Maths

  $co{s}^{3}3x+co{s}^{3}5x={\left(2cos4xcosx\right)}^3$  

$={\left(cos5x+cos3x\right)}^3$            

$co{s}^{3}3x+co{s}^{3}5x$            

$\Rightarrow cosx.cos3x.cos4x.cos5x=0$            

$\therefore x=\left(2n+1\right)\frac{\pi }{2},\left(2n+1\right)\frac{\pi }{6},\left(2n+1\right)\frac{\pi }{8},\left(2n+1\right)\frac{\pi }{10}$            

$\Rightarrow$ Smallest +ve values of x is $\frac{\pi }{10}$  i.e. 18°.

Mean $=np$ , variance $=npq$

$np-npq=1;p+q=1$

$n^2{p}^2-n^2{p}^2{q}^2=11$

We get   $q=\frac{5}{6},p=\frac{1}{6},n=36\mathrm{}$  Probability of ' 3 ' success $={}^{36}{C}_3{\left(\frac{1}{6}\right)}^3{\left(\frac{5}{6}\right)}^{33}$

$\bar{a}+\bar{b}+\bar{c}+\bar{d}=0$

Magnitude of all vectors will be same as well as angle between these vectors will also be same. $\bar{a}+\bar{b}+\bar{c}=-\bar{d}$

$\begin{array}{c}|\bar{a}{|}^2+|\bar{b}{|}^2+|\bar{c}{|}^2+2\bar{a}\cdot \bar{b}+2\bar{b}\cdot \bar{c}+2\bar{c}\cdot \bar{a}=|d{|}^2\\ \mathrm{}1+1+1+2\mathrm{c}\mathrm{o}\mathrm{s}?\alpha +2\mathrm{c}\mathrm{o}\mathrm{s}?\alpha +2\mathrm{c}\mathrm{o}\mathrm{s}?\alpha =1\\ 6\mathrm{c}\mathrm{o}\mathrm{s}?\alpha =-2\\ \mathrm{c}\mathrm{o}\mathrm{s}?\alpha =-\frac{1}{3}\\ \alpha ={\mathrm{c}\mathrm{o}\mathrm{s}}^{-1}?\left(\frac{-1}{3}\right)=\pi -{\mathrm{c}\mathrm{o}\mathrm{s}}^{-1}?\frac{1}{3}\end{array}$

In the integral J, substitute x + 1 = t

$\Rightarrow dx=dtand{x}^2+2x=\left(t^2-1\right)$            

Now   $J={\displaystyle \underset{1}{\overset{e}{\int }}\frac{e^{\frac{t^2-2}{2}}}{t}}dtandK={\displaystyle \underset{1}{\overset{e}{\int }}t}lnt{e}^{\frac{t^2-2}{2}}dt$

Hence   $\left(J+K\right)={\displaystyle \underset{1}{\overset{e}{\int }}{e}^{\frac{t^2-2}{2}}}\left(\frac{1}{t}+tlnt\right)dt$

$={\left(e^{\frac{t^2-2}{2}}lnt\right)}_{t=1}^{t=e}=e^{\frac{e^2-2}{2}}={\left(\sqrt[]{e}\right)}^{e^2-2}$

TSA  of cube $=6a^2$

$\frac{d}{dt}\left(6a^2\right)=4.8;12a\frac{da}{dt}=4.8;a\frac{da}{dt}=0.4;\frac{dv}{dt}=\frac{d}{dt}\left(a^3\right)\Rightarrow 3a\left(a\frac{da}{dt}\right)$

$3*15*0.4=3*15*\frac{04}{10}\Rightarrow 18$

$?arg\left(z\right)=\pi ifz=x+iy$

$\therefore y=0andx<0$

$\therefore z=-3+i.0\Rightarrow \left|z\right|=3$

P(W) = $\frac{1}{3};$  P(B) =   $\frac{2}{3}$

$\Rightarrow p=\frac{1}{3};q=\frac{2}{3}$           and r = 4 or 5 and n = 5

Use   $P\left(r\right){=}^n{C}_r{p}^r{q}^{n-r}$

 P(4) + P(5)

${=}^5{C}_4{\left(\frac{1}{3}\right)}^4{\left(\frac{2}{3}\right)}^1{+}^5{C}_5{\left(\frac{1}{3}\right)}^5$            

$=\frac{10}{3^5}+\frac{1}{3^5}=\frac{11}{3^5}=\frac{11}{243}$  

${\mathrm{l}\mathrm{o}\mathrm{g}}_{(2x+3)}?x^2<{\mathrm{l}\mathrm{o}\mathrm{g}}_{(2x+3)}?(2x+3)$

Case-I: $0<2x+3<1$  

$x^2>2x+3$

Case-II: $2x+3>1$

$0<x^2<2x+3$  

By solving (1) & (2) $\begin{array}{rr}& \frac{-3}{2}<x<-1\&(x-3)(x+1)>0.\\ & \frac{-3}{2}<x<-1,x>3\end{array}$

Equation (4) has no solution $\frac{-3}{2}<x<-1,x<-1$

.(5) By equation (5) . $x\in \left(\frac{-3}{2},-1\right)$

We obtain solving equation (2) $x>-1\mathrm{}x>-1\mathrm{}\Rightarrow x\in (-\mathrm{1,3})$

or $(x-3)(x+1)<0\mathrm{}-1<x<3,x^2>0\Rightarrow x\ne 0$

$x\in \left(\frac{-3}{2},-1\right)\cup (-\mathrm{1,3})-\left\{0\right\}$

Note that D^r < 0, hence given inequality (1) is true only if $N^r\ge 0$  

  i.e.   $\left(x-8\right)\left(x-2\right)\le 0and{2}^{x-3}>31$

i.e.    $2\le x\le 8and{2}^{x-3}\ge 31$

only x = 8 satisfies both the inequality.