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In a radioactive decay chain, ${}_{90}^{232}Th$ nucleus decays to ${}_{82}^{212}Pb$ nucleus. Let $N_{\alpha }$ and $N_{\beta }$ be the number of $\alpha$ and ${\beta }^{-}$ particles respectively, emitted in this decay process. Which of the following statements is (are) true?

Four particles of masses $1\text{g},2\text{g},3\text{g}$ and $4\text{g}$ are kept at points $(0,0),(0,3\text{cm}),(3\text{cm},3\text{cm}),(3\text{cm},0\text{cm})$ respectively. Find the moment of inertia of the system of particles (in ${\text{g-cm}}^2$) about $y$ axis.

Starting from rest, a particle moves on a circle with constant angular acceleration $\alpha$. The time at which its tangential and normal acceleration become equal in magnitude is?

A rod of mass m and length l is lying on a horizontal table. The work done in making it stand vertically on one end will be

There are three different liquids (liquid 1, liquid 2 and liquid 3) having density 13.6 gm/cc., 1.3 gm/cc and 0.8 gm/cc. Liquid 1 is poured in a uniform U-tube, which is kept vertical. Liquid 2 & liquid 3 are poured separately into the two arms till upper surface of liquid 2 & 3 are same. What is the height of liquid 3 if the height of liquid 2 is 16cm?

A ball is thrown vertically up (taken as + z - axis) from the ground. The correct momentum - height (p - h) diagram is:(Given p=mv)

The value of $\underset{0}{\overset{1}{\int }}(x{e}^x+\sin \left(\frac{\pi x}{4}\right))dx$ is:

A particle of mass $m$ is moving with constant velocity $v$ parallel to $x$-axis in $x-y$ plane as shown in figure. Calculate its angular momentum with respect to origin at anyinstant '$t$'.

A network consisting of three resistor, three batteries and a capacitor is shown in the figure. Charge on the capacitor C in steady state is

The figure shows two blocks $\text{A}$ and $\text{B}$, each having a mass of $320\text{g}$ connected by a light string passing over a smooth light pulley. The horizontal surface on which the block $\text{A}$ can slide is smooth. The block $\text{A}$ is attached to a spring of spring constant $40\text{N/m}$ whose other end is fixed to a support $40\text{cm}$ above the horizontal surface. Initially, the spring is vertical and unstretched when the system is released to move. Find the horizontal distance covered by the block $\text{A}$ at the instant it breaks off the surface below it.

Take $g=10{\text{m/s}}^2$.

A conducting sphere of radius 10 cm has an unknown charge. If
the electric field 20 cm from the centre of the sphere is $1.5\times {10}^{3}N/C$ and points radially inward, what is the net charge on the sphere?

1. 1

A vernier calliper has 25 divisions on its vernier scale for every 35 divisions on its main scale. If 1 cm on the main scale is split into 12 units, the least count of the calliper will be.

In Fresnel diffraction, if the distance between the disc and the screen is decreased, the intensity of central bright spot will

Consider two uniform spheres $S_1$ and $S_2$ of equal masses and radii $r$ and $\frac{r}{2}$ respectively. Gravitational force due to $S_1$ and $S_2$ on a mass at a distance $R$ from their centres is $G_1$ and $G_2$ respectively.

Which of the following is/are correct.

The vector sum of two forces P and Q is minimum when the angle between them is

Find the derivative of $y=\sqrt{x^2+1}$

A conddenser consist of two parallel plates 0.01m² area and 10^-3m apart in air. If the plates are connected to a battery of emf 100V, calculate the charge on the condenser and the energy stored in it.

A stone dropped from the top of a tower of height 300 m high splashes into the water of a pond near the base of the tower. When is the splash heard at the top given that the speed of sound in air is $340m{s}^{-1}$ ? $(g=9.8m{s}^{-2})$

If $I_1={\int }_0^1(1-x^{50}{)}^{100}dx$ and $I_2={\int }_0^1(1-x^{50}{)}^{101}dx$ such that $I_2=\alpha I_1$ then $\alpha$ equals to:

For an ideal gas, the ratio of specific heat capacities is $\frac{C_P}{C_V}=\gamma$. If the gas undergoes a polytropic process according to the equation $P{V}^n=k$, find the range of $n$ for which the temperature of the gas increases when it rejects heat to the surroundings.

One quarter sector is cut from a uniform circular disc of radius $R$. This sector has mass $M$. It is made to rotate about a line perpendicular to its plane and passing through the centre of the original disc. Its moment of inertia about the axis of rotation is

A string breaks if its tension exceeds 10 newtons. A stone of mass 250 gm tied to this string of length 10 cm is rotated in a horizontal circle. The maximum angular velocity of rotation can be

A wheel rotates so that the angle of rotation is proportional to the square of time. The first revolution was performed by the wheel for 8 sec. Find the angular velocity 32 sec after the wheel started.

In an $LR$ circuit connected to a battery, the rate at which energy is stored in the inductor is plotted against time during the growth of the current in the circuit. Which of the following curves best represents the resulting curve?

The shortest wavelength of Lyman series of Hydrogen atom is $\lambda$. If an electron jumps from a higher energy level to ground state and emits a photon of wavelength 9$\lambda$/8, the principal quantum number of that higher energy level is

What is the maximum value of the force $\text{F}$ such that the block shown in the arrangement, does not move ? (Take $g=10{\text{m/s}}^2$)

Two bodies A and B of masses $m$ and $2m$ respectively are kept a distance $d$ apart. Where should a small particle be placed, so that the net gravitational force on it due to the bodies A and B is zero?