An electric field of \(1000\,V/m\) is applied to an electric dipole at angle of \(45^o\). The value of electric dipole moment is \(10^{-29}\, C.m.\) What is the potential energy of the electric dipole?

A fully loaded boeing aircraft has a mass of \(5.4 \times 10^5\,kg\). Its total wing area is \(500\,m ^2\). It is in level flight with a speed of \(1080\,km / h\). If the density of air \(\rho\) is \(1.2\,kg\,m ^{-3}\), the fractional increase in the speed of the air on the upper surface of the wing relative to the lower surface in percentage will be \(\left( g =10\,m / s ^2\right)\)

Two sources of light emit X-rays of wavelength \(1\, nm\) and visible light of wavelength \(500\, nm ,\) respectively. Both the sources emit light of the same power \(200\, W\). The ratio of the number density of photons of X-rays to the number densitty of photons of the visible light of the given wavelengths is

Half lives of two radioactive nuclei \(A\) and \(B\) are \(10\, minutes\) and \(20\, minutes\), respectively. If, initially a sample has equal number of nuclei, then after \(60\) \(minutes\) , the ratio of decayed numbers of nuclei \(A\) and \(B\) will be

A particle having charge \(10^{-9}\) C moving in \(x\)-\(y\) plane in fields of \(0.4\hat{j}\) N/C and \(4 \times 10^{-3} \hat{k}\) T experiences a force of \((4\hat{i} + 2\hat{j}) \times 10^{-10}\) N. The velocity of the particle at that instant is _______ m/s.

A pendulum is executing simple harmonic motion and its maximum kinetic energy is \(K_1\). If the length of the pendulum is doubled and it performs simple harmonic motion with the same amplitude as in the first case, its maximum kinetic energy is \(K_2\) then

A particle of mass \(10\,g\) moves in a straight line with retarcation \(2x\), where \(x\) is the displacement in \(SI\) units. Its loss of kinetic energy for above displacement is \(\left(\frac{10}{x}\right)^{- n }\, J\). The value of \(n\) will be \(............\).

Let \({a_1},{a_2},\;.\;.\;.\;.,{a_{49}}\) be in \(A.P.\) such that \(\mathop \sum \limits_{k = 0}^{12} {a_{4k + 1}} = 416\) and \({a_9} + {a_{43}} = 66\). If \(a_1^2 + a_2^2 + \ldots + a_{17}^2 = 140m,\) then \(m = \;\;..\;.\;.\;.\;\)

Let a point \(P\) be such that its distance from the point \((5,0)\) is thrice the distance of \(P\) from the point \((-5,0) .\) If the locus of the point \(P\) is a circle of radius \(r\), then \(4 r ^{2}\) 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

Let \(y=y(x)\) be the solution of the differential equation \(\frac{d y}{d x}=(y+1)\left((y+1) e^{x^{2} / 2}-x\right), y(2)=0\) then \(y'(1)\)  equal to . . .  .

All the students of a class performed poorly in Mathematics. The teacher decided to give grace marks of \(10\) to each of the students. Which of the following statistical measures will not change even after the grace marks were given \(?\)

Let \(a , b , c\) be in arithmetic progression. Let the centroid of the triangle with vertices \(( a , c ),(2, b)\) and \((a, b)\) be \(\left(\frac{10}{3}, \frac{7}{3}\right)\). If \(\alpha, \beta\) are the roots of the equation \(ax ^{2}+ bx +1=0\), then the value of \(\alpha^{2}+\beta^{2}-\alpha \beta\) is ....... .