Multiple choice

$p,:q,:r,:$ and $:s$ are integers. If the $AM$ of the roots of the quadratic equation $\displaystyle x^{2}-px+q^{2}=0$ and $GM$ of the roots of the quadratic equation $\displaystyle x^{2}-rx+s^{2}=0$ are equal then

  1. $q$ is an odd integer
  2. $r$ is an even integer
  3. $p$ is an even integer
  4. $s$ is an odd integer
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C Correct answer
Explanation

AM of roots of x^2 - px + q^2 = 0 is p/2. GM of roots of x^2 - rx + s^2 = 0 is sqrt(s^2) = |s|. So p/2 = |s|, p = 2|s|. Since s is an integer, p must be an even integer.

AI explanation

For the first equation, the arithmetic mean (AM) of its roots is half the sum of the roots, giving $p/2$. Because the second equation has real integer roots, the geometric mean (GM) is the square root of their product, giving $s$ and implying it is an integer. Equating the two yields $p/2 = s$, which rearranges to $p = 2s$. Since $s$ is an integer, $p$ must be an even integer.

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