| Exam Board | Edexcel |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2021 |
| Session | October |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Arithmetic Sequences and Series |
| Type | Show quadratic equation in n |
| Difficulty | Standard +0.3 This is a straightforward arithmetic sequence problem requiring standard formulas. Part (a) uses the nth term formula directly, part (b) applies the sum formula and simplifies to the given quadratic (algebraic manipulation is routine), and part (c) solves the quadratic. All steps are textbook-standard with no novel insight required, making it slightly easier than average. |
| Spec | 1.02f Solve quadratic equations: including in a function of unknown1.04h Arithmetic sequences: nth term and sum formulae1.04i Geometric sequences: nth term and finite series sum |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Attempts \(480 + 13\times(-15)\) | M1 | Use \(a+(n-1)d\) with at least two of \(a\), \(d\), \(n\) correct; allow \(a+nd\) if both \(a\) and \(d\) correct |
| \(= 285\) tonnes | A1 | Condone omission of units |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Sets \(\frac{N}{2}\{2\times480+(N-1)\times-15\} = 7770\) | M1A1 | Correct sum formula with \(S=7770\) using their \(a\) and \(d\), condoning slips; A1 for correct equation for \(N\) |
| \(\frac{N}{2}\{960-15N+15\} = 7770 \Rightarrow N^2 - 65N + 1036 = 0\) | A1* | Achieves \(N^2-65N+1036=0\) (including "=0") following correct intermediate line, no errors |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| States 28 only | B1 | The 37 should be rejected if found. 29 is B0 |
## Question 5:
### Part (a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Attempts $480 + 13\times(-15)$ | M1 | Use $a+(n-1)d$ with at least two of $a$, $d$, $n$ correct; allow $a+nd$ if both $a$ and $d$ correct |
| $= 285$ tonnes | A1 | Condone omission of units |
### Part (b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Sets $\frac{N}{2}\{2\times480+(N-1)\times-15\} = 7770$ | M1A1 | Correct sum formula with $S=7770$ using their $a$ and $d$, condoning slips; A1 for correct equation for $N$ |
| $\frac{N}{2}\{960-15N+15\} = 7770 \Rightarrow N^2 - 65N + 1036 = 0$ | A1* | Achieves $N^2-65N+1036=0$ (including "=0") following correct intermediate line, no errors |
### Part (c):
| Answer | Mark | Guidance |
|--------|------|----------|
| States 28 only | B1 | The 37 should be rejected if found. 29 is B0 |
---5. A company that owned a silver mine
\begin{itemize}
\item extracted 480 tonnes of silver from the mine in year 1
\item extracted 465 tonnes of silver from the mine in year 2
\item extracted 450 tonnes of silver from the mine in year 3\\
and so on, forming an arithmetic sequence.
\begin{enumerate}[label=(\alph*)]
\item Find the mass of silver extracted in year 14
\end{itemize}
After a total of 7770 tonnes of silver was extracted, the company stopped mining.
Given that this occurred at the end of year $N$,
\item show that
$$N ^ { 2 } - 65 N + 1036 = 0$$
\item Hence, state the value of $N$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel P2 2021 Q5 [6]}}