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{r

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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

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