If α, β are the roots of the equation x^2 – px + q = 0, then the quadratic equation whose roots are α/β and α/β is …………………… 

Correct option is (A) qx² + (2q – p²)x + q = 0

Given that \(\alpha\;\&\;\beta\) are roots of equation \(x^2-px+q=0.\)

\(\therefore\) Sum of roots \(=\frac{-(-p)}1=p\)

\(\Rightarrow\) \(\alpha+\beta\) = p      ______________(1)

& Product of roots \(=\frac q1=q\)

\(\Rightarrow\) \(\alpha\beta=q\)        ______________(2)

Now, \(\frac{\alpha}{\beta}+\frac{\beta}{\alpha}\) \(=\frac{\alpha^2+\beta^2}{\alpha\beta}\)

\(=\frac{(\alpha+\beta)^2-2\alpha\beta}{\alpha\beta}\)

\(\Rightarrow\) \(\frac{\alpha}{\beta}+\frac{\beta}{\alpha}\) \(=\frac{p^2-2q}q\)       ______________(3)   (From (1) & (2))

And \(\frac{\alpha}{\beta}.\frac{\beta}{\alpha}\) \(=\frac{\alpha\beta}{\alpha\beta}\) = 1    ______________(4)

\(\therefore\) Quadratic equation whose roots are \(\frac{\alpha}{\beta}\;and\;\frac{\beta}{\alpha}\) is

\(x^2-(\frac{\alpha}{\beta}+\frac{\beta}{\alpha})x+\frac{\alpha}{\beta}.\frac{\beta}{\alpha}=0\)

\(\Rightarrow x^2-(\frac{p^2-2q}q)x+1=0\)       (From (3) & (4))

\(\Rightarrow\) \(qx^2-(p^2-2q)\,x+q=0\)

\(\Rightarrow\) \(qx^2+ (2q-p^2)\,x+q=0\)

Correct option is A) qx² + (2q – p² ) x + q = 0