Albert Einstein was confirmed with the Nobel prize in physics for his work on

Let \(S_1\) be the amount of Rayleigh scattered light of wavelength \(\lambda_1\) and \(S_2\) that of light of wavelength \(\lambda_2\) from a particle of size \(a\). Which of the following statement is true?

A cell of emf \(10 \mathrm{~V}\) and internal resistance \(3 \Omega\) is connected in parallel with another cell of emf \(7 \mathrm{~V}\) and internal resistance \(\frac{3}{5} \Omega\), such that their positive terminals are joined together and so their are negative terminals. Their positive terminals are joined with the negative terminal and their negative terminal is joined with the positive terminal of a third cell of emf \(20 \mathrm{~V}\) with internal resistance \(2 \Omega\). The combination can be replaced by a battery of emf \(E\) and internal resistance \(r\), then the values of \(E\) and \(R\) are respectively

If a body dropped freely from a height of 20 m reaches the surface of a planet with a velocity of \(31.4 \mathrm{~ms}^{-1}\), then the length of a simple pendulum that ticks seconds on the planet is

\(n\) conducting wires of same dimensions but having restivities \(1,2,3, . . n\) are connected in series. The equivalent resistivity of the combination is

A solenoid of length \(2 \mathrm{~m}\) carries a current of \(20 \mathrm{~A}\). The diameter of the solenoid is \(3 \mathrm{~cm}\). If the magnetic field inside the solenoid is \(20 \mathrm{mT}\), then the length of wire forming the solenoid is (assume, \(\mu_0=4 \pi \times 10^{-7} \mathrm{H} / \mathrm{m}\) )

As shown in the figure, a dielectric of constant K is placed between the plates of a parallel plate capacitor and is charged to a potential V using a battery. If the dielectric is pulled out after disconnecting the battery from the capacitor, the final potential difference across the plates of the capacitor is

A projectile is launched with an initial speed of $40 \mathrm{m} {\mathrm{s}}^{-1}$ at an angle $30°$ above the ground. The projectile lands on a hillside $2.0 \mathrm{s}$ later. The net displacement from where the projectile was launched to where it hits the target is (Take $g=10 \mathrm{m} {\mathrm{s}}^{-2}$)

A small disc of mass \(1 \mathrm{~g}\) slides down a smooth hill of height \(10 \mathrm{~m}\) from rest and gets on to a plank of mass \(100 \mathrm{~g}\) as shown in the figure. Due to friction between the disc and the plank. The disc slows down and moves as one piece with the plank. The work done by the frictional force is approximately (Use, \(g=10 \mathrm{~ms}^2\) )

A cyclist leans with the horizontal at angle \(30^{\circ}\), while negotiating round a circular road of radius \(20 \sqrt{3} \mathrm{~m}\). The speed of the cycle should be

Identify the logic gate from the following with the same truth table characteristics of the logic circuit below.

In the circuit shown below, the ammeter reading is zero. Then the value of the resistance \(R\) is

Momentum before collision,
\(
p_1=4 m v_1-2 m v_2
\)

Momentum after collision,
\(
p_2=6 m\left(-5 v_2\right)=-30 m v_2
\)


Conservation of momentum gives,
\(
\begin{gathered}
p_1=p_2 \\
\Rightarrow \quad 4 m v_1-2 m v_2=-30 m v_2 \quad \Rightarrow 4 v_1=-32 v_2 \\
\Rightarrow \quad \frac{v_1}{v_2}=\frac{-32}{4}=-8 \text { or }\left|\frac{v_1}{v_2}\right|=8
\end{gathered}
\)