Class 12th

$\begin{array}{l}{\displaystyle 1_{}^{}\frac{dx}{1+x^2}}={\left[ta{n}^{-1}x\right]}_0^1\\ =ta{n}^{-1}\left(1\right)-ta{n}^{-1}\left(0\right)\\ =\; \pi /4-0\\ =\; \pi /4\end{array}$

This is a multiple choice answer as classified in NCERT Exemplar

(a, b, c) Here, magnetic flux linked with the isolated coil change when the coil being in a time varying magnetic field, the coil moving in a constant magnetic field or in time varying magnetic field.

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This is a multiple choice answer as classified in NCERT Exemplar

(a, b, d) A metal plate is getting heated when a DC or AC current is passed through the plate, known as heating effect of current. Also, when metal plate is subjected to time varying magnetic field, the magnetic flux linked with the plate changes and eddy currents comes into existence which make the plate hot.

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This is a multiple choice answer as classified in NCERT Exemplar

(b) The self-inductance of a long solenoid of cross-sectional area A and length l, having n turns per unit length, filled the inside of the solenoid with a material of relative

permeability (e g. ., soft iron, which has a high value of relative permeability) is given by the relation L = ${\mu }_r{\mu }_o$ n²A/l

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This is a multiple choice answer as classified in NCERT Exemplar

(a) When the current in B (at t= 0) is counter-clockwise and the coil A is considered above to it. The counterclockwise flow of the current in B is equivalent to north pole of magnet and magnetic field lines are emanating upward to coil A. When coil A start rotating at (t= 0), the current in A is constant along clockwise direction by Lenz's rule.

${\displaystyle {\int }_4^5{e}^{x}dx}={\left[e^x\right]}_4^5=e^5-e^4=e^4\left(e-1\right)$

$\begin{array}{l}{{\displaystyle {\int }_0^{\pi /2}cos2xdx=\left[\frac{sin2x}{2}\right]}}_0^{\pi /2}=\frac{sin2*\pi /2}{2}-\frac{sin2*0}{2}\\ =\frac{si\mathrm{n\pi }}{2}-0\\ =0\end{array}$