Question

If $\alpha$ and $\beta$ are the roots of the equations $ax ^2+ bx + c =0, \alpha \beta=3$ and $a , b , c$ are in A.P., then $\left(\alpha^2+\beta^2\right)$ is equal to

• A. 2
• B. -2
• C. $\alpha^3+\beta^3$
• D. $\alpha+\beta$

Answer 1

Answer: (B), (D)

Let $\alpha+\beta=-\frac{b}{a} ; \alpha \beta=\frac{c}{a}=3 \Rightarrow c=3 a$...(i)
$\text { and } \quad 2 b = a + c $....(ii)
$\Rightarrow \quad b =2 a \text { from(i) and (ii) } $
$\therefore \alpha+\beta=-2 $
$\alpha^2+\beta^2=(\alpha+\beta)^2-2 \alpha \beta $
$=4-2 \cdot 3=4-6=-2$