Class 12th

$A=\left[\begin{array}{cc}\hfill -1\hfill & \hfill 2\hfill \\ \hfill 1\hfill & \hfill -1\hfill \end{array}\right]$ $EA=\left[\begin{array}{cc}\hfill a\hfill & \hfill c\hfill \\ \hfill b\hfill & \hfill d\hfill \end{array}\right]\left[\begin{array}{cc}\hfill -1\hfill & \hfill 2\hfill \\ \hfill 1\hfill & \hfill -1\hfill \end{array}\right]$

$=\left[\begin{array}{cc}\hfill -a+c\hfill & \hfill 2a-c\hfill \\ \hfill -b+d\hfill & \hfill 2b-d\hfill \end{array}\right]$

For a = c For $\begin{array}{cc}\hfill -a+c=0\hfill & \hfill 2a-c=1\hfill \end{array}]\to a=1,c=1E=\left[\begin{array}{cc}\hfill 1\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 1\hfill \end{array}\right]$

d = b + 1, d = 1, b = 0

$\begin{array}{cc}\hfill b+d=1\hfill & \hfill 2b-d=-1\hfill \end{array}]\to b=0,d=1R_1\to R_1\to R_2\left[\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill \end{array}\right]$

For $\begin{array}{cc}\hfill -a+c=1\hfill & \hfill 2a-c=1\hfill \end{array}]\to a=0,c=1$

$For\begin{array}{cc}\hfill -a+c=-1\hfill & \hfill 2a-c=2\hfill \end{array}]\to a=1,c=0$

$\begin{array}{cc}\hfill -b+d=-2\hfill & \hfill 2b-d=7\hfill \end{array}]\to b=5,d=3\left[\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \hfill 5\hfill & \hfill 3\hfill \end{array}\right]\left[\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill \end{array}\right]$

R₂ → 5R₁ + 3R₂

For $For\begin{array}{cc}\hfill -a+c=-1\hfill & \hfill 2a-c=2\hfill \end{array}]\to a=1,c=1$

$\begin{array}{cc}\hfill -b+d=-1\hfill & \hfill 2b-d=3\hfill \end{array}]\to b=2,d=1$

(A) R₁ → R₁ + R₂

(B) R₂ → R₂ + 2R₁ $\left[\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \hfill 2\hfill & \hfill 1\hfill \end{array}\right]\left[\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill \end{array}\right]$

(C) R₂ → 3R₂ + 5R₁

${\mathcal{l}}_1:$

$\overrightarrow{r}=\frac{1}{8}\widehat{i}+\lambda \left(-\frac{1}{8},\frac{1}{4\sqrt[]{2}},0\right)$

$\overrightarrow{r}=-\frac{1}{8}\widehat{i}+\mu \left(\frac{1}{8},0,-\frac{-1}{6\sqrt[]{3}}\right)$

$\overrightarrow{n}=\left|\begin{array}{ccc}\hfill \widehat{i}\hfill & \hfill \widehat{j}\hfill & \hfill \widehat{k}\hfill \\ \hfill \frac{-1}{8}\hfill & \hfill \frac{1}{4\sqrt[]{2}}\hfill & \hfill 0\hfill \\ \hfill \frac{1}{8}\hfill & \hfill 0\hfill & \hfill \frac{-1}{6\sqrt[]{3}}\hfill \end{array}\right|$

$=\left(-\frac{1}{24\sqrt[]{6}},\frac{-1}{48\sqrt[]{3}},-\frac{1}{32\sqrt[]{2}}\right)$

$d=projectionofA\overrightarrow{C}on\overrightarrow{n}=\left(-\frac{2}{8},0,0\right).\frac{\left(4,2\sqrt[]{2},3\sqrt[]{3}\right)}{\sqrt[]{16}+8+27}=\frac{-1}{\sqrt[]{16+8+27}}$

d²⁼ $\frac{1}{51}$                                       

$f\left(x\right)=\left|\left(x-1\right)\left(x+1\right)\left(x-3\right)\right|+x-3$  

For    $x\in \left[-1,1\right]\cup \left[3,\infty \right)$

y = x² (x – 3)

  $\frac{dy}{dx}=2x\left(x-3\right)+x^2=3x^2-6x=3x\left(x-2\right)$

Number of max = 2

Number of min = 1

  $\overrightarrow{a}*\overrightarrow{b}=\left|\begin{array}{ccc}\hfill \widehat{i}\hfill & \hfill \widehat{j}\hfill & \hfill \widehat{k}\hfill \\ \hfill a_1\hfill & \hfill a_2\hfill & \hfill a_3\hfill \\ \hfill 1\hfill & \hfill 1\hfill & \hfill \lambda \hfill \end{array}\right|$              

$=\left(\lambda a_2-a_3,a_3-a_1\lambda ,a_1-a_2\right)$

$=\left(13,-1,-4\right)$

$\overrightarrow{a}.\overrightarrow{b}=-21$

$\Rightarrow a_1+a_2+a_3\lambda =-21$

$\lambda a_2-a_3=13\Rightarrow \lambda =\frac{13+a_3}{a_2}=\frac{a_3+1}{a_1}$

$\overrightarrow{b}-\overrightarrow{a}=\left(3,-1,10\right)$

$\overrightarrow{b}+\overrightarrow{a}=\left(-1,3,-4\right)$

= 11 + 3 + 14               

  ${\displaystyle \underset{3}{\overset{b}{\int }}\frac{1}{3}\left(\frac{1}{x^2-4}+\frac{1}{x^2-1}\right)}dx$

$=\frac{1}{3}{\left[\frac{1}{4}\mathcal{l}n\frac{x-2}{x+2}-\frac{1}{2}\mathcal{l}n\frac{x-1}{x+1}\right]}_3^6$

$=\mathcal{l}n{\left(\frac{49}{40}\right)}^{\frac{1}{12}}$

$\frac{5\left(b-2\right)}{b+2}/\frac{4{\left(b-1\right)}^2}{{\left(b+1\right)}^2}=\frac{49}{40}$

$\frac{4b}{2\left(b^2+1\right)}=\frac{-b+198}{99b-2}$

->b³ – 3b – 198 = 0

b = 6

  $f\left(x\right)=\left[2x^2+1\right]$

  $g\left(x\right)=\{\begin{array}{l}2x-3,x<0\\ 2x+3,x\ge 0\end{array}$

  $f\left(g\left(x\right)\right)=\left[2{\left(g\left(x\right)\right)}^2\right]+1$              

 for 0 < x < 1

-3 < 2x 3 < -1

2 < 2 (2x 3)² < 18

31 + 15 = 46

  $ta{n}^{-1}\left(\frac{\frac{1}{\sqrt[]{2}-1}}{\frac{1}{\sqrt[]{2}}}\right)$

$=ta{n}^{-1}\left(1-\sqrt[]{2}\right)$

$=ta{n}^{-1}\left(\sqrt[]{2}-1\right)$

$=-\theta$

$tan\theta =\sqrt[]{2}-1$

$tan2\theta =\frac{2\left(\sqrt[]{2}-1\right)}{1-{\left(\sqrt[]{2}-1\right)}^2}$

$=\frac{2\left(\sqrt[]{2}-1\right)}{2\left(\sqrt[]{2}-1\right)}$

tan 2 θ = 1

$\theta =\frac{\pi }{8}$                

$\sim \left(\left(\sim \left(\sim q\right)\wedge p\right)\right)\vee \left(\left(\sim p\right)\vee q\right)$

$=-\left(\left(q\vee \sim p\right)\vee \left(\sim p\vee q\right)\right)$

$=p\vee \sim q$

$\sim \left(p\to q\right)$

$=\sim \left(\sim p\vee q\right)=p\wedge \sim q$

x + y + z = 48

x, y, z  $\in$ {2, 4, 8, 16, 32, 32}

16 + 16 + 16 -> P_{1 $={\left(\frac{1}{16}\right)}^3$}  

 $32+8+8\to +P_2=\frac{1}{32}.{\left(\frac{1}{8}\right)}^2.\frac{3!}{2!}$

$P=\frac{1}{2^{12}}+\frac{3}{2^{11}}$

$=\frac{1+6}{2^{12}}=\frac{7}{2^{12}}$  

$2sin12°-sin72°$

$=sin12°-\left(sin72°-sin12°\right)$

$=sin12°-2cos42°sin30°$

$=2cos30°\left(-sin18°\right)$

$=-\sqrt[]{3}sin18°$

$=-\sqrt[]{3}\left(\frac{\sqrt[]{5}-1}{4}\right)$