| Exam Board | OCR MEI |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2006 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Roots of polynomials |
| Type | Sum of powers of roots |
| Difficulty | Moderate -0.3 Part (i) is direct recall of Vieta's formulas for cubic equations (standard FP1 content). Part (ii) requires the standard algebraic manipulation (α+β+γ)² = α²+β²+γ² + 2(αβ+βγ+γα), which is a textbook exercise. This is routine application of memorized techniques with no problem-solving or insight required, making it slightly easier than average. |
| Spec | 4.05a Roots and coefficients: symmetric functions |
Question 3:
3
Section B (36 marks)
x2
7 Acurve has equation y (cid:4) .
(x(cid:3)2)(x(cid:1)1)
(i) Write down the equations of the three asymptotes. [3]
(ii) Determine whether the curve approaches the horizontal asymptote from above or from below
for
(A) large positive values of x,
(B) large negative values of x. [3]
(iii) Sketch the curve. [4]
x2
(iv) Solve the inequality (cid:7) 0. [3]
(x(cid:3)2)(x(cid:1)1)
8 (i) Verify that 2(cid:1)jis a root of the equation 2x3(cid:3)11x2(cid:1)22x(cid:3)15 (cid:4) 0. [5]
(ii) Write down the other complex root. [1]
(iii) Find the third root of the equation. [4]
9 (i) Show that r(r(cid:1)1)(r(cid:1)2)(cid:3)(r(cid:3)1)r(r(cid:1)1) (cid:2) 3r(r(cid:1)1). [2]
n
(ii) Hence use the method of differences to find an expression for S r(r(cid:1)1). [6]
r(cid:4)1
n
(iii) Show that you can obtain the same expression for S r(r(cid:1)1) using the standard formulae
n n r(cid:4)1
for S r and S r2. [5]
r(cid:4)1 r(cid:4)1The cubic equation $z^3 + 4z^2 - 3z + 1 = 0$ has roots $\alpha$, $\beta$ and $\gamma$.
\begin{enumerate}[label=(\roman*)]
\item Write down the values of $\alpha + \beta + \gamma$, $\alpha\beta + \beta\gamma + \gamma\alpha$ and $\alpha\beta\gamma$. [3]
\item Show that $\alpha^2 + \beta^2 + \gamma^2 = 22$. [3]
\end{enumerate}
\hfill \mbox{\textit{OCR MEI FP1 2006 Q3 [6]}}