Question

$\alpha$,$\beta$ are the roots of $x^2-px+q=0$ and $\alpha ,\beta ,\gamma$ are the roots of $x^3-a{x}^2+bx-c=0$, then

• A. $pq+c=aq$
• B. $p+\gamma =a$
• C. $p\gamma +q=b$
• D. $\gamma q=c$

Answer 1

Answer: (A), (B), (C), (D)

The correct options are
A $p\gamma +q=b$
B $pq+c=aq$
C $\gamma q=c$
D $p+\gamma =a$

As $\alpha ,\beta$ are roots of $x^2-px+q=0$
$\alpha +\beta =p$ ...(1)
$\alpha \beta =q$ ...(2)
And as $\alpha ,\beta ,\gamma$ are roots of $x^3-a{x}^2+bx-c=0$
$\alpha +\beta +\gamma =a$ ...(3)
$\alpha \beta +\beta \gamma +\gamma \alpha =b$ ...(4)
$\alpha \beta \gamma =c$ ...(5)
Now
Multiplying (3) by (2), we get
$(\alpha +\beta +\gamma )\alpha \beta =a\alpha \beta \Rightarrow (\alpha +\beta )\alpha \beta +\alpha \beta \gamma =a\alpha \beta \Rightarrow pq+c=aq$
From (3) and (1)
$\alpha +\beta +\gamma =a\Rightarrow p+\gamma =a$
From (1),(2) and (4), we have
$\alpha \beta +\beta \gamma +\gamma \alpha =b\Rightarrow \gamma (\alpha +\beta )+\alpha \beta =b\Rightarrow \gamma p+q=b$
And from (2) and (5), we have
$\alpha \beta \gamma =c\Rightarrow q\gamma =c$
Hence, options 'A','B','C' and 'D' are correct.