| Exam Board | SPS |
|---|---|
| Module | SPS FM (SPS FM) |
| Year | 2022 |
| Session | October |
| Marks | 20 |
| Topic | Inequalities |
| Type | Solve linear inequality |
| Difficulty | Challenging +1.8 Part (a) requires recognizing a telescoping pattern by rewriting fractions as (x-a)/b = 1 - (a+b)/b, which is non-standard insight for A-level. Part (b) involves a symmetric system requiring substitution u=1/x, v=1/y, w=1/z and solving three simultaneous equations, which is challenging but methodical. Both parts demand problem-solving beyond routine techniques, placing this well above average difficulty. |
| Spec | 1.02c Simultaneous equations: two variables by elimination and substitution1.02g Inequalities: linear and quadratic in single variable |
\begin{enumerate}
\item a) Solve the inequality:
\end{enumerate}
$$\frac { x - 9 } { 2012 } + \frac { x - 8 } { 2013 } + \frac { x - 7 } { 2014 } + \frac { x - 6 } { 2015 } + \frac { x - 5 } { 2016 } \leq \frac { x - 2012 } { 9 } + \frac { x - 2013 } { 8 } + \frac { x - 2014 } { 7 } + \frac { x - 2015 } { 6 } + \frac { x - 2016 } { 5 }$$
b) Find all ( $x , y , z$ ) such that:
$$\frac { 1 } { x } + \frac { 1 } { y + z } = \frac { 1 } { 3 } , \quad \frac { 1 } { y } + \frac { 1 } { z + x } = \frac { 1 } { 5 } , \quad \frac { 1 } { z } + \frac { 1 } { x + y } = \frac { 1 } { 7 }$$
[Question 1 - Continued]\\[0pt]
[Question 1 - Continued]\\
\hfill \mbox{\textit{SPS SPS FM 2022 Q1 [20]}}