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Question: Starting from the origin at time $t=0$, with initial velocity $5 \hat{\mathrm{j}} \mathrm{ms}^{-1}$, a particle moves in the $\mathrm{x}-\mathrm{y}$ plane with a constant acceleration of $(10 \hat{\mathrm{i}}+4 \hat{\mathrm{j}}) \mathrm{ms}^{-2}$. At time $\mathrm{t}$, its coordinates are $\left(20 \mathrm{~m}, \mathrm{y}_{0} \mathrm{~m}\right)$. The values of $\mathrm{t}$ and $\mathrm{y}_{0}$, are respectively : $4 \mathrm{~s}$ and $52 \mathrm{~m}$$2 \mathrm{~s}$ and $24 \mathrm{~m}$$...
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Question: On the $x$-axis and a dsitance $x$ from the origin, the gravitational field due to a mass distribution is given by $\frac{\mathrm{Ax}}{\left(\mathrm{x}^{2}+\mathrm{a}^{2}\right)^{3 / 2}}$ in the $\mathrm{x}$-direction. The magnitude of gravitational potential on the $x$-axis at a distance $x$, taking its value to be zero at infinity, is : $\frac{A}{\left(x^{2}+a^{2}\right)^{1 / 2}}$$\frac{A}{\left(x^{2}+a^{2}\right)^{3 / 2}}$$\mathrm{A}\left(\mathrm{x}^{2}+\mathrm{a}^{2}\right)^{3 / 2}...
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Question: Dimensional formula for thermal conductivity is (here $\mathrm{K}$ denotes the temperature) $\mathrm{MLT}^{-3} \mathrm{~K}$$\mathrm{MLT}^{-2} \mathrm{~K}$$\mathrm{MLT}^{-2} \mathrm{~K}^{-2}$$\mathrm{MLT}^{-3} \mathrm{~K}^{-1}$Correct Option: , 4 Solution: $\because \frac{\mathrm{d} \theta}{\mathrm{dt}}=\mathrm{kA} \frac{\mathrm{dT}}{\mathrm{dx}}$ $\mathrm{k}=\frac{\left(\frac{\mathrm{d} \theta}{\mathrm{dt}}\right)}{\mathrm{A}\left(\frac{\mathrm{dT}}{\mathrm{dx}}\right)}$ $[\mathrm{k}]=...
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Question: A air bubble of radius $1 \mathrm{~cm}$ in water has an upward acceleration $9.8 \mathrm{~cm} \mathrm{~s}^{-2}$. The density of water is $1 \mathrm{gm} \mathrm{cm}^{-3}$ and water offers negligible drag force on the bubble. The mass of the bubble is $\left(\mathrm{g}=980 \mathrm{~cm} / \mathrm{s}^{2}\right)$ $3.15 \mathrm{gm}$$4.51 \mathrm{gm}$$4.15 \mathrm{gm}$$1.52 \mathrm{gm}$Correct Option: , 3 Solution: Volume $\mathrm{V}=\frac{4 \pi}{3} \mathrm{r}^{3}=\frac{4 \pi}{3} \times(1)^{3...
Read More →Choose the correct option relating wavelengths of differnet parts of electromagnetic wave spectrum :
Question: Choose the correct option relating wavelengths of differnet parts of electromagnetic wave spectrum : $\lambda_{\text {x-rays }}\lambda_{\text {micro waves }}\lambda_{\text {radio waves }}\lambda_{\text {visible }}$$\lambda_{\text {visible }}\lambda_{\text {x-rays }}\lambda_{\text {radio waves }}\lambda_{\text {micro waves }}$$\lambda_{\text {radio waves }}\lambda_{\text {micro waves }}\lambda_{\text {visible }}\lambda_{x \text {-rays }}$$\lambda_{\text {visible }}\lambda_{\text {micro ...
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Question: Match the $C_{P} / C_{V}$ ratio for ideal gases with different tune of molecules. $\mathrm{A}-\mathrm{IV}, \mathrm{B}-\mathrm{I}, \mathrm{C}-\mathrm{II}, \mathrm{D}-\mathrm{III}$A-IV, B-II, C-I, D-IIIA-III, B-IV, C-II, D-IA-II, B-III, C-I, D-IVCorrect Option: 1 Solution: $\gamma=\frac{\mathrm{C}_{\mathrm{p}}}{\mathrm{C}_{\mathrm{v}}}=1+\frac{2}{\mathrm{f}}$ where ' $f$ ' is degree of freedom (A) Monoatomic $\mathrm{f}=3, \gamma=1+\frac{2}{3}=\frac{5}{3}$ (B) Diatomic rigid molecules, $...
Read More →A beam of plane polarised light of large cross sectional area and uniform intensity of
Question: A beam of plane polarised light of large cross sectional area and uniform intensity of $3.3 \mathrm{Wm}^{-2}$ falls normally on a polariser (cross sectional area $3 \times 10^{-4} \mathrm{~m}^{2}$ ) which rotates about its axis with an angular speed of $31.4 \mathrm{rad} / \mathrm{s}$. The energy of light passing through the polariser per revolution, is close to : $1.0 \times 10^{-5} \mathrm{~J}$$5.0 \times 10^{-4} \mathrm{~J}$$1.0 \times 10^{-4} \mathrm{~J}$$1.5 \times 10^{-4} \mathrm...
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Question: A wire of density $9 \times 10^{-3} \mathrm{~kg} \mathrm{~cm}^{-3}$ is stretched between two clamps $1 \mathrm{~m}$ apart. The resulting strain in the wire is $4.9 \times 10^{-4}$. The lowest frequency of the transverse vibrations in the wire is (Young's modulus of wire $\mathrm{Y}=9 \times 10^{10} \mathrm{Nm}^{-2}$, (to the nearest integer), Solution: $\rho_{\text {wire }}=9 \times 10^{-3} \frac{\mathrm{kg}}{\mathrm{cm}^{3}}=\frac{9 \times 10^{-3}}{10^{-6}} \mathrm{~kg} / \mathrm{m}^{...
Read More →A square shaped hole of side
Question: A square shaped hole of side $l=\frac{\mathrm{a}}{2}$ is carved out at a distance $\mathrm{d}=\frac{\mathrm{a}}{2}$ from the centre ' $\mathrm{O}$ ' of a uniform circular disk of radius a. If the distance of the centre of mass of the remaining portion from $\mathrm{O}$ is $-\frac{\mathrm{a}}{\mathrm{X}}$, value of $\mathrm{X}$ (to the nearest integer) is Solution: $X_{c o m}=\frac{m_{1} x_{1}-m_{2} x_{2}}{m_{1}-m_{2}}$ where: - $\mathrm{m}_{1}=$ mass of complete disc - $\mathrm{m}_{2}=...
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Question: A particle of mass $m$ is moving along the $x$-axis with initial velocity u\hat{i. It collides elastically } with a particle of mass $10 \mathrm{~m}$ at rest and then moves with half its initial kinetic energy (see figure). If $\sin \theta_{1}=\sqrt{\mathrm{n}} \sin \theta_{2}$ then value of $\mathrm{n}$ is Solution: By momentum conservation along $y$ : $\mathrm{m}_{1} \mathrm{u}_{1} \sin \theta_{1}=\mathrm{m}_{2} \mathrm{u}_{2} \sin \theta_{2}$ i.e. $\quad m u_{1} \sin \theta_{1}=10 m...
Read More →A light ray enters a solid glass sphere of refractive index
Question: A light ray enters a solid glass sphere of refractive index $\mu=\sqrt{3}$ at an angle of incidence $60^{\circ}$. The ray is both reflected and refracted at the farther surface of the sphere. The angle (in degrees) between the reflected and refracted rays at this surface is Solution: By Snell's law at $\mathrm{A}$ : $1 \times \sin 60^{\circ}=\sqrt{3} \times \sin \theta$ $\frac{\sqrt{3}}{2}=\sqrt{3} \sin \theta$ $\sin \theta=\frac{1}{2} \Rightarrow \theta=30^{\circ}$ So at B : $\theta+6...
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Question: An ideal cell of emf $10 \mathrm{~V}$ is connected in circuit shown in figure. Each resistance is $2 \Omega$. The potential difference (in V) across the capacitor when it is fully charged is Solution: - $\mathrm{R}_{1}$ to $\mathrm{R}_{5} \rightarrow$ each $2 \Omega$ - Cap. is fully charged - So no current is there in branch ADB - Effective circuit of current flow : $\mathrm{R}_{e q}=\left(\frac{4 \times 2}{4+2}\right)+2$ $\mathrm{R}_{e q}=\frac{4}{3}+2=\frac{10}{3} \Omega$ $\mathrm{i}...
Read More →A particle is moving 5 times as fast as an electron. The ratio of the de-Broglie wavelength of the particle to that of the electron
Question: A particle is moving 5 times as fast as an electron. The ratio of the de-Broglie wavelength of the particle to that of the electron is $1.878 \times 10^{-4}$. The mass of the particle is close to : $4.8 \times 10^{-27} \mathrm{~kg}$$1.2 \times 10^{-28} \mathrm{~kg}$$9.1 \times 10^{-31} \mathrm{~kg}$$9.7 \times 10^{-28} \mathrm{~kg}$Correct Option: , 4 Solution: Let mass of particle $=\mathrm{m}$ Let speed of $\mathrm{e}^{-}=\mathrm{V}$ $\Rightarrow$ speed of particle $=5 \mathrm{~V}$ D...
Read More →In a plane electromagnetic wave, the directions of electric field and magnetic field are represented
Question: In a plane electromagnetic wave, the directions of electric field and magnetic field are represented by $\hat{k}$ and $2 \hat{i}-2 \hat{j}$, respectively. What is the unit vector along direction of propagation of the wave. $\frac{1}{\sqrt{2}}(\hat{i}+\hat{j})$$\frac{1}{\sqrt{5}}(\hat{i}+2 \hat{j})$$\frac{1}{\sqrt{5}}(2 \hat{\mathrm{i}}+\hat{\mathrm{j}})$$\frac{1}{\sqrt{2}}(\hat{\mathrm{j}}+\hat{\mathrm{k}})$Correct Option: 1 Solution: $\hat{\mathrm{E}}=\hat{\mathrm{k}}$ $\overrightarro...
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Question: When the temperature of a metal wire is increased from $0^{\circ} \mathrm{C}$ to $10^{\circ} \mathrm{C}$, its length increases by $0.02 \%$. The percentage change in its mass density will be closest to: $0.008$$0.06$$0.8$$2.3$Correct Option: , 2 Solution: Given $\frac{\Delta \mathrm{L}}{\mathrm{L}}=0.02 \%$ $\therefore \Delta \mathrm{L}=\mathrm{L} \alpha \Delta \mathrm{T} \Rightarrow \frac{\Delta \mathrm{L}}{\mathrm{L}}=\alpha \Delta \mathrm{T}=0.02 \%$ $\therefore \beta=2 \alpha$ (Are...
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Question: A $10 \mu \mathrm{F}$ capacitor is fully charged to a potential difference of $50 \mathrm{~V}$. After removing the source voltage it is connected to an uncharged capacitor in parallel. Now the potential difference across them becomes $20 \mathrm{~V}$. The capacitance of the second capacitor is: $10 \mu \mathrm{F}$$15 \mu \mathrm{F}$$20 \mu \mathrm{F}$$30 \mu \mathrm{F}$Correct Option: 2, Solution: - Charge on capacitor $10 \mu \mathrm{F}$ $\mathrm{Q}=\mathrm{CV}=(10 \mu \mathrm{F})(50 ...
Read More →A small point mass carrying some positive charge on it, is released from the edge of a table
Question: A small point mass carrying some positive charge on it, is released from the edge of a table. There is a uniform electric field in this region in the horizontal direction. Which of the following options then correctly describe the trajectory of the mass ? (Curves are drawn schematically and are not to scale). Correct Option: 4, Solution: Since initial velocity is zero and acceleration of particle will be constant, so particle will travel on a straight line path....
Read More →wo uniform circular discs are rotating independently in the same direction around their common axis passing through their centres.
Question: Two uniform circular discs are rotating independently in the same direction around their common axis passing through their centres. The moment of inertia and angular velocity of the first disc are $0.1 \mathrm{~kg}-\mathrm{m}^{2}$ and $10 \mathrm{rad} \mathrm{s}^{-1}$ respectively while those for the second one are $0.2 \mathrm{~kg}-\mathrm{m}^{2}$ and $5 \mathrm{rad} \mathrm{s}^{-1}$ respectively. At some instant they get stuck together and start rotating as a single system about thei...
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Question: An inductance coil has a reactance of $100 \Omega$. When an $\mathrm{AC}$ signal of frequency $1000 \mathrm{~Hz}$ is applied to the coil, the applied voltage leads the current by $45^{\circ}$. The self-inductance of the coil is :$1.1 \times 10^{-2} \mathrm{H}$$1.1 \times 10^{-1} \mathrm{H}$$5.5 \times 10^{-5} \mathrm{H}$$6.7 \times 10^{-7} \mathrm{H}$Correct Option: 1 Solution: - Reactance of inductance coil $=\sqrt{\mathrm{R}^{2}+\mathrm{x}_{\mathrm{L}}^{2}}=100$ .............(I) - $\...
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Question: A charge $Q$ is distributed over two concentric conducting thin spherical shells radii $r$ and $R$ $(\mathrm{R}\mathrm{r})$. If the surface charge densities on the two shells are equal, the electric potential at the common centre is : $\frac{1}{4 \pi \varepsilon_{0}} \frac{(R+2 r) Q}{2\left(R^{2}+r^{2}\right)}$$\frac{1}{4 \pi \varepsilon_{0}} \frac{(\mathrm{R}+\mathrm{r})}{2\left(\mathrm{R}^{2}+\mathrm{r}^{2}\right)} \mathrm{Q}$$\frac{1}{4 \pi \varepsilon_{0}} \frac{(\mathrm{R}+\mathrm...
Read More →An ideal gas in a closed container is slowly heated. As its temperature increases, which of the following statements are true ?
Question: An ideal gas in a closed container is slowly heated. As its temperature increases, which of the following statements are true ? (A) the mean free path of the molecules decreases. (B) the mean collision time between the molecules decreases. (C) the mean free path remains unchanged. (D) the mean collision time remains unchanged. (C) and (D)(A) and (B)(A) and (D)(B) and (C)Correct Option: , 4 Solution: The mean free path of molecules of an ideal gas is given as: $\lambda=\frac{\mathrm{V}}...
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Question: The height ' $h$ ' at which the weight of a body will be the same as that at the same depth 'h' from the surface of the earth is (Radius of the earth is $R$ and effect of the rotation of the earth is neglected):$\frac{\sqrt{5} R-R}{2}$$\frac{\sqrt{5}}{2} R-R$$\frac{\mathrm{R}}{2}$$\frac{\sqrt{3} R-R}{2}$Correct Option: 1 Solution: - $\mathrm{M}=$ mass of earth $\mathrm{M}_{1}=$ mass of shaded portion $\mathrm{R}=$ Radius of earth $\mathrm{M}_{1}=\frac{\mathrm{M}}{\frac{4}{3} \pi \mathr...
Read More →A capillary tube made of glass of radius
Question: A capillary tube made of glass of radius $0.15 \mathrm{~mm}$ is dipped vertically in a beaker filled with methylene iodide (surface tension $=0.05 \mathrm{Nm}^{-1}$, density $=667 \mathrm{~kg} \mathrm{~m}^{-3}$ ) which rises to height $h$ in the tube. It is observed that the two tangents drawn from liquid-glass interfaces (from opp. sides of the capillary) make an angle of $60^{\circ}$ with one another. Then $\mathrm{h}$ is close to $\left(\mathrm{g}=10 \mathrm{~ms}^{-2}\right)$. $0.13...
Read More →If momentum (P), area (A) and time (T) are taken to be the fundamental quantities then the dimensional formula for energy is :
Question: If momentum (P), area (A) and time (T) are taken to be the fundamental quantities then the dimensional formula for energy is : $\left[\mathrm{PA}^{-1} \mathrm{~T}^{-2}\right]$$\left[\mathrm{PA}^{1 / 2} \mathrm{~T}^{-1}\right]$$\left[\mathrm{P}^{2} \mathrm{AT}^{-2}\right]$$\left[\mathrm{P}^{1 / 2} \mathrm{AT}^{-1}\right]$Correct Option: , 2 Solution: Let $[\mathrm{E}]=[\mathrm{P}]^{\mathrm{x}}[\mathrm{A}]^{\mathrm{y}}[\mathrm{T}]^{\mathrm{z}}$ $\mathrm{ML}^{2} \mathrm{~T}^{-2}=\left[\ma...
Read More →In the following digital circuit, what will be the output at 'Z',
Question: In the following digital circuit, what will be the output at 'Z', when the input (A,B) are $(1,0)$, $(0,0),(1,1),(0,1):$ $1,0,1,1$$0,1,0,0$$0,0,1,0$$1,1,0,1$Correct Option: , 3 Solution: $Z=(\overline{P+R})$ $Z=(\overline{P+P Q})$ $Z=(\overline{P(1+Q)})$ $Z=(\bar{P})$ [Using Identity $(1+A)=1$ ] $Z=\overline{(\overline{\mathrm{AB}})}$ $Z=\overline{\mathrm{AB}}$ Truth table for $\mathrm{Z}=\mathrm{AB}$...
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