Question

The value of $k$ for which the system of equations $x+ky=0, 2x-y=0$ has a unique solution is:

• A. $k \neq -1/2$
• B. $k = -1/2$
• C. $k = 0$
• D. All real $k$

Hint:

For homogeneous equations, a unique solution always means $x=0, y=0$.

Answer 1

The Correct Option is A

Step 1: Concept
A system of linear equations $a_{1}x + b_{1}y = 0$ and $a_{2}x + b_{2}y = 0$ has a unique solution (the trivial solution) if $\frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}}$.
Step 2: Analysis
Here $a_{1}=1, b_{1}=k, a_{2}=2, b_{2}=-1$.
Condition: $\frac{1}{2} \neq \frac{k}{-1}$.
Step 3: Conclusion
$-1 \neq 2k \Rightarrow k \neq -1/2$.
Final Answer: (A)