| Exam Board | OCR MEI |
|---|---|
| Module | Further Pure Core AS (Further Pure Core AS) |
| Year | 2021 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Roots of polynomials |
| Type | Roots with special relationships |
| Difficulty | Challenging +1.2 This is a Further Maths question requiring use of Vieta's formulas with a special relationship between roots (one root is the ratio of the other two). While it requires systematic algebraic manipulation and understanding of symmetric functions, the approach is relatively standard for Further Pure: use the product of roots to find α, then sum of roots to find k. The algebra is straightforward once the method is identified, making it moderately above average difficulty but not requiring deep insight. |
| Spec | 4.05a Roots and coefficients: symmetric functions4.05b Transform equations: substitution for new roots |
| Answer | Marks |
|---|---|
| 8 | DR |
| Answer | Marks |
|---|---|
| Sum of roots: −k =5+1+2i+1−2i=7⇒k =−7 | M1 |
| Answer | Marks |
|---|---|
| B1 | 3.1a |
| Answer | Marks |
|---|---|
| 2.2a | or 5β2 −10β+25=0 |
| Or equivalent use of formula | SCB1 if ow |
| Alternative solution | M1 |
| Answer | Marks |
|---|---|
| A1 | Factorising |
Question 8:
8 | DR
Product of roots =α2 =25
⇒α=5
α α 25
αβ+β× + ×α=15⇒5β+5+ =15
β β β
⇒β2−2β+5=0
⇒(β−1)2+4=0
⇒β=1±2i so roots are 5, 1+2i, 1−2i
Sum of roots: −k =5+1+2i+1−2i=7⇒k =−7 | M1
A1
M1
A1
M1
A1
B1 | 3.1a
1.1
1.1
1.1
1.1
2.2a
2.2a | or 5β2 −10β+25=0
Or equivalent use of formula | SCB1 if ow
Alternative solution | M1
A1
M1
A1
M1
M1
A1 | Factorising
Solving the resulting quadratic
Product of roots =α2 =25
⇒α=5
Substituting x=5: 125+25k+75−25=0
⇒k =−7
x3−7x2+15x−25=(x−5)(x2−2x+5)
x2−2x+5=0
Other roots are given by
⇒x=1±2i
[7]
M1
A1
M1
A1
M1
M1
A1
Factorising
Solving the resulting quadratic8 In this question you must show detailed reasoning.
The equation $\mathrm { x } ^ { 3 } + \mathrm { kt } ^ { 2 } + 15 \mathrm { x } - 25 = 0$ has roots $\alpha , \beta$ and $\frac { \alpha } { \beta }$. Given that $\alpha > 0$, find, in any order,
\begin{itemize}
\item the roots of the equation,
\item the value of $k$.
\end{itemize}
\hfill \mbox{\textit{OCR MEI Further Pure Core AS 2021 Q8 [7]}}