Question

If $\alpha ,\beta$ are the roots of $x^2+x+1=0$ and $\gamma ,\delta$ are the roots $x^2+3x+1=0,$ then $(\alpha -\gamma )(\beta +\delta )(\alpha +\delta )(\beta -\gamma )=$

• A. $2$`
• B. $4$
• C. $6$
• D. $8$

Answer 1

Answer: (D)

$8$

Using $\alpha \beta =1$ and $\gamma \delta =1$

$(\alpha \beta +\alpha \delta -\beta \gamma -\gamma \delta )(\alpha \beta -\alpha \gamma +\beta \delta -\gamma \delta )$

$\Rightarrow (1+\alpha \delta -\beta \gamma -1)(1-\alpha \gamma +\beta \delta -1)$

$\Rightarrow (\alpha \delta -\beta \gamma )(\beta \delta -\alpha \gamma )$

$\Rightarrow {\delta }^2-{\alpha }^2-{\beta }^2+{\gamma }^2$

$\Rightarrow -({\alpha }^2+{\beta }^2)+({\gamma }^2+{\delta }^2)$

$\Rightarrow -(-1)+7=8$

Final answer is 8