| Exam Board | OCR |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Arithmetic Sequences and Series |
| Type | Arithmetic progression with parameters |
| Difficulty | Moderate -0.3 This is a straightforward arithmetic progression question requiring basic formula application. Parts (a) and (b) involve simple algebraic manipulation to find the common difference and solve a quadratic equation. Parts (c) and (d) are direct applications of standard AP formulas once t is known. While it requires multiple steps, each step uses routine techniques with no novel problem-solving or insight required, making it slightly easier than average. |
| Spec | 1.04h Arithmetic sequences: nth term and sum formulae1.04i Geometric sequences: nth term and finite series sum |
| Answer | Marks |
|---|---|
| (a)(i) \(= (t^2 - 5) - (t - 1) = t^2 - t - 4\) | M1 A1 |
| (a)(iii) \(= (t^2 - 5) + (t^3 - t - 4) = 2t^2 - t - 9\) | M1 A1 |
| Answer | Marks |
|---|---|
| \((2t + 7)(t - 4) = 0\) | M1 |
| \(t > 0 \therefore t = 4\) | A1 |
| (c) \(a = 4 - 1 = 3\), \(d = 16 - 4 - 4 = 8\) | B1 |
| \(u_{10} = 3 + (9 \times 8) = 3 + 72 = 75\) | M1 A1 |
| (d) \(= \frac{40}{2}[6 + (39 \times 8)] = 20 \times 318 = 6360\) | M1 A1 (11) |
**(a)(i)** $= (t^2 - 5) - (t - 1) = t^2 - t - 4$ | M1 A1 |
**(a)(iii)** $= (t^2 - 5) + (t^3 - t - 4) = 2t^2 - t - 9$ | M1 A1 |
**(b)** $2t^2 - t - 9 = 19$
$2t^2 - t - 28 = 0$
$(2t + 7)(t - 4) = 0$ | M1 |
$t > 0 \therefore t = 4$ | A1 |
**(c)** $a = 4 - 1 = 3$, $d = 16 - 4 - 4 = 8$ | B1 |
$u_{10} = 3 + (9 \times 8) = 3 + 72 = 75$ | M1 A1 |
**(d)** $= \frac{40}{2}[6 + (39 \times 8)] = 20 \times 318 = 6360$ | M1 A1 (11) |\begin{enumerate}
\item The first two terms of an arithmetic progression are $( t - 1 )$ and $\left( t ^ { 2 } - 5 \right)$ respectively, where $t$ is a positive constant.\\
(a) Find and simplify expressions in terms of $t$ for\\
(i) the common difference,\\
(ii) the third term.
\end{enumerate}
Given also that the third term is 19 ,\\
(b) find the value of $t$,\\
(c) show that the 10th term is 75,\\
(d) find the sum of the first 40 terms.\\
\hfill \mbox{\textit{OCR C2 Q8 [11]}}