Class 12th

  $f\left(x\right)+f\left(x+k\right)=n\forall x\in R,\&k>0............\left(i\right)$

Replace x by x + k.          

$f\left(x+k\right)+f\left(x+2k\right)=n...............\left(ii\right)$

From (i) & (ii), f(x + 2k) = f(x).

$\therefore f\left(x\right)$  is periodic with period = 2k.

$I_1={\displaystyle \underset{0}{\overset{4nk}{\int }}f\left(x\right)}dx=2x{\displaystyle \underset{0}{\overset{2k}{\int }}f\left(x\right)dx}...................\left(ii\right)$

$I_2={\displaystyle \underset{-k}{\overset{3k}{\int }}f\left(x\right)}dx$ put x = t + k

$={\displaystyle \underset{-2k}{\overset{2k}{\int }}t\left(t+k\right)dt}=2{\displaystyle \underset{0}{\overset{2k}{\int }}f\left(t+k\right)dt}$

$=2n{\displaystyle \underset{0}{\overset{2k}{\int }}ndx}=4n^{2}k.$

I_R = I_B (due to magnetic field)

(due to Radiation)

$\frac{P}{A}*Efficiency=\frac{B_0^2}{2{\mu }_0}c$

$\Rightarrow \frac{200}{4\pi *4^2}*\frac{3.5}{100}=\frac{B_0^2}{2*4\pi *10^{-7}}*3*10^8$

$\Rightarrow B_0=1.71*10^{-8}T$

$F_C=F_q$

$\Rightarrow \frac{m{v}^2}{R}=\frac{\rho R}{2{\in }_0}q$

$m{v}^2=\frac{\rho R^{2}q}{2{\epsilon }_0}\Rightarrow \frac{1}{2}m{v}^2=k.E.=\frac{1}{4}\frac{\rho R^{2}q}{{\in }_0}$

$U_i=\frac{Q^2}{2C}$

Uf = 1.44 Ui

Qf = Q + 2

$1.44U_i=\frac{{\left(Q+2\right)}^2}{2C}$

$\Rightarrow 1.44\frac{Q^2}{2C}=\frac{{\left(Q+2\right)}^2}{2C}$

$\Rightarrow 1.2Q=Q+2$

0.2 Q = 2

Q = 10 C

  $f\left(\frac{r}{4}\right)=\sqrt[]{2},f\left(\frac{r}{2}\right)=0\&f\text{'}\left(\frac{r}{2}\right)=1$

g(x)   ${\displaystyle \underset{}{\overset{}{\int }}\left(f\text{'}\left(t\right)sect+tantsectf\left(t\right)\right)dt}$

$={\left[f\left(t\right)sect\right]}_x^{\pi /4}$

=   $\sqrt[]{2}*\sqrt[]{2}-f\left(x\right)secx$

$=\frac{2cosx-f\left(x\right)}{cosx}$

=   $\frac{2sinx+1}{sinx}=3$

Sphalarite           ZnS

Calamine            ZnCO₃

Galena                PbS

Siderite               FeCO₃

$f\left(x\right)=\{\begin{array}{cc}\hfill \left[x\right],x<0\hfill & \hfill \left|1-x\right|,x\ge 0\hfill \end{array}$ $g\left(x\right)=\{\begin{array}{cc}\hfill c^x-x,x<0\hfill & \hfill {\left(x-1\right)}^2-1,x\ge 0\hfill \end{array}$

fog

$fog(\begin{array}{}\end{array}\left[e^x-x\right],\left(e^x-x<0\right)\left(x<0\right)\left[{\left(x-1\right)}^2-1\right]{\left(x-1\right)}^2-1<0x\ge 0\left|1-e^x+x\right|,e^x-x\ge 0x<0\left|1-{\left(x+1\right)}^2+1\right|,{\left(x-1\right)}^2-1\ge 0x\ge 0,x\in \left(2,\infty \right)($

 Not possible as of inequalities give $?.$  

e^x – x < 0, x < 0                 (x 1)² – 1 < 0

Not possible                     (x – 1 + 1)(x – 1 – 1) < 0

(x)(x – 2) < 0

x   $\in \left(0,2\right)$

continuous  $\forall$ x < 0

$fog\{\begin{array}{ccc}\hfill \left|1-e^x+x\right|,x<0\hfill & \hfill \left|1-{\left(x-1\right)}^2+1\right|,x=0\hfill & \hfill \left|1-{\left(x-1\right)}^2+1\right|,x\ge 2\hfill \end{array}$ Discontinuous at 0

continuous   $\forall x$

  $\therefore$  fog is discontinuous at 0

Emulsions gets separated into two layers on standing.

For stabilization of emulsion. Emulsifying agents added into it but not electrolyte.

Z = $\sqrt[]{R^2+{\left(x_C-x_L\right)}^2},$ if only L & C are present then R = 0 then p = 0

$r=\frac{mv}{qB}=\frac{\sqrt[]{2mk.E}}{qB}$  

$r\propto \frac{\sqrt[]{m}}{q}$                

m_p = m

q_p = q

m_d = 2m

q_d = q

m_a = 4m

q_a = 2q

$\frac{r_p}{r_d}=\frac{\sqrt[]{m_p}*q_d}{q_P*\sqrt[]{m_d}}$

$\frac{r_d}{r_{\alpha }}=\sqrt[]{2}$

$r_p:r_d:r_{\alpha }=1:\sqrt[]{2}:1$