Question

The equations $4x^2-x-1=0$ and $3x^2+(\lambda +\mu )x+\lambda -\mu =0$ have at least one common root , where $\lambda$ and $\mu$ are rational numbers. Find the value of $\lambda$ and $\mu$.

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

Answer 1

Answer: (B)

$\lambda =-\frac{3}{4},\mu =0$

Given,

$4x^2-x-1=0$ and $3x^2+(\lambda +\mu )x+\lambda -\mu =0$

$4x^2-x-1=0$

$\Rightarrow x=\frac{-(-1)\pm \sqrt{1-4(-4)}}{2\times 4}$

$=\frac{1\pm \sqrt{17}}{8}$

Hence, the roots of the first equation are irrational.

Now, let us assume that only one root is common between the two equations. If the other root of the second equation is not common, then, either the product or the sum of the roots in the second equation will be irrational.

However, the coefficients of the second equation are rational. Hence, our assumption that only root is common is incorrect. Hence, both the roots must be common.

Hence,

$\lambda +\mu =-\frac{3}{4}$

$\lambda -\mu =-\frac{3}{4}$

On solving we get,

$\lambda =-\frac{3}{4},\mu =0$

Hence, option B is correct.