Download App

JUNE 2024 — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsJUNE 20244 Marks
Question
Solve system of linear equation, using matrix method :
$\begin{aligned}
x+y+z & =6 \\
y+3 z & =11 \\
x+z & =2 y
\end{aligned}$
✓

Answer

Let first, second and third numbers be denoted by $x, y$ and $z$, respectively.
Then, according to given conditions, we have
$\begin{array}{l}
x+y+z=6 \\
y+3 z=11 \\
x+z=2 y \text { or } x-2 y+z=0
\end{array}$
This system can be written as $AX = B$, where,
$A =\left[\begin{array}{ccc}1 & 1 & 1 \\ 0 & 1 & 3 \\ 1 & -2 & 1\end{array}\right], X =\left[\begin{array}{l}x \\ y \\ z\end{array}\right]$ and $B =\left[\begin{array}{c}6 \\ 11 \\ 0\end{array}\right]$
Hence, $| A |=1(1+6)-1(0-3)+1(0-1)=9 \neq 0$.
Now we find adj A.
$\begin{array}{l}
A_{11}=1(1+6)=7, \\
A_{12}=-(0-3)=3, \\
A_{13}=-1 \\
A_{21}=-(1+2)=-3,
\end{array}$
$\begin{array}{l}A_{22}=0 \\ A_{23}=-(-2-1)=3 \\ A_{31}=(3-1)=2, A_{32}=-(3-0)=-3, \\ A_{33}=(1-0)=1\end{array}$
Hence, adj $A =\left[\begin{array}{ccc}7 & -3 & 2 \\ 3 & 0 & -3 \\ -1 & 3 & 1\end{array}\right]$
Thus, $\quad A ^{-1}=\frac{1}{|A|} \operatorname{adj}( A )$
$=\frac{1}{9}\left[\begin{array}{ccc}7 & -3 & 2 \\ 3 & 0 & -3 \\ -1 & 3 & 1\end{array}\right]$
Since $\quad X=A^{-1} B$
$X =\frac{1}{9}\left[\begin{array}{ccc}7 & -3 & 2 \\ 3 & 0 & -3 \\ -1 & 3 & 1\end{array}\right]\left[\begin{array}{c}6 \\ 11 \\ 0\end{array}\right]$
or $\left[\begin{array}{l}x \\ y \\ z\end{array}\right]=\frac{1}{9}\left[\begin{array}{c}42-33+0 \\ 18+0+0 \\ -6+33+0\end{array}\right]$
$=\frac{1}{9}\left[\begin{array}{c}9 \\ 18 \\ 27\end{array}\right]=\left[\begin{array}{l}1 \\ 2 \\ 3\end{array}\right]$
Thus, $\quad x=1, y=2, z=3$.

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "4 Marks" from JUNE 2024→JUNE 2024 — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Find the area of the ragion bounded by x2 = 16y, y = 1, y = 4 and the parabola y-axis and the first quadrant.
→
verify that $\text{y}=\text{e}^{\text{m}\cos^{-1}}$ is a solution of the differential equation $(1+\text{x}^2)\frac{\text{d}^2\text{y}}{\text{dx}^2}-\text{x}\frac{\text{dy}}{\text{dx}}-\text{m}^2\text{y}=0$
→
Find the equation of the plane passing through the line of intersection of the planes x + 2y +3z - 4 = 0 and 2x + y - z + 5 = 0 and perpendicular to the plane 5x + 3y - 6z + 8 = 0.
→
Show that the set of all prime numbers is infinite.
→
Prove that function $\sin ^2 x(1+\cos x)$, at $\cos x$ $=\frac{1}{3}$ is maximum.
→
Differentiate the following functions with respect to x:
$\sin\Big(\frac{1+\text{x}^2}{1-\text{x}^2}\Big)$
→
Evaluate the following integrals:
$ \int\sqrt{\cot}\theta\text{d}\theta$
→
Solve the following linear programming problem using graphical method. Maximize $Z=60 x+40 y$, Under the constraints
$
\begin{array}{l}
x+2 y \leq 12 \\
2 x+y \leq 12 \\
x+\frac{5}{4} y \geq 5 ; x \geq 0, y \geq 0
\end{array}
$
→
Solve the following systems of homogeneous linear equations by matrix method:
x + y - z = 0
x - 2y + z = 0
3x + 6y - 5z = 0
→
If $(\text{x}-\text{y})\text{e}^{\frac{\text{x}}{\text{x}-\text{y}}}=\text{a},$ prove that $\text{y}\frac{\text{dy}}{\text{dx}}+\text{x}=2\text{y}$
→