Question

Let $sin\alpha ,cos\alpha$ be the roots of the equation $x^2-bx+c=0$.Then which of the following statements is / are correct?

• A. $c\le \frac{1}{2}$
• B. $b\le \sqrt{2}$
• C. $c>\frac{1}{2}$
• D. $b>\sqrt{2}$

Answer 1

Answer: (A), (B)

The correct options are
A $c\le \frac{1}{2}$
B $b\le \sqrt{2}$

Sum of roots=$\text{sin}\alpha +\text{cos}\alpha =b$

Product of roots =$sin\alpha +cos\alpha =c$

$sin\alpha +cos\alpha$ lies between $[-\sqrt{1^2+1^2},+\sqrt{1{+}^2+1^2}]=[-\sqrt{2},\sqrt{2}]$

$\therefore b\le \sqrt{2}$

And $sin\alpha cos\alpha =sin2\alpha /2$

$sin2\alpha \epsilon [-1,1]\Rightarrow sin2\alpha /2 \

epsilon [-1/2,1/2]$

$\therefore c\le 1/2$