Force acts for \(20\,s\) on a body of mass \(20\,kg\), starting from rest, after which the force ceases and then body describes \(50\,m\) in the next \(10\,s\). The value of force will be \(..........\,N\)

Shown in the figure is a hollow icecream cone (it is open at the top). If its mass is \(M,\) radius of its top, \(R\) and height, \(H\), then its moment of inertia about its axis is

Heat energy of \(184\,kJ\) is given to ice of mass \(600\,g\) at \(-12^{\circ}\,C\), Specific heat of ice is \(2222.3\,J\,kg ^{-1^{\circ}}\,C ^{-1}\) and latent heat of ice in \(336\,kJ / kg ^{-1}\) \((A)\) Final temperature of system will be \(0^{\circ} C\). \((B)\) Final temperature of the system will be greater than \(0^{\circ} C\). \((C)\) The final system will have a mixture of ice and water in the ratio of \(5: 1\). \((D)\) The final system will have a mixture of ice and water in the ratio of \(1: 5\). \((E)\) The final system will have water only. Choose the correct answer from the options given below:

The engine of a train moving with speed \(10\,ms ^{-1}\) towards a platform sounds a whistle at frequency \(400\,Hz\). The frequency heard by a passenger inside the train is \(........\,Hz\) (neglect air speed. Speed of sound in air \(330\,ms ^{-1}\) )

Calculate the amount of charge on capacitor of \(4\, \mu\, {F}\). The internal resistance of battery is \(1\, \Omega\) : (in \(\mu {C}\))

A particle moving in a straight line covers half the distance with speed \(6 \mathrm{~m} / \mathrm{s}\). The other half is covered in two equal time intervals with speeds 9 \(\mathrm{m} / \mathrm{s}\) and \(15 \mathrm{~m} / \mathrm{s}\) respectively. The average speed of the particle during the motion is _______.

A body weights \(49\,N\) on a spring balance at the north pole. ..... \(N\) will be its weight recorded on the same weighing machine, if it is shifted to the equator? (Use \(g=\frac{G M}{R^{2}}=9.8 \,ms ^{-2}\) and radius of earth, \(R =6400\, km .]\)

Let \(O\) be the origin and the position vector of the point \(P\) be \(-\hat{i}-2 \hat{j}+3 \hat{k}\). If the position vectors of the points \(A , B\) and \(C\) are \(-2 \hat{i}+\hat{j}-3 \hat{k}, 2 \hat{i}+4 \hat{j}-2 \hat{k}\) and \(-4 \hat{i}+2 \hat{j}-\hat{k}\) respectively then the projection of the vector \(\overline{O P}\) on a vector perpendicular to the vectors \(\overline{A B}\) and \(\overline{A C}\) is \(......\).

If the angle of elevation of a cloud from a point \(P\) which is \(25\,m\) above a lake be \(30^o\) and the angle of depression of reflection of the cloud in the lake from \(P\) be \(60^o,\) then the height of the cloud (in meters) from the surface of the lake is

If \(S=\frac{7}{5}+\frac{9}{5^{2}}+\frac{13}{5^{3}}+\frac{19}{5^{4}}+\ldots .\), then \(160 \mathrm{~S}\) is equal to....... .

In a building there are \(15\) bulbs of \(45\; \mathrm{W}, 15\) bulbs of \(100\; \mathrm{W}, 15\) small fans of \(10 \;\mathrm{W}\) and \(2\) heaters of \(1 \;\mathrm{kW}\). The voltage of electric main is \(220\; \mathrm{V}\). The minimum fuse capacity (rated value) of the building will be: .......... \(A\)

If the mean of the frequency distribution

Class:	\(0-10\)	\(10-20\)	\(20-30\)	\(30-40\)	\(40-50\)
Frequency	\(2\)	\(3\)	\(x\)	\(5\)	\(4\)

is \(28\) , then its variance is \(........\).

Let \(\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}\) and \(\overrightarrow{\mathrm{c}}\) be three unit vectors such that \(|\vec{a}-\vec{b}|^{2}+|\vec{a}-\vec{c}|^{2}=8\) Then \(|\vec{a}+2 \vec{b}|^{2}+|\vec{a}+2 \vec{c}|^{2}\) is equal to