A parallel plate capacitor
Question: A parallel plate capacitor with plate area 'A' and distance of separation 'd' is filled with a dielectric. What is the capacity of the capacitor when permittivity of the dielectric varies as : $\varepsilon(x)=\varepsilon_{0}+k x$, for $\left(0x \leq \frac{d}{2}\right)$ $\varepsilon(x)=\varepsilon_{0}+k(d-x)$, for $\left(\frac{d}{2} \leq x \leq d\right)$$\left(\varepsilon_{0}+\frac{\mathrm{kd}}{2}\right)^{2 / \mathrm{kA}}$$\frac{\mathrm{kA}}{2 \ln \left(\frac{2 \varepsilon_{0}+\mathrm{k...
Read More →What will be the projection of vector
Question: What will be the projection of vector $\overrightarrow{\mathrm{A}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}$ on vector $\overrightarrow{\mathrm{B}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}$ ?$\sqrt{2}(\hat{i}+\hat{j}+\hat{k})$$2(\hat{i}+\hat{j}+\hat{k})$$\sqrt{2}(\hat{i}+\hat{j})$$(\hat{i}+\hat{j})$Correct Option: , 4 Solution: $(A \cos \theta) \hat{B}=A\left(\frac{\vec{A} \cdot \vec{B}}{A B}\right) \hat{B}=\frac{\vec{A} \cdot \vec{B}}{B} \hat{B}$ $=\frac{2}{\sqrt{2}}\left(\frac{\ha...
Read More →A person whose mass is 100 kg travels from Earth to Mars in a spaceship.
Question: A person whose mass is $100 \mathrm{~kg}$ travels from Earth to Mars in a spaceship. Neglect all other objects in sky and take acceleration due to gravity on the surface of the Earth and Mars as $10 \mathrm{~m} / \mathrm{s}^{2}$ and $4 \mathrm{~m} / \mathrm{s}^{2}$ respectively. Identify from the below figures, the curve that fits best for the weight of the passenger as a function of time. (c)(a)(d)(b)Correct Option: 1 Solution: At neutral point $g=0$ so graph $(C)$ is correct Hence op...
Read More →Match List I with List II.
Question: Match List I with List II. Choose the correct answer from the options given below :(a) $\rightarrow$ (iv), (b) $\rightarrow$ (i) , (c) $\rightarrow$ (iii), (d) $\rightarrow$ (ii)$(\mathrm{a}) \rightarrow(\mathrm{iv})$, (b) $\rightarrow($ iii $),(\mathrm{c}) \rightarrow(\mathrm{i}),(\mathrm{d}) \rightarrow(\mathrm{ii})$(a) $\rightarrow$ (iii), (b) $\rightarrow$ (ii), (c) $\rightarrow$ (iv), (d) $\rightarrow$ (i)$(\mathrm{a}) \rightarrow(\mathrm{i}),(\mathrm{b}) \rightarrow(\mathrm{iv})$...
Read More →The relative permittivity of distilled water is 81. The velocity of light in it will be :
Question: The relative permittivity of distilled water is 81. The velocity of light in it will be : (Given $\mu_{\mathrm{r}}=1$ )$4.33 \times 10^{7} \mathrm{~m} / \mathrm{s}$$2.33 \times 10^{7} \mathrm{~m} / \mathrm{s}$$3.33 \times 10^{7} \mathrm{~m} / \mathrm{s}$$5.33 \times 10^{7} \mathrm{~m} / \mathrm{s}$Correct Option: , 3 Solution: $\mathrm{V}=\frac{\mathrm{c}}{\sqrt{\mu_{\mathrm{r}} \varepsilon_{\mathrm{r}}}}$ $=3.33 \times 10^{7} \mathrm{~m} / \mathrm{sec}$...
Read More →Solve this following
Question: A Copper $(\mathrm{Cu})$ rod of length $25 \mathrm{~cm}$ and crosssectional area $3 \mathrm{~mm}^{2}$ is joined with a similar Aluminium (Al) rod as shown in figure. Find the resistance of the combination between the ends $\mathrm{A}$ and $B$. (Take Resistivity of Copper $=1.7 \times 10^{-8} \Omega \mathrm{m}$ Resistivity of Aluminium $=2.6 \times 10^{-8} \Omega \mathrm{m}$ ) $2.170 \mathrm{~m} \Omega$$1.420 \mathrm{~m} \Omega$$0.0858 \mathrm{~m} \Omega$$0.858 \mathrm{~m} \Omega$Correc...
Read More →Some nuclei of a radioactive
Question: Some nuclei of a radioactive material are undergoing radioactive decay. The time gap between the instances when a quarter of the nuclei have decayed and when half of the nuclei have decayed is given as : (where $\lambda$ is the decay constant)$\frac{1}{2} \frac{\ln 2}{\lambda}$$\frac{\ln 2}{\lambda}$$\frac{2 \ln 2}{\lambda}$$\frac{\ln \frac{3}{2}}{\lambda}$Correct Option: , 4 Solution: $\frac{3 \mathrm{~N}_{0}}{4}=\mathrm{N}_{0} \mathrm{e}^{-\lambda \mathrm{t}_{1}}$ $\frac{\mathrm{N}_{...
Read More →Region I and II are separated by a spherical surface
Question: Region I and II are separated by a spherical surface of radius $25 \mathrm{~cm}$. An object is kept in region $\mathrm{I}$ at a distance of $40 \mathrm{~cm}$ from the surface. The distance of the image from the surface is : $55.44 \mathrm{~cm}$$9.52 \mathrm{~cm}$$18.23 \mathrm{~cm}$$37.58 \mathrm{~cm}$Correct Option: , 4 Solution: $\frac{\mu_{2}}{\mathrm{~V}}-\frac{\mu_{1}}{\mathrm{u}}=\frac{\mu_{2}-\mu_{1}}{\mathrm{R}}$ $\frac{1.4}{v}-\frac{1.25}{-40}=\frac{1.4-1.25}{-25}$ $\frac{1.4}...
Read More →The number of molecules in one litre of an ideal gas at 300 K and 2 atmospheric pressure
Question: The number of molecules in one litre of an ideal gas at $300 \mathrm{~K}$ and 2 atmospheric pressure with mean kinetic energy $2 \times 10^{-9} \mathrm{~J}$ per molecules is :$0.75 \times 10^{11}$$3 \times 10^{11}$$1.5 \times 10^{11}$$6 \times 10^{11}$Correct Option: , 3 Solution: $\mathrm{KE}=\frac{3}{2} \mathrm{kT}$ $\mathrm{PV}=\frac{\mathrm{N}}{\mathrm{N}_{\mathrm{A}}} \mathrm{RT}$ $\mathrm{N}=\frac{\mathrm{PV}}{\mathrm{kT}}$ $=\mathrm{N}=1.5 \times 10^{11}$...
Read More →A particle of mass
Question: A particle of mass $4 \mathrm{M}$ at rest disintegrates into two particles of mass $\mathrm{M}$ and $3 \mathrm{M}$ respectively having non zero velocities. The ratio of de-Broglie wavelength of particle of mass $M$ to that of mass $3 \mathrm{M}$ will be:$1: 3$$3: 1$$1: \sqrt{3}$$1: 1$Correct Option: , 4 Solution: $\lambda=\frac{h}{p}$ both the particles will move with momentum same in magnitude \ opposite in direction. So De-Broglie wavelength of both will be same i.e. ratio $1: 1$...
Read More →In a circuit consisting of a capacitance and a generator with alternating emf
Question: In a circuit consisting of a capacitance and a generator with alternating emf $E_{g}=E_{g_{0}} \sin \omega t, V_{C}$ and $\mathrm{I}_{\mathrm{C}}$ are the voltage and current. Correct phasor diagram for such circuit is : Correct Option: , 3 Solution: In capacitor, current lead voltage by $\frac{\pi}{2}$...
Read More →AC voltage V(t) = 20 sin ωt of frequency
Question: AC voltage $V(t)=20 \sin \omega t$ of frequency $50 \mathrm{~Hz}$ is applied to a parallel plate capacitor. The separation between the plates is $2 \mathrm{~mm}$ and the area is $1 \mathrm{~m}^{2}$. The amplitude of the oscillating displacement current for the applied AC voltage is_________. [Take $\varepsilon_{0}=8.85 \times 10^{-12} \mathrm{~F} / \mathrm{m}$ ]$21.14 \mu \mathrm{A}$$83.37 \mu \mathrm{A}$$27.79 \mu \mathrm{A}$$55.58 \mu \mathrm{A}$Correct Option: , 3 Solution: From the...
Read More →Identify the logic operation carried out.
Question: Identify the logic operation carried out. ORANDNORNANDCorrect Option: , 2 Solution:...
Read More →In the given figure, a battery of emf E is connected across a conductor PQ of length l
Question: In the given figure, a battery of emf $E$ is connected across a conductor PQ of length $l '$ and different area of cross-sections having radii $r_{1}$ and $r_{2}\left(r_{2}r_{1}\right)$. Choose the correct option as one moves from P to Q :Drift velocity of electron increases.Electric field decreases.Electron current decreasesAll of theseCorrect Option: 1 Solution: Current is constant in conductor i = constant Resistance of element $\mathrm{dR}=\frac{\rho \mathrm{dx}}{\pi \mathrm{r}^{2}...
Read More →What should be the order
Question: What should be the order of arrangement of de-Broglie wavelength of electron $\left(\lambda_{e}\right)$, an $\alpha$-particle $\left(\lambda_{\alpha}\right)$ and proton $\left(\lambda_{p}\right)$ given that all have the same kinetic energy ?$\lambda_{\mathrm{e}}=\lambda_{\mathrm{p}}=\lambda_{\alpha}$$\lambda_{\mathrm{e}}\lambda_{\mathrm{p}}\lambda_{\alpha}$$\lambda_{\mathrm{e}}\lambda_{\mathrm{p}}\lambda_{\alpha}$$\lambda_{\mathrm{e}}=\lambda_{\mathrm{p}}\lambda_{\alpha}$Correct Option...
Read More →Given below are two statements :
Question: Given below are two statements : one is labelled as Assertion $\mathbf{A}$ and the other is labelled as Reason $\mathbf{R}$. Assertion A : Moment of inertia of a circular disc of mass ' $M$ ' and radius ' $R$ ' about $X, Y$ axes (passing through its plane) and Z-axis which is perpendicular to its plane were found to be $I_{x}, I_{y}$ and $I_{z}$ respectively. The respective radii of gyration about all the three axes will be the same. Reason $\mathbf{R}$ : A rigid body making rotational...
Read More →A steel block of 10kg
Question: A steel block of $10 \mathrm{~kg}$ rests on a horizontal floor as shown. When three iron cylinders are placed on it as shown, the block and cylinders go down with an acceleration $0.2 \mathrm{~m} / \mathrm{s}^{2}$. The normal reaction $R^{\prime}$ by the floor if mass of the iron cylinders are equal and of $20 \mathrm{~kg}$ each, is N. [Take $g=10 \mathrm{~m} / \mathrm{s}^{2}$ and $\left.\mu_{\mathrm{s}}=0.2\right]$ 716686714684Correct Option: , 2 Solution: Writing force equation in ve...
Read More →For a gas
Question: For a gas $\mathrm{C}_{\mathrm{P}}-\mathrm{C}_{\mathrm{V}}=\mathrm{R}$ in a state $\mathrm{P}$ and $\mathrm{C}_{\mathrm{P}}-\mathrm{C}_{\mathrm{V}}=1.10 \mathrm{R}$ in a state $\mathrm{Q}, \mathrm{T}_{\mathrm{P}}$ and $\mathrm{T}_{\mathrm{Q}}$ are the temperatures in two different states $P$ and $Q$ respectively. Then$T_{P}=T_{Q}$$T_{P}T_{Q}$$T_{P}=0.9 T_{Q}$$T_{P}T_{Q}$Correct Option: , 4 Solution: $\mathrm{C}_{\mathrm{P}}-\mathrm{C}_{\mathrm{V}}=\mathrm{R}$ for ideal gas and gas beha...
Read More →Show that A B=B A in each of the following cases:
Question: Show that $A B=B A$ in each of the following cases: $A=\left[\begin{array}{lll}1 2 1 \\ 3 4 2 \\ 1 3 2\end{array}\right]$ and $B=\left[\begin{array}{ccc}10 -4 -1 \\ -11 5 0 \\ 9 -5 1\end{array}\right]$ Solution: Given : $\mathrm{A}=\left[\begin{array}{lll}1 2 1 \\ 3 4 2 \\ 1 3 2\end{array}\right]$ and $\mathrm{B}=\left[\begin{array}{ccc}10 -4 -1 \\ -11 5 0 \\ 9 -5 1\end{array}\right]$ Matrix A is of order $3 \times 3$ and Matrix $B$ is of order $3 \times 3$ To show : matrix $\mathrm{AB...
Read More →Evaluate the following integrals:
Question: Evaluate $\int \frac{1}{\sin ^{4} x+\cos ^{4} x} d x$ Solution: Consider $\int \frac{1}{\sin ^{4} x+\cos ^{4} x} d x$, Divide num and denominator by $\cos ^{4} x$ to get, $\int \frac{1}{\sin ^{4} x+\cos ^{4} x} d x=\int \frac{\frac{1}{\cos ^{4} x}}{\frac{\sin ^{4} x}{\cos ^{4} x}+\frac{\cos ^{4} x}{\cos ^{4} x}} d x$ $=\int \frac{\sec ^{4} x}{\tan ^{4} x+1} d x$ $=\int \frac{\sec ^{2} x \cdot \sec ^{2} x}{\tan ^{4} x+1} d x$ $=\int \frac{\sec ^{2} x\left(1+\tan ^{2} x\right)}{\tan ^{4}...
Read More →A radioactive material decays by simultaneous emissions
Question: A radioactive material decays by simultaneous emissions of two particles with half lives of 1400 years and 700 years respectively. What will be the time after the which one third of the material remains ? (Take $\ln 3=1.1$ )1110 years700 years340 years740 yearsCorrect Option: , 4 Solution: Given $\lambda_{1}=\frac{\ell \mathrm{n} 2}{700} /$ year,$\lambda_{2}=\frac{\ell \mathrm{n} 2}{1400} /$ year $\therefore \lambda_{\text {net }}=\lambda_{1}+\lambda_{2}=\ell \operatorname{n} 2\left[\f...
Read More →Show that A B=B A in each of the following cases:
Question: Show that $A B=B A$ in each of the following cases: $A=\left[\begin{array}{cc}\cos \theta -\sin \theta \\ \sin \theta \cos \theta\end{array}\right]$ and $B=\left[\begin{array}{cc}\cos \phi -\sin \phi \\ \sin \phi \cos \phi\end{array}\right]$ Solution: Given : $A=\left[\begin{array}{cc}\cos \theta -\sin \theta \\ \sin \theta \cos \theta\end{array}\right]$ and $B=\left[\begin{array}{cc}\cos \phi -\sin \phi \\ \sin \phi \cos \phi\end{array}\right]$ Matrix $A$ is of order $2 \times 2$ and ...
Read More →Evaluate the following integrals:
Question: Evaluate $\int \frac{1}{\sin x+\sin 2 x} d x$ Solution: To solve this type of solution, we are going to substitute the value of $\sin x$ and $\cos x$ in terms of $\tan (x / 2)$ $\sin x=\frac{2\left[\tan \frac{x}{2}\right]}{1+\tan ^{2} \frac{x}{2}}$ $\cos x=\frac{\left(1-\frac{\tan ^{2} x}{2}\right)}{1+\frac{\tan ^{2} x}{2}}$ $I=\int \frac{1}{\frac{2 \tan x / 2}{1+\tan ^{2} \frac{x}{2}}\left(1+2 \cdot \frac{\left.1-\tan ^{2} \frac{x}{2}\right)}{1+\tan \frac{2}{2}}\right)} d x$ $I=\int \...
Read More →Evaluate the following integrals:To solve this type of solution, we are going to substitute the value of $sin x$ and $cos x$ in terms of $ an (x / 2)$
Question: Evaluate $\int \frac{1}{\sin x+\sin 2 x} d x$ Solution: To solve this type of solution, we are going to substitute the value of $\sin x$ and $\cos x$ in terms of $\tan (x / 2)$ $\sin x=\frac{2\left[\tan \frac{x}{2}\right]}{1+\tan ^{2} \frac{x}{2}}$ $\cos x=\frac{\left(1-\frac{\tan ^{2} x}{2}\right)}{1+\frac{\tan ^{2} x}{2}}$ $I=\int \frac{1}{\frac{2 \tan x / 2}{1+\tan ^{2} \frac{x}{2}}\left(1+2 \cdot \frac{\left.1-\tan ^{2} \frac{x}{2}\right)}{1+\tan \frac{2}{2}}\right)} d x$ $I=\int \...
Read More →The normal reaction 'N' for a vehicle of
Question: The normal reaction ' $\mathrm{N}$ ' for a vehicle of $800 \mathrm{~kg}$ mass, negotiating a turn on a $30^{\circ}$ banked road at maximum possible speed without skidding is__________ $\times 10^{3} \mathrm{~kg} \mathrm{~m} / \mathrm{s}^{2}$.$10.2$$7.2$$12.4$$6.96$Correct Option: 1 Solution: At $\mathrm{v}_{\max }, \mathrm{f}$ will be limiting in nature. $\therefore$ Balancing force in vertical direction, $N \cos 30^{\circ}-\mathrm{mg}-\mu \mathrm{N} \cos 60^{\circ}=0$ $\Rightarrow \ma...
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