Two projectile thrown at \(30^{\circ}\) and \(45^{\circ}\) with the horizontal respectively, reach the maximum height in same time. The ratio of their initial velocities is

The pitch and the number of divisions, on the circular scale, for a given screw gauge are \(0.5\,mm\) and \(100\) respectively. When the screw gauge is fully tightened without any object, the zero of its circular scale lies \(3\) divisions below the mean line. The readings of the main scale and the circular scale for a thin sheet, are \(5.5\,mm\) and \(48\) respectively, the thickness of this sheet is

The displacement x versus time graph is shown below.

(A) The average velocity during 0 to 3 s is \(10 \mathrm{~m} / \mathrm{s}\)
(B) The average velocity during 3 to 5 s is \(0 \mathrm{~m} / \mathrm{s}\)
(C) The instantaneous velocity at \(\mathrm{t}=2 \mathrm{~s}\) is \(5 \mathrm{~m} / \mathrm{s}\)
(D) The average velocity during 5 to 7 s and instantaneous velocity at \(\mathrm{t}=6.5 \mathrm{~s}\) are equal
(E) The average velocity from \(t=0\) to \(t=9 \mathrm{~s}\) is zero
Choose the correct answer from the options given below:

The distance of centre of mass from end \(A\) of a one dimensional rod \(( AB )\) having mass density \(\rho=\rho_{0}\left(1-\frac{ x ^{2}}{ L ^{2}}\right)\,kg / m\) and length \(L\) (in meter) is \(\frac{3 L }{\alpha} m\). The value of \(\alpha\) is \(\ldots \ldots \ldots . .\) (where \(x\) is the distance form end \(A\) )

A musician using an open flute of length \(50\,cm\) produces second harmonic sound waves. A person runs towards the musician from another end of a hall at a speed of \(10 \,km/h.\) If the wave speed is \(330\,m/s,\) the frequency heard by the running person shall be close to...... \(Hz\)

If the magnetic field in a plane electromagnetic wave is given by \(\overrightarrow{\mathrm{B}}=3 \times 10^{-8} \sin \left(1.6 \times 10^{3} \mathrm{x}+48 \times 10^{10} \mathrm{t}\right) \hat{\mathrm{j}}\; \mathrm{T}\) then what will be expression for electric field?

Intensity of sunlight is observed as \(0.092\, {Wm}^{-2}\) at a point in free space. What will be the peak value of magnetic field at that point? \(\left(\sigma_{0}=8.85 \times 10^{-12}\, {C}^{2} \,{N}^{-1} \,{m}^{-2}\right.\) )

A string of area of cross-section \(4\,mm ^{2}\) and length \(0.5\) is connected with a rigid body of mass \(2\,kg\). The body is rotated in a vertical circular path of radius \(0.5\,m\). The body acquires a speed of \(5\,m / s\) at the bottom of the circular path. Strain produced in the string when the body is at the bottom of the circle is \(\ldots . . \times 10^{-5}\). (Use Young's modulus \(10^{11}\,N / m ^{2}\) and \(g =10\,m / s ^{2}\) )

The nuclear activity of a radioactive element becomes \(\left(\frac{1}{8}\right)^{\text {th }}\) of its initial value in \(30\, years.\) The half-life of radioactive element is \(....\,years.\)

The distance between object and its \(3\) times magnified virtual image as produced by a convex lens is \(20 \mathrm{~cm}\). The focal length of the lens used is _______ \(\mathrm{cm}\).

Match List\(-I\) with List\(-II.\)

List\(-I\)	List\(-II\)
\((a)\) Torque	\((i)\) \({MLT}^{-1}\)
\((b)\) Impulse	\((ii)\) \({MT}^{-2}\)
\((c)\) Tension	\((iii)\) \({ML}^{2} {T}^{-2}\)
\((d)\) Surface Tension	\((iv)\) \({MI} {T}^{-2}\)

Choose the most appropriate answer from the option given below :