Maths

89 – k + k + 98 – k + x = 100

=> $87\le k\le 89$

$f\left(x\right)=\frac{1-x}{1+x},0<x<1$ (on simplification)

$f\text{'}\left(x\right)=\frac{-2}{{\left(1+x\right)}^2}$

${\left(1-x\right)}^{2}f\text{'}\left(x\right)=-2{\left(f\left(x\right)\right)}^2$

Given equation

$\Rightarrow \frac{cosx}{1+sinx}=\frac{2\left|sinx\right|cosx}{\left|co{s}^{2}x-si{n}^{2}x\right|}$

$\Rightarrow \left|1-2si{n}^{2}x\right|=2\left|sinx\right|\left(1+sinx\right)$

Put sin x = t

then equation,

$\left|1-2t^2\right|=2\left|t\right|\left(1+t\right),t\in \left(-1,1\right)-\left\{\pm \frac{1}{\sqrt[]{2}}\right\}$

Case I : For $\ge \frac{1}{\sqrt[]{2}}$ $\frac{}{\text{exception 25:}}$

the equation has no solution

Case II : For $0\le t<\frac{1}{\sqrt[]{2}}$ $\frac{}{\text{exception 25:}}$

Equation t = $\frac{\sqrt[]{5}-1}{4}\Rightarrow x=18°$ $\frac{\text{exception 25:}}{}$

Case III : $For\frac{-1}{\sqrt[]{2}}<t<0$ $\frac{}{\text{exception 25:}}$

Equation t = $\frac{-1}{2}\Rightarrow x=-30°$ $\frac{}{}$

Case IV : For $T<-\frac{1}{\sqrt[]{2}}$ $\frac{}{\text{exception 25:}}$

Equation t = $\frac{\sqrt[]{5}+1}{4}\Rightarrow$ $\frac{\text{exception 25:}}{}$ x = 54°

Sum of solutions = 18° - 30° - 54° = 66°

Total number = 20 + 8 + 24 = 52

$f\left(0^{-}\right)=f\left(0\right)=-a$

$f\left(0^{+}\right)=\underset{x\to 0^{+}}{lim}\frac{sin2x\left(\frac{1}{cos2x}-1\right)}{b{x}^3}$ $=\frac{4}{b}$

ab = 4 10 – ab = 14

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

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

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

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

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

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

$=-8+21=13$

$x^2+y^2+{\left(x-1\right)}^2+y^2+x^2+{\left(y-1\right)}^2+{\left(x-1\right)}^2+{\left(y-1\right)}^2=18$

$\Rightarrow x^2+y^2-x-y-\frac{7}{2}=0$

$r^2=\frac{1}{4}+\frac{1}{4}+\frac{7}{2}=4$

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

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

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

$d^2=\frac{1950}{75}=26$  

$LHS={\displaystyle \sum _{r=1}^{15}r\left(r!\right)}$

$=\sum \left(r+1-1\right)r!={\displaystyle \sum _{r=1}^{15}\left(r+1\right)!-r!}$

$=\left(2!-1!\right)+\left(3!-2!\right)+....+\left(16!-15!\right)$

$\begin{array}{l}=16!-1\\ {=}^{16}{P}_{16}-1\end{array}$