A thin infinite sheet charge and an infinite line charge of respective charge densities \(+\sigma\) and \(+\lambda\) are placed parallel at \(5\,m\) distance from each other. Points \(P\) and \(Q\), are at \(\frac{3}{\pi} m\) and \(\frac{4}{\pi} m\) perpendicular distance from line charge towards sheet charge, respectively. \(E_P\) and \(E_Q\) are the magnitudes of resultant electric field intensities at point \(P\) and \(Q\), respectively. If \(\frac{E_p}{E_Q}=\frac{4}{a}\) for \(2|\sigma|=|\lambda|\). Then the value of \(a\) is ...........

A block of metal weighing \(2 \,kg\) is resting on a frictionless plane (as shown in figure). It is struck by a jet releasing water at a rate of \(1 \,kgs ^{-1}\) and at a speed of \(10 \,ms ^{-1}\). Then, the initial acceleration of the block, in \(ms ^{-2}\), will be .................

Match List - I with List - II.

List - I	List - II
A. Boltzmann constant	I. \([M^{-1}L^3T^{-2}]\)
B. Stefan's constant	II. \([ML^2T^{-1}]\)
C. Planck's constant	III. \([ML^2T^{-2}K^{-1}]\)
D. Gravitational constant	IV. \([ML^0T^{-3}K^{-4}]\)

Choose the correct answer from the options given below :

A particle of charge \(q\) and mass \(m\) is projected from origin with an initial velocity \(\vec{v} = \left(\dfrac{v_0}{\sqrt{2}}\hat{x} + \dfrac{v_0}{\sqrt{2}}\hat{y}\right)\). There exists a uniform magnetic field \(\vec{B} = B_0 \hat{z}\) and a space varying electric field \(\vec{E} = E_0 e^{-\lambda x}\hat{x}\) within the region \(0 \leq x \leq L\). After travelling a distance such that \(x\)-coordinate has changed from \(x=0\) to \(x=L\), the change in the kinetic energy is _______.

Consider a block and trolley system as shown in figure. If the coefficient of kinetic friction between the trolley and the surface is \(0.04\) , the acceleration of the system in \(\mathrm{ms}^{-2}\) is _______. (Consider that the string is massless and unstretchable and the pulley is also massless and frictionless)

A physical quantity C is related to four other quantities \(\mathrm{p}, \mathrm{q}, \mathrm{r}\) and s as follows \(\mathrm{C}=\frac{\mathrm{pq}^2}{\mathrm{r}^3 \sqrt{\mathrm{~s}}}\)
The percentage errors in the measurement of \(\mathrm{p}, \mathrm{q}, \mathrm{r}\) and s are \(1 \%, 2 \% 3 \%\) and \(2 \%\) respectively. The percentage error in the measurement of C will be _____\(\%\).

Two blocks of masses \(m\) and \(M,(M \gt m)\), are placed on a frictionless table as shown in figure. A massless spring with spring constant k is attached with the lower block. If the system is slightly displaced and released then
(\(\mu=\) coefficient of friction between the two blocks)

(A) The time period of small oscillation of the two blocks is \(\mathrm{T}=2 \pi \sqrt{\frac{(\mathrm{~m}+\mathrm{M})}{\mathrm{k}}}\)
(B) The acceleration of the blocks is \(\mathrm{a}=\frac{\mathrm{kx}}{\mathrm{M}+\mathrm{m}}\) (\(\mathrm{x}=\) displacement of the blocks from the mean position)
(C) The magnitude of the frictional force on the upper block is \(\frac{m \mu|x|}{M+m}\)
(D) The maximum amplitude of the upper block, if it does not slip, is \(\frac{\mu(M+m) g}{k}\)
(E) Maximum frictional force can be \(\mu(\mathrm{M}+\mathrm{m}) \mathrm{g}\).
Choose the correct answer from the options given below:

If \(2.5 \times 10^{-6\,} N\) average force is exerted by a light wave on a non-reflecting surface of \(30\, cm ^{2}\) area during \(40\) minutes of time span, the energy flux of light just before it falls on the surface is .................. \(W / cm ^{2}\) (Round off to the Nearest Integer) (Assume complete absorption and normal incidence conditions are there)

In the given figure an ammeter \(A\) consists of a \(240 \Omega\) coil connected in parallel to a \(10 \Omega\) shunt. The reading of the ammeter is _______ \(\mathrm{mA}\).

Electrons are accelerated through a potential difference \(V\) and protons are accelerated through a potential difference \(4\, V\). The de-Broglie wavelengths are \(\lambda_e \) and \(\lambda_p \) for electrons and protons respectively. The ratio of \(\frac{{{\lambda _e}}}{{{\lambda _p}}}\) is given by : (given \(m_e\) is mass of electron and \(m_p\) is mass of proton).

A solid cylinder length is suspended symmetrically through two massless strings, as shown in the figure. The distance from the initial rest position, the cylinder should by unbinding the strings to achieve a speed of \(4\,ms ^{-1}\), is\(........cm\). \(\left(\right.\) take \(\left.g=10\,ms ^{-2}\right)\)