Two bodies, each of mass $M$, are kept fixed with a separation of $2L$. A particle of mass $m$ is projected from the midpoint of the line joining their centers, perpendicular to the line. The gravitational constant is $G$. The correct statement (s) is (are) :

Light of wavelength ${\lambda }_{ph}$ falls on a cathode plate inside a vacuum tube as shown in the figure. The work function of the cathode surface is $\varphi$ and the anode is a wire mesh of conducting material kept at a distance d from the cathode. A potential difference V is maintained between the electrodes. If the minimum de Broglie wavelength of the electrons passing through the anode is ${\lambda }_e,$ which of the following statement(s) is(are) true ?

An incompressible liquid is kept in a container having a weightless piston with a hole. A capillary tube of inner radius $0.1\mathrm{mm}$ is dipped vertically into the liquid through the airtight piston hole, as shown in the figure. The air in the container is isothermally compressed from its original volume $V_0$to$\frac{100}{101}{V}_0$with the movable piston.Considering air as an ideal gas, the height $\left(h\right)$ of the liquid column in the capillary above the liquid level in $\mathrm{cm}$is _____.
[Given: Surface tension of the liquid is $0.075\mathrm{N}{\mathrm{m}}^{-1}$, atmospheric pressure is ${10}^5\mathrm{N}{\mathrm{m}}^{-2}$, acceleration due to gravity $\left(g\right)$ is $10\mathrm{m}{\mathrm{s}}^{-2}$, density of the liquid is ${10}^3\mathrm{kg}{\mathrm{m}}^{-3}$and contact angle of capillary surface with the liquid is zero]

Consider a Vernier callipers in which each 1 cm on the main scale is divided into 8 equal divisions and a screw gauge with 100 divisions on its circular scale. In the Vernier callipers, 5 divisions of the Vernier scale coincide with 4 divisions on the main scale and in the screw gauge, one complete rotation of the circular scale moves it by two divisions on the linear scale. Then:

Young’s modulus of elasticity $Y$ is expressed in terms of three derived quantities, namely, the gravitational constant $G$, Planck’s constant $h$ and the speed of light $c$, as $Y=c^{\alpha }{h}^{\beta }{G}^{\gamma }$. Which of the following is the correct option?

A biconvex lens of focal length \(f\) forms a circular image of radius \(r\) of sun in focal plane. Then, which option is correct?

A radioactive sample \(S_1\) having an activity of \(5 \mu \mathrm{Ci}\) has twice the number of nuclei as another sample \(S_2\) which has an activity of \(10 \mu \mathrm{Ci}\). The half lives of \(S_1\) and \(S_2\) can be

A hyperbola, having the transverse axis of length \(2 \sin \theta\), is confocal with the ellipse \(3 x^2+4 y^2=12\). Then, its equation is

A pulse of light of duration 100 ns is absorbed completely by a small object initially at rest. Power of the pulse is 30 mW and the speed of light is 3 ×10⁸ ms^-1. The final momentum of the object is

Paragraph:
Read the following passage and answer the questions.
Question:
The value of \(|U|\) is

Consider the $6\times 6$ square in the figure. Let $A_1,A_2,\dots ,A_{49}$ be the points of intersections (dots in the picture) in some order. We say that $A_i$ and $A_j$ are friends if they are adjacent along a row or along a column. Assume that each point $A_i$ has an equal chance of being chosen.



(There are two questions based on $\mathrm{PARAGRAPH} "\mathrm{II}"$, the question given below is one of them)

Let $p_i$ be the probability that a randomly chosen point has $i$ many friends, $i=0,1,2,3,4$. Let $X$ be a random variable such that for $i=0,1,2,3,4$, the probability $P(X=i)=p_i$. Then the value of $7 E\left(X\right)$ is

Area of the region $\{\left(x,y\right)\in R^2: y\ge \sqrt{\left|x+3\right|},$ $5y\le x+9\le 15\}$ is equal to

Let \(E\) and \(F\) be two independent events. The probability that exactly one of them occurs is \(11 / 25\) and the probability of none of them occurring is \(2 / 25\). If \(P(T)\) denotes the probability of occurrence of the event \(T\), then