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4-2.Quadratic Equations and Inequations
easy
If $S$ is a set of $P(x)$ is polynomial of degree $ \le 2$ such that $P(0) = 0,$$P(1) = 1$,$P'(x) > 0{\rm{ }}\forall x \in (0,\,1)$, then
A
$S = 0$
B
$S = ax + (1 - a){x^2}{\rm{ }}\forall a \in (0,\infty )$
C
$S = ax + (1 - a){x^2}{\rm{ }}\forall a \in R$
D
$S = ax + (1 - a){x^2}{\rm{ }}\forall a \in (0,2)$
(IIT-2005)
Solution
(d) Let $P(x) = b{x^2} + ax + c$
$\therefore$ As $P(0) = 0 \Rightarrow c = 0$
$\therefore$ As $P(1) = 1 \Rightarrow a + b = 1$
$\because$ $P(x) = ax + (1 – a){x^2}$
Now $P'(x) = a + 2(1 – a)x$
as $P'(x) > 0$ for $x \in (0,\,1)$
Only option $(d)$ satisfies above condition
Standard 11
Mathematics
