Maths

  $\Delta =\left|\begin{array}{ccc}\hfill 3sin3\theta \hfill & \hfill -1\hfill & \hfill 1\hfill \\ \hfill 3cos2\theta \hfill & \hfill 4\hfill & \hfill 3\hfill \\ \hfill 6\hfill & \hfill 7\hfill & \hfill 7\hfill \end{array}\right|$

= 3 sin3θ (28 – 21) + (21 cos 2θ - 18) + 1 (21 cos 2θ - 24)

$\Delta =21sin3\theta +42cos2\theta -42$          

for no solution

sin 3θ + 2 cos 2θ = 2

$\theta =\pi ,2\pi ,3\pi ,\frac{\pi }{6},\frac{5\pi }{6},\frac{13\pi }{6},\frac{17\pi }{6}$

$S_n=\left\{z\in C:\left|z-3+2i\right|=\frac{n}{4}\right\}$

represents a circle with centre C1 (3, 2) and radius

$r_1=\frac{n}{4}$

Similarly Tn represents circle with centre C2 (2, 3) and radius

$r_1=\frac{1}{n}$

$\sqrt[]{2}<\left|\frac{n}{4}-\frac{1}{n}\right|$

n take infinite values.

From properties of nth root of unity

$1^{2021}+{\alpha }^{2021}+{\beta }^{2021}+{\gamma }^{2021}+{\delta }^{2021}=0$

$\Rightarrow {\alpha }^{2021}+{\beta }^{2021}+{\gamma }^{2021}+{\delta }^{2021}=-1$

Case 1 : If f (3) = 3 then f (1) and f (2) take 1 or 2

No. of ways = 2.6 = 12

Case 2 : If f (3) = 5 then f (1) and f (2) take 2 or 3

OR 1 and 4

No. of ways = 2.6.2 = 24

Similarly for all other cases

Total no. of ways 90.

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 Þ x = 0, -1

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

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

$\frac{{\left(x+1\right)}^2}{\lambda +5}=\frac{{\left(y+1\right)}^2}{\frac{\lambda +5}{4}}=1$  

length of latus rectum = $\frac{2b^2}{a}=\frac{2\left(\frac{\lambda +5}{4}\right)}{\sqrt[]{5+\lambda }}=4$  

$\sqrt[]{\lambda +5}=8\Rightarrow \lambda =59$                

Major axis = $2\sqrt[]{\lambda +5}=16$  

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 = 

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

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

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

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

= 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.