Two radioactive substances A and B of mass numbers \(200\) and \(212\) respectively, shows spontaneous \(\alpha\)-decay with same \(Q\) value of \(1\) MeV. The ratio of energies of \(\alpha\)-rays produced by A and B is ________.

An elastic spring under tension of \(3 \mathrm{~N}\) has a lengtha. Its length is \(b\) under tension \(2 \mathrm{~N}\). For its length\((3 a-2 b)\), the value of tension will be_______. \(\mathrm{N}\).

If \(Y, K\) and \(\eta\) are the values of Young's modulus, bulk modulus and modulus of rigidity of any material respectively. Choose the correct relation for these parameters.

A wire of resistance \(R\) is bent to form a square \(ABCD\) as shown in the figure. The effective resistance between \(E\) and \(C\) is ( \(E\) is mid-point of arm \(CD\) )

The explosive in a Hydrogen bomb is a mixture of \({ }_1 \mathrm{H}^2,{ }_1 \mathrm{H}^3\) and \({ }_3 \mathrm{Li}^6\) in some condensed form. The chain reaction is given by \( { }_3 \mathrm{Li}^6+{ }_0 \mathrm{n}^1 \rightarrow{ }_2 \mathrm{He}^4+{ }_1 \mathrm{H}^3 \) \( { }_1 \mathrm{H}^2+{ }_1 \mathrm{H}^3 \rightarrow{ }_2 \mathrm{He}^4+{ }_0 \mathrm{n}^1\) During the explosion the energy released is approximately [Given : \(\mathrm{M}(\mathrm{Li})=6.01690\  \mathrm{amu} . \mathrm{M}\left({ }_1 \mathrm{H}^2\right)=2.01471 \  amu.\) \(\mathrm{M}\left({ }_2 \mathrm{He}^4\right)=4.00388\  \mathrm{amu}\),\( and\ \)\(1 \ \mathrm{amu}=931.5\) \(\mathrm{MeV}]\)

Light source having wavelength \(331\) nm is used to generate photo-electrons whose stopping potential is \(0.2\) V. The work function of the used metal in the experiment is \(\alpha \times 10^{-19}\) J. The value of \(\alpha\) is _____.
(\(h = 6.62 \times 10^{-34}\) J s, \(e = 1.6 \times 10^{-19}\) C and \(c = 3 \times 10^8\) m/s)

A uniform thin rod \(AB\) of length \(L\) has linear mass density \(\mu \left( x \right) = a + \frac{{bx}}{L}\) , where \(x\) is measured from \(A\). If the \(CM\) of the rod lies at a distance of \(\left( {\frac{7}{12}} \right)L\) from \(A\), then \(a\) and \(b\) are related as

Let the centre of the circle \(x^2 + y^2 + 2gx + 2fy + 25 = 0\) be in the first quadrant and lie on the line \(2x - y = 4\). Let the area of an equilateral triangle inscribed in the circle be \(27\sqrt{3}\). Then the square of the length of the chord of the circle on the line \(x = 1\) is _______.

If the sum of the coefficients of all the positive even powers of \(x\) in the binomial expansion of \(\left(2 x^{3}+\frac{3}{x}\right)^{10}\) is \(5^{10}-\beta \cdot 3^{9}\), then \(\beta\) is equal to

A pressure-pump has a horizontal tube of cross-sectional area \(10\,cm ^{2}\) for the outflow of water at a speed of \(20\,m / s\). The force exerted on the vertical wall just in front of the tube which stops water horizontally flowing out of the tube, is \(...N\) [given : density of water \(=1000\,kg / m ^{3}\) ]

A point particle of charge \(Q\) is located at \(P\) along the axis of an electric dipole 1 at a distance \(r\) as shown in the figure. The point P is also on the equatorial plane of a second electric dipole 2 at a distance r . The dipoles are made of opposite charge q separated by a distance \(2 a\). For the charge particle at P not to experience any net force, which of the following correctly describes the situation?

The equation of motion of a particle is given by \(x = a \sin\left(50t + \dfrac{\pi}{3}\right)\) cm. The particle will come to rest at time \(t_1\) and it will have zero acceleration at time \(t_2\). The \(t_1\) and \(t_2\) respectively are _______.

Let the foci of the ellipse \(\frac{x^{2}}{16}+\frac{y^{2}}{7}=1\) and the hyperbola \(\frac{ x ^{2}}{144}-\frac{ y ^{2}}{\alpha}=\frac{1}{25}\) coincide. Then the length of the latus rectum of the hyperbola is:-