Maths

 $\left(\stackrel{?}{a}.\stackrel{?}{c}\right)\stackrel{?}{b}?\left(\stackrel{?}{a}.\stackrel{?}{b}\right)\stackrel{?}{c}$

$=\stackrel{?}{b}+?\stackrel{?}{c}\left(given\right)$

$\stackrel{?}{a}.\stackrel{?}{c}=1,?=?\stackrel{?}{a}.\stackrel{?}{b}$

$?=?\left(3,1,0\right).\left(1,2,1\right)$

= (3 + 2)= 5

A (1, 4, 3)

2x + my + nz = 4

$M\Rightarrow -4+\frac{7m}{2}+\frac{3n}{2}=4$

7m + 3n = 16

$\overrightarrow{AM}=\left(-1,\frac{-1}{2},\frac{-3}{2}\right)$

$\frac{2}{-1}=\frac{m}{\frac{-1}{2}}=\frac{n}{\frac{-3}{2}}$

m = 1, n = 3

Plane : 2x + y + 3z = 4

$cos\theta =\frac{\left|6-1-12\right|}{\sqrt[]{26}\sqrt[]{14}}=\frac{7}{2\sqrt[]{91}}$

AM = $\frac{\left|-2+4+9-4\right|}{\sqrt[]{14}}=\frac{7}{\sqrt[]{14}}$

$c$$o$$s$$\theta$$=$$\frac{AM}{AB}$$\Rightarrow$$A$$B$$=$$\frac{7/\text{exception 25:}}{7/2\text{exception 25:}}$$=$$\frac{2\text{exception 25:}}{\text{exception 25:}}$$=$$2$$\text{exception 25:}$$=$

y – 1 = m(x – 1)

mx – y + 1 – m = 0

$\frac{\left|1-m\right|}{\sqrt[]{m^2+1}}=\frac{4}{\sqrt[]{17}}$

$17{\left(1-m\right)}^2=16\left(m^2+1\right)$

m² – 3m + 1 = 0

$\begin{array}{l}bx+10y-8=0\\ y=\frac{2}{3}x\end{array}\}\Rightarrow \left(b+\frac{20}{3}\right)x=8$

$x=\frac{24}{3b+20},y=\frac{16}{3b+20}$

b = $\frac{2\left(3-2\sqrt[]{2}\right)\left(4+3\sqrt[]{2}\right)}{-2}$

$=\frac{12-8\sqrt[]{2}+9\sqrt[]{2}}{-1}=-\sqrt[]{2}$

b² = 2

$\frac{x^2}{5}+\frac{y^2}{2}=1$

2 = 5 (1 – e²)

$e^2=\frac{3}{5}$

$e$$=$

IF =e^x

$y{e}^x={\displaystyle \int \frac{e^x}{1+e^{2x}}dx}$

${\displaystyle \int \frac{dt}{1+t^2}}$

= tan^-1 (t) + c

$\underset{x\to \infty }{lim}{e}^{x}y=\underset{x\to \infty }{lim}ta{n}^{-1}{e}^x+\frac{\pi }{4}=\frac{3\pi }{4}$

 ${\displaystyle \underset{0}{\overset{\frac{\pi }{2}}{\int }}\frac{dx}{3+2sinx+cosx}Puttan\frac{x}{2}=t\Rightarrow \frac{1}{2}se{c}^2\frac{x}{2}dx=dt}$

$={\displaystyle \underset{0}{\overset{1}{\int }}\frac{2dt}{2t^2+4t+4}={\displaystyle \underset{0}{\overset{1}{\int }}\frac{dt}{{\left(t+1\right)}^2+1}={\left[ta{n}^{-1}\left(t+1\right)\right]}_0^1}=ta{n}^{-1}2-ta{n}^{-1}1}$

 $\underset{x\to 0}{lim}\frac{\alpha e^x+\beta e^{-x}+\gamma sinx}{xsi{n}^{2}x}$

$=\underset{x\to 0}{lim}\frac{\alpha e^x+\beta e^{-x}+\gamma sinx}{x^3}$

$\Rightarrow \alpha +\beta =0,\alpha -\beta +\gamma =0,\frac{\alpha +\beta }{2}=0,\frac{\alpha }{6}-\frac{\beta }{6}-\frac{\gamma }{6}=\frac{2}{3}$

$\Rightarrow \beta =-\alpha \gamma =-2\alpha \alpha -\beta -\gamma =4\Rightarrow \alpha +\alpha +2\alpha =4\Rightarrow \gamma =1$

$\alpha =1,\beta =-1,\gamma =-2$

 $\frac{1}{\left(20-a\right)\left(40-a\right)}+\frac{1}{\left(40-a\right)\left(60-a\right)}+\frac{1}{\left(60-a\right)\left(80-a\right)}+.....+\frac{1}{\left(180-a\right)\left(200-a\right)}=\frac{1}{256}$

LHS = $\frac{1}{20}\left(\frac{1}{20-a}-\frac{1}{40-a}\right)+\frac{1}{20}\left(\frac{1}{40-a}-\frac{1}{60-a}\right)+...+\frac{1}{20}\left(\frac{1}{180-a}-\frac{1}{200-a}\right)$

$=\frac{1}{20}\left(\frac{1}{20-a}-\frac{1}{200-a}\right)=\frac{1}{20}.\frac{180}{\left(20-a\right)\left(200-a\right)}$

$\Rightarrow a^2-220a+4000-2304=0\Rightarrow a^{2}220a+1696=0$

$a=\frac{220\pm 204}{8}=212^2$

 $A_{3*3},B_{3*3}\&AB=0$

$\ne 0\ne 0\Rightarrow \left|A\right|=0\&\left|B\right|=0$ only .

 $z^5+{\left(\overline{z}\right)}^5$

$={\left(2+3i\right)}^5+{\left(2-3i\right)}^5$

$=2\left(32-720+810\right)=244$

R = { (a, b): b = pq, where p, q $\ge 3$ are prime}

$\Rightarrow 60*11=660$

p, q $\in$  {3, 5, 7, 11, 13, 17, 19, 23, 29, 3, 37, 41, 43, 47, 53, 59} total 16

p, q $\in$  {3, 5, 7, 11, 13, 17, 19}

{3, 5, 7, 11, 13, 17, 19}

7 + 3 + 1 = 11