A body is projected vertically upwards from the surface of earth with a velocity sufficient enough to carry it to infinity.
Question: A body is projected vertically upwards from the surface of earth with a velocity sufficient enough to carry it to infinity. The time taken by it to reach height $h$ is$\sqrt{\frac{\mathrm{R}_{\mathrm{e}}}{2 \mathrm{~g}}}\left[\left(1+\frac{\mathrm{h}}{\mathrm{R}_{\mathrm{e}}}\right)^{3 / 2}-1\right]$$\sqrt{\frac{2 \mathrm{R}_{\mathrm{e}}}{\mathrm{g}}}\left[\left(1+\frac{\mathrm{h}}{\mathrm{R}_{\mathrm{e}}}\right)^{3 / 2}-1\right]$$\frac{1}{3} \sqrt{\frac{\mathrm{R}_{\mathrm{e}}}{2 \mat...
Read More →A heat engine
Question: A heat engine has an efficiency of $\frac{1}{6}$. When the temperature of $\operatorname{sink}$ is reduced by $62^{\circ} \mathrm{C}$, its efficiency get doubled. The temperature of the source is :$124^{\circ} \mathrm{C}$$37^{\circ} \mathrm{C}$$62^{\circ} \mathrm{C}$$99^{\circ} \mathrm{C}$Correct Option: , 4 Solution: $\eta=1-\frac{\mathrm{T}_{\mathrm{L}}}{\mathrm{T}_{\mathrm{H}}} \ldots$ (i) $2 \eta=1-\frac{\left(\mathrm{T}_{\mathrm{L}}-62\right)}{\mathrm{T}_{\mathrm{H}}}=1-\frac{\mat...
Read More →The amplitude of wave disturbance propagating in
Question: The amplitude of wave disturbance propagating in the positive $x$-direction is given by $y=\frac{1}{(1+x)^{2}}$ at time $t=0$ and $y=\frac{1}{1+(x-2)^{2}}$ at $t=1 s$, where $x$ and $\mathrm{y}$ are in meres. The shape of wave does not change during the propagation. The velocity of the wave will be______ $\mathrm{m} / \mathrm{s}$. Solution: At $t=0, y=\frac{1}{1+x^{2}}$ At time $\mathrm{t}=\mathrm{t}, \mathrm{y}=\frac{1}{1+(\mathrm{x}-\mathrm{vt})^{2}}$ At $\mathrm{t}=1, \mathrm{y}=\fr...
Read More →A prism of refractive index
Question: A prism of refractive index $\mu$ and angle of prism A is placed in the position of minimum angle of deviation. If minimum angle of deviation is also $\mathrm{A}$, then in terms of refractive index$2 \cos ^{-1}\left(\frac{\mu}{2}\right)$$\sin ^{-1}\left(\frac{\mu}{2}\right)$$\sin ^{-1}\left(\sqrt{\frac{\mu-1}{2}}\right)$$\cos ^{-1}\left(\frac{\mu}{2}\right)$Correct Option: 1 Solution: $\mu=\frac{\sin \left(\frac{\mathrm{A}+\delta_{\min }}{2}\right)}{\sin \left(\frac{\mathrm{A}}{2}\righ...
Read More →A ball is thrown up with a certain velocity so that it reaches a height ' h '.
Question: A ball is thrown up with a certain velocity so that it reaches a height ' $h$ '. Find the ratio of the two different times of the ball reaching $\frac{\mathrm{h}}{3}$ in both the directions.$\frac{\sqrt{2}-1}{\sqrt{2}+1}$$\frac{1}{3}$$\frac{\sqrt{3}-\sqrt{2}}{\sqrt{3}+\sqrt{2}}$$\frac{\sqrt{3}-1}{\sqrt{3}+1}$Correct Option: , 3 Solution: $\mathbf{u}=\sqrt{2 \mathrm{gh}}$ Now, $S=\frac{h}{3} \quad a=-g$ $\mathrm{S}=\mathrm{ut}+\frac{1}{2} \mathrm{at}^{2}$ $\frac{\mathrm{h}}{3}=\sqrt{2 \...
Read More →In an LCR series circuit, an inductor
Question: In an LCR series circuit, an inductor $30 \mathrm{mH}$ and a resistor $1 \Omega$ are connected to an AC source of angular frequency $300 \mathrm{rad} / \mathrm{s}$. The value of capacitance for which, the current leads the voltage by $45^{\circ}$ is $\frac{1}{x} \times 10^{-3} \mathrm{~F}$. Then the value of $x$ is______. Solution: $\tan \phi=\frac{\mathrm{x}_{\mathrm{C}}-\mathrm{x}_{\mathrm{L}}}{\mathrm{R}}$ $\tan 45=\frac{x_{C}-x_{L}}{R}$ $x_{C}-x_{L}=R$ $\frac{1}{\omega \mathrm{C}}-...
Read More →An electric dipole is placed on
Question: An electric dipole is placed on $x$-axis in proximity to a line charge of linear charge density $3.0 \times 10^{-6} \mathrm{C} / \mathrm{m}$. Line charge is placed on $\mathrm{z}$-axis and positive and negative charge of dipole is at a distance of $10 \mathrm{~mm}$ and $12 \mathrm{~mm}$ from the origin respectively. If total force of $4 \mathrm{~N}$ is exerted on the dipole, find out the amount of positive or negative charge of the dipole.$815.1 \mathrm{nC}$$8.8 \mu \mathrm{C}$$0.485 \...
Read More →A force
Question: A force $\overrightarrow{\mathrm{F}}=(40 \hat{\mathrm{i}}+10 \hat{\mathrm{j}}) \mathrm{N}$ acts on a body of mass $5 \mathrm{~kg}$. If the body starts from rest, its position vector $\overrightarrow{\mathrm{r}}$ at time $\mathrm{t}=10 \mathrm{~s}$, will be :$(100 \hat{\mathrm{i}}+400 \hat{\mathrm{j}}) \mathrm{m}$$(100 \hat{i}+100 \hat{j}) m$$(400 \hat{\mathrm{i}}+100 \hat{\mathrm{j}}) \mathrm{m}$$(400 \hat{\mathrm{i}}+400 \hat{\mathrm{j}}) \mathrm{m}$Correct Option: , 3 Solution: $\fra...
Read More →In a spring gun having spring constant
Question: In a spring gun having spring constant $100 \mathrm{~N} / \mathrm{m}$ a small ball 'B' of mass $100 \mathrm{~g}$ is put in its barrel (as shown in figure) by compressing the spring through $0.05 \mathrm{~m}$. There should be a box placed at a distance 'd' on the ground so that the ball falls in it. If the ball leaves the gun horizontally at a height of $2 \mathrm{~m}$ above the ground. The value of $d$ is_______m. $\left(g=10 \mathrm{~m} / \mathrm{s}^{2}\right)$ Solution: $\frac{1}{2} ...
Read More →In a simple harmonic oscillation,
Question: In a simple harmonic oscillation, what fraction of total mechanical energy is in the form of kinetic energy, when the particle is midway between mean and extreme position.$\frac{1}{2}$$\frac{3}{4}$$\frac{1}{3}$$\frac{1}{4}$Correct Option: , 2 Solution: $\mathrm{K}=\frac{1}{2} \mathrm{m \omega}{ }^{2}\left(\mathrm{~A}^{2}-\mathrm{x}^{2}\right)$ $=\frac{1}{2} m \omega^{2}\left(\mathrm{~A}^{2}-\frac{\mathrm{A}^{2}}{4}\right)$ $=\frac{1}{2} \mathrm{~m} \omega^{2}\left(\frac{3 \mathrm{~A}^{...
Read More →Two capacitors of capacities 2C and C are joined in parallel and charged up to potential V.
Question: Two capacitors of capacities 2C and C are joined in parallel and charged up to potential V. The battery is removed and the capacitor of capacity C is filled completely with a medium of dielectric constant K. The potential difference across the capacitors will now be :$\frac{\mathrm{V}}{\mathrm{K}+2}$$\frac{\mathrm{V}}{\mathrm{K}}$$\frac{3 \mathrm{~V}}{\mathrm{~K}+2}$$\frac{3 \mathrm{~V}}{\mathrm{~K}}$Correct Option: , 3 Solution: $V_{C}=\frac{2 C V+C V}{K C+2 C}$ $=\frac{3 V}{K+2}$...
Read More →The relation between
Question: The relation between time $\mathrm{t}$ and distance $\mathrm{x}$ for a moving body is given as $\mathrm{t}=\mathrm{mx}^{2}+\mathrm{nx}$, where $\mathrm{m}$ and $\mathrm{n}$ are constants. The retardation of the motion is : (When $v$ stands for velocity)$2 \mathrm{mv}^{3}$$2 \mathrm{mnv}^{3}$$2 \mathrm{nv}^{3}$$2 n^{2} v^{3}$Correct Option: 1 Solution: $\mathrm{t}=\mathrm{mx} \mathrm{x}^{2}+\mathrm{nx}$ $\frac{1}{\mathrm{~V}}=\frac{\mathrm{dt}}{\mathrm{dx}}=2 \mathrm{mx}+\mathrm{n}$ $\m...
Read More →The motion of a mass on a spring, with spring constant
Question: The motion of a mass on a spring, with spring constant $\mathrm{K}$ is as shown in figure. The equation of motion is given by $x(t)=A \sin \omega t+$ $B \cos \omega t$ with $\omega=\sqrt{\frac{\mathrm{K}}{\mathrm{m}}}$ Suppose that at time $\mathrm{t}=0$, the position of mass is $x(0)$ and velocity $v(0)$, then its displacement can also be represented as $x(t)=\operatorname{Cos}(\omega t-\phi)$, where $C$ and $\phi$ are :$C=\sqrt{\frac{2 v(0)^{2}}{\omega^{2}}+x(0)^{2}}, \phi=\tan ^{-1}...
Read More →The radiation corresponding to
Question: The radiation corresponding to $3 \rightarrow 2$ transition of a hydrogen atom falls on a gold surface to generate photoelectrons. These electrons are passed through a magnetic field of $5 \times 10^{-4} \mathrm{~T}$. Assume that the radius of the largest circular path followed by these electrons is $7 \mathrm{~mm}$, the work function of the metal is: $\left(\right.$ Mass of electron $\left.=9.1 \times 10^{-31} \mathrm{~kg}\right)$$1.36 \mathrm{eV}$$1.88 \mathrm{eV}$$0.16 \mathrm{eV}$$...
Read More →A particle of mass
Question: A particle of mass ' $m$ ' is moving in time ' $t$ ' on a trajectory given by $\overrightarrow{\mathrm{r}}=10 \alpha \mathrm{t}^{2} \hat{\mathrm{i}}+5 \beta(\mathrm{t}-5) \hat{\mathrm{j}}$ Where $\alpha$ and $\beta$ are dimensional constants. The angular momentum of the particle becomes the same as it was for $\mathrm{t}=0$ at time $t=\ldots$ seconds. Solution: $\overrightarrow{\mathrm{r}}=10 \alpha \mathrm{t}^{2} \hat{\mathrm{i}}+5 \beta(\mathrm{t}-5) \hat{\mathrm{j}}$ $\overrightarro...
Read More →A pendulum bob has a speed
Question: A pendulum bob has a speed of $3 \mathrm{~m} / \mathrm{s}$ at its lowest position. The pendulum is $50 \mathrm{~cm}$ long. The speed of bob, when the length makes an angle of $60^{\circ}$ to the vertical will be $\left(\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^{2}\right)$_________ $\mathrm{m} / \mathrm{s}$. Solution: Applying work energy theorem : $\mathrm{w}_{\mathrm{g}}+\mathrm{w}_{\mathrm{T}}=\Delta \mathrm{K}$ $-\mathrm{mgl}\left(1-\cos 60^{\circ}\right)=\frac{1}{2} \mathrm{mv}^{2}-\f...
Read More →The entropy of any system is given by
Question: The entropy of any system is given by $S=\alpha^{2} \beta \ln \left[\frac{\mu \mathrm{kR}}{\mathrm{J} \beta^{2}}+3\right]$ where $\alpha$ and $\beta$ are the constants. $\mu, J, k$ and $R$ are no. of moles, mechanical equivalent of heat, Boltzmann constant and gas constant respectively. $\left[\operatorname{Take} S=\frac{\mathrm{dQ}}{\mathrm{T}}\right]$ Choose the incorrect option from the following :$\alpha$ and $J$ have the same dimensions.$\mathrm{S}, \beta, \mathrm{k}$ and $\mu \ma...
Read More →Intensity of sunlight is observed as
Question: Intensity of sunlight is observed as $0.092 \mathrm{Wm}^{-2}$ at a point in free space. What will be the peak value of magnetic field at that point ? $\left(\varepsilon_{0}=8.85 \times 10^{-12} \mathrm{C}^{2} \mathrm{~N}^{-1} \mathrm{~m}^{-2}\right)$$2.77 \times 10^{-8} \mathrm{~T}$$1.96 \times 10^{-8} \mathrm{~T}$$8.31 \mathrm{~T}$$5.88 \mathrm{~T}$Correct Option: 1 Solution: $\mathrm{I}_{\text {avg }}=\frac{\mathrm{B}_{0}^{2} \mathrm{C}}{2 \mu_{0}} \ \frac{1}{\mu_{0}}=\epsilon_{0} \m...
Read More →An electric bulb rated
Question: An electric bulb rated as $200 \mathrm{~W}$ at $100 \mathrm{~V}$ is used in a circuit having $200 \mathrm{~V}$ supply. The resistance ' $R$ ' that must be put in series with the bulb so that the bulb delivers the same power is________$\Omega$ Solution: Power, $\mathrm{P}=\frac{\mathrm{V}^{2}}{\mathrm{R}_{\mathrm{B}}}$ $\mathrm{R}_{\mathrm{B}}=\frac{\mathrm{V}^{2}}{\mathrm{P}}=\frac{100 \times 100}{200}$ $\mathrm{R}_{\mathrm{B}}=50 \Omega$ To produce same power, same voltage (i.e. $100 ...
Read More →If ' f ' denotes the ratio of the number of nuclei decayed
Question: If ' $f$ ' denotes the ratio of the number of nuclei decayed $\left(\mathrm{N}_{\mathrm{d}}\right)$ to the number of nuclei at $t=0\left(\mathrm{~N}_{0}\right)$ then for a collection of radioactive nuclei, the rate of change of ' $f$ ' with respect to time is given as : $[\lambda$ is the radioactive decay constant]$-\lambda\left(1-e^{-\lambda t}\right)$$\lambda\left(1-e^{-\lambda t}\right)$$\lambda \mathrm{e}^{-\lambda \mathrm{t}}$$-\lambda \mathrm{e}^{-\lambda t}$Correct Option: , 3 S...
Read More →The arm PQ of a rectangular conductor is moving
Question: The arm $\mathrm{PQ}$ of a rectangular conductor is moving from $x=0$ to $x=2 b$ outwards and then inwards from $x=2 b$ to $x=0$ as shown in the figure. $A$ uniform magnetic field perpendicular to the plane is acting from $\mathrm{x}=0$ to $\mathrm{x}=\mathrm{b}$. Identify the graph showing the variation of different quantities with distance: A-Flux, B-Power dissipated, C-EMFA-Power dissipated, B-Flux, C-EMFA-Flux, B-EMF, C-Power dissipatedA-EMF, B-Power dissipated, C-FluxCorrect Optio...
Read More →A particle of mass
Question: A particle of mass $1 \mathrm{mg}$ and charge $\mathrm{q}$ is lying at the mid-point of two stationary particles kept at a distance ' $2 \mathrm{~m}$ ' when each is carrying same charge ' $q$ '. If the free charged particle is displaced from its equilibrium position through distance ' $x$ ' $(x1 \mathrm{~m})$. The particle executes SHM. Its angular frequency of oscillation will be __________$-\times 10^{5} \mathrm{rad} / \mathrm{s}$ if $\mathrm{q}^{2}=10 \mathrm{C}^{2}$. Solution: Net ...
Read More →What will be the average value of energy for a monoatomic gas in thermal equilibrium at temperature T?
Question: What will be the average value of energy for a monoatomic gas in thermal equilibrium at temperature T?$\frac{2}{3} \mathrm{k}_{\mathrm{B}} \mathrm{T}$$\mathrm{k}_{\mathrm{B}} \mathrm{T}$$\frac{3}{2} \mathrm{k}_{\mathrm{B}} \mathrm{T}$$\frac{1}{2} \mathrm{k}_{\mathrm{B}} \mathrm{T}$Correct Option: , 3 Solution: As per Equi-partition law : Each degree of freedom contributes $\frac{1}{2} \mathrm{k}_{\mathrm{B}} \mathrm{T}$ Average Energy In monoatomic gas D.O.F. $=3$ $\Rightarrow$ Average...
Read More →The value of aluminium
Question: The value of aluminium susceptibility is $2.2 \times 10^{-5}$. The percentage increase in the magnetic field if space within a current carrying toroid is filled with aluminium is $\frac{x}{10^{4}}$. Then the value of $x$ is ______ Solution: $\mathrm{B}=\mu .(\mathrm{H}+\mathrm{I})$ $\mathrm{B}=\mu . \mathrm{H}\left(1+\frac{\mathrm{I}}{\mathrm{H}}\right)$ $\mathrm{B}=\mathrm{B}_{0}(1+\mathrm{x})$ $\mathrm{B}-\mathrm{B}_{0}=\mathrm{B}_{0} \mathrm{x}$ $\frac{\mathrm{B}-\mathrm{B}_{0}}{\ma...
Read More →A particle starts executing simple harmonic motion (SHM) of amplitude 'a' and total energy E. At any instant,
Question: A particle starts executing simple harmonic motion (SHM) of amplitude 'a' and total energy E. At any instant, its kinetic energy is $\frac{3 \mathrm{E}}{4}$ then displacement ' $y$ ' is given by :y = a$y=\frac{a}{\sqrt{2}}$$y=\frac{a \sqrt{3}}{2}$$y=\frac{a}{2}$Correct Option: , 4 Solution: $\mathrm{E}=\frac{1}{2} \mathrm{Ka}^{2}$ $\frac{3 \mathrm{E}}{4}=\frac{1}{2} \mathrm{~K}\left(\mathrm{a}^{2}-\mathrm{y}^{2}\right)$ $\frac{3}{4} \times \frac{1}{2} \mathrm{Ka}^{2}=\frac{1}{2} \mathr...
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