Q87 / 170 · Manipal (MET) 2017 · Mathematics
If $x^2+p x+q=0$ is the quadratic equation whose roots are $a-2$ and $b-2$, where $a$ and $b$ are the roots of $x^2-3 x+1=0$, then
- A$p=1, q=5$
- B$p=5, q=1$
- C$p=1, q=1$
- Correct answerD$p=1, q=-1$
Solution
Since, $a$ and $b$ are the roots of equation
$\begin{array}{rlrl}
x^2-3 x+1 & =0 \\
\end{array}$
$\therefore a+b=3$ and $a b=1$
Now, the given roots are $a-2$ and $b-2$
$\begin{aligned}
\therefore \text { Sum of roots } & =a-2+b-2 \\
& =3-4=-1
\end{aligned}$
and products of roots $=(a-2)(b-2)$
$\begin{aligned}
& =a b-2(a+b)+4 \\
& =1-6+4=-1
\end{aligned}$
$\therefore$ Required equation is $x^2+x-1=0$
Which is equivalent to $x^2+p x+q=0$
$\therefore p=1 \text { and } q=-1$
Hence, the correct answer is option: 'D'.