Question

If the roots of the quadratic equation $4x^2-kx-1=0$ are $\sin \theta$ and $\cos \theta$ , then $\left|k\right|$ has the value equal to

• A. $2$
• B. $8$
• C. $4\sqrt{2}$
• D. $2\sqrt{2}$

Answer 1

Answer: (D)

$2\sqrt{2}$

The equation can be written as $4x^2-kx-1=0$

The roots are $\sin \theta$ and $\cos \theta$

Sum of roots $=\sin \theta +\cos \theta =\frac{k}{4}$ ,

Product of roots $=\sin \theta \times \cos \theta =\frac{-1}{4}$ ,

$(\sin \theta +\cos \theta {)}^2={\sin }^{2}a+{\cos }^{2}a+2\sin a\times \cos a$,

${\sin }^2\theta +{\cos }^2\theta =1$

$\sin \theta +\cos \theta =\sqrt{1-\frac{2}{4}}$

$\sqrt{\frac{1}{2}}=\frac{k}{4}$

$\therefore |k|=2\sqrt{2}$