An electron is travelling with a velocity \(v\) in free space and when it enters a medium, its velocity is reduced by \(20\%\). The de Broglie wavelength of electron in the medium is \(\alpha\lambda_0\), where \(\lambda_0\) is its de Broglie wavelength in free space. The value of \(\alpha\) is _______.

Least count of a vernier caliper is \(\frac{1}{20 \mathrm{~N}} \mathrm{~cm}\). The value of one division on the main scale is \(1 \mathrm{~mm}\). Then the number of divisions of main scale that coincide with \(\mathrm{N}\) divisions of vernier scale is _______.

A star has \(100 \%\) helium composition. It starts to convert three \({ }^4 \mathrm{He}\) into one \({ }^{12} \mathrm{C}\) via triple alpha process as \({ }^4 \mathrm{He}+{ }^4 \mathrm{He}+{ }^4 \mathrm{He} \rightarrow{ }^{12} \mathrm{C}+\mathrm{Q}\). The mass of the star is \(2.0 \times 10^{32} \mathrm{~kg}\) and it generates energy at the rate of \(5.808 \times 10^{30} \mathrm{~W}\). The rate of converting these \({ }^4 \mathrm{He}\) to \({ }^{12} \mathrm{C}\) is \(\mathrm{n} \times 10^{42} \mathrm{~s}^{-1}\), where \(\mathrm{n}\) is _______. [Take, mass of \({ }^4 \mathrm{He}=4.0026 \mathrm{u}\), mass of \({ }^{12} \mathrm{C}=12 \mathrm{u}\)]

In hydrogen atom spectrum, (R is Rydberg's constant):
A. the maximum wavelength of the radiation of Lyman series is \(\frac{4}{3 R}\)
B. the Balmer series lies in the visible region of the spectrum
C. the minimum wavelength of the radiation of Paschen series is \(\frac{9}{R}\)
D. the minimum wavelength of Lyman series is \(\frac{5}{4 R}\)
Choose the correct answer from the options given below :

The density \(\rho\) of a uniform cylinder is determined by measuring its mass \(m\), length \(l\) and diameter \(d\). The measured values of \(m\), \(l\) and \(d\) are \(97.42 \pm 0.02\) g, \(8.35 \pm 0.05\) mm and \(20.20 \pm 0.02\) mm, respectively. Calculated percentage fractional error in \(\rho\) is _______.

The velocity of the bullet becomes one third after it penetrates \(4\,cm\) in a wooden block. Assuming that bullet is facing a constant resistance during its motion in the block. The bullet stops completely after travelling at \((4+x)\,cm\) inside the block. The value of \(x\) is\(.....\)

A cubical block of side \(0.5\,m\) floats on water with \(30\%\) of its volume under water. ....... \(kg\) is the maximum weight that can be put on the block without fully submerging it under water ? [Take density of water \(= 10^3\,kg/m^3\) ]

A dipole with two electric charges of \(2 \mu \mathrm{C}\) magnitude each, with separation distance \(0.5 \mu \mathrm{~m}\), is placed between the plates of a capacitor such that its axis is parallel to an electric field established between the plates when a potential difference of 5 V is applied. Separation between the plates is 0.5 mm. If the dipole is rotated by \(30^{\circ}\) from the axis, it tends to realign in the direction due to a torque. The value of torque is :

As shown in the figure, forces of \(10^5\,N\) each are applied in opposite directions, on the upper and lower faces of a cube of side \(10\,cm\), shifting the upper face parallel to itself by \(0.5\,cm\) . If the side of another cube of the same material is, \(20\,cm\) then under similar conditions as above, the displacement will be......... \(cm\)

Two wires each of radius \(0.2\,cm\) and negligible mass, one made of steel and other made of brass are loaded as shown in the figure. The elongation of the steel wire is \(.........\times 10^{-6}\,m\). [Young's modulus for steel \(=2 \times 10^{11}\,Nm ^{-2}\) and \(g =10\,ms ^{-2}\) ]