Class 12th

$S=21+{\displaystyle \sum _{k=1}^{21}{\left(z^k+\frac{1}{z^k}\right)}^3}=\sum {\left(z^k+{\overline{z}}^k\right)}^3=8\sum {\left(Re\left(z^k\right)\right)}^3$

$\left(As\left|z\right|=1\Rightarrow \frac{1}{z}=\overline{z}\right)$

$S=2{\displaystyle \sum _{k=1}^{21}cosk\pi +6}{\displaystyle \sum _{k=1}^{21}cos\frac{k\pi }{3}+21}$

= $2\left(-1\right)+6Re\left(\alpha +{\alpha }^2+{\alpha }^3\right)+21$

Where $\alpha =e^{i\left(2\pi /6\right)}$

$=2\left(-1\right)+6\left(\frac{1}{2}-\frac{1}{2}-1\right)+21$

$=-8+21=13$

Number of total car owners of TATA = 30 + 20 + 24 + 42 + 23 + 33 + 26 + 50 = 251 lakh

Now, 13% of 251 = 32.63 lakh

Those states which enjoy more than 34.45 Lacs car owners of TATA are Pune and Guwahati.

(x + y) = 4xy

$\Rightarrow \frac{dy}{dx}=\frac{4xy+4y^2-1}{1-4x^2-4xy},$

$\frac{d^{2}y}{d{x}^2}=\frac{\left(1-4x^2-4xy\right)\left(4y+4xy\text{'}+8yy\text{'}\right)-\left(4xy+4y^2-1\right)\left(-8x-4y-4xy\text{'}\right)}{{\left(1-4x^2-4xy\right)}^2}$

$B\left(1+2\lambda ,3+\lambda ,4+2\lambda \right)$

x – 2y – z = 3 => $\lambda =-6\Rightarrow B\left(-11,-3,-8\right)$

$N\left(1+t,3-2t,4-t\right)$  

$x-2y-z=3\Rightarrow t=2\Rightarrow N\left(3,-1,2\right)$  

Line NB : $\frac{x-3}{7}=\frac{y+1}{1}=\frac{3-2}{5}$

$d^2=P{M}^2=\frac{{\left|P\overrightarrow{Q}*\overrightarrow{R}\right|}^2}{{\left|\overrightarrow{R}\right|}^2}$

${\left|\overrightarrow{R}\right|}^2=75$

$d^2=\frac{1950}{75}=26$  

$E_{C{\mathcal{l}}_2/C{\mathcal{l}}^{-}}^0=+1.36V$

$E_{l_2/l^{-}}^0=+0.54V$

$E_{A{g}^{+}/Ag}^0=+0.80V$

$E_{N{a}^{+}/Na}^0=-2.71V$

$E_{L{i}^{+}/Li}^0=-3.05V$

$\therefore E_{C{\mathcal{l}}_2/C{\mathcal{l}}^{-}}^0>E_{A{g}^{+}/Ag}^0>E_{N{a}^{+}/Na}^0>E_{L{i}^{+}/Li}^0$

Required area

$={\displaystyle \underset{-2}{\overset{4}{\int }}\left(\frac{3x}{2}+6-\frac{3x^2}{4}\right)dx=27}$

$\frac{3x^2}{4}=\frac{3x}{2}+6\Rightarrow x=-2,4$

(Sum of the incomes in 2019-20) – (Sum of income in 2018-19)

= 4851.86 – 4139.43

= 712.43

Percentage change

$=\frac{\begin{array}{l}\text{\hspace{0.17em}sumofincomein}2004-05)-\\ \text{\hspace{0.17em}sumofincomein}2005-06\end{array}}{2004-05}*100=14.68\mathrm{\%}$

For AB₃ interhalogen compound, which is T-shaped, only two lone pair electrons are available on central atom.

Except electron gain enthalpy all the physical properties like sublimation enthalpy, ionization enthalpy and hydration energy affected the value of reduction potential.

Ans. = 3