| Exam Board | OCR MEI |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2011 |
| Session | June |
| Marks | 17 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Geometric Sequences and Series |
| Type | Compare two growth models |
| Difficulty | Moderate -0.3 This is a straightforward multi-part question testing standard arithmetic and geometric sequence formulas. Part (i) uses basic arithmetic sequence calculations (nth term, sum formula). Part (ii) applies geometric sequence formulas with clear setup. The comparison in (ii)(B) requires simple substitution, and (ii)(D) involves rearranging the sum formula—all routine procedures with no novel insight required. Slightly easier than average due to clear structure and standard applications. |
| Spec | 1.04h Arithmetic sequences: nth term and sum formulae1.04i Geometric sequences: nth term and finite series sum1.04j Sum to infinity: convergent geometric series |r|<1 |
| Answer | Marks | Guidance |
|---|---|---|
| (i)(A) \(390\) | B2 | M1 for \(500 - 11 \times 10\) |
| (i)(B) \(S_{24} = \frac{24}{2}(2 \times 500 + (24 - 1) \times -10)\) o.e. i.s.w. or \(S_{24} = \frac{24}{2}(500 + 270)\) o.e. i.s.w. or \([= 9240]\) (answer given) | B2 | nothing simpler than \(12(1000 + 23x-10)\) or \(\frac{24}{2}(1000 - 230)\) or \(12(2 \times 500 - 230)\) or \(12(2 \times 500 - 230)\); if B2 not awarded, then M1 for use of a.p. formula for \(S_{24}\) with \(n = 24, a = 500\) and \(d = -10\) or M1 for \(l = 270\) s.o.i. |
| (ii)(A) \(368.33(…)\) or \(368.34\) | B2 | M1 for \(460 \times 0.98^{11}\) |
| (ii)(B) \(J_{20} = 310\) | B3 | B3 for all 4 values correct or B2 for 3 values correct or B1 for 2 values correct |
| \(M_{20} = 313.36(…), 313.4, 313.3, 313.37\) or \(313\) | ||
| \(J_{19} = 320\) | ||
| \(M_{19} = 319.76(…), 319.8\) or \(319.7\) | ||
| (ii)(C) \(8837\) to \(8837.06\) | B2 | M1 for \(S_{24} = \frac{460(1-0.98^{24})}{1-0.98}\) o.e. |
| (ii)(D) \(\frac{a(1-0.98^{24})}{(1-0.98)} = 9240\) o.e. | M1 | f.t. their power of 24 from (ii)C |
| \(480.97\) to \(480.98\) | A1 |
**(i)(A)** $390$ | B2 | M1 for $500 - 11 \times 10$
**(i)(B)** $S_{24} = \frac{24}{2}(2 \times 500 + (24 - 1) \times -10)$ o.e. i.s.w. or $S_{24} = \frac{24}{2}(500 + 270)$ o.e. i.s.w. or $[= 9240]$ (answer given) | B2 | nothing simpler than $12(1000 + 23x-10)$ or $\frac{24}{2}(1000 - 230)$ or $12(2 \times 500 - 230)$ or $12(2 \times 500 - 230)$; if B2 not awarded, then M1 for use of a.p. formula for $S_{24}$ with $n = 24, a = 500$ and $d = -10$ or M1 for $l = 270$ s.o.i. | condone omission of final bracket or "(23)-10" if recovered in later work; if they write the sum out, all the terms must be listed for 2 marks; $12 \times (1000 - 230)$ or $12 \times 770$ on its own do not score; 12 × (1000 − 230) or 12 × 770 on its own do not score
**(ii)(A)** $368.33(…)$ or $368.34$ | B2 | M1 for $460 \times 0.98^{11}$
**(ii)(B)** $J_{20} = 310$ | B3 | B3 for all 4 values correct or B2 for 3 values correct or B1 for 2 values correct | values which are clearly wrongly attributed do not score
$M_{20} = 313.36(…), 313.4, 313.3, 313.37$ or $313$ | |
$J_{19} = 320$ | |
$M_{19} = 319.76(…), 319.8$ or $319.7$ | |
**(ii)(C)** $8837$ to $8837.06$ | B2 | M1 for $S_{24} = \frac{460(1-0.98^{24})}{1-0.98}$ o.e.
**(ii)(D)** $\frac{a(1-0.98^{24})}{(1-0.98)} = 9240$ o.e. | M1 | f.t. their power of 24 from (ii)C
$480.97$ to $480.98$ | A1 |12 Jim and Mary are each planning monthly repayments for money they want to borrow.
\begin{enumerate}[label=(\roman*)]
\item Jim's first payment is $\pounds 500$, and he plans to pay $\pounds 10$ less each month, so that his second payment is $\pounds 490$, his third is $\pounds 480$, and so on.\\
(A) Calculate his 12th payment.\\
(B) He plans to make 24 payments altogether. Show that he pays $\pounds 9240$ in total.
\item Mary's first payment is $\pounds 460$ and she plans to pay $2 \%$ less each month than the previous month, so that her second payment is $\pounds 450.80$, her third is $\pounds 441.784$, and so on.\\
(A) Calculate her 12th payment.\\
(B) Show that Jim's 20th payment is less than Mary's 20th payment but that his 19th is not less than her 19th.\\
(C) Mary plans to make 24 payments altogether. Calculate how much she pays in total.\\
(D) How much would Mary's first payment need to be if she wishes to pay $2 \%$ less each month as before, but to pay the same in total as Jim, $\pounds 9240$, over the 24 months?
\end{enumerate}
\hfill \mbox{\textit{OCR MEI C2 2011 Q12 [17]}}