June 2022 Paper 3 Q4
4 The positive integers \(x\), \(y\) and \(z\) are the first, second and third terms, respectively, of an arithmetic progression with common difference \(-4\).
Also, \(x\), \(\dfrac{15}{y}\) and \(z\) are the first, second and third terms, respectively, of a geometric progression.
| Scheme | Marks | AO |
|---|---|---|
| GP: \(x, \dfrac{15}{y}, z \Rightarrow \dfrac{\frac{15}{y}}{x} = \dfrac{z}{\frac{15}{y}}\) | M1* | 3.1a |
| AP: \(x, y, z \Rightarrow y - x = -4\) or \(z - y = -4\) | M1* | 1.1 |
| \((y + 4)y^2(y - 4) = 225\) | M1dep* | 1.1 |
| \(y^2(y^2 - 4y + 4y - 16) = 225 \Rightarrow y^4 - 16y^2 - 225 = 0\) | A1 | 2.2a |
| [4] |
Notes
M1*: M1 for un-simplified correct use of \(r = \dfrac{u_n}{u_{n-1}}\) so allow this mark for stating that \(r = \dfrac{15}{xy}\) or \(r = \dfrac{yz}{15}\) or \(r^2 x = z\) oe
Or for the terms of the GP as \(y + 4, \dfrac{15}{y}, y - 4\) in \(y\) only
M1*: M1 for \(y - x = \pm 4\) or \(z - y = \pm 4\) or \(z - x = \pm 8\) oe
Or for the terms of the AP as \(y + 4, y, y - 4\) in \(y\) only
For reference: \(xy^2z = 225\), \(x = y + 4\) and \(z = y - 4\)
M1dep*: Eliminate \(x\) and \(z\) to form an equation in \(y\) only (must be equivalent to a quartic in \(y\))
Or for \(\dfrac{\;y - 4\;}{\frac{15}{y}} = \dfrac{\frac{15}{y}}{y + 4}\) \(\left(\Rightarrow y^2 - 16 = \dfrac{225}{y^2}\right)\)
A1: AG so sufficient working must be shown www – note that \(y = x + 4\) and \(y = z - 4\) (from \(d = +4\)) can lead to the correct equation (which can score the M marks only)
| Scheme | Marks | AO |
|---|---|---|
| \(y^4 - 16y^2 - 225 = 0 \Rightarrow y = \pm 5\) but \(y \gt 0 \Rightarrow y = 5\) | B1 | 1.1 |
| \(\Rightarrow r^2 = \dfrac{z}{x} = \dfrac{1}{9} \Rightarrow r = \dfrac{1}{3}\) | M1 | 1.1 |
| \(S_\infty = \dfrac{x}{1 - r} = \dfrac{9}{\frac{2}{3}}\) | M1 | 1.1 |
| \(S_\infty = 13.5\) | A1 | 1.1 |
| [4] |
Notes
B1: BC
Allow implied e.g. \(y = \pm 5\) then using \(y = 5\) only (B0 if the sum to infinity for other values of \(y\) are not rejected)
\(x = 9\), \(z = 1\)
M1: Calculate \(r\) (soi) corresponding to \(y = 5\)
allow unsimplified e.g. \(r = \dfrac{(5)(1)}{15}\) and allow if more than one value of \(y\) stated
Possibly done implicitly in formula for \(S_\infty\)
M1: Using the correct formula for the sum to infinity of a GP with their value of \(x\) (= their \(y \pm 4\)) and a value of \(r\) where \(-1 \lt r \lt 1\)
A1: cao oe eg \(\dfrac{27}{2}\) - do not award this mark if more than one value for \(S_\infty\) stated
A0 for a triple-decker fraction e.g. \(\dfrac{9}{\frac{2}{3}}\)