Exercise 8.1
Question 7
निम्नलिखित समीकरण निकायों को आव्यूहों का प्रयोग कर हल करें।
[Solve the following system of equations using matrices]
(i)
x-y-z=4
2c+y-3z=0
x+y-z=2
Sol :
(ii)
x-y+2z=7
3x+4y-5z=-5
2x-y+3z=12
Sol :
(iii)
3x-2y+3z=8
2x+y-z=1
4x-3y+2z=4
Sol :
(iv)
2x+y+z=1
$x-2y-z=\frac{3}{2}$
3y-5z=9
Sol :
(v)
2x+3y+3z=5
x-2y+z=-4
3x-y-2z=3
Sol :
(vi)
3x+y+z=3
2x-y-z=2
-x-y+z=1
Sol :
(vii)
2x+y+2z=3
x+y+2z=2
2x+3y-z=-2
Sol :
Question 8
निम्नलिखित समीकरण निकाय की संगतता की जाँच करें
[Examine the consistency of the following system of equations:]
(i)
x+3 y=5
2x+6y=8
Sol :
<to be added>
Let $A=\left[\begin{array}{ll}1 & 3 \\ 2 & 6\end{array}\right]$ , $x=\left[\begin{array}{l}x \\ y\end{array}\right]$,$B=\left[\begin{array}{l}5 \\ 8\end{array}\right]$
|A|=6-6=0
$a d j A=\left[\begin{array}{cc}6 & -2 \\ -3 & 1\end{array}\right]^{\prime}=\left[\begin{array}{cc}6 & -3 \\ -2 & 1\end{array}\right]$
$(a dj A) B=\left[\begin{array}{cc}6 & -3 \\ -2 & 1\end{array}\right]\left[\begin{array}{l}5 \\ 8\end{array}\right]$
$=\left[\begin{array}{cc}30 -24 \\ -10+8\end{array}\right]=\left[\begin{array}{c}6 \\ -2\end{array}\right]$
≠0
<to be added>
(ii)
x+2y=2
2x+3y=3
Sol :
<to be added>
माना
$A=\left[\begin{array}{ll}1 & 2 \\ 2 & 3\end{array}\right]$, $B=\left[\begin{array}{l}2 \\ 3\end{array}\right]$,$x=\left[\begin{array}{l}x \\ y\end{array}\right]$
|A|=3-4=-1≠0
∴संतता, अद्वितीय हल
(iii)
2x-y=5
x+y=4
(iv)
x+y=2
5 x+5y=10
Sol :
<to be added>
माना
$A=\left[\begin{array}{ll}1 & 1 \\ 5 & 5\end{array}\right]$ , $B=\left[\begin{array}{l}2 \\ 10\end{array}\right]$ , $X=\left[\begin{array}{l}x \\ y\end{array}\right]$
|A|=5-5=0
$a d{j} A=\left[\begin{array}{cc}5 & -5 \\ -1 & 1\end{array}\right]^{\prime}=\left[\begin{array}{cc}5 & -1 \\ -5 & 1\end{array}\right]$
$(a dj A) B=\left[\begin{array}{cc}5 & -1 \\ -5 & 1\end{array}\right]\left[\begin{array}{l}2 \\ 10\end{array}\right]$
$=\left[\begin{array}{r}10-10 \\ -10+10\end{array}\right]=\left[\begin{array}{l}0 \\ 0\end{array}\right]=0$
<to be added>
(v)
5x-y+4z=5
2x+3y+5z=2
5x-2y+6z=-1
(vi)
x+y+z=1
2x+3y+2z=2
ax+ay+2az=4
Sol :
<to be added>
माना
$|A|=\left|\begin{array}{lll}1 & 1 & 1 \\ 2 & 3 & 2 \\ a & a & 2 a\end{array}\right|$
$=\left|\begin{array}{ccc}0 & 0 & 1 \\ -1 & 1 & 2 \\ 0 & -a & 2 a\end{array}\right|$
$=1\left|\begin{array}{cc}-1 & 1 \\ 0 & -a\end{array}\right|$
=a≠0
$C_{1} \rightarrow C_{1}-C_{2}$ ,$C_{2}\rightarrow C_{2}-C_{3}$
∴संतता, अद्वितीय हल
(i)
x+3 y=5
2x+6y=8
Sol :
<to be added>
Let $A=\left[\begin{array}{ll}1 & 3 \\ 2 & 6\end{array}\right]$ , $x=\left[\begin{array}{l}x \\ y\end{array}\right]$,$B=\left[\begin{array}{l}5 \\ 8\end{array}\right]$
|A|=6-6=0
$a d j A=\left[\begin{array}{cc}6 & -2 \\ -3 & 1\end{array}\right]^{\prime}=\left[\begin{array}{cc}6 & -3 \\ -2 & 1\end{array}\right]$
$(a dj A) B=\left[\begin{array}{cc}6 & -3 \\ -2 & 1\end{array}\right]\left[\begin{array}{l}5 \\ 8\end{array}\right]$
$=\left[\begin{array}{cc}30 -24 \\ -10+8\end{array}\right]=\left[\begin{array}{c}6 \\ -2\end{array}\right]$
≠0
<to be added>
(ii)
x+2y=2
2x+3y=3
Sol :
<to be added>
माना
$A=\left[\begin{array}{ll}1 & 2 \\ 2 & 3\end{array}\right]$, $B=\left[\begin{array}{l}2 \\ 3\end{array}\right]$,$x=\left[\begin{array}{l}x \\ y\end{array}\right]$
|A|=3-4=-1≠0
∴संतता, अद्वितीय हल
(iii)
2x-y=5
x+y=4
(iv)
x+y=2
5 x+5y=10
Sol :
<to be added>
माना
$A=\left[\begin{array}{ll}1 & 1 \\ 5 & 5\end{array}\right]$ , $B=\left[\begin{array}{l}2 \\ 10\end{array}\right]$ , $X=\left[\begin{array}{l}x \\ y\end{array}\right]$
|A|=5-5=0
$a d{j} A=\left[\begin{array}{cc}5 & -5 \\ -1 & 1\end{array}\right]^{\prime}=\left[\begin{array}{cc}5 & -1 \\ -5 & 1\end{array}\right]$
$(a dj A) B=\left[\begin{array}{cc}5 & -1 \\ -5 & 1\end{array}\right]\left[\begin{array}{l}2 \\ 10\end{array}\right]$
$=\left[\begin{array}{r}10-10 \\ -10+10\end{array}\right]=\left[\begin{array}{l}0 \\ 0\end{array}\right]=0$
<to be added>
(v)
5x-y+4z=5
2x+3y+5z=2
5x-2y+6z=-1
(vi)
x+y+z=1
2x+3y+2z=2
ax+ay+2az=4
Sol :
<to be added>
माना
$|A|=\left|\begin{array}{lll}1 & 1 & 1 \\ 2 & 3 & 2 \\ a & a & 2 a\end{array}\right|$
$=\left|\begin{array}{ccc}0 & 0 & 1 \\ -1 & 1 & 2 \\ 0 & -a & 2 a\end{array}\right|$
$=1\left|\begin{array}{cc}-1 & 1 \\ 0 & -a\end{array}\right|$
=a≠0
$C_{1} \rightarrow C_{1}-C_{2}$ ,$C_{2}\rightarrow C_{2}-C_{3}$
∴संतता, अद्वितीय हल
Question 9
निम्नलिखित समीकरण निकाय की संगतता की जाँच करें तथा यदि ये संगत हैं तो हल करें।
[Examine the consistency of the following system of equations and consistent solve them.
(i)
2x-y+3z=5
3x+2y-z=7
4x+5y-5z=9
Sol :
(ii)
3x-y-2z=2
2y-z=-1
3x-5y=3
Sol :
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