The magnetic moment of a bar magnet is \(0.5 \mathrm{Am}^2\). It is suspended in a uniform magnetic field of \(8 \times 10^{-2} \mathrm{~T}\). The work done in rotating it from its most stable to most unstable position is _______.

Which of the following nuclear fragments corresponding to nuclear fission between neutron \(\left({ }_0^1 \mathrm{n}\right)\) and uranium isotope \(\left({ }_{92}^{235} \mathrm{U}\right)\) is correct:

Three small identical bubbles of water having same charge on each coalesce to form a bigger bubble. Then the ratio of the potentials on one initial bubble and that on the resultant bigger bubble is:

Two waves executing simple harmonic motion travelling in the same direction with same amplitude and frequency are superimposed. The resultant amplitude is equal to the \(\sqrt{3}\) times of amplitude of individual motions. The phase difference between the two motions is \(.....(degree)\)

A uniform magnetic field of \(2 \times 10^{-3} \mathrm{~T}\) acts along positive \(\mathrm{Y}\)-direction. A rectangular loop of sides \(20\) \(\mathrm{cm}\) and \(10 \mathrm{~cm}\) with current of \(5 \mathrm{~A}\) is \(\mathrm{Y}-\mathrm{Z}\) plane. The current is in anticlockwise sense with reference to negative \(\mathrm{X}\) axis. Magnitude and direction of the torque is _______.

A rectangular loop of sides \(12 \mathrm{~cm}\) and \(5 \mathrm{~cm}\), with its sides parallel to the \(x\)-axis and \(y\)-axis respectively moves with a velocity of \(5 \mathrm{~cm} / \mathrm{s}\) in the positive \(\mathrm{x}\) axis direction, in a space containing a variable magnetic field in the positive \(z\) direction. The field has a gradient of \(10^{-3} \mathrm{~T} / \mathrm{cm}\) along the negative \(\mathrm{x}\) direction and it is decreasing with time at the rate of \(10^{-5} \mathrm{~T} / \mathrm{s}\). If the resistance of the loop is \(6 \mathrm{~m} \Omega\), the power dissipated by the loop as heat is _______ \(\times 10^{-9} \mathrm{~W}\).

A body cools from \(60^{\circ} C\) to \(40^{\circ} C\) in \(6\) minutes. If, temperature of surroundings is \(10^{\circ} C\). Then, after the next 6 minutes, its temperature will be \(.........{ }^{\circ} C\).

If \(A\) and \(B\) are the points of intersection of the circle \(x^2+y^2-8 x=0\) and the hyperbola \(\frac{x^2}{9}-\frac{y^2}{4}=1\) and a point P moves on the line \(2 x-3 y+4=0\), then the centroid of \(\triangle \mathrm{PAB}\) lies on the line :

For \(a , b \in Z\) and \(| a - b | \leq 10\), let the angle between the plane \(P : ax + y - z = b\) and the line \(l: x -1= a\) \(-y=z+1\) be \(\cos ^{-1}\left(\frac{1}{3}\right)\). If the distance of the point \((6,-6,4)\) from the plane \(P\) is \(3 \sqrt{6}\), then \(a^4+b^2\) is equal to

A screw gauge has \(50\) divisions on its circular scale. The circular scale is \(4\) units ahead of the pitch scale marking, prior to use. Upon one complete rotation of the circular scale, a displacement of \(0.5\, mm\) is noticed on the pitch scale. The nature of zero error involved, and the least count of the screw gauge, are respectively

A bob of mass ' \(m\) ' is suspended by a light string of length ' \(L\) '. It is imparted a minimum horizontal velocity at the lowest point \(A\) such that it just completes half circle reaching the top most position B. The ratio of kinetic energies \(\frac{(\text { K.E. })_A}{(\text { K.E. })_B}\) is _______.

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

	List \(-I\)		List \(-II\)
\(A\).	Coefficient of Viscosity	\(I\).	\([M L^2T^{–2}]\)
\(B\).	Surface Tension	\(II\).	\([M L^2T^{–1}]\)
\(C\).	Angular momentum	\(III\).	\([M L^{-1}T^{–1}]\)
\(D\).	Rotational Kimeatic energy	\(IV\).	\([M L^0T^{–2}]\)

If force \(\vec{F}=3 \hat{i}+4 \hat{j}-2 \hat{k}\) acts on a particle having position vector \(2 \hat{i}+\hat{j}+2 \hat{k}\) then, the torque about the origin will be