Question

If $\sin x+\cos x=y^2-y+a$ has no value of $x$ for any value of $y$ then $a$ belongs to

• A. $(-\infty ,\frac{4\sqrt{2}+1}{4})$
• B. $(-\infty ,\frac{4\sqrt{2}-1}{4})$
• C. $(\frac{4\sqrt{2}-1}{4},\infty )$
• D. $(\frac{4\sqrt{2}+1}{4},\infty )$

Answer 1

The correct option is D $(\frac{4\sqrt{2}+1}{4},\infty )$

$y^2-y+a={(y-\frac{1}{2})}^2+a-\frac{1}{4}$
Since $-\sqrt{2}\le \sin x+\cos x\le \sqrt{2}$
the given equation will have no real value $x$ for any $y$ if $a-\frac{1}{4}>\sqrt{2}$
$a\in (\frac{4\sqrt{2}+1}{4},\infty )$