If \(M_0\) is the mass of isotope \({ }_5^{12} B, M_p\) and \(M_n\) are the masses of proton and neutron, then nuclear binding energy of isotope is _______.

In an a.c. circuit, voltage and current are given by: \(V=100 \sin (100 t)\  V\) and \(I=100 \sin \left(100 t+\frac{\pi}{3}\right)\  \mathrm{mA}\) respectively. The average power dissipated in one cycle is _______.

In thermodynamics, heat and work are

For \(z=a^{2} x^{3} y^{\frac{1}{2}}\), where \(a\) is a constant. If percentage error in measurement of \(x\) and \(y\) are \(4 \%\) and \(12 \%\), respectively, then the percentage error for \(z\) will be \(%\)

An elliptical loop having resistance \(R\), of semi major axis \(a,\) and semi  inor axis \(b\) is placed in a magnetic field as shown in the figure. If the loop is rotated about the \(x-\) axis with angular frequency \(\omega\), the average power loss in the loop due to Joule heating is :

A parallel plate capacitor has a capacitance \(\mathrm{C}=200 \mathrm{pF}\). It is connected to \(230 \mathrm{~V}\) ac supply with an angular frequency \(300 \mathrm{rad} / \mathrm{s}\). The \(\mathrm{rms}\) value of conduction current in the circuit and displacement current in the capacitor respectively are _______.

A pole is vertically submerged in swimming pool, such that it gives a length of shadow \(2.15 \,m\) within water when sunlight is incident at an angle of \(30^{\circ}\) with the surface of water. If swimming pool is filled to a height of \(1.5\,m\), then the height of the pole above the water surface in centimetres is \(\left(\eta_W=4 / 3\right)........\)

The point \(A\) moves with a uniform speed along the circumference of a circle of radius \(0.36\, m\) and covers \(30^{\circ}\) in \(0.1\, s\). The perpendicular projection \('P'\) from \('A'\) on the diameter \(MN\) represents the simple harmonic motion of \('P'.\) The restoration force per unit mass when \(P\) touches \(M\) will be ...... \(N\)

Let \(L_1\) be the length of the common chord of the curves \(x^2 + y^2\, = 9\) and \(y^2\, = 8x\), and \(L_2\) be the length of the latus rectum of \(y^2\, = 8x\), then

In a survey of \(220\) students of a higher secondary school, it was found that at least \(125\) and at most \(130\) students studied Mathematics; at least \(85\) and at most \(95\) studied Physics; at least \(75\) and at most \(90\) studied Chemistry; \(30\) studied both Physics and Chemistry; \(50\) studied both Chemistry and Mathematics; \(40\) studied both Mathematics and Physics and \(10\) studied none of these subjects. Let \(\mathrm{m}\) and \(\mathrm{n}\) respectively be the least and the most number of students who studied all the three subjects. Then \(\mathrm{m}+\mathrm{n}\) is equal to ...........

Let P be the foot of the perpendicular from the point \((1,2,2)\) on the line \(\mathrm{L}: \frac{x-1}{1}=\frac{y+1}{-1}=\frac{z-2}{2}\). Let the line \(\vec{r}=(-\hat{i}+\hat{j}-2 \hat{k})+\lambda(\hat{i}-\hat{j}+\hat{k}), \lambda \in \mathbf{R}\), intersect the line L at Q . Then \(2(\mathrm{PQ})^2\) is equal to:

Let ABC be a triangle. Consider four points \(p _1, p _2\), \(p _3, p _4\) on the side AB , five points \(p _5, p _6, p _7, p _8, p _9\) on the side BC and four points \(p _{10}, p _{11}, p _{12}, p _{13}\) on the side AC . None of these points is a vertex of the triangle ABC . Then the total number of pentagons, that can be formed by taking all the vertices from the points \(p _1, p _2, \ldots . p _{13}\), is ___ .

The coefficient of \(x^{70}\) in \(x^2(1+x)^{98}+x^3(1+x)^{97}+\) \(x^4(1+x)^{96}+\ldots \ldots . .+x^{54}(1+x)^{46}\) is \({ }^{99} \mathrm{C}_p-{ }^{46} \mathrm{C}_{\mathrm{q}}\). Then a possible value to \(\mathrm{p}+\mathrm{q}\) is :