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4-2.Quadratic Equations and Inequations
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Let $P(x) = x^3 - ax^2 + bx + c$ where $a, b, c \in R$ has integral roots such that $P(6) = 3$, then $' a '$ cannot be equal to
A
$13$
B
$15$
C
$17$
D
$21$
Solution
$\mathrm{P}(\mathrm{x})=\mathrm{x}^{3}-\mathrm{ax}^{2}+\mathrm{bx}+\mathrm{c}$
$=(\mathrm{x}-\alpha)(\mathrm{x}-\beta)(\mathrm{x}-\gamma)$
Put $x=6,$ then
$3=(6-\alpha)(6-\beta)(6-\gamma)$
Various possibilities are
$(i)$ $\alpha=3 ; \beta=5 ; \gamma=5 \Rightarrow \mathrm{a}=13$
$(ii)$ $\alpha=3 ; \beta=7 ; \gamma=7 \Rightarrow \mathrm{a}=17$
$(iii)$ $\alpha=9 ; \beta=5 ; \gamma=7 \Rightarrow \mathrm{a}=21$
Standard 11
Mathematics
