An electromagnetic wave of frequency
Question: An electromagnetic wave of frequency $5 \mathrm{GHz}$, is travelling in a medium whose relative electric permittivity and relative magnetic permeability both are 2 . Its velocity in this medium is _______ $\times 10^{7} \mathrm{~m} / \mathrm{s}$ Solution: (15) Given : $\mathrm{f}=5 \mathrm{GHz}$ $\varepsilon_{\mathrm{r}}=2$ $\mu_{\mathrm{r}}=2$ velocity of wave $\Rightarrow v=\frac{c}{n} \ldots(1)$ where, $\mathrm{n}=\sqrt{\mu_{\mathrm{r}} \varepsilon_{\mathrm{r}}}$ and $\mathrm{c}=$ s...
Read More →for x
Question: For $x \in\left(0, \frac{3}{2}\right)$, let $f(x)=\sqrt{x}, \mathrm{~g}(x)=\tan x$ and $h(x)=\frac{1-x^{2}}{1+x^{2}}$. If $\phi(x)=((h o f) \log )(x)$, then $\phi\left(\frac{\pi}{3}\right)$ is equal to: (1) $\tan \frac{\pi}{12}$(2) $\tan \frac{11 \pi}{12}$(3) $\tan \frac{7 \pi}{12}$(4) $\tan \frac{5 \pi}{12}$Correct Option: , 2 Solution: $\because \quad \phi(x)=(($ hof $) \circ g)(x)$ $\because \phi\left(\frac{\pi}{3}\right)=h\left(f\left(g\left(\frac{\pi}{3}\right)\right)\right)=h(f(\...
Read More →Solve this
Question: $2 x+3 y=2$ $x-2 y=8$ Solution: On a graph paper, draw a horizontal lineX'OXand a vertical lineYOY'representing thex-axis andy-axis, respectively. 2x+ 3y= 2⇒ 3y= (2 2x)⇒ 3y= 2(1 x) $\Rightarrow y=\frac{2(1-x)}{3}$ .......(i) Puttingx= 1, we gety= 0Puttingx= 2, we gety= 2Puttingx= 4, we gety= 2Thus, we have the following table for the equation 2x+ 3y= 2. Now, plot the pointsA(1, 0),B( 2 , 2) andC(4, 2) on the graph paper.JoinABandACto get the graph lineBC. Extend it on both ways.Thus, t...
Read More →A plane electromagnetic wave propagating along y-direction can have the
Question: A plane electromagnetic wave propagating along y-direction can have the following pair of electric field $(\vec{E})$ and magnetic field $(\vec{B})$ components.(1) $\mathrm{E}_{\mathrm{y}}, \mathrm{B}_{\mathrm{y}}$ or $\mathrm{E}_{\mathrm{z}}, \mathrm{B}_{\mathrm{z}}$(2) $\mathrm{E}_{\mathrm{y}}, \mathrm{B}_{\mathrm{x}}$ or $\mathrm{E}_{\mathrm{x}}, \mathrm{B}_{\mathrm{y}}$(3) $\mathrm{E}_{\mathrm{x}}, \mathrm{B}_{\mathrm{z}}$ or $\mathrm{E}_{\mathrm{z}}, \mathrm{B}_{\mathrm{x}}$(4) $\m...
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Question: For $x \in \mathrm{R}$, let $[x]$ denote the greatest integer $\leq x$, then the sum of the series $\left[-\frac{1}{3}\right]+\left[-\frac{1}{3}-\frac{1}{100}\right]+\left[-\frac{1}{3}-\frac{2}{100}\right]+\cdots+\left[-\frac{1}{3}-\frac{99}{100}\right]_{\mathrm{i}}$(1) $-153$(2) $-133$(3) $-131$(4) $-135$Correct Option: 2, Solution: $\because[x]+\left[x+\frac{1}{n}\right]+\left[x+\frac{2}{n}\right] \ldots\left[x+\frac{n-1}{n}\right]=[n x]$ and $[x]+[-x]=-1(x \notin z)$ $\therefore\lef...
Read More →Let f(x)=
Question: Let $f(x)=\log _{\mathrm{e}}(\sin x),(0x\pi)$ and $g(x)=\sin ^{-1}\left(e^{-x}\right)$, $(x \geq 0)$. If $\alpha$ is a positive real number such that $a=(f \circ g)^{\prime}(\alpha)$ and $b=(f \circ g)(\alpha)$, then:(1) $a \alpha^{2}+b \alpha+a=0$(2) $a \alpha^{2}-b \alpha-a=1$(3) $a \alpha^{2}-b \alpha-\mathrm{a}=0$(4) $a \alpha^{2}+b \alpha-a=-2 a^{2}$Correct Option: , 2 Solution: $f(x)=\ln (\sin x), g(x)=\sin ^{-1}\left(e^{-x}\right)$ $\Rightarrow f(g(x))=\ln \left(\sin \left(\sin ...
Read More →The electric field intensity produced by the radiation coming from
Question: The electric field intensity produced by the radiation coming from a $100 \mathrm{~W}$ bulb at a distance of $3 \mathrm{~m}$ is $\mathrm{E}$. The electric field intensity produced by the radiation coming from $60 \mathrm{~W}$ at the same distance is $\sqrt{\frac{\mathrm{x}}{5} \mathrm{E} \text {. Where the value }}$ of $x=$ ___________ Solution: (3) $c \in_{0} E^{2}=\frac{100}{4 \pi \times 3^{2}}$ $c \in_{0}\left(\sqrt{\frac{x}{5}} E\right)^{2}=\frac{60}{4 \pi \times 3^{2}}$ $\Rightarr...
Read More →which one of the following statements is not true?
Question: Let $f(x)=x^{2}, x \in \mathrm{R} .$ For any $\mathrm{A} \subseteq \mathrm{R}$, define $\mathrm{g}(\mathrm{A})=$ $\{x \in \mathrm{R}: f(x) \in \mathrm{A}\}$. If $\mathrm{S}=[0,4]$, then which one of the following statements is not true? (1) $g(f(S)) \neq S$(2) $\mathrm{f}(\mathrm{g}(\mathrm{S}))=\mathrm{S}$(3) $\mathrm{g}(\mathrm{f}(\mathrm{S}))=\mathrm{g}(\mathrm{S})$(d) $\mathrm{f}(\mathrm{g}(\mathrm{S})) \neq \mathrm{f}(\mathrm{S})$Correct Option: , 3 Solution: $f(x)=x^{2} ; x \in \...
Read More →Red light differs from blue light as they have :
Question: Red light differs from blue light as they have :(1) Different frequencies and different wavelengths(2) Different frequencies and same wavelengths(3) Same frequencies and same wavelengths(4) Same frequencies and different wavelengthsCorrect Option: 1 Solution: (1) Red light and blue light have different wavelength and different frequency....
Read More →The domain of the definition
Question: The domain of the definition of the function $f(x)=\frac{1}{4-x^{2}}+\log _{10}\left(x^{3}-x\right)$ is: (1) $(-1,0) \cup(1,2) \cup(3, \infty)$(2) $(-2,-1) \cup(-1,0) \cup(2, \infty)$(3) $(-1,0) \cup(1,2) \cup(2, \infty)$(4) $(1,2) \cup(2, \infty)$Correct Option: , 3 Solution: To determine domain, denominator $\neq 0$ and $x^{3}-x0$ i.e., $4-x^{2} \neq 0 x \neq \pm 2$ .......(1) and $x(x-1)(x+1)0$ $x \in(-1,0) \cup(1, \infty)$ .......(2) Hence domain is intersection of $(1) \(2)$. i.e....
Read More →For an electromagnetic wave travelling in free space,
Question: For an electromagnetic wave travelling in free space, the relation between average energy densities due to electric $\left(\mathrm{U}_{\mathrm{e}}\right)$ and magnetic $\left(\mathrm{U}_{\mathrm{m}}\right)$ fields is :(1) $U_{e}=U_{m}$(2) $U_{e}U_{m}$(3) $\mathrm{U}_{\mathrm{e}}\mathrm{U}_{\mathrm{m}}$(4) $\mathrm{U}_{\mathrm{e}} \neq \mathrm{U}_{\mathrm{m}}$Correct Option: 1 Solution: (1) In EMW, Average energy density due to electric $\left(\mathrm{U}_{\mathrm{e}}\right)$ and magneti...
Read More →A plane electromagnetic wave of frequency
Question: A plane electromagnetic wave of frequency $500 \mathrm{MHz}$ is travelling in vacuum along y-direction. At a particular point in space and time, $\overrightarrow{\mathrm{B}}=8.0 \times 10^{-8} \hat{\mathrm{z}} \mathrm{T}$. The value of electric field at this point is: (speed of light $=3 \times 10^{8} \mathrm{~ms}^{-1}$ ) $\hat{\mathrm{x}}, \hat{\mathrm{y}}, \hat{\mathrm{z}}$ are unit vectors along $\mathrm{x}, \mathrm{y}$ and $\mathrm{Z}$ direction.(1) $-24 \hat{\mathrm{x}} \mathrm{V}...
Read More →A plane electromagnetic wave of frequency
Question: A plane electromagnetic wave of frequency $500 \mathrm{MHz}$ is travelling in vacuum along y-direction. At a particular point in space and time, $\overrightarrow{\mathrm{B}}=8.0 \times 10^{-8} \hat{\mathrm{z}} \mathrm{T}$. The value of electric field at this point is: (speed of light $=3 \times 10^{8} \mathrm{~ms}^{-1}$ ) $\hat{\mathrm{x}}, \hat{\mathrm{y}}, \hat{\mathrm{z}}$ are unit vectors along $\mathrm{x}, \mathrm{y}$ and $\mathrm{Z}$ direction.(1) $-24 \hat{\mathrm{x}} \mathrm{V}...
Read More →Let
Question: Let $\sum_{k=1}^{10} f(a+k)=16\left(2^{10}-1\right)$, where the function $f$ satisfies $f(x+y)=f(x) f(y)$ for all natural numbers $x, y$ and $f(1)=2$. Then the natural number ' $\mathrm{a}$ ' is:(1) 2(2) 16(3) 4(4) 3Correct Option: , 4 Solution: $\because f(x+y)=f(x) \cdot f(y)$ $\Rightarrow$ Let $f(x)=t^{x}$ $\because \mathrm{f}(1)=2$ $\therefore t=2$ $\Rightarrow \mathrm{f}(\mathrm{x})=2^{\mathrm{x}}$ Since, $\sum_{k=1}^{10} f(a+k)=16\left(2^{10}-1\right)$ Then, $\sum_{k=1}^{10} 2^{a...
Read More →Use remainder theorem to find the value of k, it being given that when
Question: Use remainder theorem to find the value of $k$, it being given that when $x^{3}+2 x^{2}+k x+3$ is divided by $(x-3)$, then the remainder is 21 . Solution: Let $p(x)=x^{3}+2 x^{2}+k x+3$ Now, $p(3)=(3)^{3}+2(3)^{2}+3 k+3$ $=27+18+3 k+3$ $=48+3 k$ It is given that the remainder is 21 $\therefore 3 k+48=21$ $\Rightarrow 3 k=-27$ $\Rightarrow k=-9$...
Read More →If the function
Question: If the function $\mathrm{f}: \mathrm{R}-\{1,-1\} \rightarrow \mathrm{A}$ defined by $\mathrm{f}(x)=\frac{x^{2}}{1-x^{2}}$, is suriective, then $A$ is equal to:(1) $\mathrm{R}-\{-1\}$(2) $[0, \infty)$(3) $\mathrm{R}-[-1,0)$(4) $\mathrm{R}-(-1,0)$Correct Option: , 3 Solution: $f(x)=\frac{x^{2}}{1-x^{2}}$ $\Rightarrow f(-x)=\frac{x^{2}}{1-x^{2}}=f(x)$ $f^{\prime}(-x)=\frac{2 x}{\left(1-x^{2}\right)^{2}}$ $\therefore \mathrm{f}(\mathrm{x})$ increases in $\mathrm{x} \in(10, \infty)$ Also $\...
Read More →Find the quotient when p(x)
Question: Find the quotient when $p(x)=3 x^{4}+5 x^{3}-7 x^{2}+2 x+2$ is divided by $\left(x^{2}+3 x+1\right)$. Solution: Given : $p(x)=3 x^{4}+5 x^{3}-7 x^{2}+2 x+2$ Dividing $p(x)$ by $\left(x^{2}+3 x+1\right)$, we have : $\therefore$ The quotient is $3 x^{2}-4 x+2$...
Read More →If two zeroes of the polynomial p(x)
Question: If two zeroes of the polynomial $p(x)=2 x^{4}-3 x^{3}-3 x^{2}+6 x-2$ are $\sqrt{2}$ and $-\sqrt{2}$, find its other two zeroes. Solution: Given: $\mathrm{p}(\mathrm{x})=2 x^{4}-3 x^{3}-3 x^{2}+6 x-2$ and the two zeroes, $\sqrt{2}$ and $-\sqrt{2}$ So, the polynomial is $(x+\sqrt{2})(x-\sqrt{2})=x^{2}-2$. Let us divide $p(x)$ by $\left(x^{2}-2\right)$. Here, $2 \mathrm{x}^{4}-3 \mathrm{x}^{3}-3 \mathrm{x}^{2}+6 \mathrm{x}-2=\left(x^{2}-2\right)\left(2 x^{2}-3 x+1\right)$ $=\left(x^{2}-2\...
Read More →A 10 m long horizontal wire extends from North East to South West.
Question: A $10 \mathrm{~m}$ long horizontal wire extends from North East to South West. It is falling with a speed of $5.0 \mathrm{~ms}^{-1}$, at right angles to the horizontal component of the earth's magnetic field, of $0.3 \times 10^{-4} \mathrm{~Wb} / \mathrm{m}^{2}$. The value of the induced emf in wire is :(1) $1.5 \times 10^{-3} \mathrm{~V}$(2) $1.1 \times 10^{-3} \mathrm{~V}$(3) $2.5 \times 10^{-3} \mathrm{~V}$(4) $0.3 \times 10^{-3} \mathrm{~V}$Correct Option: 1 Solution: (1) Induced e...
Read More →let f(x)
Question: Let $f(x)=a^{x}(a0)$ be written as $f(x)=f_{1}(x)+f_{2}(x)$, where $f_{1}(x)$ is an even function and $f_{2}(x)$ is an odd function. Then $f_{1}(x+y)+f_{1}(x-y)$ equals :(1) $2 f_{1}(x) f_{1}(y)$(2) $2 f_{1}(x+y) f_{1}(x-y)$(3) $2 f_{1}(x) f_{2}(y)$(4) $2 f_{1}(x+y) f_{2}(x-y)$Correct Option: 1 Solution: Given function can be written as $f(x)=a^{x}=\left(\frac{a^{x}+a^{-x}}{2}\right)+\left(\frac{a^{x}-a^{-x}}{2}\right)$ where $f_{1}(x)=\frac{a^{x}+a^{-x}}{2}$ is even function $f_{2}(x)...
Read More →If one zero of the polynomial p(x)
Question: If one zero of the polynomial $p(x)=x 3-6 x^{2}+11 x-6$ is 3, find the other two zeroes. Solution: Given: $p(x)=x^{3}-6 x^{2}+11 x-6$ and its factor, $x+3$ Let us divide $p(x)$ by $(x-3)$. Here, $x^{3}-6 x^{2}+11 x-6=(x-3)\left(x^{2}-3 x+2\right)$ $=(x-3)\left[x^{2}-(2+1) x+2\right]$ $=(x-3)\left(x^{2}-2 x-x+2\right)$ $=(x-3)[x(x-2)-1(x-2)]$ $=(x-3)(x-1)(x-2)$ $\therefore$ The other two zeroes are 1 and 2 ....
Read More →A copper wire is wound on a wooden frame,
Question: A copper wire is wound on a wooden frame, whose shape is that of an equilateral triangle. If the linear dimension of each side of the frame is increased by a factor of 3 , keeping the number of turns of the coil per unit length of the frame the same, then the self inductance of the coil:(1) decreases by a factor of 9(2) increases by a factor of 27(3) increases by a factor of 3(4) decreases by a factor of $9 \sqrt{3}$Correct Option: , 3 Solution: (3) As total length $\mathrm{L}$ of the ...
Read More →Show that the polynomial f(x)
Question: Show that the polynomial $f(x)=x^{4}+4 x^{2}+6$ has no zeroes. Solution: Let $t=x^{2}$ So, $f(t)=t^{2}+4 t+6$ Now, to find the zeroes, we will equate $f(t)=0$. $\Rightarrow t^{2}+4 t+6=0$ Now, $t=\frac{-4 \pm \sqrt{16-24}}{2}$ $=\frac{-4 \pm \sqrt{-8}}{2}$ $=-2 \pm \sqrt{-2}$ i. e., $x^{2}=-2 \pm \sqrt{-2}$ $\Rightarrow x=\sqrt{-2 \pm \sqrt{-2}}$, which is not a real number. The zeroes of a polynomial should be real number $s$. $\therefore$ The given $f(x)$ has no zeroes....
Read More →if f(x) = loge
Question: If $f(x)=\log _{e}\left(\frac{1-x}{1+x}\right),|x|1$, then $f\left(\frac{2 x}{1+x^{2}}\right)$ is equal to: (1) $2 f(x)$(2) $2 f\left(x^{2}\right)$(3) $(f(x))^{2}$(4) $-2 f(x)$Correct Option: 1 Solution: $f(x)=\log \left(\frac{1-x}{1+x}\right),|x|1$ $f\left(\frac{2 x}{1+x^{2}}\right)=\log \left(\frac{1-\frac{2 x}{1+x^{2}}}{1+\frac{2 x}{1+x^{2}}}\right)$ $=\log \left(\frac{1+x^{2}-2 x}{1+x^{2}+2 x}\right)$ $=\log \left(\frac{1-x}{1+x}\right)^{2}$ $=2 \log \left(\frac{1-x}{1+x}\right)$ $...
Read More →If α, β are the zeros of the polynomial f(x)
Question: If $\alpha, \beta$ are the zeros of the polynomial $f(x)=x^{2}-5 x+k$ such that $\alpha-\beta=1$, find the value of $k$. Solution: Given: $f(x)=x^{2}-5 x+k$ The co-efficients are $a=1, b=-5$ and $c=k$. $\therefore \alpha+\beta=\frac{-b}{a}$ $=\alpha+\beta=-\frac{(-5)}{1}$ $=\alpha+\beta=5 \quad \ldots(1)$ Also, $\alpha-\beta=1 \quad \ldots(2)$ From (1) (2), we get: $2 \alpha=6$ $=\alpha=3$ Pu tting the value of $\alpha$ in $(1)$, we get $\beta=2$. Now, $\alpha \beta=\frac{c}{a}$ $=3 \t...
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