| Exam Board | OCR |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2010 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Geometric Sequences and Series |
| Type | Form and solve quadratic in parameter |
| Difficulty | Standard +0.3 This is a standard C2 geometric/arithmetic progression problem requiring algebraic manipulation to form a cubic equation, then using convergence conditions to find r, and finally applying the sum to infinity formula. While it involves multiple steps, each technique is routine for C2 level—forming equations from sequence properties, solving by factorization, and applying standard formulas. The cubic factorizes straightforwardly as (r-1)(r²+r-1)=0, and the convergence condition |r|<1 immediately eliminates most solutions. Slightly above average difficulty due to the multi-part nature and cubic equation, but no novel insight required. |
| Spec | 1.02f Solve quadratic equations: including in a function of unknown1.04i Geometric sequences: nth term and finite series sum1.04j Sum to infinity: convergent geometric series |r|<1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(ar = a + d\), \(ar^3 = a + 2d\) | M1 | Attempt to link terms of AP and GP, implicitly or explicitly |
| \(2ar - ar^3 = a\) | M1 | Attempt to eliminate \(d\), implicitly or explicitly, to show given equation |
| \(r^3 - 2r + 1 = 0\) A.G. | A1 | Show \(r^3 - 2r + 1 = 0\) convincingly |
| Answer | Marks | Guidance |
|---|---|---|
| \(f(r) = (r-1)(r^2 + r - 1)\) | B1 | Identify \((r-1)\) as factor or \(r = 1\) as root |
| M1* | Attempt to find quadratic factor | |
| \(r = \frac{-1 \pm \sqrt{5}}{2}\) | A1 | Obtain \(r^2 + r - 1\) |
| M1d* | Attempt to solve quadratic | |
| Hence \(r = \frac{-1+\sqrt{5}}{2}\) | A1 | Obtain \(r = \frac{-1+\sqrt{5}}{2}\) only |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{a}{1-r} = 3 + \sqrt{5}\) | M1 | Equate \(S_\infty\) to \(3 + \sqrt{5}\) |
| \(a = \left(\frac{3}{2} - \frac{\sqrt{5}}{2}\right)(3+\sqrt{5})\) | A1 | Obtain \(\frac{a}{1-\left(\frac{-1+\sqrt{5}}{2}\right)} = 3+\sqrt{5}\) |
| \(a = \frac{9}{2} - \frac{5}{2}\) | M1 | Attempt to find \(a\) |
| \(a = 2\) | A1 | Obtain \(a = 2\) |
## Question 9:
### Part (i)
| $ar = a + d$, $ar^3 = a + 2d$ | M1 | Attempt to link terms of AP and GP, implicitly or explicitly |
| $2ar - ar^3 = a$ | M1 | Attempt to eliminate $d$, implicitly or explicitly, to show given equation |
| $r^3 - 2r + 1 = 0$ **A.G.** | A1 | Show $r^3 - 2r + 1 = 0$ convincingly |
### Part (ii)
| $f(r) = (r-1)(r^2 + r - 1)$ | B1 | Identify $(r-1)$ as factor or $r = 1$ as root |
| | M1* | Attempt to find quadratic factor |
| $r = \frac{-1 \pm \sqrt{5}}{2}$ | A1 | Obtain $r^2 + r - 1$ |
| | M1d* | Attempt to solve quadratic |
| Hence $r = \frac{-1+\sqrt{5}}{2}$ | A1 | Obtain $r = \frac{-1+\sqrt{5}}{2}$ only |
### Part (iii)
| $\frac{a}{1-r} = 3 + \sqrt{5}$ | M1 | Equate $S_\infty$ to $3 + \sqrt{5}$ |
| $a = \left(\frac{3}{2} - \frac{\sqrt{5}}{2}\right)(3+\sqrt{5})$ | A1 | Obtain $\frac{a}{1-\left(\frac{-1+\sqrt{5}}{2}\right)} = 3+\sqrt{5}$ |
| $a = \frac{9}{2} - \frac{5}{2}$ | M1 | Attempt to find $a$ |
| $a = 2$ | A1 | Obtain $a = 2$ |9 A geometric progression has first term $a$ and common ratio $r$, and the terms are all different. The first, second and fourth terms of the geometric progression form the first three terms of an arithmetic progression.\\
(i) Show that $r ^ { 3 } - 2 r + 1 = 0$.\\
(ii) Given that the geometric progression converges, find the exact value of $r$.\\
(iii) Given also that the sum to infinity of this geometric progression is $3 + \sqrt { 5 }$, find the value of the integer $a$.
\hfill \mbox{\textit{OCR C2 2010 Q9 [12]}}