Without solving the equation, find the value of

(i)     \(\alpha  + \beta \);

(ii)     \(\alpha \beta \).

Another quadratic equation \({x^2} + px + q = 0,{\text{ }}p,{\text{ }}q \in \mathbb{Z}\) has roots \(\frac{2}{\alpha }\) and \(\frac{2}{\beta }\).

Find the value of \(p\) and the value of \(q\).

using the formulae for the sum and product of roots:

(i)     \(\alpha  + \beta  = 4\)     A1

(ii)     \(\alpha \beta  = \frac{1}{2}\)     A1

Note:     Award A0A0 if the above results are obtained by solving the original equation (except for the purpose of checking).

[2 marks]

METHOD 1

required quadratic is of the form \({x^2} - \left( {\frac{2}{\alpha } + \frac{2}{\beta }} \right)x + \left( {\frac{2}{\alpha }} \right)\left( {\frac{2}{\beta }} \right)\)     (M1)

\(q = \frac{4}{{\alpha \beta }}\)

\(q = 8\)     A1

\(p =  - \left( {\frac{2}{\alpha } + \frac{2}{\beta }} \right)\)

\( =  - \frac{{2(\alpha  + \beta )}}{{\alpha \beta }}\)     M1

\( =  - \frac{{2 \times 4}}{{\frac{1}{2}}}\)

\(p =  - 16\)     A1

Note:     Accept the use of exact roots

METHOD 2

replacing \(x\) with \(\frac{2}{x}\)     M1

\(2{\left( {\frac{2}{x}} \right)^2} - 8\left( {\frac{2}{x}} \right) + 1 = 0\)

\(\frac{8}{{{x^2}}} - \frac{{16}}{x} + 1 = 0\)     (A1)

\({x^2} - 16x + 8 = 0\)

\(p =  - 16\) and \(q = 8\)     A1A1

Note:     Award A1A0 for \({x^2} - 16x + 8 = 0\) ie, if \(p =  - 16\) and \(q = 8\) are not explicitly stated.

[4 marks]

Total [6 marks]

Most candidates obtained full marks.

Many candidates obtained full marks, but some responses were inefficiently expressed. A very small minority attempted to use the exact roots, usually unsuccessfully.