350 \mathrm{Hz}$. The phase difference between the voltage maximum and the current maximum is

\(\begin{aligned}
&\text { Mean deviation about the mean for the following data is }\\
&\begin{array}{|l|c|c|c|c|c|}
\hline \text { Class Interval } & 0-6 & 6-12 & 12-18 & 18-24 & 24-30 \\
\hline \text { Frequency } & 1 & 2 & 3 & 2 & 1 \\
\hline
\end{array}
\end{aligned}\)

3-hydroxybutanal is formed when \((X)\) reacts with \((Y)\) in dilute \((Z)\) solution. What are \(X, Y\) and \(Z\) ?
\(\begin{array}{lll}X & Y & Y\end{array}\)

\(\tan 9^{\circ}-\tan 27^{\circ}-\tan 63^{\circ}+\tan 81^{\circ}=\)