| Exam Board | OCR |
|---|---|
| Module | Further Pure Core 1 (Further Pure Core 1) |
| Year | 2021 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Second order differential equations |
| Type | Series solution from differential equation |
| Difficulty | Standard +0.8 This is a Further Maths FP1 question requiring multiple techniques: using Vieta's formulas to relate sum of squares to elementary symmetric polynomials, recognizing that a negative sum of squares of real numbers is impossible (thus complex roots exist), and transforming a cubic equation. While each part is methodical, the combination of complex number theory, algebraic manipulation, and the insight needed for part (a) places this above average difficulty but not exceptionally hard for FP1 students. |
| Spec | 4.05a Roots and coefficients: symmetric functions4.05b Transform equations: substitution for new roots |
| Answer | Marks | Guidance |
|---|---|---|
| 3 | (a) | e.g. |
| Answer | Marks |
|---|---|
| real. | B1 |
| Answer | Marks |
|---|---|
| B1 | 2.4 |
| Answer | Marks |
|---|---|
| (b) | α+β+γ=3 |
| Answer | Marks |
|---|---|
| ⇒k =7 | B1 |
| Answer | Marks |
|---|---|
| A1 | 1.1 |
| Answer | Marks |
|---|---|
| 1.1 | Attempt to obtain identity and substitute |
| Answer | Marks |
|---|---|
| (c) | 3 2 |
| Answer | Marks |
|---|---|
| ⇒−5u3 +7u2 −3u+1=0 oe | M1 |
| A1 | 1.1 |
| 1.1 | For the substitution |
| Answer | Marks |
|---|---|
| Alternate method | M1 |
| A1 | For calculating the sum, product and sum of product of |
Question 3:
3 | (a) | e.g.
α2 +β2 +γ2 =−5 means that at least one root is
complex
But complex roots come in complex pairs so there
are 2 complex roots.
Given that there are 3 roots and 2 are complex one is
real. | B1
B1
B1 | 2.4
2.4
2.4
[3]
(b) | α+β+γ=3
(α+β+γ)2 =α2 +β2 +γ2 +2 (αβ+βγ+γα)
9=−5+2 (αβ+βγ+γα)
But k =αβ+βγ+γα
⇒k =7 | B1
M1
A1 | 1.1
3.1a
1.1 | Attempt to obtain identity and substitute
Condone missing 2 and sign errors
[3]
(c) | 3 2
1 1 1
−3 +7 −5=0
u u u
⇒−5u3 +7u2 −3u+1=0 oe | M1
A1 | 1.1
1.1 | For the substitution
“=0” not necessary here but needed for A1
Allow in terms of z Allow ft from their k in (b)
Alternate method | M1
A1 | For calculating the sum, product and sum of product of
pairs of reciprocals of α, β, γ
1 1 1 7 1 1 1 1 1 1 3 1 1 1 1
+ + = , + + = , =
α β γ 5 αβ βγ γα 5 αβγ 5
Answer as above
[2]
M1
A1
For calculating the sum, product and sum of product of
pairs of reciprocals of α, β, γA function $\mathrm{f}(z)$ is defined on all complex numbers $z$ by $\mathrm{f}(z) = z^3 - 3z^2 + kz - 5$ where $k$ is a real constant. The roots of the equation $\mathrm{f}(z) = 0$ are $\alpha$, $\beta$ and $\gamma$. You are given that $\alpha^2 + \beta^2 + \gamma^2 = -5$.
\begin{enumerate}[label=(\alph*)]
\item Explain why $\mathrm{f}(z) = 0$ has only one real root. [3]
\item Find the value of $k$. [3]
\item Find a cubic equation with integer coefficients that has roots $\frac{1}{\alpha}$, $\frac{1}{\beta}$ and $\frac{1}{\gamma}$. [2]
\end{enumerate}
\hfill \mbox{\textit{OCR Further Pure Core 1 2021 Q3 [8]}}