Quadratic EquationHard

Question

If the equation $sinx + cosx = y^{2} - y +$ a has no solution in $x$ and $y$, then

Options

A.$a < - 2$
B.$a \in ( - 1,1)$
C.$a < \frac{1}{4} - \sqrt{2}$
D.$a > \sqrt{2} + \frac{1}{4}$

Solution

$$sinx + cosx = y^{2} - y + a$$

$$\begin{matrix} & \Rightarrow & y^{2} - y + a > \sqrt{2}\ \forall y \in R \\ \Rightarrow & & \frac{1}{4} - \frac{1}{2} + a > \sqrt{2} \\ \Rightarrow & & a > \sqrt{2} + \frac{1}{4} \end{matrix}$$

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