Question

Consider the system of equations:
$x+y+z=0$
$\alpha x+\beta y+\gamma z=0$
${\alpha }^{2}x+{\beta }^{2}y+{\gamma }^{2}z=0$
Then the system of equations has

• A. A unique solution for all values $\alpha ,\beta ,\gamma$
• B. Infinite number of solutions if any two of $\alpha ,\beta ,\gamma$ are equal
• C. A unique solution if $\alpha ,\beta ,\gamma$ are distinct
• D. More than one, but finite number of solutions depending on values of $\alpha ,\beta ,\gamma$

Answer 1

Answer: (B), (C)

The correct options are
B Infinite number of solutions if any two of $\alpha ,\beta ,\gamma$ are equal
C A unique solution if $\alpha ,\beta ,\gamma$ are distinct

$x+y+z=0$

$\alpha x+\beta y+\gamma z=0$

${\alpha }^{2}x+{\beta }^{2}y+{\gamma }^{2}z=0$

$△=\left|\begin{array}{ccc}1& 1& 1\\ \alpha & \beta & \gamma \\ {\alpha }^2& {\beta }^2& {\gamma }^2\end{array}\right|$

If any of the two values $(\alpha ,\beta )$ or $(\alpha ,\gamma )$ or $(\beta ,\gamma )$ are equal then $△=0$

Infinite solution

Option B

For all different values of $\alpha ,\beta ,\gamma$

$△\ne 0$

Unique solution

Option C