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Consider the following data and choose the correct option.

$1.$ The time period of a simple pendulum, whose bob is a hollow metallic sphere is $T$.
$2.$ The time period of a simple pendulum is $T_1,$ when the bob is filled with sand,
$3.$ The time period of a simple pendulum is $T_2,$ when it is filled with mercury and
$4.$ The time period of a simple pendulum is $T_3,$ when it is half-filled with mercury as shown.

The magnification produced by an astronomical telescope when the final image is formed at infinity is $10$ and the length of the telescope is $1.1\text{m}$. The magnification when the final image is formed at least distance of distinct vision is -

Which of the following correctly represents the electric field lines for a spherical conductor having a spherical cavity with a charge $Q$ (not at the center) ?

In a single slit diffraction experiment, intensity at $1^{st}$ maxima is equal to $10{\text{Wm}}^{-2}$. The intensity at the central maxima is approximately

A simple supported beam of length 2L is subjected to a moment N at the mid-point x=0 as shown in figure. The deflection in the domain $0\le x\le L$ is

$W=\frac{-Mx}{12EIL}(L-x)(x+c)$

where E is the Young's modulus, I is the area moment of inertia and C is a constant (to be determined).

The slope at the center $x=0$ is

Two charges of charge $+1\text{C}$ are connected by a spring of spring constant $100\text{N/m}$ and length $1\text{m}$. A small displacement in charge $2$ results into oscillation of the system. Calculate the time period of oscillation.

(mass of charge is $1\text{gm})$

The$K_{\alpha }$ X -rays of aluminium (Z= 13) and zinc (Z=30) have a wavelengths 887 pm and 146 pm respectively. Use Moseley's law $\sqrt{v}$ = a(Z-b) to find the wavelength of the $K_{\alpha }$ X-ray of iron (Z= 26).

Two bodies of masses 1 kg and 2 kg have equal momentum. Then, the ratio of their kinetic energies is

Two vessels separately contains two ideal gases $A$ and $B$ at the same temperature. The pressure of $A$ being twice that of $B$. Under such conditions, the density of $A$ is found to be $1.5$ times the density of $B$. The ratio of molecular weight of $A$ and $B$ is

A particle of mass $M$ orginally at rest is subjected to a force whose direction is constant but magnitude varies with time according to the relation $F=F_0[1-{\left(\frac{t-T}{T}\right)}^2]$

where $F_0$ and $T$ are constants. The force acts only for the time interval $2T$. The velocity $v$ of the particle after time $2T$ is :

A graph of acceleration versus time of a particle starting from rest at $t=0$ is as shown in figure. The speed of the particle at $t=14\text{s}$ is

Two blocks $A$ and $B$ of mass $M$ and $2M$ respectively, are hanging from a ceiling by means of a massless spring and a light string as shown. If the string between the blocks is suddenly cut and the downward acceleration is taken positive, then what will be the acceleration of the block $A$ and $B$?

A uniform thin bar of mass $6\text{kg}$ and length $2.4\text{m}$ is bent to make an equilateral hexagon. The moment of inertia about an axis passing through the centre of mass and perpendicular to the plane is $\times {10}^{-1}{\text{kg-m}}^2$

Two point A and B Lie on two extremes of a diameter AB as shown in figure. The sphere rotates with constant angular velocity $\omega$ about an axis as shown. What is the angular velocity of Q with respect to P?

In Young's double- slit experiment, the $y-$ coordinate of central maxima and ${10}^{th}$ maxima are $2\text{cm}$ and $5\text{cm}$ respectively. When the $\text{YDSE}$ apparatus is immersed in a liquid of refractive index $({\mu }_w=1.5)$, the corresponding $y-$ coordinate will be:

Three forces 20√2N,20√2N , 40N are acting along X,Y&Z axes respectively in a 5√2kg mass at rest at the origin . The magnitude of its displacement after 5 sec is

Vector A has magnitude of 3 units and makes an angle of ${30}^{\circ }$ with the positive x-axis.

Vector B has magnitude of 4 units and makes an angle of ${120}^{\circ }$ with the positive x-axis.

Find the magnitude of the resultant and the angle that it makes with the positive x-axis.

$\left[ta{n}^{-1}\right(\frac{4}{3})={53}^{\circ }]$

A body is projected at ${30}^{\circ }$ with velocity $v_1$ from point $A$ on the ground. At the same time, another body is projected vertically upwards from $B$ with velocity $v_2$ from the ground. The point $B$ lies vertically below the highest point of the trajectory of ball $A$. For both the bodies to collide, what should be the ratio of $\frac{v_1}{v_2}$ ?

A $3\frac{1}{2}$ digit multi meter has an accuracy specification of 0.5% of reading plus 5 counts. The value of an unknown resistance is read as $50.0\Omega$ on the $200\Omega$ scale of the meter, the value of the resistance is

Find derivative of $f\left(x\right)=e^{\cos x}+\ln x$

What is the natural frequency of an LC oscillator?

In the circuit shown, the $5\Omega$ resistance develops $45{\text{Js}}^{-1}$ of heat due to the flow of current through it. The current $I$ is

The distance of the centres of moon and earth is D. The mass of earth is 81 times the mass of the moon.
At what distance from the centre of the earth, the gravitational force will be zero

A
particle starts from the origin at t
= 0 s with a velocity of

and moves in the x-y
plane with a constant
acceleration of
.

(a) At what time is the
x-coordinate
of the particle 16 m? What is the y-coordinate
of the particle at that time?

(b) What is the speed of
the particle at the time?

Two balls $A$ and $B$ are projected simultaneously as shown in the figure. Find the closest distance between the balls.

A particle is fastened at the end of a string and is whirled in a vertical circle with the other end of the string being fixed. The motion of the particle is

A constant current I is maintained in a solenoid. Which of the following quantities will not increase if an iron rod is inserted in the solenoid along its axis?

Calculate the load current and load voltage in the following transformer circuit. Soyrce : 30v AC, Np = 100, Ns =100, Rs =100 Ω