| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2012 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Roots of polynomials |
| Type | Quadratic with transformed roots |
| Difficulty | Standard +0.3 This is a standard Further Maths question on transformed roots requiring systematic application of sum/product formulas. Part (a) is direct recall, part (b) is routine algebraic manipulation, and part (c) follows a well-practiced technique of finding sum and product of new roots. While it requires multiple steps and careful algebra, it's a textbook exercise with no novel insight needed—slightly easier than average for FP1 material. |
| Spec | 4.05a Roots and coefficients: symmetric functions4.05b Transform equations: substitution for new roots |
| Answer | Marks | Guidance |
|---|---|---|
| 1(a) | \(\alpha + \beta = \frac{7}{5} (=1.4)\) | B1 |
| 1(a) | \(\alpha\beta = \frac{1}{5} (=0.2)\) | B1 |
| 1(b) | \(\frac{\alpha}{\beta} + \frac{\beta}{\alpha} = \frac{\alpha^2 + \beta^2}{\alpha\beta}\) | M1 |
| 1(b) | \(= \frac{(\alpha + \beta)^2 - 2\alpha\beta}{\alpha\beta} = \frac{(\frac{7}{5})^2 - 2(\frac{1}{5})}{\frac{1}{5}}\) | m1 |
| 1(b) | \(= \frac{\frac{49}{25} - 2(\frac{1}{5})}{\frac{1}{5}} = \frac{\frac{49}{25} - \frac{2}{5}}{\frac{1}{5}} = \frac{\frac{39}{25}}{\frac{1}{5}} = \frac{39}{5}\) | A1 |
| 1(c) | (Sum=) \(\alpha + \frac{1}{\alpha} + \beta + \frac{1}{\beta} = \alpha + \beta + \frac{\alpha + \beta}{\alpha\beta}\) | M1 |
| 1(c) | \(= \frac{7}{5} + \frac{\frac{7}{5}}{\frac{1}{5}}\) | |
| 1(c) | (Product =) \(\alpha\beta + \frac{\alpha}{\beta} + \frac{\beta}{\alpha} + \frac{1}{\alpha\beta}\) | M1 |
| 1(c) | \(= \frac{1}{5} + \frac{39}{5} + 5\) | |
| 1(c) | Sum = \(\frac{42}{5}\), Product = 13 | A1 |
| 1(c) | \(x^2 - Sx + P (=0)\) | M1 |
| 1(c) | Equation is \(5x^2 - 42x + 65 = 0\) | A1 |
1(a) | $\alpha + \beta = \frac{7}{5} (=1.4)$ | B1 | Accept correct equivalent decimals in place of some/all fractions in the scheme
1(a) | $\alpha\beta = \frac{1}{5} (=0.2)$ | B1 |
1(b) | $\frac{\alpha}{\beta} + \frac{\beta}{\alpha} = \frac{\alpha^2 + \beta^2}{\alpha\beta}$ | M1 | OE eg $\frac{\alpha}{\beta} + \frac{\beta}{\alpha} = \frac{1}{5}[7(\alpha + \beta) - 1 - 1]$ scores M1 m1
1(b) | $= \frac{(\alpha + \beta)^2 - 2\alpha\beta}{\alpha\beta} = \frac{(\frac{7}{5})^2 - 2(\frac{1}{5})}{\frac{1}{5}}$ | m1 | Correct expression for $\frac{\alpha}{\beta} + \frac{\beta}{\alpha}$ in terms of either $(\alpha+\beta)$ and $\alpha\beta$ or with numerical substitution of correct/c's values from (a). CSO AG must see some intermediate evaluation, must see one of the first three expressions
1(b) | $= \frac{\frac{49}{25} - 2(\frac{1}{5})}{\frac{1}{5}} = \frac{\frac{49}{25} - \frac{2}{5}}{\frac{1}{5}} = \frac{\frac{39}{25}}{\frac{1}{5}} = \frac{39}{5}$ | A1 | A0 if $\alpha + \beta$ has wrong sign
1(c) | (Sum=) $\alpha + \frac{1}{\alpha} + \beta + \frac{1}{\beta} = \alpha + \beta + \frac{\alpha + \beta}{\alpha\beta}$ | M1 | Writing $\alpha + \frac{1}{\alpha} + \beta + \frac{1}{\beta}$ in a correct form or with numerical values
1(c) | $= \frac{7}{5} + \frac{\frac{7}{5}}{\frac{1}{5}}$ | |
1(c) | (Product =) $\alpha\beta + \frac{\alpha}{\beta} + \frac{\beta}{\alpha} + \frac{1}{\alpha\beta}$ | M1 | Correct expression for product into which substitution of numbers attempted for all terms, at least one either correct/correct ft
1(c) | $= \frac{1}{5} + \frac{39}{5} + 5$ | |
1(c) | Sum = $\frac{42}{5}$, Product = 13 | A1 | OE Both. SC If B0 for $\alpha + \beta = -\frac{7}{5}$ in (a), and (c) $S= -\frac{42}{5}$ oe, P = 13 award this A1
1(c) | $x^2 - Sx + P (=0)$ | M1 | Using correct general form of LHS of equation with ft substitution of c's S and P values. PI. M0 if either $S = \alpha + \beta$ or $P = \alpha\beta$ values
1(c) | Equation is $5x^2 - 42x + 65 = 0$ | A1 | CSO Integer coefficients and '= 0' needed. Dependent on B1B1 in (a) and previous 4 marks in (c) scored1 The quadratic equation
$$5 x ^ { 2 } - 7 x + 1 = 0$$
has roots $\alpha$ and $\beta$.
\begin{enumerate}[label=(\alph*)]
\item Write down the values of $\alpha + \beta$ and $\alpha \beta$.
\item Show that $\frac { \alpha } { \beta } + \frac { \beta } { \alpha } = \frac { 39 } { 5 }$.
\item Find a quadratic equation, with integer coefficients, which has roots
$$\alpha + \frac { 1 } { \alpha } \quad \text { and } \quad \beta + \frac { 1 } { \beta }$$
(5 marks)
\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2012 Q1 [10]}}