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(A) Given the quadratic equation $x^{2}-bx+c=0$ with roots $\sin \alpha$ and $\cos \alpha$.From the relation between roots and coefficients:$\sin \alp

Vedclass · Accepted Answer

$c \leq \frac{1}{2}$

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(A) Given the quadratic equation $x^{2}-bx+c=0$ with roots $\sin \alpha$ and $\cos \alpha$.From the relation between roots and coefficients:$\sin \alpha + \cos \alpha = b$ $(i)$$\sin \alpha \cdot \cos \alpha = c$ (ii)We know that $(\sin \alpha + \cos \alpha)^{2} = \sin^{2} \alpha + \cos^{2} \alpha + 2 \sin \alpha \cos \alpha$.Substituting the values: $b^{2} = 1 + 2c$, which implies $c = \frac{b^{2}-1}{2}$.Since $-1 \leq \sin \alpha \leq 1$ and $-1 \leq \cos \alpha \leq 1$, we have $-\sqrt{2} \leq \sin \alpha + \cos \alpha \leq \sqrt{2}$, so $-\sqrt{2} \leq b \leq \sqrt{2}$.Also, $\sin 2\alpha = 2 \sin \alpha \cos \alpha = 2c$.Since $-1 \leq \sin 2\alpha \leq 1$, we have $-1 \leq 2c \leq 1$, which implies $c \leq \frac{1}{2}$.

If $\sin \alpha$ and $\cos \alpha$ are the roots of the equation $x^{2}-bx+c=0$, then which of the following statements is/are correct?

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