\(0.08 \mathrm{~kg}\) air is heated at constant volume through \(5^{\circ} \mathrm{C}\). The specific heat of air at constant volume is \(0.17 \mathrm{kcal} / \mathrm{kg}^{\circ} \mathrm{C}\) and \(\mathrm{J}=4.18\) joule \(/ \mathrm{cal}\). The change in its internal energy is approximately.

Two closed vessels of same volume are joined through a narrow tube and both vessels are filled with air of pressure \(90\) kPa and temperature \(400\) K. Keeping the temperature of one vessel constant at \(400\) K the second vessel temperature is raised to \(500\) K. The final pressure in the vessels is _______ kPa.

Three identical particle \(A , B\) and \(C\) of mass \(100\, kg\) each are placed in a straight line with \(AB = BC =13 \,m\). The gravitational force on a fourth particle \(P\) of the same mass is \(F\), when placed at a distance \(13 \,m\) from the particle \(B\) on the perpendicular bisector of the line \(AC\). The value of \(F\) will be approximately\(....G\)

If the time period \(t\) of the oscillation of a drop of liquid of density \(d\), radius \(r\), vibrating under surface tension \(s\) is given by the formula \(t = \sqrt {{r^{2b}}\,{s^c}\,{d^{a/2}}} \) . It is observed that the time period is directly proportional to \(\sqrt {\frac{d}{s}} \) . The value of \(b\) should therefore be

The electric potential at the centre of two concentric half rings of radii \(R_1\) and \(R_2\), having same linear charge density \(\lambda\) is

Which of the following relations is true for two unit vectors \(\hat{ A }\) and \(\hat{ B }\) making an angle \(\theta\) to each other\(?\)

Using a battery, a 100 pF capacitor is charged to 60 V and then the battery is removed. After that, a second uncharged capacitor is connected to the first capacitor in parallel. If the final voltage across the second capacitor is 20 V, its capacitance is : (in pF)

\( \text { If } S(x)=(1+x)+2(1+x)^2+3(1+x)^3+\ldots . \) \( +60(1+x)^{60}, x \neq 0 \text {, and }(60)^2 S(60)=a(b)^b+b\) where \(a, b N\), then \((a+b)\) equal to ...............

Let \( \vec{a}=\hat{i}-2\hat{j}+3\hat{k},\vec{b}=2\hat{i}+\hat{j}-\hat{k},\vec{c}=\lambda\hat{i}+\hat{j}+\hat{k} \) and \( \vec{v}=\vec{a}\times\vec{b} \). If \( \vec{v} \cdot \vec{c}=11 \) and the length of the projection of \( \vec{b} \) on \( \vec{c} \) is \( p \), then \( 9p^{2} \) is equal to:

Given below are two statements: Statement \(I\) : In hydrogen atom, the frequency of radiation emitted when an electron jumps from lower energy orbit \(\left( E _{1}\right)\) to higher energy orbit \(\left( E _{2}\right)\), is given as \(hf = E _{1}- E _{2}\). Statement \(II\) : The jumping of electron from higher energy orbit \(\left(E_{2}\right)\) to lower energy orbit \(\left(E_{1}\right)\) is associated with frequency of radiation given as \(f\) \(=\left( E _{2}- E _{1}\right) / h\) This condition is Bohr's frequency condition. In the light of the above statements, choose the correct answer from the options given below

A fighter plane of length \(20\, m\), wing span (distance from tip of one wing to the tip of the other wing) of \(15\,m\) and height \(5\,m\) is lying towards east over Delhi. Its speed is \(240\, ms^{-1}\) . The earth's magnetic field over Delhi is \(5 \times 10^{-5}\,T\) with the declination angle \( \sim {0^o}\) and dip of \(\theta\) such that \(\sin \,\theta  = \frac{2}{3}\). If the voltage developed is \(V_B\) between the lower and upper side of the plane and \(V_W\) between the tips of the wings then \(V_B\) and \(V_W\) are close to

Let a variable line of slope \(m>0\) passing through the point \((4,-9)\) intersect the coordinate axes at the points \(A\) and \(B\). the minimum value of the sum of the distances of \(\mathrm{A}\) and \(\mathrm{B}\) from the origin is

Let \(A(2,3,5)\) and \(C(-3,4,-2)\) be opposite vertices of a parallelogram \(A B C D\) if the diagonal \(\overrightarrow{B D}=\hat{i}+2 \hat{j}+3 \hat{k}\) then the area of the parallelogram is equal to