It is given that the difference between the zeroes of
Question: It is given that the difference between the zeroes of $4 x^{2}-8 k x+9$ is 4 and $k0$. Then, $k=$ ? (a) $\frac{1}{2}$ (b) $\frac{3}{2}$ (c) $\frac{5}{2}$ (d) $\frac{7}{2}$ Solution: (c) $\frac{5}{2}$ Let the zeroes of the polynomial be $\alpha$ and $\alpha+4$. Here, $p(x)=4 x^{2}-8 k x+9$ Comparing the given polynomial with $a x^{2}+b x+c$, we get: $a=4, b=-8 k$ and $c=9$ Now, sum of the roots $=-\frac{b}{a}$ $=\alpha+\alpha+4=\frac{-(-8 k)}{4}$ $=2 \alpha+4=2 k$ $=\alpha+2=k$ $=\alpha...
Read More →The domain of the function
Question: The domain of the function $f(x)=\sin ^{-1}\left(\frac{|x|+5}{x^{2}+1}\right)$ is $(-\infty,-a] \cup[a, \infty] .$ Then $a$ is equal to :(1) $\frac{\sqrt{17}}{2}$(2) $\frac{\sqrt{17}-1}{2}$(3) $\frac{1+\sqrt{17}}{2}$(4) $\frac{\sqrt{17}}{2}+1$Correct Option: , 3 Solution: $\because f(x)=\sin ^{-1}\left(\frac{|x|+5}{x^{2}+1}\right)$ $\therefore-1 \leq \frac{|x|+5}{x^{2}+1} \leq 1$ $\Rightarrow|x|+5 \leq x^{2}+1$$\left[\because x^{2}+1 \neq 0\right]$ $\Rightarrow x^{2}-|x|-4 \geq 0$ $\Ri...
Read More →If α, β are the zeroes of
Question: If $\alpha, \beta$ are the zeroes of $k x^{2}-2 x+3 k$ such that $\alpha+\beta=\alpha \beta$, then $k=?$ (a) $\frac{1}{3}$ (b) $\frac{-1}{3}$ (c) $\frac{2}{3}$ (d) $\frac{-2}{3}$ Solution: (c) $\frac{2}{3}$ Here, $\mathrm{p}(x)=x^{2}-2 x+3 k$ Comparing the given polynomial with $a x^{2}+b x+c$, we get: $a=1, b=-2$ and $c=3 k$ It is given that $\alpha$ and $\beta$ are the roots of the polynomial. $\therefore \alpha+\beta=-\frac{b}{a}$ $=\alpha+\beta=-\left(\frac{-2}{1}\right)$ $=\alpha+...
Read More →The total number of turns and cross-section area in a solenoid is fixed.
Question: The total number of turns and cross-section area in a solenoid is fixed. However, its length $\mathrm{L}$ is varied by adjusting the separation between windings. The inductance of solenoid will be proportional to:(1) $\mathrm{L}$(2) $\mathrm{L}^{2}$(3) $1 / \mathrm{L}^{2}$(4) $1 / \mathrm{L}$Correct Option: , 4 Solution: (4) Inductance $=\frac{\mu_{0} N^{2} A}{L}$...
Read More →A 20 Henry inductor coil is connected to a 10 ohm resistance in series as shown in figure.
Question: A 20 Henry inductor coil is connected to a 10 ohm resistance in series as shown in figure. The time at which rate of dissipation of energy (Joule's heat) across resistance is equal to the rate at which magnetic energy is stored in the inductor, is : (1) $\frac{2}{\ln 2}$(2) $\frac{1}{2} \ln 2$(3) $2 \ln 2$(4) $\ln 2$Correct Option: 3, Solution: (3) $i^{2} R=\left(\tau \frac{d i}{d t}\right) i$ $\Rightarrow \frac{d i}{d t}=\frac{i}{\tau}$ $\Rightarrow \mathrm{t}=\tau \ln 2=2 \ln 2\left[...
Read More →Let f(x)=
Question: Let $f(x)=\sin ^{-1} x$ and $g(x)=\frac{x^{2}-x-2}{2 x^{2}-x-6}$. If $g(2)=\lim _{x \rightarrow 2} g(x)$, then the domain of the function fog is :(1) $(-\infty,-2) \cup\left[-\frac{4}{3}, \infty\right)$(2) $(-\infty,-1] \cup[2, \infty)$(3) $(-\infty,-2] \cup[-1, \infty)$(4) $(-\infty,-2] \cup\left[-\frac{3}{2}, \infty\right)$Correct Option: 1 Solution: $g(2)=\lim _{x \rightarrow 2} \frac{(x-2)(x+1)}{(2 x+3)(x-2)}=\frac{3}{7}$ For domain of fog $(\mathrm{x})\left|\frac{x^{2}-x-2}{2 x^{2...
Read More →If α, β, γ are the zeros of the polynomial
Question: If , , are the zeros of the polynomialx3 6x2x+ 30, then the value of ( + + ) is(a) 1(b) 1(c) 5(d) 30 Solution: (a) 1 Here, $p(x)=x^{3}-6 x^{2}-x+3$ Comparing the given polynomial with $x^{3}-(\alpha+\beta+\gamma) x^{2}+(\alpha \beta+\beta \gamma+\gamma \alpha) x-\alpha \beta \gamma$, we get: $(\alpha \beta+\beta \gamma+\gamma \alpha)=-1$...
Read More →Let A
Question: Let $\mathrm{A}=\{1,2,3 \ldots \ldots, 10\}$ and $\mathrm{f}: \mathrm{A} \rightarrow \mathrm{A}$ be defined as $f(k)=\left\{\begin{array}{cc}k+1 \text { if } k \text { is odd } \\ k \text { if } k \text { is even }\end{array}\right.$ Then the number of possible functions $g: A \rightarrow A$ such that gof $=f$ is : (1) $10^{5}$(2) ${ }^{10} \mathrm{C}_{5}$(3) $5^{5}$(4) $5 !$Correct Option: 1 Solution: $g(f(x))=f(x)$ $\Rightarrow g(x)=x$, when $x$ is even 5 elements in A can be mapped ...
Read More →A thin strip 10 cm long is on a $U$ shaped wire
Question: A thin strip $10 \mathrm{~cm}$ long is on a $U$ shaped wire of negligible resistance and it is connected to a spring of spring constant $0.5 \mathrm{Nm}^{-1}$ (see figure). The assembly is kept in a uniform magnetic field of $0.1 \mathrm{~T}$. If the strip is pulled from its equilibrium position and released, the number of oscillations it performs before its amplitude decreases by a factor of $e$ is $\mathrm{N}$. If the mass of strip is 50 grams, its resistance $10 \Omega$ and air drag...
Read More →Zeros of p(x) = x2 − 2x − 3 are
Question: Zeros of $p(x)=x^{2}-2 x-3$ are (a) 1, 3(b) 3, 1(c) 3, 1(d) 1, 3 Solution: (b) 3,-1 Here, $\mathrm{p}(\mathrm{x})=x^{2}-2 x-3$ Let $x^{2}-2 x-3=0$ $=x^{2}-(3-1) x-3=0$ $=x^{2}-3 x+x-3=0$ $=x(x-3)+1(x-3)=0$ $=(x-3)(x+1)=0$ $=x=3,-1$...
Read More →A function f(x) is given by f(x)
Question: A function $f(x)$ is given by $f(x)=\frac{5^{x}}{5^{x}+5}$, then the sum of the series $f\left(\frac{1}{20}\right)+f\left(\frac{2}{20}\right)+f\left(\frac{3}{20}\right)+\ldots \ldots+f\left(\frac{39}{20}\right)$ is equal to:(1) $\frac{19}{2}$(2) $\frac{49}{2}$(3) $\frac{39}{2}$(4) $\frac{29}{2}$Correct Option: , 3 Solution: $f(x)=\frac{5^{x}}{5^{x}+5} \ldots$ (i) $f(2-x)=\frac{5^{2-x}}{5^{2-x}+5}$ $f(2-x)=\frac{5}{5^{x}+5} \ldots \ldots$ Adding equation (i) and(ii) $f(x)+f(2-x)=1$ $f\l...
Read More →Which of the following is a true statement?
Question: Which of the following is a true statement?(a)x2+ 5x 3 is a linear polynomial.(b)x2+ 4x 1 is a binomial.(c)x+ 1 is a monomial.(d) 5x3is a monomial. Solution: (d) $5 x^{2}$ is a monomial. $5 x^{2}$ consists of one term only. So, it is a monomial....
Read More →On dividing a polynomial p(x) by a non-zero polynomial q(x),
Question: On dividing a polynomialp(x) by a non-zero polynomialq(x), letg(x) be the quotient andr(x) be the remainder, thanp(x) =q(x)g(x) +r(x), where(a)r(x) = 0 always(b) degr(x) degg(x) always(c) eitherr(x) = 0 or degr(x) degg(x)(d)r(x) =g(x) Solution: (c) either $r(x)=0$ or $\operatorname{deg} r(x)\operatorname{deg} g(x)$ By division algorithm on polynomials, either $r(x)=0$ or $\operatorname{deg} r(x)\operatorname{deg} g(x)$....
Read More →If α, β be the zero of the polynomial
Question: If $\alpha, \beta$ be the zero of the polynomial $2 x^{2}+5 x+k$ such that $\alpha^{2}+\beta^{2}+\alpha \beta=\frac{21}{4}$, then $k=?$ (a) 3(b) 3(c) 2(d) 2 Solution: $(\mathrm{d}) 2$ Since $\alpha$ and $\beta$ are the zeroes of $2 \mathrm{x}^{2}+5 x+k$, we have : $\alpha+\beta=\frac{-5}{2}$ and $\alpha \beta=\frac{k}{2}$ Also, it is given that $\alpha^{2}+\beta^{2}+\alpha \beta=\frac{21}{4}$. $=(\alpha+\beta)^{2}-\alpha \beta=\frac{21}{4}$ $=\left(\frac{-5}{2}\right)^{2}-\frac{k}{2}=\...
Read More →Let x denote the total number of one-one functions from
Question: Let $x$ denote the total number of one-one functions from a set $A$ with 3 elements to a set $B$ with 5 elements and $y$ denote the total number of one-one functions from the set $A$ to the set $A \times B$. Then:(1) $y=273 x$(2) $2 y=91 x$(3) $y=91 x$(4) $2 y=273 x$Correct Option: , 2 Solution: Number of elements in $A=3$ Number of elements in $\mathrm{B}=5$ Number of elements in $A \times B=15$ Number of one-one function $x=5 \times 4 \times 3$ $x=60$ Number of one-one function $y=15...
Read More →In a fluorescent lamp choke (a small transformer)
Question: In a fluorescent lamp choke (a small transformer) $100 \mathrm{~V}$ of reverse voltage is produced when the choke current changes uniformly from $0.25$ A to 0 in a duration of $0.025$ ms. The self-inductance of the choke (in $\mathrm{mH}$ ) is estimated to be Solution: (10) Given $d I=0.25-0=0.25 \mathrm{~A}$ $d t=0.025 \mathrm{~ms}$ Induced voltage $E_{\text {ind }}=100 \mathrm{v}$ Self-inductance, $L=$ ? Using, $E_{\text {ind }}=\frac{\Delta \phi}{\Delta t}$ $\Rightarrow 100=\frac{L(...
Read More →If one of the zeros of the cubic polynomial
Question: If one of the zeros of the cubic polynomial $x^{3}+a x^{2}+b x+c$ is $-1$, then the product of the other two zeros is (a)ab 1(b)ba 1(c) 1 a+b(d) 1 +ab Solution: (c) $1-a+b$ Since $-1$ is a zero of $x^{3}+a x^{2}+b x+c$, we have: $(-1)^{3}+a \times(-1)^{2}+b \times(-1)+c=0$ $=a-b+c-1=0$ $=c=1-a+b$ Also, product of all zeroes is given by $\alpha \beta \times(-1)=-c$ $=\alpha \beta=c$ $=\alpha \beta=1-a+b$...
Read More →Let f
Question: Let $\mathrm{f}, \mathrm{g}: \mathrm{N} \rightarrow \mathrm{N}$ such that $\mathrm{f}(\mathrm{n}+1)=\mathrm{f}(\mathrm{n})+\mathrm{f}(a) \quad \forall \mathrm{n} \in \mathrm{N}$ and $\mathrm{g}$ be any arbitrary function. Which of the following statements is NOT true?(1) $\mathrm{f}$ is one-one(2) If fog is one-one, then $g$ is one-one(3) If $\mathrm{g}$ is onto, then fog is one-one(4) If $\mathrm{f}$ is onto, then $\mathrm{f}(\mathrm{n})=\mathbf{n} \forall \mathbf{n} \in \mathbf{N}$Co...
Read More →At time t=0 magnetic field of 1000 Gauss is passing perpendicularly through the area defined
Question: At time $t=0$ magnetic field of 1000 Gauss is passing perpendicularly through the area defined by the closed loop shown in the figure. If the magnetic field reduces linearly to 500 Gauss, in the next $5 \mathrm{~s}$, then induced EMF in the loop is: (1) $56 \mu \mathrm{V}$(2) $28 \mu \mathrm{V}$(3) $48 \mu \mathrm{V}$(4) $36 \mu \mathrm{V}$Correct Option: 1 Solution: (1) According to question, $d B=1000-500=500$ gauss $=500 \times 10^{-4} \mathrm{~T}$ Time $d t=5 \mathrm{~s}$ Using far...
Read More →If one of the zeros of the cubic polynomial
Question: If one of the zeros of the cubic polynomial $a x^{3}+b \times 2+c x+d$ is 0 , then the product of the other two zeros is (a) $\frac{-c}{a}$ (b) $\frac{c}{a}$ (c) 0 (d) $\frac{-b}{a}$ Solution: (b) $\frac{c}{a}$ Let $\alpha, \beta$ and 0 be the zeroes of $a x^{3}+b x^{2}+c x+d$. Then, sum of the products of zeroes taking two at at a time is given by $(\alpha \beta+\beta \times 0+\alpha \times 0)=\frac{c}{a}$ $=\alpha \beta=\frac{c}{a}$ $\therefore$ The product of the other two zeroes is...
Read More →if a+
Question: If $a+\alpha=1, b+\beta=2$ and $a f(x)+\alpha f\left(\frac{1}{x}\right)=b x+\frac{\beta}{x}, x \neq 0$, then the value of the expression $\frac{f(x)+f\left(\frac{1}{x}\right)}{x+\frac{1}{x}}$ is Solution: $\operatorname{af}(x)+\alpha f\left(\frac{1}{x}\right)=b x+\frac{\beta}{x}$.............. $x \rightarrow \frac{1}{x}$ af $\left(\frac{1}{x}\right)+\operatorname{af}(x)=\frac{b}{x}+\beta x$ ........... $(i)+(i i)$ $(a+\alpha)\left[f(x)+f\left(\frac{1}{x}\right)\right]=\left(x+\frac{1}{...
Read More →If two of the zeros of the cubic polynomial
Question: If two of the zeros of the cubic polynomial $a x^{3}+b x^{2}+c x+d$ is 0 , then the third zeros is (a) $\frac{-b}{a}$ (b) $\frac{b}{a}$ (C) $\frac{c}{a}$ (d) $\frac{-d}{a}$ Solution: (a) $\frac{-b}{a}$ Let $\alpha, 0$ and 0 be the zeroes of $a x^{3}+b x^{2}+c x+d=0$. Then sum of the zeroes $=\frac{-b}{a}$ $=\alpha+0+0=\frac{-b}{a}$ $\Rightarrow \alpha=\frac{-b}{a}$ Hence, the third zero is $\frac{-b}{a}$....
Read More →A planar loop of wire rotates in a uniform magnetic field.
Question: A planar loop of wire rotates in a uniform magnetic field. Initially, at $t=0$, the plane of the loop is perpendicular to the magnetic field. If it rotates with a period of $10 \mathrm{~s}$ about an axis in its plane then the magnitude of induced emf will be maximum and minimum, respectively at:(1) $2.5 \mathrm{~s}$ and $7.5 \mathrm{~s}$(2) $2.5 \mathrm{~s}$ and $5.0 \mathrm{~s}$(3) $5.0 \mathrm{~s}$ and $7.5 \mathrm{~s}$(4) $5.0 \mathrm{~s}$ and $10.0 \mathrm{~s}$Correct Option: , 4 S...
Read More →If α, β, γ be the zeros of the polynomial p(x) such that
Question: If $\alpha, \beta, y$ be the zeros of the polynomial $p(x)$ such that $(\alpha+\beta+\gamma)=3,(\alpha \beta+\beta y+y \alpha)=-$ 10 and $\alpha \beta y=-24$, then $p(x)=$ ? (a) $x^{3}+3 x^{2}-10 x+24$ (b) $x^{3}+3 x^{2}+10 x-24$ (c) $x^{3}-3 x^{2}-10 x+24$ (d) None of these Solution: (c) $x^{3}-3 x^{2}-10 x+24$ Given: $\alpha, \beta$ and $\gamma$ are the zeroes of polynomial $p(x)$. Also, $(\alpha+\beta+\gamma)=3,(\alpha \beta+\beta \gamma+\gamma \alpha)=-10$ and $\alpha \beta \gamma=...
Read More →A long solenoid of radius R carries a time t
Question: A long solenoid of radius $R$ carries a time $(t)$ - dependent current $I(t)=I_{0} t(1-t)$. A ring of radius $2 R$ is placed coaxially near its middle. During the time interval $0 \leq t \leq$ 1 , the induced current $\left(I_{R}\right)$ and the induced $\operatorname{EMF}\left(V_{R}\right)$ in the ring change as:(1) Direction of $I_{R}$ remains unchanged and $V_{R}$ is maximum at $t=0.5$(2) At $t=0.25$ direction of $I_{R}$ reverses and $V_{R}$ is maximum(3) Direction of $I_{R}$ remain...
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