Question

The cost of $4kg$ onion, $3kg$ wheat and $2kg$ rice is ₹ 60 . The cost of $2kg$ onion, $4kg$ wheat and $6kg$ rice is ₹ 90 . The cost of $6kg$ onion, $2kg$ wheat and $3kg$ rice is $₹70$. The costs (in $₹$ ) of onion, wheat and rice per kilogram are respectively

• A. $5,9,10$
• B. $5,8,8$
• C. $8,5,8$
• D. $8,8,5$

Answer 1

Answer: (B)

Let the prices (per $kg$ ) of onion, wheat and rice be $₹x$, $₹y$ and $₹z$, respectively. Then,
$4x+3y+2z=60$
$2x+4y+6z=90$
$6x+2y+3z=70$
This system of equations can be written as $AX=B$, where
$A=\left[\begin{array}{ccc}4& 3& 2\\ 2& 4& 6\\ 6& 2& 3\end{array}\right],X=\left[\begin{array}{c}x\\ y\\ z\end{array}\right]\text{ and }B=\left[\begin{array}{c}60\\ 90\\ 70\end{array}\right]$
Here, $|A|=\left|\begin{array}{ccc}4& 3& 2\\ 2& 4& 6\\ 6& 2& 3\end{array}\right|$
$=4(12-12)-3(6-36)+2(4-24).$
$=0+90-40=50\ne 0$
Thus, $A$ is non-singular. Therefore, its inverse exists. Therefore, the given system is consistent and has a unique solution given by $X=A^{-1}B$.
Cofactors of $A$ are
$A_{11}=12-12=0,A_{12}=-(6-36)=30,A_{13}=4-24=-20,$
$A_{21}=-(9-4)=-5,A_{22}=12-12=0,A_{23}=-(8-18)=10,$
$A_{31}=(18-8)=10,A_{32}=-(24-4)=-20,A_{33}=16-6=10$
$\mathrm{adj}(A)={\left[\begin{array}{ccc}0& 30& -20\\ -5& 0& 10\\ 10& -20& 10\end{array}\right]}^{\prime }=\left[\begin{array}{ccc}0& -5& 10\\ 30& 0& -20\\ -20& 10& 10\end{array}\right]$
$\therefore A^{-1}=\frac{1}{|A|}(\mathrm{adj}(A)=\frac{1}{50}\left[\begin{array}{ccc}0& -5& 10\\ 30& 0& -20\\ -20& 10& 10\end{array}\right]$
Now,$X=A^{-1}B\Rightarrow \left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\frac{1}{50}\left[\begin{array}{ccc}30& -5& 10\\ -20& 10& -20\\ 0-450+700& & \end{array}\right]\left[\begin{array}{c}60\\ 90\\ 70\end{array}\right]$
$=\frac{1}{50}\left[\begin{array}{c}0-450+700\\ 1800+0-1400\\ -1200+900+700\end{array}\right]$
$\Rightarrow \left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\frac{1}{50}\left[\begin{array}{c}250\\ 400\\ 400\end{array}\right]=\left[\begin{array}{c}5\\ 8\\ 8\end{array}\right]$
$\Rightarrow \left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}5\\ 8\\ 8\end{array}\right]$
$\therefore x=5,y=8\text{ and }z=8.$
Hence, price of onion per $kg$ is $₹5$, price of wheat per $kg$ is ₹ 8 and price of rice per $kg$ is $₹8$.