Let f(x) =
Question: Let $f(x)=\left(\sin \left(\tan ^{-1} x\right)+\sin \left(\cot ^{-1} x\right)\right)^{2}-1,|x|1$. If $\frac{d y}{d x}=\frac{1}{2} \frac{d}{d x}\left(\sin ^{-1}(f(x))\right)$ and $y(\sqrt{3})=\frac{\pi}{6}$, then $y(-\sqrt{3})$ is equal to:(1) $\frac{2 \pi}{3}$(2) $-\frac{\pi}{6}$(3) $\frac{5 \pi}{6}$(4) $\frac{\pi}{3}$Correct Option: , 2 Solution: $\frac{d y}{d x}=\frac{1}{2} \frac{d}{d x}\left(\sin ^{-1} f(x)\right)$ $2 y=\sin ^{-1} f(x)+\mathrm{C}=\sin ^{-1}\left(\sin \left(2 \tan ^{...
Read More →In a meter bridge experiment,
Question: In a meter bridge experiment, the circuit diagram and the corresponding observation table are shown in figure. Which of the reading is consistent?(1) 3(2) 2(3) 4(4) 1Correct Option: , 3 Solution: (3) For a balanced bridge $\frac{\mathrm{R}_{1}}{\mathrm{R}_{2}}=\frac{l_{2}}{l_{1}}$ So $\frac{\mathrm{R}}{\mathrm{X}}=\frac{l}{100-l}$ Using the above expression $\mathrm{X}=\frac{\mathrm{R}(100-l)}{l}$ for observation (1) $\mathrm{X}=\frac{100 \times 40}{60}=\frac{2000}{3} \Omega$ for obser...
Read More →Let y=y(x) be a function of x satisfying
Question: Let $y=y(x)$ be $a$ function of $x$ satisfying $y \sqrt{1-x^{2}}=k-x \sqrt{1-y^{2}}$ where $k$ is $a$ constant and $y\left(\frac{1}{2}\right)=-\frac{1}{4}$. Then $\frac{d y}{d x}$ at $x=\frac{1}{2}$, is equal to:(1) $-\frac{\sqrt{5}}{4}$(2) $-\frac{\sqrt{5}}{2}$(3) $\frac{2}{\sqrt{5}}$(4) $\frac{\sqrt{5}}{2}$Correct Option: , 2 Solution: Given, $x=\frac{1}{2}, y=\frac{-1}{4} \Rightarrow x y=\frac{-1}{8}$ $y \cdot \frac{1 \cdot(-2 x)}{2 \sqrt{1-x^{2}}}+y^{\prime} \sqrt{1-x^{2}}$ $=-\lef...
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Question: If $x=\frac{2}{3}$ and $x=-3$ are the roots of the quadratic equation $a x^{2}+7 x+b=0$ then find the values of $a$ and $b$. Solution: Given: $a x^{2}+7 x+b=0$ Since, $x=\frac{2}{3}$ is the root of the above quadratic equation Hence, It will satisfy the above equation.Therefore, we will get $a\left(\frac{2}{3}\right)^{2}+7\left(\frac{2}{3}\right)+b=0$ $\Rightarrow \frac{4}{9} a+\frac{14}{3}+b=0$ $\Rightarrow 4 a+42+9 b=0$ $\Rightarrow 4 a+9 b=-42$ .....(1) Since, $x=-3$ is the root of ...
Read More →In the given circuit, an ideal voltmeter connected across
Question: In the given circuit, an ideal voltmeter connected across the $10 \Omega$ resistance reads $2 \mathrm{~V}$. The internal resistance $\mathrm{r}$, of each cell is : (1) $1 \Omega$(2) $0.5 \Omega$(3) $1.5 \Omega$(4) $0 \Omega$Correct Option: , 2 Solution: (2) For the given circuit $\mathrm{i}=\frac{3}{8+2 \mathrm{r}}$ Now voltage across $\mathrm{AB}$ $\mathrm{i} \times 6=\frac{3}{8+2 \mathrm{r}} \times 6=2$ $\Rightarrow 9=8+2 \mathrm{r}$ $\Rightarrow \mathrm{r}=\frac{1}{2} \Omega$...
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Question: If $y(\alpha)=\sqrt{2\left(\frac{\tan \alpha+\cot \alpha}{1+\tan ^{2} \alpha}\right)+\frac{1}{\sin ^{2} \alpha}}, \alpha \in\left(\frac{3 \pi}{4}, \pi\right)$, then $\frac{d y}{d \alpha}$ at $\alpha=\frac{5 \pi}{6}$ is:(1) 4(2) $\frac{4}{3}$(3) $-4$(4) $-\frac{1}{4}$Correct Option: 1 Solution: $y(\alpha)=\sqrt{\frac{\frac{2 \sin \alpha}{\cos \alpha}+\frac{\cos \alpha}{\sin \alpha}}{\sec ^{2} \alpha}}=\sqrt{\frac{2 \cos ^{2} \alpha}{\sin \alpha \cos \alpha}+\frac{1}{\sin ^{2} \alpha}}$ ...
Read More →Solve this
Question: If $(x+a)$ is a factor of the polynomial $2 x^{2}+2 a x+5 x+10$, find the value of $a .$ Solution: Given: $(x+a)$ is a factor of $2 x^{2}+2 a x+5 x+10$ So, we have $x+a=0$ $\Rightarrow x=-a$ Now, It will satisfy the above polynomial.Therefore, we will get $2(-a)^{2}+2 a(-a)+5(-a)+10=0$ $\Rightarrow 2 a^{2}-2 a^{2}-5 a+10=0$ $\Rightarrow-5 a=-10$ $\Rightarrow a=2$...
Read More →Find the quadratic polynomial whose zeros are
Question: Find the quadratic polynomial whose zeros are $\frac{2}{3}$ and $\frac{-1}{4}$. Verify the relation between the coefficients and the zeros of the polynomial. Solution: Let $\alpha=\frac{2}{3}$ and $\beta=\frac{-1}{4}$. Sum of the zeroes $=(\alpha+\beta)=\frac{2}{3}+\left(\frac{-1}{4}\right)=\frac{8-3}{12}=\frac{5}{12}$ Product of the zeroes $=\alpha \beta=\frac{2}{3} \times\left(\frac{-1}{4}\right)=\frac{-\frac{2^{1}}{12^{6}}}{12^{2}}=\frac{-1}{6}$ $\therefore$ Required polynomial $=x^...
Read More →In a conductor, if the number of conduction electrons per unit volume
Question: In a conductor, if the number of conduction electrons per unit volume is $8.5 \times 10^{28} \mathrm{~m}^{-3}$ and mean free time is $25 \mathrm{fs}$ (femto second), it's approximate resistivity is: $\left(\mathrm{m}_{\mathrm{e}}=9.1 \times\right.$ $10^{-31} \mathrm{~kg}$ )(1) $10^{-6} \Omega m$(2) $10^{-7} \Omega \mathrm{m}$(3) $10^{-8} \Omega \mathrm{m}$(4) $10^{-5} \Omega \mathrm{m}$Correct Option: , 3 Solution: (3) $\rho=\frac{m}{n e^{2} \tau}$ $=\frac{9.1 \times 10^{-31}}{8.5 \tim...
Read More →The derivative
Question: The derivative of $\tan ^{-1}\left(\frac{\sqrt{1+x^{2}}-1}{x}\right)$ with respect to $\tan ^{-1}\left(\frac{2 x \sqrt{1-x^{2}}}{1-2 x^{2}}\right)$ at $x=\frac{1}{2}$ is :(1) $\frac{2 \sqrt{3}}{5}$(2) $\frac{\sqrt{3}}{12}$(3) $\frac{2 \sqrt{3}}{3}$(4) $\frac{\sqrt{3}}{10}$Correct Option: , 4 Solution: Let $u=\tan ^{-1}\left(\frac{\sqrt{1+x^{2}}-1}{x}\right)$ Put $x=\tan \theta \Rightarrow \theta=\tan ^{-1} x$ $\therefore u=\tan ^{-1}\left(\frac{\sec \theta-1}{\tan \theta}\right)=\tan ^...
Read More →Find the quadratic polynomial whose zeros are 2 and −6.
Question: Find the quadratic polynomial whose zeros are 2 and 6. Verify the relation between the coefficients and the zeros of the polynomial. Solution: Let $\alpha=2$ and $\beta=-6$ Sum of the zeroes, $(\alpha+\beta)=2+(-6)=-4$ Product of the zeroes, $\alpha \beta=2 \times(-6)=-12$ $\therefore$ Required polynomial $=x^{2}-(\alpha+\beta) x+\alpha \beta=x^{2}-(-4) x-12$ $=x^{2}+4 x-12$ Sum of the zeroes $=-4=\frac{-4}{1}=\frac{-(\text { coefficient of } x)}{\text { (coefficient of } x^{2} \text {...
Read More →The resistance of a galvanometer
Question: The resistance of a galvanometer is $50 \mathrm{ohm}$ and the maximum current which can be passed through it is $0.002 \mathrm{~A}$. What resistance must be connected to it order to convert it into an ammeter of range $0-0.5 \mathrm{~A}$ ?(1) $0.5 \mathrm{ohm}$(2) $0.002 \mathrm{ohm}$(3) $0.02 \mathrm{ohm}$(4) $0.2 \mathrm{ohm}$Correct Option: , 4 Solution: (4) Using, $i_{g}=i \frac{S}{S+G}$ $0.002=0.5 \frac{S}{S+50}$ On solving, we get $S=\frac{100}{498} \simeq 0.2 \Omega$...
Read More →Find the quadratic polynomial, the sum of whose zeros is
Question: Find the quadratic polynomial, the sum of whose zeros is $\left(\frac{5}{2}\right)$ and their product is 1 . Hence, find the zeros of the polynomial. Solution: Let $\alpha$ and $\beta$ be the zeros of the required polynomial $f(x)$. Then $(\alpha+\beta)=\frac{5}{2}$ and $\alpha \beta=1$ $\therefore f(x)=x^{2}-(\alpha+\beta) x+\alpha \beta$ $=f(x)=x^{2}-\frac{5}{2} x+1$ $=f(x)=2 x^{2}-5 x+2$ Hence, the required polynomial is $f(x)=2 x^{2}-5 x+2$. $\therefore f(x)=0=2 x^{2}-5 x+2=0$ $=2 ...
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Question: If $(a+\sqrt{2} b \cos x)(a-\sqrt{2} b \cos y)=a^{2}-b^{2}$, where $ab0$, then $\frac{d x}{d y}$ at $\left(\frac{\pi}{4}, \frac{\pi}{4}\right)$ is:(1) $\frac{a-2 b}{a+2 b}$(2) $\frac{a-b}{a+b}$(3) $\frac{a+b}{a-b}$(4) $\frac{2 a+b}{2 a-b}$Correct Option: , 3 Solution: $(a+\sqrt{2} b \cos x)(a-\sqrt{2} b \cos y)=a^{2}-b^{2}$ Differentiating both sides, $(-\sqrt{2} b \sin x)(a-\sqrt{2} b \cos y)+(a+\sqrt{2} b \cos x)$ $(\sqrt{2} b \sin y) \frac{d y}{d x}=0$ $\Rightarrow \frac{d y}{d x}=\...
Read More →Find the quadratic polynomial, the sum of whose zeros is 0 and their product is −1.
Question: Find the quadratic polynomial, the sum of whose zeros is 0 and their product is 1. Hence, find the zeros of the polynomial. Solution: Let $\alpha$ and $\beta$ be the zeros of the required polynomial $f(x)$. Then $(\alpha+\beta)=0$ and $\alpha \beta=-1$ $\therefore f(x)=x^{2}-(\alpha+\beta) x+\alpha \beta$ $=f(x)=x^{2}-0 x+(-1)$ $=f(x)=x^{2}-1$ Hence, the required polynomial is $f(x)=x^{2}-1$. $\therefore f(x)=0=x^{2}-1=0$ $=(x+1)(x-1)=0$ $=(x+1)=0$ or $(x-1)=0$ $=x=-1$ or $x=1$ So, the...
Read More →A metal wire of resistance
Question: A metal wire of resistance $3 \Omega$ is elongated to make a uniform wire of double its previous length. This new wire is now bent and the ends joined to make a circle. If two points on the circle make an angle $60^{\circ}$ at the centre, the equivalent resistance between these two points will be:(1) $\frac{12}{5} \Omega$(2) $\frac{5}{2} \Omega$(3) $\frac{5}{3} \Omega$(4) $\frac{7}{2} \Omega$Correct Option: , 3 Solution: (3) When length becomes double its resistance becomes $\left(R \p...
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Question: If $y^{2}+\log _{e}\left(\cos ^{2} x\right)=y, x \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$, then :(1) $y^{\prime \prime}(0)=0$(2) $\left|y^{\prime}(0)\right|+\left|y^{\prime \prime}(0)\right|=1$(3) $\left|y^{\prime \prime}(0)\right|=2$(4) $\left|y^{\prime}(0)\right|+\left|y^{\prime \prime}(0)\right|=3$Correct Option: , 3 Solution: $y^{2}+2 \log _{e}(\cos x)=y$........(1) $\Rightarrow 2 y y^{\prime}-2 \tan x=y^{\prime}$.....(2) From(1), $y(0)=0$ or 1 $\therefore y^{\prime}(0)=0$ Ag...
Read More →Find the quadratic polynomial, sum of whose zeros is 8 and their product is 12.
Question: Find the quadratic polynomial, sum of whose zeros is 8 and their product is 12. Hence, find the zeros of the polynomial. Solution: Let $\alpha$ and $\beta$ be the zeroes of the required polynomial $f(x)$. Then $(\alpha+\beta)=8$ and $\alpha \beta=12$ $\therefore f(x)=x^{2}-(\alpha+\beta) x+\alpha \beta$ $=f(x)=x^{2}-8 x+12$ Hence, required polynomial $f(x)=x^{2}-8 x+12$ $\therefore f(x)=0=x^{2}-8 x+12=0$ $=x^{2}-(6 x+2 x)+12=0$ $=x^{2}-6 x-2 x+12=0$ $=x(x-6)-2(x-6)=0$ $=(x-2)(x-6)=0$ $...
Read More →If α and β are the zeros of the polynomial
Question: If $\alpha$ and $\beta$ are the zeros of the polynomial $p(x)=2 x^{2}+5 x+k$ satisfying the relation $\alpha^{2}+\beta^{2}+\alpha \beta=\frac{21}{4}$ then find the value of $k$. Solution: Let $\alpha$ and $\beta$ be the zeroes of the polynomial $p(x)=2 x^{2}+5 x+k$. Sum of zeroes $=-\frac{b}{a}$ $\Rightarrow \alpha+\beta=-\frac{5}{2} \quad \ldots(1)$ and Product of zeroes $=\frac{c}{a}$ $\Rightarrow \alpha \beta=\frac{k}{2} \quad \ldots(2)$ Now, using (1) $\alpha+\beta=-\frac{5}{2}$ Sq...
Read More →A moving coil galvanometer has resistance
Question: A moving coil galvanometer has resistance $50 \Omega$ and it indicates full deflection at $4 \mathrm{~mA}$ current. A voltmeter is made using this galvanometer and a $5 \mathrm{k} \Omega$ resistance. The maximum voltage, that can be measured using this voltmeter, will be close to:(1) $40 \mathrm{~V}$(2) $15 \mathrm{~V}$(3) $20 \mathrm{~V}$(4) $10 \mathrm{~V}$Correct Option: , 3 Solution: (3) $V=i_{g}(G+R)=4 \times 10^{-3}$ $(50+5000)=20 \mathrm{~V}$...
Read More →Let f(x) be a differentiable function at
Question: Let $f(x)$ be a differentiable function at $x=a$ with $f^{\prime}(a)=2$ and $f(a)=4$. Then lim $_{z \rightarrow a} \frac{x f(a)-a f(x)}{x-a}$ equals:(1) $2 \mathrm{a}+4$(2) $2 a-4$(3) $4-2 a$(4) $a+4$Correct Option: , 3 Solution: By L-H rule $L=\lim _{x \rightarrow a} \frac{f(a)-a f^{\prime}(x)}{1}$ $\therefore L=4-2 a$...
Read More →Let f(x) be a differentiable function at
Question: Let $f(x)$ be a differentiable function at $x=a$ with $f^{\prime}(a)=2$ and $f(a)=4$. Then lim $_{z \rightarrow a} \frac{x f(a)-a f(x)}{x-a}$ equals:(1) $2 \mathrm{a}+4$(2) $2 a-4$(3) $4-2 a$(4) $a+4$Correct Option: , 3 Solution: By L-H rule...
Read More →A wire of resistance R is bent to form a square A B C D as shown in the figure.
Question: A wire of resistance $R$ is bent to form a square $A B C D$ as shown in the figure. The effective resistance between $\mathrm{E}$ and $C$ is: $(E$ is mid-point of arm $C D)$ (1) $\mathrm{R}$(2) $\frac{7}{64} \mathrm{R}$(3) $\frac{3}{4} \mathrm{R}$(4) $\frac{1}{16} R$Correct Option: , 2 Solution: (2) Here $\mathrm{R}_{\mathrm{DA}}=\mathrm{R}_{\mathrm{AB}}=\mathrm{R}_{\mathrm{BC}}=\mathrm{R} / 4$ and $\mathrm{R}_{\mathrm{DE}}=\mathrm{R}_{\mathrm{EC}}=\mathrm{R} / 8$ Now $R_{E D}, R_{D A}...
Read More →Let f be any function defined on R
Question: Let $f$ be any function defined on $\mathrm{R}$ and let it satisfy the condition : $|f(x)-f(y)| \leq\left|(x-y)^{2}\right|, \forall(x, y) \in R$ If $f(0)=1$, then:(1) $f(x)0, \forall x \in R$(2) $f(x)$ can take any value in $R$(3) $f(x)=0, \forall x \in R$(4) $f(x)0, \forall x \in R$Correct Option: , 4 Solution: $|f(x)-f(y)| \leq\left|(x-y)^{2}\right|, \forall(x, y) \in R$ $\left|\frac{f(x)-f(y)}{x-y}\right| \leq|x-y|$ $\lim _{x \rightarrow y}\left|\frac{f(x)-f(y)}{x-y}\right| \leq 0$ ...
Read More →If f(x)= sin
Question: If $f(x)=\sin \left(\cos ^{-1}\left(\frac{1-2^{2 x}}{1+2^{2 x}}\right)\right)$ and its first derivative with respect to $x$ is $-\frac{b}{a} \log _{e} 2 w h e n x=1$, where $a$ and $b$ are integers, then the minimum value of $\left|a^{2}-b^{2}\right|$ is__________. Solution: $f(x)=\sin \left(\cos ^{-1}\left(\frac{1-2^{2 x}}{1+2^{2 x}}\right)\right)$ at $x=1 ; 2^{2 x}=4$ for $\sin \left(\cos ^{-1}\left(\frac{1-x^{2}}{1+x^{2}}\right)\right)$ Let $\tan ^{-1} \mathrm{x}=\theta ; \quad \the...
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