mathematics-class-11-unit-07-chapter-06-linear-inequalities-l-6-6-tfnfb3-ebec

### <Success>Solution</Success>
<div style='float: left; width:99%;font-size:24px;text-align:left'>
<p>$\begin{aligned} & \begin{array}{l}R_1 \rightarrow R_1-R_3 \\  R_2 \rightarrow R_2+R_3 \\  R_4 \rightarrow R_4-R_3\end{array}\left[\begin{array}{cccc|c}1 & 0 & 0 & -1 & 0 \\  0 & 1 & 0 & 1 & 8 \\  0 & 0 & 1 & 1 & 13 \\  0 & 0 & 0 & -2 & -8\end{array}\right] \\  & R_4 \rightarrow \frac{-1}{2} R_4\left[\begin{array}{cccc|c}1 & 0 & 0 & -1 & 0 \\  0 & 1 & 0 & 1 & 8 \\  0 & 0 & 1 & 1 & 13 \\  0 & 0 & 0 & 1 & 4\end{array}\right] \\  & \begin{array}{l}R_1 \rightarrow R_1+R_4 \\  R_2 \rightarrow R_2-R_4 \\  R_3 \rightarrow R_3-R_4\end{array}\left[\begin{array}{llll|l}1 & 0 & 0 & 0 & 4 \\  0 & 1 & 0 & 0 & 4 \\  0 & 0 & 1 & 0 & 9 \\  0 & 0 & 0 & 1 & 4\end{array}\right] \\  & \end{aligned}$          <br>
$a=4, b=4, c=9, d=4$</p>
</div>
<div style='float: right; width:01%;'>
<p><img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/mathematics-class-11-Unit-07-chapter-06-linear-inequalities-l-6-6-tfnfb3-ebec-4.jpg">
<img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/mathematics-class-11-Unit-07-chapter-06-linear-inequalities-l-6-6-tfnfb3-ebec-4a.jpg">

### <Success>Problem 2</Success>
<div style='float: left; width:99%;font-size:25px;text-align:left'>
<p>Solve the system</p>
<p>$
\begin{gathered}
x-3 y+2 z=0 \\ 2 x-5 y-2 z=0 \\ 4 x-11 y+2 z=0
\end{gathered}
$</p>
<p><strong>Solution</strong> <br>
$\begin{aligned} & \left[\begin{array}{ccc}1 & -3 & 2 \\  2 & -5 & -2 \\  4 & -11 & 2\end{array}\right] \\  & \begin{array}{l}R_2 \rightarrow R_2-2 R_1 \\  R_3 \longrightarrow R_3-4 R_1\end{array}\left[\begin{array}{ccc}1 & -3 & 2 \\  0 & 1 & -6 \\  0 & 1 & -6\end{array}\right] \\  & \end{aligned}$</p>
</div>
<div style='float: right; width:01%;'>
<p><img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/mathematics-class-11-Unit-07-chapter-06-linear-inequalities-l-6-6-tfnfb3-ebec-5.jpg">
<img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/mathematics-class-11-Unit-07-chapter-06-linear-inequalities-l-6-6-tfnfb3-ebec-5a.jpg"></p>

### <Success>Solution</Success>
<div style='float: left; width:99%;font-size:27px;text-align:left'>
<p>$\begin{aligned}
& \left[\begin{array}{cccc|c}
1 & -1 & 2 & -3 & 9 \\
4 & 0 & 11 & -10 & 46 \\
3 & -1 & 4 & -6 & 27
\end{array}\right] \\
& R_2 \rightarrow R_2-4 R_1 , \quad R_3 \rightarrow R_3-3R_1 \left[\begin{array}{cccc|c}
1 & -1 & 2 & -3 & 9 \\
0 & 4 & 3 & 2 & 10 \\
0 & 2 & 2 & 3 & 0
\end{array}\right]
\end{aligned}$</p>
<p>$R_2 \rightarrow \frac{1}{4} R_2\left[\begin{array}{cccc|c}1 & -1 & 2 & -3 & 9 \\  0 & 1 & 3 / 4 & 1 / 2 & 5 / 2 \\  0 & 2 & 2 & 3 & 0\end{array}\right]$</p>
<p>$\begin{aligned} & R_1 \rightarrow R_1+R_2 \\  & R_3 \rightarrow R_3-2 R_2\end{aligned}\left[\begin{array}{cccc|c}1 & 0 & 11 / 4 & -5 / 2 & 23 / 2 \\  0 & 1 & 3 / 4 & 1 / 2 & 5 / 2 \\  0 & 0 & 1 / 2 & 2 & -5\end{array}\right]$</p>
</div>
<div style='float: right; width:01%;'>
<p><img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/mathematics-class-11-Unit-07-chapter-06-linear-inequalities-l-6-6-tfnfb3-ebec-8.jpg"></p>

### <Success>Solution</Success>
<div style='float: left; width:99%;font-size:25px;text-align:left'>
<p>$\begin{aligned} & R_3 \rightarrow 2 R_3\left[\begin{array}{cccc|c}1 & 0 & 11 / 4 & -5 / 2 & 23 / 2 \\  0 & 1 & 3 / 4 & 1 / 2 & 5 / 2 \\  0 & 0 & 1 & 4 & -10\end{array}\right] \\  & R_1 \rightarrow R_1-\frac{11}{4} R_3 ;R_2 \rightarrow R_2-\frac{3}{4} R_3\left[\begin{array}{cccc|c}1 & 0 & 0 & -\frac{5}{2}-11 & \frac{23}{2}+\frac{55}{2} \\  0 & 1 & 0 & \frac{1}{2}-3 & \frac{5}{2}-\frac{15}{2} \\  0 & 0 & 1 & 4 & -10\end{array}\right] \\  & \left[\begin{array}{cccc|c}1 & 0 & 0 & -27 / 2 & 39 \\  0 & 1 & 0 & -5 / 2 & -5 \\  0 & 0 & 1 & 4 & -10\end{array}\right]\end{aligned}$</p>
</div>
======
### <Success>Solution</Success>
<div style='float: left; width:99%;font-size:25px;text-align:left'>
<p>$w$ - idependent variable. Let $w=\lambda$.</p>
<p>$
\begin{aligned}
& t-\frac{27}{2} w=39 \Rightarrow t=\frac{27}{2} \lambda+39 \\
& u-\frac{5}{2} w=-5 \Rightarrow u=\frac{5}{2} \lambda-5 \\
& v+4 w=-10 \Rightarrow v=-4 \lambda-10 \\
& \left(\frac{27}{2} \lambda+39, \frac{5}{2} \lambda-5,-4 \lambda-10, \lambda\right) .
\end{aligned}
$</p>
</div>
<div style='float: right; width:01%;'>
<p><img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/mathematics-class-11-Unit-07-chapter-06-linear-inequalities-l-6-6-tfnfb3-ebec-9.jpg">
<img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/mathematics-class-11-Unit-07-chapter-06-linear-inequalities-l-6-6-tfnfb3-ebec-9a.jpg"></p>

### <Success>Solution</Success>
<div style='float: left; width:99%;font-size:25px;text-align:left'>
<p>$\begin{aligned}
& R_2 \rightarrow \frac{-1}{3} R_2\left[\begin{array}{ccc|c}
1 & 2 & 3 & 1 \\
0 & 1 & 1 & 0 \\
0 & -5 & -6 & -1
\end{array}\right] \\
& \begin{array}{l}
R_1 \rightarrow R_1-2 R_2 \\
R_3 \rightarrow R_3+5 R_2
\end{array}\left[\begin{array}{ccc|c}
1 & 0 & 1 & 1 \\
0 & 1 & 1 & 0 \\
0 & 0 & -1 & -1
\end{array}\right] \\
& R_3 \rightarrow-R_3\left[\begin{array}{lll|l}
1 & 0 & 1 & 1 \\
0 & 1 & 1 & 0 \\
0 & 0 & 1 & 1
\end{array}\right] \\
& \begin{array}{l}
R_1 \rightarrow R_1-R_3 \\
R_2 \rightarrow R_2-R_3
\end{array}\left[\begin{array}{ccc|c}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & -1 \\
0 & 0 & 1 & 1
\end{array}\right]
&
\end{aligned}$</p>
<p>$x=0, y=-1, z=1 .$</p>
</div>
<div style='float: right; width:01%;'>
<p><img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/mathematics-class-11-Unit-07-chapter-06-linear-inequalities-l-6-6-tfnfb3-ebec-11.jpg">
<img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/mathematics-class-11-Unit-07-chapter-06-linear-inequalities-l-6-6-tfnfb3-ebec-11a.jpg"></p>

### <Success>Problem 5</Success>
<div style='float: left; width:99%;font-size:25px;text-align:left'>
Solve the systems
<p>$
\begin{aligned}
x+i y & =0 \\
-i x+z & =0 \\
y-z & =0
\end{aligned}
$</p>
<p><strong>Solution</strong>   $\left[\begin{array}{ccc}1 & i & 0 \\  -i & 0 & 1 \\  0 & 1 & -1\end{array}\right]$
$
R_2 \rightarrow R_2+i R_1\left[\begin{array}{ccc}
1 & i & 0 \\
0 & -1 & 1 \\
0 & 1 & -1
\end{array}\right]
$</p>
<p>$
\begin{aligned}
& R_2 \rightarrow-R_2\left[\begin{array}{ccc}
1 & i & 0 \\
0 & 1 & -1 \\
0 & 1 & -1
\end{array}\right] \\
& R_1 \rightarrow R_1-i R_2 ;  R_3 \rightarrow R_3-R_2\left[\begin{array}{ccc}
1 & 0 & i \\
0 & 1 & -1 \\
0 & 0 & 0
\end{array}\right]
\end{aligned}
$</p>
</div>
<p>======</p>
<h3 id="successsolutionsuccess"><Success>Solution</Success></h3>
<div style='float: left; width:99%;font-size:30px;text-align:left'>
<p>$z$ - independent variable.</p>
<p>$
\begin{aligned}
& z=\lambda . \\
& x+i z=0 \Rightarrow x=-i \lambda \\
& y-z=0 \Rightarrow y=\lambda . \\
& \quad(-i \lambda, \lambda, \lambda), \quad\lambda \in \mathbb{1R} .
\end{aligned}
$</p>
</div>
<div style='float: right; width:01%;'>
<p><img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/mathematics-class-11-Unit-07-chapter-06-linear-inequalities-l-6-6-tfnfb3-ebec-12.jpg">
<img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/mathematics-class-11-Unit-07-chapter-06-linear-inequalities-l-6-6-tfnfb3-ebec-12a.jpg"></p>

### <Success>Solution</Success>
<div style='float: left; width:99%;font-size:25px;text-align:left'>
$\begin{aligned} & \begin{array}{l}R_1 \rightarrow R_3-2 R_1 \\\  R_2 \rightarrow R_3-2 R_2\end{array}\left[\begin{array}{ccc|c}0 & 1 & \lambda-6 & \mu-12 \\\  0 & -1 & \lambda-10 & \mu-18 \\\  2 & 5 & \lambda & \mu\end{array}\right] \\\  & R_1 \rightarrow R_1+R_2\left[\begin{array}{ccc|c}0 & 0 & 2 \lambda-16 & 2 \mu-30 \\\  0 & -1 & \lambda-10 & \mu-18 \\\  2 & 5 & \lambda & \mu\end{array}\right] \\\  & \end{aligned}$
<p>$
(2 \lambda-16) z=2 \mu-30
$
(or) $(\lambda-8) z=\mu-15$.</p>
<ul>
<li>$\lambda=8 \quad$ & $\mu \in \mathbb{R}$ \ $\{15\}$, the system has no soultion.</li>
<li>$\lambda \neq 8\quad$  & $\mu \in \mathbb{R}$, the system has a unique solution.</li>
<li>$\lambda=8\quad$ & $\mu=15$, the system has infinite number of solutions.</li>
</ul>
</div>
<div style='float: right; width:01%;'>
<p><img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/mathematics-class-11-Unit-07-chapter-06-linear-inequalities-l-6-6-tfnfb3-ebec-14.jpg"></p>

### <Success>Solution</Success>
<div style='float: left; width:99%;font-size:26px;text-align:left'>
<p>$\left[\begin{array}{ccc}
4 \beta^2+\gamma^2 & 2 \beta^2-\gamma^2 & -2 \beta^2+\gamma^2 \\
2 \beta^2-\gamma^2 & \alpha^2+\beta^2+\gamma^2 & \alpha^2-\beta^2-\gamma^2 \\
-2 \beta^2+\gamma^2 & \alpha^2-\beta^2-\gamma^2 & \alpha^2+\beta^2+\gamma^2
\end{array}\right]=\left[\begin{array}{lll}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array}\right]$</p>
<p>$\begin{aligned}
& 4 \beta^2+\gamma^2=1 \\
& 2 \beta^2-\gamma^2=0 \\
& \alpha^2+\beta^2+\gamma^2=1 \\
& \alpha^2-\beta^2-\gamma^2=0 \\
& {\left[\begin{array}{ccc|c}
0 & 4 & 1 & 1 \\
0 & 2 & -1 & 0 \\
1 & 1 & 1 & 1 \\
1 & -1 & -1 & 0
\end{array}\right]}
\end{aligned}$</p>
</div>
<p>======</p>
<h3 id="successsolutionsuccess-1"><Success>Solution</Success></h3>
<div style='float: left; width:99%;font-size:30px;text-align:left'>
<p>$\begin{aligned}
& R_1 \longleftrightarrow R_3 , R_2 \longleftrightarrow R_4\left[\begin{array}{ccc|c}
1 & 1 & 1 & 1 \\
1 & -1 & -1 & 0 \\
0 & 4 & 1 & 1 \\
0 & 2 & -1 & 0
\end{array}\right] \\
& R_2 \rightarrow R_2-R_1\left[\begin{array}{ccc|c}
1 & 1 & 1 & 1 \\
0 & -2 & -2 & -1 \\
0 & 4 & 1 & 1 \\
0 & 2 & -1 & 0
\end{array}\right] \\
& R_2 \rightarrow \frac{1}{-2} R_2\left[\begin{array}{ccc|c}
1 & 1 & 1 & 1 \\
0 & 1 & 1 & 1 / 2 \\
0 & 4 & 1 & 1 \\
0 & 2 & -1 & 0
\end{array}\right] \\
&
\end{aligned}$</p>
</div>
<div style='float: right; width:01%;'>
<p><img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/mathematics-class-11-Unit-07-chapter-06-linear-inequalities-l-6-6-tfnfb3-ebec-19.jpg"></p>
<p><img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/mathematics-class-11-Unit-07-chapter-06-linear-inequalities-l-6-6-tfnfb3-ebec-19a.jpg"></p>

### <Success>Solution</Success>
<div style='float: left; width:99%;font-size:22px;text-align:left'>
<p>$\begin{aligned} & \begin{array}{l}R_1 \rightarrow R_1-R_2 \\  R_3 \rightarrow R_3-4 R_2 \\  R_4 \rightarrow R_4-2 R_2\end{array}\left[\begin{array}{ccc|c}1 & 0 & 0 & 1 / 2 \\  0 & 1 & 1 & 1 / 2 \\  0 & 0 & -3 & -1 \\  0 & 0 & -3 & -1\end{array}\right] \\  & R_3 \rightarrow \frac{1}{-3} R_3\left[\begin{array}{ccc|c}1 & 0 & 0 & 1 / 2 \\  0 & 1 & 1 & 1 / 2 \\  0 & 0 & 1 & 1 / 3 \\  0 & 0 & -3 & -1\end{array}\right] \\  & \begin{array}{l}R_2 \rightarrow R_2-R_3 \\ R_4 \rightarrow R_4+3 R_3\end{array}\left[\begin{array}{lll|l}1 & 0 & 0 & 1 / 2 \\  0 & 1 & 0 & 1 / 6 \\  0 & 0 & 1 & 1 / 3 \\  0 & 0 & 0 & 0\end{array}\right] \\  & \end{aligned}$</p>
<p>$\begin{array}{ll}
\alpha^2=1 / 2, & \beta^2=1 / 6, \quad \gamma^2=1 / 3 \\
\alpha= \pm 1 / \sqrt{2} & \beta= \pm 1 / \sqrt{6} \quad \gamma= \pm 1 / \sqrt{3}
\end{array}$</p>
</div>
<div style='float: right; width:01%;'>
<p><img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/mathematics-class-11-Unit-07-chapter-06-linear-inequalities-l-6-6-tfnfb3-ebec-20.jpg">
<img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/mathematics-class-11-Unit-07-chapter-06-linear-inequalities-l-6-6-tfnfb3-ebec-20a.jpg"></p>
</div>
<p>======</p>
<h3 id="successthank-yousuccess"><Success>Thank you</Success></h3>
<div style='float: left; width:99%;font-size:26px;text-align:left'>
</div>