Multiple choice

If $\displaystyle \alpha : and: \beta $ are the roots of $\displaystyle x^{2}+p=0 $ where p is a prime then which equation has the roots $\displaystyle \frac{1}{\alpha }: &: \frac{1}{\beta } $?

  1. $\displaystyle \frac{1}{x^{2}}+\frac{1}{p}=0 $
  2. $\displaystyle px^{2}+1=0$
  3. $\displaystyle px^{2}-1=0$
  4. $\displaystyle \frac{1}{x}+\frac{1}{p^2}=0 $
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B Correct answer
Explanation

Roots of x^2 + p = 0 are alpha and beta. alpha + beta = 0, alpha * beta = p. New roots are 1/alpha and 1/beta. New sum = (alpha + beta) / (alpha * beta) = 0/p = 0. New product = 1 / (alpha * beta) = 1/p. Equation: x^2 - (sum)x + product = 0 => x^2 + 1/p = 0 => px^2 + 1 = 0.

AI explanation

To find the equation whose roots are the reciprocals of the roots of x^2 + p = 0, we use the transformation method of replacing x with 1/x. Substituting 1/x into the equation gives (1/x)^2 + p = 0. Multiplying the entire equation by x^2 to clear the denominator results in 1 + px^2 = 0, or px^2 + 1 = 0.

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