Question

If equations $x^2-3x+4=0$ & $4x^2-2[3a+b]x+b=0(a,bϵR)$ have a common root then the complete set of values of a, is

Note:[k] denotes the largest integer less than or equal to [k.]

• A. $[\frac{-11}{3},\frac{-10}{3})$
• B. $[\frac{-10}{3},-3)$
• C. Both A and B
• D. None of these

Answer 1

The correct option is B $[\frac{-10}{3},-3)$

$x^2-3x+4=0$

$D=9-16=-7<0$

So, both roots will be common

$\alpha +\beta =3$

$\alpha -\beta =4$

Similarly,

$\alpha +\beta ==\frac{2}{4}[3a+b]=3$
$\alpha -\beta =\frac{b}{4}=4$
$\Rightarrow b=16$
$\frac{2}{4}[3a+16]=3$
$[3a+16]=6$
$\left[3a\right]=6-16$
$\left[3a\right]=-10$
$-10\le 3a<-9\Rightarrow \frac{-10}{3}\le a<-3$
$aϵ[\frac{-10}{3},-3)$