Maths

$l={\displaystyle {\int }_{-3}^{103}\left(\left[sin\left(\pi x\right)\right]+e^{\left[cos\left(2\pi x\right)\right]}\right)}dx$

$\left[sin\pi x\right]$ is periodic with period 2 and

$e^{\left[cos2\pi x\right]}$ is periodic with period 1.

So,

$I=52{\displaystyle {\int }_0^2\left(\left[sin\left(\pi x\right)\right]+e^{\left[cos2\pi x\right]}\right)}dx$

= $\frac{52}{e}$

$P\left(A\cup B\right)=P\left(A\right)+P\left(B\right)-P\left(A\cap B\right)$

$\Rightarrow \frac{1}{2}=\frac{1}{3}+\frac{1}{5}-P\left(A\cap B\right)$

$P\left(\frac{A}{B\text{'}}\right)+P\left(\frac{B}{A\text{'}}\right)$

$=\frac{P\left(A\right)-P\left(A\cap B\right)}{1-P\left(B\right)}+\frac{P\left(B\right)-P\left(A\cap B\right)}{1-P\left(A\right)}=\frac{5}{8}$

$l=\underset{n\to x}{lim}\frac{1}{2^n}\left(\frac{1}{\sqrt[]{1-\frac{1}{2^n}}}+\frac{1}{\sqrt[]{1-\frac{2}{2^n}}}+\frac{1}{\sqrt[]{1-\frac{3}{2^n}}}+.....+\frac{1}{\sqrt[]{1-\frac{2^n-1}{2^n}}}\right)$

Let 2n = t and if n $\infty$ then t $\infty$

$l=\underset{n\to x}{lim}\frac{1}{t}\left({\displaystyle \sum _{r=1}^{t=1}\frac{1}{\sqrt[]{1+\frac{r}{t}}}}\right)$

$={\left[2x^{\frac{1}{2}}\right]}_0^1=2$

$\underset{x\to \frac{x}{4}}{lim}\frac{8\sqrt[]{2}-{\left(cosx+sinx\right)}^7}{\sqrt[]{2}-\sqrt[]{2}sin2x}\left(\frac{0}{0}form\right)$

$=\underset{x\to \frac{x}{4}}{lim}\frac{-7{\left(cosx+sinx\right)}^6\left(-sinx+cosx\right)}{-2\sqrt[]{2}cos2x}$ $\underset{x\to \frac{\pi }{4}}{Lt}\frac{{\left(cosx+sinx\right)}^5\left(co{s}^{2}x-si{n}^{2}x\right)}{2\sqrt[]{2}cos2x}=\frac{7{\left(\sqrt[]{2}\right)}^5}{2\sqrt[]{2}}=14$

${\displaystyle \sum _{n=1}^{21}\frac{3}{\left(4n-1\right)\left(4n+3\right)}}$

$=\frac{3}{4}{\displaystyle \sum _{n=1}^{21}\frac{\left(4n+3\right)-\left(4n-1\right)}{\left(4n-1\right)\left(4n+3\right)}}$

$=\frac{3}{4}\frac{4n+3-2}{3\left(4n+3\right)}=\frac{n}{4n+3}$

for n = 21

$S_{21}=\frac{21}{84+3}=\frac{21}{87}=\frac{7}{29}$

$11\equiv 2mod\left(9\right)$

${\left(11\right)}^{1011}\equiv 2^{1011}mod\left(9\right)$

Again 23 $\equiv$ 1 mod (9)

$\Rightarrow {\left(2^3\right)}^{337}\equiv {\left(-1\right)}^{337}mod\left(9\right)$

$\therefore {\left(11\right)}^{1011}+{\left(1011\right)}^{11}\equiv 8mod\left(9\right)$

$\therefore$ Remainder = 8

f : {1, 3, 5, 7, ….,99} {2, 4, 6, 8, …….100}

$f\left(3\right)\ge f\left(9\right)\ge f\left(15\right)......\ge f\left(99\right)$

$3+\left(n-1\right)6=99\Rightarrow n=17$

cases f (3) > f (9) > f (15) ……. > f (99)

$\therefore$ from the set {2, 4, 6, …., 100}

17 distinct numbers can be selected in 50C17 ways again remaining {1, 5, 7, 11, ….} can map in 33! ways

$\therefore$ total number of such required functions

${=}^{50}{C}_{17}*33!$

$=\frac{50!}{33!17!}*33!{=}^{50}{P}_{33}$

System of equation can be written as

$\left(\begin{array}{ccc}\hfill 2\hfill & \hfill -3\hfill & \hfill 5\hfill \\ \hfill 1\hfill & \hfill 3\hfill & \hfill -1\hfill \\ \hfill 3\hfill & \hfill -1\hfill & \hfill {\lambda }^2-\left|\lambda \right|\hfill \end{array}\right)\left(\begin{array}{l}x\\ y\\ z\end{array}\right)=\left(\begin{array}{l}9\\ -18\\ 16\end{array}\right)$

for no solution

$\frac{9}{10}\left({\lambda }^2-\left|\lambda \right|+3\right)-7=0$

$\Rightarrow 9{\lambda }^2-9\left|\lambda \right|-43=0$

$\Rightarrow \left|\lambda \right|=\frac{9+\sqrt[]{81+36*43}}{18}>0$

$\therefore$ only two possible value of $\lambda$ are there.

$\left|z-3\sqrt[]{2}\right|+\left|z-p\sqrt[]{2}i\right|$ is minimum for z,  $3\sqrt[]{2}\&p\sqrt[]{2}i$ are collinear.

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

$\Rightarrow 18+2p^2=50$

$2p^2=32$

$p=\pm 4$

$\frac{{\left(x-3\right)}^2}{16}+\frac{{\left(y-4\right)}^2}{9}\le 1,x,y\in N,{\left(x-7\right)}^2+{\left(y-4\right)}^2\ge 36,x,y\in R$        

Total number of common point = 1 + 5 + 7 + 5 + 5 + 3 + 1 = 27