Complex Numbers And Quadratic Equations question 36
Question: 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\forall x\in (0,1) $ , then [IIT Screening 2005]
Options:
A) $ S=0 $
B) $ S=ax+(1-a)x^{2}\forall a\in (0,\infty ) $
C) $ S=ax+(1-a)x^{2}\forall a\in R $
D) $ S=ax+(1-a)x^{2}\forall a\in (0,2) $
Show Answer
Answer:
Correct Answer: D
Solution:
Let $ P(x)=bx^{2}+ax+c $ As $ P(0)=0\Rightarrow c=0 $ As $ P(1)=1\Rightarrow a+b=1 $ $ 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
BETA
