Question

If the system of equations $2x-3y+5z=12,3x+y+pz=q$ and $x-7y+8z=17$ is consistent, then which of the following is not true?

• A. $p=2,q=7$
• B. $p\ne 2,q=7$
• C. $p\ne 2,q\ne 7$
• D. $p=2,q\ne 7$

Answer 1

Answer: (A)

Using Cramer’s rule:
$\Delta =\left|\begin{array}{ccc}2& -3& 5\\ 3& 1& p\\ 1& -7& 8\end{array}\right|=2\left(8+7p\right)+3\left(24-p\right)+5\left(-21-1\right)$
$=-22+11p=11\left(p-2\right)$
${\left(\Delta \right)}_1=\left|\begin{array}{ccc}12& -3& 5\\ q& 1& p\\ 17& -7& 8\end{array}\right|=12\left(8+7p\right)+3\left(8q-17p\right)+5\left(-7q-17\right)$
$=33p-11q+11=11\left(3p-q+1\right)$
${\left(\Delta \right)}_2=\left|\begin{array}{ccc}2& 12& 5\\ 3& q& p\\ 1& 17& 8\end{array}\right|=2\left(8q-17p\right)-12\left(24-p\right)+5\left(51-q\right)$
$=11\left(q-2p-3\right)$
${\left(\Delta \right)}_3=\left|\begin{array}{ccc}2& -3& 12\\ 3& 1& q\\ 1& -7& 17\end{array}\right|=2\left(17+7q\right)+3\left(51-q\right)+12\left(-21-1\right)$
$=11q-77=11\left(q-7\right)$
For unique solution $\to p\ne 2,q\in R$
For infinite solutions $\to p=2,q=7$