Values of $k$ for which the quadratic equation $2x^2 - kx + k = 0$ has equal roots is/are:
$0$ only
$4$
$8$ only
$0, 8$
Step-by-Step Solution
Key Concept: Equal roots mean discriminant $D = b^2 - 4ac = 0$.
Stepwise Solution:
Here $a=2, b=-k, c=k$. $D = (-k)^2 - 4(2)(k) = k^2 - 8k$. [0.5 Mark]
Set $D = 0 \Rightarrow k^2 - 8k = 0 \Rightarrow k(k - 8) = 0 \Rightarrow k = 0$ or $k = 8$. [0.5 Mark]
Marking Scheme:
• Setting discriminant $D = 0$: 0.5 Mark
• Finding roots $k = 0$ and $k = 8$: 0.5 Mark
Correct Answer: $0, 8$