Solve this following
Question: If $y=x^{\log x}+(\log x)^{x}$, prove that $\frac{d y}{d x}=x^{(\log x)}\left\{\frac{2 \log x}{x}\right\}+(\log x)^{x}\left\{\frac{1}{\log x}+\log (\log x)\right\}$ Solution:...
Read More →Solve this following
Question: If $y=x^{\cos x}+(\cos x)^{x}$, prove that $\frac{d y}{d x}=x^{\cos x} \cdot\left\{\frac{\cos x}{x}-(\sin x) \log x\right\}+(\cos x)^{x}$ $[(\log \cos x)-x \tan x]$ Solution:...
Read More →Solve this following
Question: If $y=(\tan x)^{\cot x}+(\cot x)^{\tan x}, \frac{d y}{d x}=(\tan x)^{\cot x} \operatorname{cosec}^{2} x(1-\log \tan x)$ $+(\cot x)^{\tan x} \cdot \sec ^{2} x[\log (\cot x)-1]$ Solution:...
Read More →A finite ladder is constructed by connecting several sections of
Question: A finite ladder is constructed by connecting several sections of $2 \mu \mathrm{F}, 4 \mu \mathrm{F}$ capacitor combination as shown in figure. It is terminated by a capacitor of capacitance $C$. What value should be chosen for $C$, such that the equivalent capacitance of the ladder between the points $A$ and $B$ becomes independent of the number of sections in between? Solution:...
Read More →Find the equivalent capacitance of the infinite ladder
Question: Find the equivalent capacitance of the infinite ladder shown in figure between the points $\mathrm{A}$ and $\mathrm{B}$. Solution:...
Read More →Find the capacitance of the combination
Question: Find the capacitance of the combination shown in figure between $A$ and $B$ Solution:...
Read More →Solve this following
Question: If $y=(\sin x)^{\cos x}+(\cos x)^{\sin x}$, prove tha $\frac{d y}{d x}=(\sin x)^{\cos x} \cdot[\cot x \cos x-\sin x$ $(\log \sin x)]+(\cos x)^{\sin x \cdot}[\cos x(\log \cos x)-\sin x \tan x]$ Solution:...
Read More →Find the equivalent capacitance of the combination of
Question: Find the equivalent capacitance of the combination of the combination shown in figure between the indicated points. Solution:...
Read More →Solve this following
Question: If $y=(x)^{\cos x}+(\sin x)^{\tan x}$, prove that $\frac{d y}{d x}=x^{\cos x}\left\{\frac{\cos x}{x}-(\sin x) \log x\right\}+(\sin x)^{\tan x}$ $\left\{1+(\log \sin x) \sec ^{2} x\right\}$ Solution:...
Read More →Find the potential difference Va-Vb between the points a and b
Question: Find the potential difference $\mathrm{V}_{\mathrm{a}}-\mathrm{V}_{\mathrm{b}}$ between the points $\mathrm{a}$ and $\mathrm{b}$ shown in each part of the figure Solution:...
Read More →Solve this following
Question: If $y=\log \sqrt{\frac{1+\cos ^{2} x}{1-e^{2 x}}}$, show that $\frac{d y}{d x}=\frac{e^{2 x}}{\left(1-e^{2 x}\right)}-\frac{\sin x \cos x}{\left(1+\cos ^{2} x\right)}$ Solution:...
Read More →Solve this following
Question: If $y=\sqrt{\frac{1-\sin 2 x}{1+\sin 2 x}}$, show that $\frac{d y}{d x}+\sec ^{2}\left(\frac{\pi}{4}-x\right)=0$ Solution:...
Read More →Convince yourself that parts (a), (b), (c)
Question: Convince yourself that parts (a), (b), (c) of figure are identical. Find the capacitance between the points A and B of the assembly. Solution: All the circuits are in balanced wheat stone symmetry so no current flows is $5 \mu \mathrm{F}$ capacitor $C_{e q}=\frac{1 \times 3}{1+3}+\frac{2 \times 6}{2+6}=2.25 \mu F$...
Read More →Solve this following
Question: If $y=\log \tan \left(\frac{\pi}{4}+\frac{x}{2}\right)$, show that $\frac{d y}{d x}=\sec x$ Solution:...
Read More →The plates of a capacitor are 2.00cm apart.
Question: The plates of a capacitor are $2.00 \mathrm{~cm}$ apart. An electron-proton pair is released somewhere in the gap between the plates and it is found that the proton reaches the negative plate at the same time as the electron reaches the positive plate. At the distance from the negative plate was the pair released? Solution:...
Read More →Solve this following
Question: If $y=\log \sqrt{\frac{1-\cos x}{1+\cos x}}$, show that $\frac{d y}{d x}=\operatorname{cosecx}$ Solution:...
Read More →Both the capacitors shown in figure are made of square plates of edges a.
Question: Both the capacitors shown in figure are made of square plates of edges a. The separations between the plates of the capacitors are $d_{1}$ and $d_{2}$ as shown in the figure. A potential difference $V$ is applied between the point a and $b$. An electron is projected between the plates of the upper capacitor along the central line. With what minimum speed should the electron be projected so that it does not collide with any plate? Consider only the electric forces. Solution:...
Read More →Solve this following
Question: If $y=\log \sin \sqrt{x^{2}+1}$, prove that $\frac{d y}{d x}=\frac{x \cot \sqrt{x^{2}+1}}{\sqrt{x^{2}+1}}$ Solution:...
Read More →The particle P shown in figure has a mass of
Question: The particle $P$ shown in figure has a mass of $10 \mathrm{mg}$ and a charge of $-0.01 \mu \mathrm{C}$. Each plate has a surface area $100 \mathrm{~cm}^{2}$. On one side. What potential difference $\mathrm{V}$ should be applied to the combination to hold the particle $\mathrm{P}$ in equilibrium? Solution:...
Read More →Solve this following
Question: If $y=\log \left(x+\sqrt{1+x^{2}}\right)$, prove that $\frac{d y}{d x}=\frac{1}{\log \left(x+\sqrt{1+x^{2}}\right)} \cdot \frac{1}{\sqrt{1+x^{2}}}$ Solution:...
Read More →Each capacitor shown in figure has a capacitance of
Question: Each capacitor shown in figure has a capacitance of $5.0 \mu \mathrm{F}$. The emf of the battery is $50 \mathrm{~V}$. How much charge will flow through $A B$ if the switch $S$ is closed? Solution:...
Read More →Solve this following
Question: If $y=e^{\sin x}+(\tan x)^{x}$, prove that $\frac{d y}{d x}=e^{\sin x} \cos x+(\tan x)^{x}[2 x \operatorname{cosec} 2 x+\log \tan x]$ Solution:...
Read More →A 100pF capacitor is charged to a potential difference of
Question: A $100 \mathrm{pF}$ capacitor is charged to a potential difference of $24 \mathrm{~V}$. It is connected to an uncharged capacitor of capacitance $20 \mathrm{pF}$. What will be the new potential difference across the $100 \mathrm{pF}$ capacitor? Solution:...
Read More →A cylindrical capacitor is constructed using two coaxial cylinders of
Question: A cylindrical capacitor is constructed using two coaxial cylinders of the same length $10 \mathrm{~cm}$ and of radii $2 \mathrm{~mm}$ and $4 \mathrm{~mm}$. (a) Calculate the Capacitance. (b) Another capacitor of the same length is constructed with cylinders of radii $4 \mathrm{~mm}$ and $8 \mathrm{~mm}$. Calculate the capacitance. Solution:...
Read More →Solve this following
Question: If $\left(x^{x}+y^{x}\right)=1$, show that $\frac{d y}{d x}=-\left\{\frac{x^{x}(1+\log x)+y^{x}(\log y)}{x y^{x-1}}\right\}$ Solution:...
Read More →