| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Session | Specimen |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Geometric Sequences and Series |
| Type | Form and solve quadratic in parameter |
| Difficulty | Moderate -0.3 This is a straightforward two-part question testing basic definitions of arithmetic and geometric progressions. Part (i) requires forming and solving a simple quadratic equation (x² - 4x = 12), while part (ii) involves using the sum to infinity formula and finding r. Both parts are routine applications of standard formulas with no conceptual challenges, making it slightly easier than average. |
| 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 |
|---|---|---|
| Answer | Marks | Guidance |
| \(x^2 - 4x = 12\) | M1 | \(4x - x^2 = 12\) scores M1A0 |
| \(x = -2\) or \(6\) | A1 | |
| \(3^{rd}\) term \(= (-2)^2 + 12 = 16\) or \(6^2 + 12 = 48\) | A1A1 | SC1 for 16, 48 after \(x = 2, -6\) |
| Total: 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(r^2 = \frac{x^2}{4x}\left(=\frac{x}{4}\right)\) soi | M1 | |
| \(\dfrac{4x}{1-\frac{x}{4}} = 8\) | M1 | Accept use of unsimplified \(\frac{x^2}{4x}\) or \(\frac{4x}{x^2}\) or \(\frac{4}{x}\) |
| \(x = \frac{4}{3}\) or \(r = \frac{1}{3}\) | A1 | |
| \(3^{rd}\) term \(= \frac{16}{27}\) (or 0.593) | A1 | |
| Total: 4 | ||
| Alternative Method: | ||
| \(\frac{4x}{1-r} = 8 \rightarrow r = 1 - \frac{1}{2}x\) or \(\frac{4x}{1-r} = 8 \rightarrow x = 2(1-r)\) | (M1) | |
| \(x^2 = 4x\left(1 - \frac{1}{2}x\right)\) or \(r = \frac{2(1-r)}{4}\) | (M1) | |
| \(x = \frac{4}{3}\) and \(r = \frac{1}{3}\) | (A1) |
## Question 8(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $x^2 - 4x = 12$ | M1 | $4x - x^2 = 12$ scores M1A0 |
| $x = -2$ or $6$ | A1 | |
| $3^{rd}$ term $= (-2)^2 + 12 = 16$ or $6^2 + 12 = 48$ | A1A1 | SC1 for 16, 48 after $x = 2, -6$ |
| **Total: 4** | | |
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## Question 8(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $r^2 = \frac{x^2}{4x}\left(=\frac{x}{4}\right)$ soi | M1 | |
| $\dfrac{4x}{1-\frac{x}{4}} = 8$ | M1 | Accept use of unsimplified $\frac{x^2}{4x}$ or $\frac{4x}{x^2}$ or $\frac{4}{x}$ |
| $x = \frac{4}{3}$ or $r = \frac{1}{3}$ | A1 | |
| $3^{rd}$ term $= \frac{16}{27}$ (or 0.593) | A1 | |
| **Total: 4** | | |
| **Alternative Method:** | | |
| $\frac{4x}{1-r} = 8 \rightarrow r = 1 - \frac{1}{2}x$ **or** $\frac{4x}{1-r} = 8 \rightarrow x = 2(1-r)$ | (M1) | |
| $x^2 = 4x\left(1 - \frac{1}{2}x\right)$ or $r = \frac{2(1-r)}{4}$ | (M1) | |
| $x = \frac{4}{3}$ and $r = \frac{1}{3}$ | (A1) | |
---8 The first term of a progression is $4 x$ and the second term is $x ^ { 2 }$.\\
(i) For the case where the progression is arithmetic with a common difference of 12 , find the possible values of $x$ and the corresponding values of the third term.\\
(ii) For the case where the progression is geometric with a sum to infinity of 8 , find the third term.\\
\hfill \mbox{\textit{CAIE P1 Q8 [8]}}