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In an adjoining figure are shown three capacitors C₁,C₂ and C₃ joined to a

battery. The correct condition will be (Symbols have their usual meanings)

Three rings each of mass M and radius R are arranged as shown in the figure. The moment of inertia of the system about YY' will be

A 6 metre pole casts a shadow of 8 m at a certain time of the day. Find the length of the shadow cast by a 4.5 m tower at the same time.

A sniper shoots a bullet into a gasoline tank making a hole $53.0\text{m}$ below the surface of gasoline. The tank was sealed at $3.10\text{atm}$. The stored gasoline has a density of $660{\text{kg/m}}^3$. The velocity with which gasoline begins to come out of the hole is

($g=10{\text{m/s}}^2$)

Estimate the kinetic energy gained by a proton in one revolution if the potential gap between the Dees is $80\text{kV}$ in a cyclotron.

In a four dimensional space, where unit vectors along the axes are $\hat{i},\hat{j},\hat{k}$ and $\hat{l},$$\overrightarrow{a_1},\overrightarrow{a_2},\overrightarrow{a_3},\overrightarrow{a_4}$ are four non-zero vectors such that no vector can be expressed as a linear combination of others and $(\lambda -1)({\overrightarrow{a}}_1-{\overrightarrow{a}}_2)+\alpha ({\overrightarrow{a}}_2+{\overrightarrow{a}}_3)+\gamma ({\overrightarrow{a}}_3+{\overrightarrow{a}}_4-2{\overrightarrow{a}}_2)+{\overrightarrow{a}}_3+\delta {\overrightarrow{a}}_4=\overrightarrow{0}.$
Then which of the following is/are correct?

A conducting loop of area $10{\text{cm}}^2$ is placed in a magnetic field which varies sinusoidally with time as $B=2\sin 100t$. The normal to the coil makes an angle of ${60}^{\circ }$ with direction of the field. Find the maximum magnetic flux through the coil.

Consider two concentric spherical surfaces ${\text{S}}_1$ with radius $a$ and ${\text{S}}_2$ with radius $2a$, both centred on the origin. There is a charge $+q$ at the origin, and no other charges. Compare the flux ${\varphi }_1$ through ${\text{S}}_1$ with flux ${\varphi }_2$ through ${\text{S}}_2$

The magnetic flux linked with a loop changes with time according to the equation, $\varphi =a{t}^2+bt+c$ where $t$ is time in $\text{s}$ and $a,b$ and $c$ are constants. The SI unit of $a$ is

A liquid stands at the plane level in U-tube when at rest. If areas of cross-section of both the limbs are equal, what will be the difference in heights $h$ of the liquid in the two limbs of U-tube, when the system is given an acceleration $a$ in horizontal direction towards right as shown ?

A vessel containing water is given a constant acceleration a towards the right, along a straight horizontal path. Which of the following diagram represents the surface of the liquid

[IIT 1981]

In sound wave, $V_w$ is the wave velocity and $V_p$ is particle velocity. Then:

If $C_f$ is the coefficient of speed fluctuation of a flywheel then the ratio of ${\omega }_{max}/{\omega }_{min}$ will be

A car is negotiating a curve of radius R = 20 m on a banked road with banking angle $\theta$ and coefficient of friction $\mu$. Take speed of car is v.

$\begin{array}{cccccc}& \text{Column 1}& & \text{Column 2}& & \text{Column 3}\\ \left(I\right)& v=10\sqrt{2}m/s& \left(i\right)& \mu =0.2& \left(p\right)& \theta ={30}^{\circ }\\ \left(II\right)& v=20m/s& \left(ii\right)& \mu =0.4& \left(Q\right)& \theta ={45}^{\circ }\\ \left(III\right)& v=50m/s& \left(iii\right)& \mu =0.6& \left(R\right)& \theta ={37}^{\circ }\\ \left(IV\right)& v=5m/s& \left(iv\right)& \mu =0.8& \left(S\right)& \theta ={60}^{\circ }\end{array}$

In which case will friction be outwards of curve ?

Avinash practicing in a swimming pool takes enough air from his tank to fully expand his lungs before abandoning the tank at depth 'L' and swimming to the surface and failing to exhale during his ascent. At the surface ∆P between the external pressure on him and the air pressure in his lungs is 9.3 kpa. From what depth does he start?

A proton and a deutron both having the same kinetic energy, enter perpendicularly into a uniform magnetic field B. For motion of proton and deutron on circular path of radius $R_p$ and $R_d$ respectively, the correct statement is

A uniformly charged sheet is placed in front of a point charge of $6\mu C$. The area of sheet is $2{\mathrm{cm}}^2$ and charge on sheet is$100\mu C$. Find force by point charge on sheet if they are at a distance of ${10}^{-4}m$.

In the arrangement shown in figure find the tensions in the rope and accelerations of the masses $m_1$ and $m_2$ and pulleys $P_1$ and $P_2$ when the system is set free to move. Assume the pulleys to be massless and strings are light and inextensible.

p) T (i) m₁g

q) a₁ (ii) m₂g

r) a₂ (iii) (m₁ - m₂)g

s) ap₁ (iv) g downward

t) ap₂ (v) g upward

(vi) 3g downward

A substance is taken through the process abc as shown in figure. If the internal energy of the substance increases by 5000 J and a heat of 2625 cal is given to the system, calculate the value of J.

A ray of light travelling in a medium of refractive index $\mu$ is incident at an angle $i$ on a composite transparent plate consisting of three layers of refractive indices ${\mu }_1$, ${\mu }_2$ and ${\mu }_3$. The ray emerges from the composite plate into a medium of refractive index ${\mu }_4$ at angle $x$. Then

Find the angular displacement of the hour hand of a normal clock in $4\text{h}$

A conducting rod AC of length $4l$ is rotated about a point O in a uniform magnetic field directed into the paper. AO $=l$ and OC $=3l$. Then

The mass and density of a solid sphere are measured to be $(12.4\pm 0.1)\text{kg}$ and $(4.60\pm 0.2){\text{kg/m}}^3$. The volume of the sphere expressed with error limits is:

With what speed should a body be thrown upwards so that the distance traversed in $5^{th}$ second and $6^{th}$ second are equal? (Take $g=9.8m/s^2$)

A pendulum makes 400 oscillations in 10 seconds. Calculate the frequency and time period of oscillation of the pendulum in milliseconds.

In the circuit diagram.

Find
(i) total resistance
(ii) current shown by the ammeter A.

The capacitance of a parallel plate capacitor is $C_0$ when the plates have air between them. This region is now filled with a dielectric slab of dielectric constant $K$ and capacitor is connected with a battery of emf $E$ and zero internal resistance. Now slab is taken out, then during removal of slab:



$1.)$ Charge $E{C}_0(K-1)$ flows through the cell.



$2.)$ Energy $E^2{C}_0(K-1)$ is absorbed by cell.



$3.)$ The energy stored in the capacitor is reduced by $E^2{C}_0(K-1)$.



$4.)$ The external agent has to do $E^2{C}_0(K-1)$ amount of work to take out the slab.



Select the correct phenomenon happening during the process:

Find the displacement travelled by the sphere in $1$ second if the acceleration of the cube is $2{\text{m/s}}^2$. System starts from rest and the sphere remains in contact with the cube.