When a resistance of \(5\,\Omega\) is shunted with a moving coil galvanometer, it shows a full scale deflection for a current of \(250\,mA\), however when \(1050\,\Omega\) resistance is connected with it in series, it gives full scale deflection for \(25\) volt. The resistance of galvanometer is \(......\,\Omega\).

Two cars \(P\) and \(Q\) start from a point at the same time in a straight line and their positions are represented by \( x_p(t)=at+bt^2 \) and \(x_Q(t)= ft- t^2\). At what time do the cars have the same velocity \(?\) 

Consider a cylinder of mass \(M\) resting on a rough horizontal rug that is pulled out from under it with acceleration \('a'\) perpendicular to the axis of the cylinder. What is \(F_{friction}\) at point \(P\) ? It is assumed that the cylinder does not slip

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:

Given below are two statements : one is labeled as Assertion \(A\) and the other is labeled as Reason \(R\) Assertion \(A:\) Photodiodes are used in forward bias usually for measuring the light intensity. Reason \(R:\) For a \(p-n\) junction diode, at applied voltage \(V\) the current in the forward bias is more than the current in the reverse bias for \(\left|V_z\right| > \pm V \geq\left|V_0\right|\) where \(V_0\) is the threshold voltage and \(V _{ z }\) is the breakdown voltage. In the light of the above statements, choose the correct answer from the options given below

A particle moves in \(\mathrm{x}-\mathrm{y}\) plane under the influence of a force \(\vec{F}\) such that its linear momentum is \(\vec{P}(t)=\hat{i} \cos (k t)-\hat{j} \sin (k t)\). If \(k\) is constant, the angle between \(\overrightarrow{\mathrm{F}}\) and \(\overrightarrow{\mathrm{P}}\) will be _______.

Five identical cells each of internal resistance \(1\, \Omega\) and \(emf \;5\, {V}\) are connected in series and in parallel with an external resistance \('R'.\) For what value of \('R',\) current in series and parallel combination will remain the same ? (in \(\Omega\))

The first member of the Balmer series of hydrogen atom has a wavelength of \(6561\; \mathring A\). The wavelength of the second member of the Balmer series (in \(nm\)) is

If the digits are not allowed to repeat in any number formed by using the digits \(0,2,4,6,8\), then the number of all numbers greater than \(10,000\) is equal to \(....\)

Let \(A =\left[\begin{array}{ccc}2 & -1 & -1 \\ 1 & 0 & -1 \\ 1 & -1 & 0\end{array}\right]\) and \(B = A - I\). If \(\omega=\frac{\sqrt{3} i -1}{2}\) then the number of elements in the set \(\left\{ n \in\{1,2, \ldots, 100\}: A ^{ n }+(\omega B )^{ n }= A + B \right\}\) is equal to \(..........\)

An automobile, travelling at \(40\, km/h\), can be stopped at a distance of \(40\, m\) by applying brakes. If the same automobile is travelling at \(80\, km/h\), the minimum stopping distance, in metres, is (assume no skidding)..........\(m\)

An object of mass 1000 g experiences a time dependent force \(\vec{F}=\left(2 t \hat{i}+3 t^2 \hat{j}\right) N\). The power generated by the force at time \(t\) is :

Let M and m respectively be the maximum and the minimum values of
\(f(x)=\left|\begin{array}{ccc}
1+\sin ^2 x & \cos ^2 x & 4 \sin 4 x \\
\sin ^2 x & 1+\cos ^2 x & 4 \sin 4 x \\
\sin ^2 x & \cos ^2 x & 1+4 \sin 4 x
\end{array}\right|, x \in \mathrm{R}\)
Then \(M^4-m^4\) is equal to :