For a plane electromagnetic wave, the magnetic field at a point $x$ and time $t$ is
Question: For a plane electromagnetic wave, the magnetic field at a point $x$ and time $t$ is $\overrightarrow{\mathrm{B}}(x, t)=\left[1.2 \times 10^{-7} \sin \left(0.5 \times 10^{3} x+1.5 \times 10^{11} t\right) \hat{k}\right] \mathrm{T}$ is: (speed of light $\mathrm{c}=3 \times 10^{8} \mathrm{~ms}^{-1}$ )(1) $\overrightarrow{\mathrm{E}}(x, t)=\left[-36 \sin \left(0.5 \times 10^{3} x+1.5 \times 10^{11} t\right) \hat{j}\right] \frac{\mathrm{V}}{\mathrm{m}}$(2) $\overrightarrow{\mathrm{E}}(x, t)=...
Read More →Suppose that intensity of a laser
Question: Suppose that intensity of a laser is $\left(\frac{315}{\pi}\right) \mathrm{W} / \mathrm{m}^{2}$. The rms electric field, in units of $\mathrm{V} / \mathrm{m}$ associated with this source is close to the nearest integer is________ $\left(\epsilon_{0}=8.86 \times 10^{-12} \mathrm{C}^{2} \mathrm{Nm}^{-2} ; \mathrm{c}=3 \times 10^{8} \mathrm{~ms}^{-1}\right)$ Solution: (270) Using, intensity, $I=\frac{1}{2} C \epsilon_{0} E_{\mathrm{rms}}^{2}$ $\Rightarrow \frac{1}{2} C \in_{0} E_{\mathrm{...
Read More →The correct match between the entries in column I and column II are :
Question: The correct match between the entries in column I and column II are : (1) (A)-(ii), (B)-(i), (C)-(iv), (D)-(iii)(2) (A)-(i), (B)-(iii), (C)-(iv), (D)-(ii)(3) (A)-(iii), (B)-(ii), (C)-(i), (D)-(iv)(4) (A)-(iv), (B)-(ii), (C)-(i), (D)-(iii)Correct Option: , 4 Solution: (4) Energy sequence of radiations is $E_{\gamma \text {-Rays }}E_{\text {X-Rays }}E_{\text {microwave }}E_{\text {AM Radiowaves }}$ $\therefore \lambda_{\gamma \text {-Rays }}\lambda_{X \text {-Rays }}\lambda_{\text {micro...
Read More →An electron is constrained to move along
Question: An electron is constrained to move along the $y$-axis with a speed of $0.1 c$ ( $c$ is the speed of light) in the presence of electromagnetic wave, whose electric field is $\vec{E}=30 \hat{j}$ $\sin \left(1.5 \times 10^{7} t-5 \times 10^{-2} x\right) \mathrm{V} / \mathrm{m}$. The maximum magnetic force experienced by the electron will be : (given $c=3 \times 10^{8} \mathrm{~ms}^{-1} \$ electron charge $\left.=1.6 \times 10^{-19} \mathrm{C}\right)$(1) $3.2 \times 10^{-18} \mathrm{~N}$(2...
Read More →Let f and g be continuous functions on
Question: Let $f$ and $g$ be continuous functions on $[0$, a $]$ such that $f(x)=f(a-x)$ and $g(x)+g(a-x)=4$, then $\int_{0}^{a} f(x) g(x) d x$ is equal to:(1) $\quad 4 \int_{0}^{a} f(x) d x$(2) $\int_{0}^{a} f(x) d x$(3) $2 \int_{0}^{a} f(x) d x$(4) $-3 \int_{0}^{a} f(x) d x$Correct Option: , 3 Solution: $f(x)=f(a-x)$ $g(x)+g(a-x)=4$ Let, the integral, $I=\int_{0}^{a} f(x) g(x) d x$ $=\int_{0}^{a} f(a-x) \cdot g(a-x) d x$ $\left[\because \int_{a}^{b} f(x) d x=\int_{a}^{b} f(a+b-x) d x\right]$ $...
Read More →Considering only the principal values of inverse functions,
Question: Considering only the principal values of inverse functions, the set $A=\left\{x \geq 0: \tan ^{-1}(2 x)+\tan ^{-1}(3 x)=\frac{\pi}{4}\right\}$(1) contains two elements(2) contains more than two elements(3) is a singleton(4) is an empty setCorrect Option: , 3 Solution: Consider, $\tan ^{-1}(2 x)+\tan ^{-1}(3 x)=\frac{\pi}{4}$ $\Rightarrow \quad \tan ^{-1}\left(\frac{5 x}{1-6 x^{2}}\right)=\frac{\pi}{4}$ $\Rightarrow \frac{5 x}{1-6 x^{2}}=1 \Rightarrow 5 x=1-6 x^{2}$ $\Rightarrow \quad 6...
Read More →The electric field of a plane electromagnetic wave is given
Question: The electric field of a plane electromagnetic wave is given by $\vec{E}=E_{0}(\hat{x}+\hat{y}) \sin (k z-\omega t)$ Its magnetic field will be given by:(1) $\frac{E_{0}}{c}(-\hat{x}+\hat{y}) \sin (k z-\omega t)$(2) $\frac{E_{0}}{c}(\hat{x}+\hat{y}) \sin (k z-\omega t)$(3) $\frac{E_{0}}{c}(\hat{x}-\hat{y}) \sin (k z-\omega t)$(4) $\frac{E_{0}}{c}(\hat{x}-\hat{y}) \cos (k z-\omega t)$Correct Option: 1 Solution: (1) $\vec{E}=E_{0}(\hat{x}+\hat{y}) \sin (k z-\omega t)$ Direction of propaga...
Read More →Let f be a differentiable function such that
Question: Let $f$ be a differentiable function such that $f(1)=2$ and $f^{\prime}(x)=f(x)$ for all $x \in R$. If $h(x)=f(f(x))$, then $h^{\prime}(1)$ is equal to :(1) $2 \mathrm{e}^{2}$(2) $4 \mathrm{e}$(3) $2 \mathrm{e}$(4) $4 \mathrm{e}^{2}$Correct Option: , 2 Solution: Since, $\quad f^{\prime}(x)=f(x)$ Then, $\quad \frac{f^{\prime}(x)}{f(x)}=1$ $\Rightarrow \frac{f^{\prime}(x)}{f(x)}=d x \Rightarrow \frac{f^{\prime}(x)}{f(x)} \quad d x=\int d x$ $\Rightarrow \quad \ln |f(x)|=x+c$ $f(x)=\pm e^...
Read More →Chosse the correct option relating wavelengths of different parts of electromagnetic wave spectrum:
Question: Chosse the correct option relating wavelengths of different parts of electromagnetic wave spectrum:(1) $\lambda_{\text {visible }}\lambda_{\text {micro waves }}\lambda_{\text {radio waves }}\lambda_{x \text {-rays }}$(2) $\lambda_{\text {radio waves }}\lambda_{\text {micro waves }}\lambda_{\text {visible }}\lambda_{x \text {-rays }}$(3) $\lambda_{x \text {-rays }}\lambda_{\text {micro waves }}\lambda_{\text {radio waves }}\lambda_{\text {visible }}$(4) $\lambda_{\text {visible }}\lambd...
Read More →The electric field of a plane electromagnetic wave propagating
Question: The electric field of a plane electromagnetic wave propagating along the $x$ direction in vacuum is $\vec{E}=E_{0} \hat{j} \cos (\omega t-k x)$. The magnetic field $\vec{B}$, at the moment $t=0$ is :(1) $\vec{B}=\frac{E_{0}}{\sqrt{\mu_{0} \varepsilon_{0}}} \cos (k x) \hat{k}$(2) $\vec{B}=E_{0} \sqrt{\mu_{0} \varepsilon_{0}} \cos (k x) \hat{j}$(3) $\vec{B}=E_{0} \sqrt{\mu_{0} \varepsilon_{0}} \cos (k x) \hat{k}$(4) $\vec{B}=\frac{E_{0}}{\sqrt{\mu_{0} \varepsilon_{0}}} \cos (k x) \hat{j}...
Read More →Let a function
Question: Let a function $f:(0, \infty) \rightarrow(0, \infty)$ be defined by $f(x)=\left|1-\frac{1}{x}\right|$. Then $f$ is :(1) not injective but it is surjective(2) injective only(3) neither injective nor surjective(4) $f(x)$ is not a function.Correct Option: , 4 Solution: $f:(0, \infty) \rightarrow(0, \infty)$ $f(x)=\left|1-\frac{1}{x}\right|$ is not a function $\because \quad f(1)=0$ and $1 \in$ domain but $0 \notin$ co-domain Hence, $f(x)$ is not a function....
Read More →let f
Question: Let $f: R \rightarrow R$ be defined by $f(x)=\frac{x}{1+x^{2}}, x \in R$. Then the range of $f$ is :(1) $\left[-\frac{1}{2}, \frac{1}{2}\right]$(2) $R-[-1,1]$(3) $R-\left[-\frac{1}{2}, \frac{1}{2}\right]$(4) $(-1,1)-\{0\}$Correct Option: 1 Solution: $f(x)=\frac{x}{1+x^{2}}, x \in R$ Let, $y=\frac{x}{1+x^{2}}$ $\Rightarrow \quad y x^{2}-x+y=0 \quad \Rightarrow \quad x=\frac{1 \pm \sqrt{1-4 y^{2}}}{2}$ $\Rightarrow \quad 1-4 y^{2} \geq 0$ $\Rightarrow \quad 1 \geq 4 y^{2}$ $\Rightarrow \...
Read More →The magnetic field of a plane electromagnetic wave is
Question: The magnetic field of a plane electromagnetic wave is $\vec{B}=3 \times 10^{-8} \sin [200 \pi(y+c t)] \hat{i} T$ where $c=3 \times 10^{8} \mathrm{~ms}^{-1}$ is the speed of light. The corresponding electric field is :(1) $\vec{E}=9 \sin [200 \pi(y+c t)] \hat{k} \mathrm{~V} / \mathrm{m}$(2) $\vec{E}=-10^{-6} \sin [200 \pi(y+c t)] \hat{k} \mathrm{~V} / \mathrm{m}$(3) $\vec{E}=3 \times 10^{-8} \sin [200 \pi(y+c t)] \hat{k} \mathrm{~V} / \mathrm{m}$(4) $\vec{E}=-9 \sin [200 \pi(y+c t)] \ha...
Read More →let fk(x)
Question: Let $f_{k}(x)=\frac{1}{k}\left(\sin ^{k} x+\cos ^{k} x\right)$ for $\mathrm{k}=1,2,3, \ldots$ Then for all $\mathrm{x} \in \mathrm{R}$, the value of $f_{4}(x)-f_{6}(x)$ is equal to : (1) $\frac{1}{12}$(2) $\frac{1}{4}$(3) $\frac{-1}{12}$(4) $\frac{5}{12}$Correct Option: 1 Solution: $f_{k}(x)=\frac{1}{k}\left(\sin ^{k} x+\cos ^{k} x\right)$ $f_{4}(x)=\frac{1}{4}\left[\sin ^{4} x+\cos ^{4} x\right]$ $=\frac{1}{4}\left[\left(\sin ^{2} x+\cos ^{2} x\right)^{2}-\frac{(\sin 2 x)^{2}}{2}\righ...
Read More →In a plane electromagnetic wave,
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.(1) $\frac{1}{\sqrt{2}}(\hat{i}+\hat{j})$(2) $\frac{1}{\sqrt{2}}(\hat{j}+\hat{k})$(3) $\frac{1}{\sqrt{5}}(\hat{i}+2 \hat{j})$(4) $\frac{1}{\sqrt{5}}(2 \hat{i}+\hat{j})$Correct Option: 1 Solution: (1) Electromagnetic wave will propagate perpendicular to the direction o...
Read More →A plane electromagnetic wave, has frequency
Question: A plane electromagnetic wave, has frequency of $2.0 \times 10^{10}$ $\mathrm{Hz}$ and its energy density is $1.02 \times 10^{-8} \mathrm{~J} / \mathrm{m}^{3}$ in vacuum. The amplitude of the magnetic field of the wave is close to $\left(\frac{1}{4 \pi \varepsilon_{0}}=9 \times 10^{9} \frac{\mathrm{Nm}^{2}}{C^{2}}\right.$ and speed of light $\left.=3 \times 10^{8} \mathrm{~ms}^{-1}\right):$(1) $150 \mathrm{nT}$(2) $160 \mathrm{nT}$(3) $180 \mathrm{nT}$(4) $190 \mathrm{nT}$Correct Option...
Read More →A radiation is emitted by
Question: A radiation is emitted by $1000 \mathrm{~W}$ bulb and it generates an electric field and magnetic field at $\mathrm{P}_{t}$ placed at a distance of $2 \mathrm{~m}$. The efficiency of the bulb is $1.25 \%$. The value of peak electric field at $P$ is $x \times 10^{-1} \mathrm{~V} / \mathrm{m}$. Value of $x$ is (Rounded-off to the nearest integer) [Take $\left.\varepsilon_{0}=8.85 \times 10^{-12} \mathrm{C}^{2} \mathrm{~N}^{-1} \mathrm{~m}^{-2}, \mathrm{c}=3 \times 10^{8} \mathrm{~ms}^{-1...
Read More →The peak electric field produced by the radiation coming from
Question: The peak electric field produced by the radiation coming from the $8 \mathrm{~W}$ bulb at a distance of $10 \mathrm{~m}$ is $\frac{x}{10} \sqrt{\frac{\mu_{0} c}{\pi}} \frac{V}{m}$. The efficiency of the bulb is $10 \%$ and it is a point source. The value of $x$ is Solution: (2) $I=\frac{1}{2} c \in_{0} E_{0}^{2}$ $\frac{8}{4 \pi \times 10^{2}}=\frac{1}{2} \times c \times \frac{1}{\mu_{0} c^{2}} \times E_{0}^{2}$ $E_{0}=\frac{2}{10} \sqrt{\frac{\mu_{0} c}{\pi}}$ $\Rightarrow x=2$...
Read More →let N be the set of natural
Question: Let $\mathbf{N}$ be the set of natural numbers and two functions $f$ and $g$ be defined as $f, g: \mathbf{N} \rightarrow \mathbf{N}$ such that $f(\mathrm{n})= \begin{cases}\frac{\mathrm{n}+1}{2} \text { if } \mathrm{n} \text { is odd } \\ \frac{\mathrm{n}}{2} \text { if } \mathrm{n} \text { is even }\end{cases}$ and $g(\mathrm{n})=\mathrm{n}-(-1)^{\mathrm{n}} .$ Then $f o g$ is: (1) onto but not one-one.(2) one-one but not onto.(3) both one-one and onto.(4) neither one-one nor onto.Cor...
Read More →An electromagnetic wave of frequency
Question: An electromagnetic wave of frequency $3 \mathrm{GHz}$ enters a dielectric medium of relative electric permittivity $2.25$ from vacuum. The wavelength of this wave in that medium wil be_______ $\times 10^{-2} \mathrm{~cm}$ Solution: $(667)$ $\mathrm{f}=3 \mathrm{GHz}, \varepsilon_{\mathrm{r}}=2.25$ $\mathrm{V}=\lambda \mathrm{f} \Rightarrow \lambda=\frac{\mathrm{v}}{\mathrm{f}}$ $\mathrm{C}=\frac{1}{\sqrt{\mu_{0} \varepsilon_{0}}}$ $\mathrm{V}=\frac{1}{\sqrt{\mu_{0} \mu_{\mathrm{r}} \va...
Read More →Let A =
Question: Let $\mathrm{A}=\{x \in \mathbf{R}: x$ is not a positive integer $\}$. Define a function f: $\mathbf{A} \rightarrow \mathbf{R}$ as $f(x)=\frac{2 x}{x-1}$, then $f$ is:(1) not injective(2) neither injective nor surjective(3) surjective but not injective(4) injective but not surjectiveCorrect Option: , 4 Solution: As $A=\{x \in R: x$ is not a positive integer $\}$ A function $f: A \rightarrow R$ given by $f(x)=\frac{2 x}{x-1}$ $f\left(x_{1}\right)=f\left(x_{2}\right) \Leftrightarrow x_{1...
Read More →Solve this
Question: $2 x+3 y=8$ $x-2 y+3=0$ Solution: On a graph paper, draw a horizontal lineX'OXand a vertical lineYOY'as thex-axis andy-axis, respectively. Graph of $2 x+3 y=8$ 2x+ 3y= 8⇒ 3y= (8 2x) $\therefore y=\frac{8-2 x}{3}$ .............(i) Puttingx= 1, we gety= 2.Puttingx= 5, we gety= 6.Puttingx= 7, we gety= 2.Thus, we have the following table for the equation 2x+ 3y= 8. Now, plot the pointsA(1, 2),B(5, 6) andC(7, 2) on the graph paper.JoinABandACto get the graph lineBC. Extend it on both ways.T...
Read More →Match List - I with List - II. List - I List - II
Question: Match List - I with List - II. List - I List - II Choose the correct answer from the options given below :(1) $(a)-(i i),(b)-(i v),(c)-(i),(d)-(i i i)$(2) (a) - (vi), (b) - (iv), (c) - (i), (d) - (v)(3) (a) - (ii), (b) - (iv), (c) - (vi), (d) - (iii)(4) $(\mathrm{a})-(\mathrm{vi}),(\mathrm{b})-(\mathrm{v}),(\mathrm{c})-(\mathrm{i}),(\mathrm{d})-(\mathrm{i} v)$Correct Option: 1 Solution: (1) (a) Source of microwave frequency - (ii) Magnetron (b) Source of infra red frequency - (iv) Vibr...
Read More →for x
Question: For $x \in \mathbf{R}-\{0,1\}$, let $f_{1}(x)=\frac{1}{x}, f_{2}(x)=1-x$ and $f_{3}(x)$ $=\frac{1}{1-x}$ be three given functions. If a function, $\mathrm{J}(x)$ satisfies $\left(f_{2} o J o f_{1}\right)(x)=f_{3}(x)$ then $J(x)$ is equal to:(1) $f_{3}(x)$(2) $\frac{1}{x} f_{3}(x)$(3) $f_{2}(x)$(4) $f_{1}(x)$Correct Option: 1 Solution: The given relation is $\left(f_{2} o J o f_{1}\right)(x)=f_{3}(x)=\frac{1}{1-x}$ $\Rightarrow\left(f_{2} o J\right)\left(f_{1}(x)\right)=\frac{1}{1-x}$ $...
Read More →Prove that
Question: $3 x+2 y=12, x-y+1=0$ Solution: Given : $3 x+2 y=12 \quad \ldots(1)$ $x-y+1=0 \quad \ldots(2)$ $x-y+1=0$ $\Rightarrow y=x+1 \quad \ldots(3)$ Substituting the value of $y$ in $(1)$, we get $3 x+2(x+1)=12$ $\Rightarrow 3 x+2 x+2=12$ $\Rightarrow 5 x=12-2$ $\Rightarrow 5 x=10$ $\Rightarrow x=2 \quad \ldots(4)$ Substituting the value of $x$ in $(3)$, we get $y=2+1=3$ Hence, $x=2$ and $y=3$....
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