Polynomials
RD Sharma
CBSE
Grade 10
Question:
If $\alpha, \beta$ are the zeroes of the polynomial $x^2 - 6x + k$ and $3\alpha + 2\beta = 20$, then $k$ is:
(a) $-16$
(b) $8$
(c) $-8$
(d) $16$
Step-by-Step Solution
Key Concept: $\alpha + \beta = 6$. Combine with $3\alpha + 2\beta = 20 \Rightarrow \alpha = 8, \beta = -2 \Rightarrow k = \alpha \beta = -16$.
$\alpha + \beta = 6 \Rightarrow 2\alpha + 2\beta = 12$. Subtract from $3\alpha + 2\beta = 20 \Rightarrow \alpha = 8$. [0.5 Mark]
$\beta = 6 - 8 = -2$. $k = \alpha \beta = 8(-2) = -16$. [0.5 Mark]
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🎯 Official CBSE Marking Scheme:
Solving linear system for $\alpha = 8, \beta = -2$: 0.5 Mark
Calculating $k = -16$: 0.5 Mark
Correct Answer: $-16$
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