If the zeroes of the quadratic polynomial $ax^2 + bx + c$ (where $c
eq 0$) are equal, then:
(a) $c$ and $a$ have opposite signs
(b) $c$ and $b$ have opposite signs
(c) $c$ and $a$ have the same sign
(d) $c$ and $b$ have the same sign
Step-by-Step Solution
Key Concept: Equal zeroes mean discriminant $D = b^2 - 4ac = 0 \Rightarrow b^2 = 4ac$. Since $b^2 > 0$, $4ac > 0 \Rightarrow ac > 0$.
Equal zeroes $\Rightarrow D = b^2 - 4ac = 0 \Rightarrow b^2 = 4ac$. [0.5 Mark]
Since $b^2 \geq 0$ for real numbers (and $b
eq 0$ for non-zero equal zeroes), $4ac > 0 \Rightarrow ac > 0$. Thus $a$ and $c$ must have the same sign. [0.5 Mark]
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🎯 Official CBSE Marking Scheme:
Condition for equal zeroes ($b^2 = 4ac$): 0.5 Mark
Deducing $ac > 0$ and same sign: 0.5 Mark
Correct Answer: $c$ and $a$ have the same sign