NTSE Stage 2 SAT 2017 Question Paper

Reason :
Let roots of quadratic equation :

and roots of quadratic equation :

Then : we have $\frac{\mathrm{\beta }}{\mathrm{\gamma }}=\frac{3}{4}$ ...(2)
Also from (1) we get
$\frac{\mathrm{\alpha }\mathrm{\beta }}{\mathrm{\alpha }\mathrm{\gamma }}=\frac{6}{\mathrm{c}}\Rightarrow \frac{\mathrm{\beta }}{\mathrm{\gamma }}=\frac{6}{\mathrm{c}}$ ...(3)
from (2) & (3) we get $\frac{3}{4}=\frac{6}{\mathrm{c}}\Rightarrow \mathrm{c}=8$
Now using c = 8, second quadratic equation
${\mathrm{x}}^2-6\mathrm{x}+\mathrm{c}=0$ become
${\mathrm{x}}^2-6\mathrm{x}+8=0$
on solving ${\mathrm{x}}^2-6\mathrm{x}+8=0$,we get x=2 & 4.
If common root is 2 then by putting x = 2 in equation
${\mathit{x}}^2-\mathrm{bx}+6=0$ we get
${\left(2\right)}^2-\mathrm{b}\times 4+6=0$⇒ $\mathrm{b}=5$
If common root is 4 then by putting x = 4 in equation
${\mathrm{x}}^2-\mathrm{bx}+6=0$ we get
${\left(4\right)}^2-\mathrm{b}\times 4+6=0$ ⇒ $\mathrm{b}=\frac{11}{2}$
But sum of roots is possible only if common root is 2.
Hence option (2) is correct.