Question

The nature of roots of the quadratic equation $(x-\sin \alpha )(x-\cos \alpha )+\sqrt{3}=0$ are

• A. Rational and equal.
• B. Real and equal.
• C. Real and unequal.
• D. Imaginary.

Answer 1

The correct option is D Imaginary.

Given equation
$x^2-(\sin \alpha +\cos \alpha )x+(\sin \alpha \cos \alpha +\sqrt{3})=0$
For the nature of the roots
$\Delta =(\sin \alpha +\cos \alpha {)}^2-4(\sin \alpha \cos \alpha +\sqrt{3})$
$=(\sin \alpha -\cos \alpha {)}^2-4\sqrt{3}\because 0\le (\sin \alpha -\cos \alpha {)}^2\le 2$
$\therefore -4\sqrt{3}\le \Delta \le 2-4\sqrt{3}$
$\Rightarrow \Delta$ is negative.