Class 12th

$?=\overrightarrow{B}.\overrightarrow{A}$

$=\left(3t^3\widehat{j}+3t^2\widehat{k}\right).\left(\pi {\left(1\right)}^2\widehat{k}\right)$

$?=3t^2\pi$

$E_{ind}=\left|\frac{d?}{dt}\right|=6t\pi$

at t = 2, E_ind = 12

$\delta =A\left({\mu }_y-1\right)-A^1\left({\mu }_y^1-1\right)$

$=6\left(1.5-1\right)-5\left(1.55-1\right)$

$=\frac{1}{4}$

Independent of are a is case of uniform wire

P = 10% of 110W = 11W

(Visible)

$I_1-I_2=\frac{\rho }{4\pi }\left(\frac{1}{r_1^2}-\frac{1}{r_2^2}\right)$

$=\frac{11}{4*\frac{22}{7}}\left(\frac{1}{1}-\frac{1}{25}\right)$

$\frac{7}{8}*\left(\frac{24}{25}\right)=a*10^{-2}W/m^2$

$\frac{28*24}{8}*10^{-2}=a*10^{-2}$

=84 * 10^-2 = a * 10^-2 = a = 84

take z = x + iy

$z^2+\overline{z}=0$

$\Rightarrow x^2-y^2+x+i2xy-yi=0$

$\Rightarrow x^2-y^2+x=0andy\left(2x-1\right)=0$

if y = 0 x = 0, 1

$ifx=\frac{1}{2}\Rightarrow y=\pm \frac{\sqrt[]{3}}{2}$

$\Sigma \left(Re\left(z\right)+lm\left(z\right)\right)=\left(0-1+\frac{1}{2}+\frac{1}{2}\right)+\left(0+0+\frac{\sqrt[]{3}}{2}-\frac{\sqrt[]{3}}{2}\right)=0$

Let A = $\left[\begin{array}{ccc}\hfill a\hfill & \hfill b\hfill & \hfill c\hfill \\ \hfill d\hfill & \hfill e\hfill & \hfill f\hfill \\ \hfill 9\hfill & \hfill h\hfill & \hfill i\hfill \end{array}\right]$

Now A^TA

trace will be $a^2+b^2+c^2+d^2+e^2+f^2+9^2+h^2=6$

total ways = ⁹C₆.2⁶.1.1.1 = 5376

$\left|f\left(x\right)\right|\le 800\Rightarrow 2n^2-n-1\le 800$

$\Rightarrow 2n^2-n-801\le 0$

${\displaystyle \sum _{x\in S}^{}f\left(x\right)={\displaystyle \sum _{}^{}\left(2x^2-x-1\right)}}$

$=2\left(19^2+18^2+........1^2+0^2+1^2+.....+20^2\right)$

= 10620

y⁵ – 9xy + 2x = 0

differentiate 5y⁴ – 9x $\frac{dy}{dx}$ 9y + 2 = 0

$\frac{dy}{dx}=\frac{9y-2}{5y^4-9x}$

For horizontal tangent $\frac{dy}{dx}=0\Rightarrow y=\frac{2}{9}$ which does not satisfy the equation so no horizontal

For vertical tangent $5y^4-9x=0$

m = 0, N = 2

$sin\left(2x^2\right).109_e\left(tan{x}^2\right)dy+4xydx=4\sqrt[]{2}.x.\left(sin{x}^{2}cos\frac{\pi }{4}-cos{x}^{2}sin\frac{\pi }{4}\right)dx$

$\Rightarrow \mathcal{l}n\left(tan{x}^2\right)dy+\frac{4x}{sin\left(2x^2\right)}ydx=\frac{4x\left(sin{x}^2-cos{x}^2\right)}{sin\left(2x^2\right)}dx$

Integrate

$\Rightarrow y.\mathcal{l}n\left(tan{x}^2\right)=2.\mathcal{l}n\left(\frac{sin{x}^2+cos{x}^2-1}{sin{x}^2+cos{x}^2+1}\right)+C$

$x=\sqrt[]{\frac{\pi }{6}},y=1$ calculate C.

Plane through 4ax – y + 5z – 7a = 0 = 2x – 5y – z – 3

is $x\left(4a+2\lambda \right)+y\left(-1-5\lambda \right)+z\left(5-\lambda \right)=7a+3\lambda$

This plane contains 4, -1, 0

9a + 1 + 10 = 0…… (i)

Plane contains the line $\frac{x-4}{1}=\frac{y+1}{-2}=\frac{z}{1}$

$4a+11\lambda +7=0$ ……. (ii)

From (i) & (ii) a = 1,  $\lambda$ =1

Equation of plane $\pi \equiv x+2y+3z-2=0$

$\Rightarrow$$7$$P$$+$$3$$-$$2$$P$$+$$4$$-$$1$$2$$P$$+$$9$$-$$2$$=$$0$$\Rightarrow$$P$$=2$$\mathrm{}$