Maths

$x=\frac{1}{2}$

$y=2{\displaystyle \underset{y=0}{\overset{y=1}{\int }}\left(\frac{y^2+3}{4}-\frac{y^2+1}{2}\right)dy}$

$=\frac{1}{2}{\left(\frac{y^3}{3}+3y\right)}_0^1-{\left(\frac{y^3}{3}+y\right)}_0^1$

$=\frac{1}{2}\left(\frac{10}{3}\right)-\frac{4}{3}=\frac{1}{3}$

$\underset{x\to \frac{\pi }{2}}{lim}\frac{si{n}^{2}x\left(si{n}^{2}x-3sinx+2\right)}{co{s}^{2}x.\left(\sqrt[]{2si{n}^{2}x+3sinx+4}+\sqrt[]{si{n}^{2}x+6sinx+2}\right)}$  

$\downarrow$                                                       $\downarrow$                            

3                                                       3

$\underset{x\to \frac{\pi }{2}}{lim}\frac{1}{9}.\frac{\left(sinx-1\right)\left(sinx-2\right)}{\left(1-sinx\right)\left(1+sinx\right)}=\frac{1}{6}\left(-1\right).\frac{-1}{2}=\frac{1}{12}$

$\Delta =\left|\begin{array}{ccc}\hfill -k\hfill & \hfill 3\hfill & \hfill -14\hfill \\ \hfill -15\hfill & \hfill 4\hfill & \hfill -k\hfill \\ \hfill -4\hfill & \hfill 1\hfill & \hfill 3\hfill \end{array}\right|$

$=\left(-k\right)\left(12-k\right)+3\left(4k+45\right)-14\left(-15+16\right)$

$\Delta =0\Rightarrow k=\pm 11$             

 For k = -11,

->11x + 3y – 14z = 25

-4x + y + 3z = 4

$\left\{\begin{array}{l}-11x+3y-14z=25\\ -15x+4y-11z=3\\ -4x+y+3z=4\end{array}\right\}\to$ No solution for k = $\pm$ 11

${\overline{z}}_1=i{\overline{z}}_2,arg\left(\frac{z_1}{z_2}\right)=\pi$

$\Rightarrow z_1=-z_2,arg\left(\frac{z_1{z}_2}{{\left|z_2\right|}^2}\right)=\pi$

$\Rightarrow arg\left(z_1{z}_2\right)=\pi$

$20-\frac{\pi }{2}=\pi$

$\theta =\frac{3\pi }{4}$

$\theta -\frac{\pi }{2}=\frac{\pi }{4}$

$D=0\Rightarrow b^2=15a,\alpha =\frac{2b\pm 0}{2a}=\frac{b}{a}$  

  $\frac{b}{b^{\frac{2}{15}}}=\frac{15}{b}$

$\alpha +\beta =2b{\alpha }^2+{\beta }^2=4b^2-42$

$\alpha \beta =21$

$a{\left(x-\alpha \right)}^2=a\left(x^2-2\alpha x+{\alpha }^2\right)=a{x}^2-2bx+15$

a = 3                    a = -3

b = 7                    a = -7

a² + b² = 9 + 49 = 58

Kindly go through the solution

 $\begin{array}{l}-2<x+1<2\\ -3<x<1\end{array}|\begin{array}{cc}\hfill x-1\le -2orx-1\ge 2\hfill & \hfill x\le -1orx\ge 3\hfill \end{array}$

From  $\Delta APQ$ $$

$$ $\frac{x+15}{y}=tan75°$  …. (i)

From $\Delta RQA$ $$

$\frac{}{}$  $\frac{15}{y}=tan60°$ …… (ii)

From (i) and (ii)

$=5\left(2\sqrt[]{3}+3\right)m$

Number of solution of the equation |cos x| = sin x for $x\in \left[-4\pi ,4\pi \right]$ $$ will be equal to 4 times the number of solutions of the same equation for $x\in \left[0,2\pi \right]$ $$

Graphs of y = |cos x| and y = sin x are as shown below.

Hence, two solutions of given equation in [0, 2]

$$ $\Rightarrow$  total of 8 solutions in [4, 4]