Let f
Question: Let $\boldsymbol{f}: \mathrm{R} \rightarrow \mathrm{R}$ be defined as $\boldsymbol{f}(\mathrm{x})=2 \mathrm{x}-1$ and $\mathrm{g}: \mathrm{R}-\{1\} \rightarrow \mathrm{R}$ be defined as $\mathrm{g}(\mathrm{x})=\frac{\mathrm{x}-\frac{1}{2}}{\mathrm{x}-1}$. Then the composition function $f(g(x))$ is :(1) both one-one and onto(2) onto but not one-one(3) neither one-one nor onto(4) one-one but not ontoCorrect Option: , 4 Solution: $f(g(x))=2 g(x)-1$ $=2 \frac{\left(\frac{x-\frac{1}{2}}{2}\...
Read More →If α, β, γ are the zeros of the polynomial
Question: If $\alpha, \beta, \gamma$ are the zeros of the polynomial $2 x^{3}+x^{2}-13 x+6$, then $\alpha \beta y=?$ (a) 3(b) 3 (c) $\frac{-1}{2}$ (d) $\frac{-13}{2}$ Solution: (a) $-3$ Since $\alpha, \beta$ and $\gamma$ are the zeroes of $2 x^{3}+x^{2}-13 x+6$, we have: $\alpha \beta \gamma=\frac{-(\text { constant term })}{\text { co-efficient of } x^{3}}=\frac{-6}{2}=-3$...
Read More →If α, β, γ are the zeros of the polynomial
Question: If $\alpha, \beta, \gamma$ are the zeros of the polynomial $x^{3}-6 x^{2}-x+30$, then $(\alpha \beta+\beta \gamma+\gamma \alpha)=?$ (a) 1(b) 1(c) 5(d) 30 Solution: (a) $-1$ It is given that $\alpha, \beta$ and $\gamma$ are the zeroes of $x^{3}-6 x^{2}-x+30$. $\therefore(\alpha \beta+\beta \gamma+\gamma \alpha)=\frac{\text { co-efficient of } x}{\text { co-efficient of } x^{3}}=\frac{-1}{1}=-1$...
Read More →If α, β are the zeros of the polynomial x2 + 6x + 2, then
Question: If $\alpha, \beta$ are the zeros of the polynomial $x^{2}+6 x+2$, then $\left(\frac{1}{\alpha}+\frac{1}{\beta}\right)=?$ (a) 3(b) 3(c) 12(d) 12 Solution: (b) $-3$ Since $\alpha$ and $\beta$ are the zeroes of $x^{2}+6 x+2$, we have: $\alpha+\beta=-6$ and $\alpha \beta=2$ $\therefore\left(\frac{1}{\alpha}+\frac{1}{\beta}\right)=\left(\frac{\alpha+\beta}{\alpha \beta}\right)=\frac{-6}{2}=-3$...
Read More →An infinitely long straight wire carrying current I,
Question: An infinitely long straight wire carrying current $I$, one side opened rectangular loop and a conductor $C$ with a sliding connector are located in the same plane, as shown in the figure. The connector has length $l$ and resistance $R$. It slides to the right with a velocity $v$. The resistance of the conductor and the self inductance of the loop are negligible. The induced current in the loop, as a function of separation $r$, between the connector and the straight wire is : (1) $\frac...
Read More →If the sum of the zeros of the quadratic polynomial
Question: If the sum of the zeros of the quadratic polynomial $k x^{2}+2 x+3 k$ is equal to the product of its zeros, then $k=?$ (a) $\frac{1}{3}$ (b) $\frac{-1}{3}$ (c) $\frac{2}{3}$ (d) $\frac{-2}{3}$ Solution: (d) $\frac{-2}{3}$ Let $\alpha$ and $\beta$ be the zeroes of $k x^{2}+2 x+3 k$. Then $\alpha+\beta=\frac{-2}{k}$ and $\alpha \beta=\frac{3 k}{k}=3$ $=\alpha+\beta=\alpha \beta$ $=\frac{-2}{k}=3$ $=k=\frac{-2}{3}$...
Read More →Let f
Question: Let $f: \mathrm{R}-\{3\} \rightarrow \mathrm{R}-\{1\}$ be defined by $f(\mathrm{x})=\frac{\mathrm{x}-2}{\mathrm{x}-3}$. Let $\mathrm{g}: \mathrm{R} \rightarrow \mathrm{R}$ be given as $g(x)=2 x-3$. Then, the sum of all the values of $x$ for which $f^{-1}(x)+g-1(x)=\frac{13}{2}$ is equal to(1) 7(2) 2(3) 5(4) 3Correct Option: 3, Solution: $f(x)=y=\frac{x-2}{x-3}$ $\therefore x=\frac{3 y-2}{y-1}$ $\therefore f^{-1}(x)=\frac{3 x-2}{x-1}$ $\backslash \ g(x)=y=2 x-3$ $\therefore x=\frac{y+3}...
Read More →If one zero of 3x2 + 8x + k be the reciprocal
Question: If one zero of $3 x^{2}+8 x+k$ be the reciprocal of the other, then $k=?$ (a) 3(b) 3 (c) $\frac{1}{3}$ (d) $\frac{-1}{3}$ Solution: (a) $k=3$ Let $\alpha$ and $\frac{1}{\alpha}$ be the zeroes of $3 x^{2}-8 x+k$. Then product of zeroes $=\frac{k}{3}$ $=\alpha \times \frac{1}{\alpha}=\frac{k}{3}$ $=1=\frac{k}{3}$ $=k=3$...
Read More →If the functions are defined as
Question: If the functions are defined as $f(x)=\sqrt{x}$ and $\mathrm{g}(\mathrm{x})=\sqrt{1-\mathrm{x}}$, then what is the common domain of the following functions: $f+g, f-g, f / g, g / f, g-f$ where $(f \pm g)(x)=f(\mathrm{x}) \pm \mathrm{g}(\mathrm{x}),(f / \mathrm{g})(\mathrm{x})=\frac{f(\mathrm{x})}{\mathrm{g}(\mathrm{x})}$(1) $0 \leq x \leq 1$(2) $0 \leq x1$(3) $\$ 0$(4) $\$ 0$Correct Option: , 3 Solution: $f(x)+g(x)=\sqrt{x}+\sqrt{1-x}$, domain $[0,1]$ $f(x)-g(x)=\sqrt{x}-\sqrt{1-x}$, d...
Read More →Two concentric circular coils,
Question: Two concentric circular coils, $\mathrm{C}_{1}$ and $\mathrm{C}_{2}$, are placed in the XY plane. $C_{1}$ has 500 turns, and a radius of $1 \mathrm{~cm} . C_{2}$ has 200 turns and radius current $20 \mathrm{~cm} . \mathrm{C}_{2}$ carries a time dependent current $\mathrm{I}(t)=\left(5 t^{2}-2 t+3\right)$ A Where $t$ is in $\mathrm{s}$. The emf induced in $\mathrm{C}_{1}$ (in $\mathrm{mV}$ ), at the instant $t=1 \mathrm{~s}$ is $\frac{4}{x}$. The value of $x$ is______ Solution: (5) For ...
Read More →If −2 and 3 are the zeros of the quadratic polynomial
Question: If $-2$ and 3 are the zeros of the quadratic polynomial $x^{2}+(a+1) x+b$, then (a)a= 2,b= 6(b)a= 2,b= 6(c)a= 2,b= 6(d)a= 2,b= 6 Solution: (c) $a=-2, b=-6$ Given: $-2$ and 3 are the zeroes of $x^{2}+(a+1) x+b$. Now, $(-2)^{2}+(a+1) \times(-2)+b=0=4-2 a-2+b=0$ $=b-2 a=-2 \quad \ldots(1)$ Also, $3^{2}+(a+1) \times 3+b=0=9+3 a+3+b=0$ $=b+3 a=-12 \quad \ldots(2)$ On subtracting (1) from (2), we get $a=-2$ $\therefore b=-2-4=-6 \quad[\operatorname{From}(1)]$...
Read More →The real valued function
Question: The real valued function $f(x)=\frac{\operatorname{cosec}^{-1} x}{\sqrt{x-[x]}}$, where $[\mathrm{x}]$ denotes the greatest integer less than or equal to $x$, is defined for all $x$ belonging to:(1) all reals except integers(2) all non-integers except the interval $[-1,1]$(3) all integers except $0,-1,1$(4) all reals except the Interval $[-1,1]$Correct Option: , 2 Solution: $f(x)=\frac{\operatorname{cosec}^{-1} x}{\sqrt{\{x\}}}$ Domain $\in(-\infty,-1] \cup[1, \infty)$ $\{x\} \neq 0$ s...
Read More →The inverse of
Question: The inverse of $y=5^{\log x \text { is : }}$(1) $x=(1 / y)^{\log 5}$(2) $x=y^{\frac{1}{\log 5}}$(3) $x=5^{\log y}$(4) $x=5^{\frac{1}{\ln y}}$Correct Option: , 3 Solution: $y=5^{\log x}$ $y=x^{\log 5}$ $y^{\frac{1}{\log x}}=x$ Replying $x \rightarrow y$ and $y \rightarrow x$...
Read More →A uniform magnetic field B exists in a direction perpendicular to the plane of a square loop made of a metal wire.
Question: A uniform magnetic field $B$ exists in a direction perpendicular to the plane of a square loop made of a metal wire. The wire has a diameter of $4 \mathrm{~mm}$ and a total length of $30 \mathrm{~cm}$. The magnetic field changes with time at a steady rate $d B / d t=$ $0.032 \mathrm{Ts}^{-1}$. The induced current in the loop is close to (Resistivity of the metal wire is $1.23 \times 10^{-8} \Omega \mathrm{m}$ )(1) $0.43 \mathrm{~A}$(2) $0.61 \mathrm{~A}$(3) $0.34 \mathrm{~A}$(4) $0.53 ...
Read More →If one zero of the quadratic polynomial
Question: If one zero of the quadratic polynomial $(k-1) x^{2}+k x+1$ is $-4$, then the value of $k$ is (a) $\frac{-5}{4}$ (b) $\frac{5}{4}$ (c) $\frac{-4}{3}$ (d) $\frac{4}{3}$ Solution: (b) $\frac{5}{4}$ Since $-4$ is a zero of $(k-1) x^{2}+k x+1$, we have: $(k-1) \times(-4)^{2}+k \times(-4)+1=0$ $=16 k-16-4 k+1=0$ $=12 k-15=0$ $=k=\frac{15^{5}}{12^{4}}$ $=k=\frac{5}{4}$...
Read More →The number of roots of the equation,
Question: The number of roots of the equation, $(81)^{\sin ^{2} x}+(81)^{\cos ^{2} x}=30$ in the interval $[0, \pi]$ is equal to:(1) 3(2) 4(3) 8(4) 2Correct Option: , 2 Solution: $(81)^{\sin ^{2} x}+(81)^{\cos ^{2} x}=30$ $(81)^{\sin ^{2} x}+\frac{(81)^{1}}{(18)^{\sin ^{2} x}}=30$ $(81)^{\sin ^{2} x}=t$ $\mathrm{t}+\frac{81}{t}=30$ $(t-3)(t-27)=0$ $(81)^{\sin ^{2} x}=3^{1} \quad$ or $\quad(81)^{\sin ^{2} x}=3^{3}$ $3^{4 \sin ^{2} x}=3^{1} \quad$ or $\quad 3^{4 \sin ^{2} x}=3^{3}$ $\sin ^{2} x=\f...
Read More →If one zero of the quadratic polynomial
Question: If one zero of the quadratic polynomial $k x^{2}+3 x+k$ is 2 , then the value of $k$ is (a) $\frac{5}{6}$ (b) $\frac{-5}{6}$ (c) $\frac{6}{5}$ (d) $\frac{-6}{5}$ Solution: (d) $\frac{-6}{5}$ Since 2 is a zero of $k x^{2}+3 x+k$, we have: $k \times(2)^{2}+3 \times 2+k=0$ $=4 k+k+6=0$ $=5 k=-6$ $=k=\frac{-6}{5}$...
Read More →A circular coil of radius 10 cm is placed in a uniform magnetic field
Question: A circular coil of radius $10 \mathrm{~cm}$ is placed in a uniform magnetic field of $3.0 \times 10^{-5} \mathrm{~T}$ with its plane perpendicular to the field initially. It is rotated at constant angular speed about an axis along the diameter of coil and perpendicular to magnetic field so that it undergoes half of rotation in $0.2 \mathrm{~s}$. The maximum value of EMF induced (in $\mu \mathrm{V}$ ) in the coil will be close to the integer_______ Solution: (15) Here, $B=3.0 \times 10^...
Read More →If α and β are the zero of
Question: If $\alpha$ and $\beta$ are the zero of $2 x^{2}+5 x-8$, then the value of $(\alpha \beta)$ is (a) $\frac{-5}{2}$ (b) $\frac{5}{2}$ (c) $\frac{-9}{2}$ (d) $\frac{9}{2}$ Solution: (c) $\frac{-9}{2}$ Given: $\alpha$ and $\beta$ are the zeroes of $2 x^{2}+5 x-9$ If $\alpha$ and $\beta$ are the zeroes, then $x^{2}-(\alpha+\beta) x+\alpha \beta$ is the required polynomial. The polynomial will be $x^{2}-\frac{5}{2} x-\frac{9}{2}$. $\therefore \alpha \beta=\frac{-9}{2}$...
Read More →If α and β are the zero of x2 + 5x + 8, then the value of (α + β) is
Question: If and are the zero ofx2+ 5x+ 8, then the value of ( + ) is (a) 5(b) 5(c) 8(d) 8 Solution: (b) $-5$ Given: $\alpha$ and $\beta$ are the zeroes of $x^{2}+5 x+8$. If $\alpha+\beta$ is the sum of the roots and $\alpha \beta$ is the product, then the required polynimial will be $x^{2}-(\alpha+\beta)+\alpha \beta$. $\therefore \alpha+\beta=-5$...
Read More →An aeroplane, with its wings spread 10 m,
Question: An aeroplane, with its wings spread $10 \mathrm{~m}$, is flying at a speed of $180 \mathrm{~km} / \mathrm{h}$ in a horizontal direction. The total intensity of earth's field at that part is $2.5 \times 10^{-4} \mathrm{~Wb} / \mathrm{m}^{2}$ and the angle of dip is $60^{\circ}$. The emf induced between the tips of the plane wings will be(1) $88.37 \mathrm{mV}$(2) $62.50 \mathrm{mV}$(3) $54.125 \mathrm{mV}$(4) $108.25 \mathrm{mV}$Correct Option: , 4 Solution: $\sum=B \perp v \xi$ $\sin 6...
Read More →An aeroplane, with its wings spread 10 m,
Question: An aeroplane, with its wings spread $10 \mathrm{~m}$, is flying at a speed of $180 \mathrm{~km} / \mathrm{h}$ in a horizontal direction. The total intensity of earth's field at that part is $2.5 \times 10^{-4} \mathrm{~Wb} / \mathrm{m}^{2}$ and the angle of dip is $60^{\circ}$. The emf induced between the tips of the plane wings will be(1) $88.37 \mathrm{mV}$(2) $62.50 \mathrm{mV}$(3) $54.125 \mathrm{mV}$(4) $108.25 \mathrm{mV}$Correct Option: , 4 Solution: $\sum=B \perp v \xi$ $\sin 6...
Read More →The zeros of the quadratic polynomial
Question: The zeros of the quadratic polynomial $x^{2}+88 x+125$ are (a) both positive(b) both negative(c) one positive and one negative(d) both equal Solution: (b) both negative Let $\alpha$ and $\beta$ be the zeroes of $x^{2}+88 x+125$. Then $\alpha+\beta=-88$ and $\alpha \times \beta=125$ This can only happen when both the zeroes are negative....
Read More →A quadratic polynomial whose zeros are
Question: A quadratic polynomial whose zeros are $\frac{3}{5}$ and $\frac{-1}{2}$, is (a) 10x2+x+ 3(b) 10x2+x 3(c) 10x2x+ 3(d) 10x2x 3 Solution: (d) $10 x^{2}-x-3$ Here, the zeroes are $\frac{3}{5}$ and $\frac{-1}{2}$. Let $\alpha=\frac{3}{5}$ and $\beta=\frac{-1}{2}$ So, sum of the zeroes, $\alpha+\beta=\frac{3}{5}+\left(\frac{-1}{2}\right)=\frac{1}{10}$ Also, product of the zeroes, $\alpha \beta=\frac{3}{5} \times\left(\frac{-1}{2}\right)=\frac{-3}{10}$ The polynomial will be $x^{2}-(\alpha+\b...
Read More →A coil of inductance 2 H having negligible resistance
Question: A coil of inductance 2 H having negligible resistance is connected to a source of supply whose voltage is given by $\mathrm{V}=3 \mathrm{t}$ volt. (where $\mathrm{t}$ is in second). If the voltage is applied when $\mathrm{t}=0$, then the energy stored in the coil after $4 \mathrm{~s}$ is J. Solution: (144) $L \frac{d i}{d t}=\varepsilon$ $=3 t$ $L \int d \mathrm{i}=3 \int \mathrm{td} \mathrm{t}$ $\mathrm{Li}=\frac{3 t^{2}}{2}$ $i=\frac{3 t^{2}}{2 L}$ energy, $\mathrm{E}=\frac{1}{2} \ma...
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