Question

If the equations $4x^2-x-1=0$ and $3x^2+(\lambda +\mu )x+\lambda -\mu =0$ have a common root, then the rational values of $\lambda$ and $\mu$ are

• A. $\lambda =\frac{-3}{4},\mu =\frac{1}{4}$
• B. $\lambda =\frac{-3}{4},\mu =\frac{3}{4}$
• C. $\lambda =0,\mu =\frac{-3}{4}$
• D. $\lambda =\frac{-3}{4},\mu =0$

Answer 1

The correct option is D $\lambda =\frac{-3}{4},\mu =0$

$4x^2-x-1=0$
$\Rightarrow x=\frac{1\pm \sqrt{17}}{2}$
Roots of $4x^2-x-1=0$ are irrational and exist as conjugate pairs.

$\therefore$ One root common implies both roots are common as coefficients of $2$nd equation are also rational.
$\therefore \frac{4}{3}=\frac{-1}{\lambda +\mu }=\frac{-1}{\lambda -\mu }$
On solving, we get
$\lambda =\frac{-3}{4},\mu =0$