Question

Su of values of P for which equation x + y + z = 1,x + 2y 4z = P and x + 4y + 10z = $p^2$ has a solution is

• A. 1
• B. 2
• C. 3
• D. 5

Answer 1

The correct option is C 3

Given equation,

$x+y+z=1x+2y+4z=px+4y+10z=p^2$

$\Delta =\left|\begin{array}{ccc}1& 1& 1\\ 1& 2& 4\\ 1& 4& 10\end{array}\right|$

Applying $R_1\to R_1-R_2,R_2\to R_2-R_3$

$=\left|\begin{array}{ccc}0& -1& -3\\ 0& -2& -6\\ 1& 4& 10\end{array}\right|$

Since $\Delta =0$ Solution is not unique solution

The system will have infinite solution if ${\Delta }_1=0,{\Delta }_1=0,{\Delta }_3=0$

${\Delta }_1=\left|\begin{array}{ccc}1& 1& 1\\ p& 2& 4\\ p^2& 4& 10\end{array}\right|$

Applying $C_3\to C_3-C_2$

$⟹1(12-8)-1(6p-2p^2)=0$

$⟹p^2-3p+2=0$

$⟹p=1,p=2$

hence sum of values of $p=1+2=3$