| Exam Board | AQA |
|---|---|
| Module | D1 (Decision Mathematics 1) |
| Year | 2012 |
| Session | January |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Sorting Algorithms |
| Type | Deducing Original List Properties |
| Difficulty | Standard +0.8 This question requires understanding bubble sort mechanics and systematically deriving inequalities from each pass, then solving a system of 6+ inequalities to find a unique integer value. While the individual steps are accessible, the multi-stage logical reasoning and constraint satisfaction makes it moderately challenging for A-level, though still within standard D1 scope. |
| Spec | 1.02g Inequalities: linear and quadratic in single variable7.03j Sorting: bubble sort and shuttle sort |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| For the integers to be positive, the smallest must be \(> 0\) | M1 | Must consider the smallest value |
| \((x+1) > 0\) is not sufficient alone; need \((3x-5) \geq 1\) or \((4x-13) \geq 1\) | ||
| \(4x - 13 > 0 \Rightarrow x > 3.25\), so \(x \geq 4\) (since \(x\) is an integer) | A1 | Accept equivalent argument showing \(x \geq 4\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| After first pass: \((3x-5) \leq (x+1)\), giving \(x \leq 3\) — contradiction, so \(3x-5 \geq x+1\) | ||
| \((3x-5) \leq (x+1) \Rightarrow 2x \leq 6 \Rightarrow x \leq 3\) | B1 | |
| \((x+1) \leq (4x-13) \Rightarrow 14 \leq 3x \Rightarrow x \geq \frac{14}{3}\), so \(x \geq 5\) | B1 | |
| \((4x-13) \leq (2x+3) \Rightarrow 2x \leq 16 \Rightarrow x \leq 8\) | B1 | Three inequalities required |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \((x+1) \leq (4x-13) \Rightarrow x \geq \frac{14}{3}\), i.e. \(x \geq 5\) | B1 | |
| \((4x-13) \leq (3x-5) \Rightarrow x \leq 8\) | B1 | Two inequalities required |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \((3x-5) \leq (2x+3) \Rightarrow x \leq 8\) | B1 | One inequality required |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Combining inequalities: \(x \geq 5\) and \(x \leq 8\), with \(x \geq 4\) from part (a) | M1 | Must use results to find a specific value |
| Need all four integers distinct; testing values shows \(x = 7\) gives distinct integers | A1 | \(x = 7\): values are \(16, 17, 8, 15\) — all distinct |
## Question 8:
**(a)**
| Answer | Mark | Guidance |
|--------|------|----------|
| For the integers to be positive, the smallest must be $> 0$ | M1 | Must consider the smallest value |
| $(x+1) > 0$ is not sufficient alone; need $(3x-5) \geq 1$ or $(4x-13) \geq 1$ | | |
| $4x - 13 > 0 \Rightarrow x > 3.25$, so $x \geq 4$ (since $x$ is an integer) | A1 | Accept equivalent argument showing $x \geq 4$ |
**(b)(i)**
| Answer | Mark | Guidance |
|--------|------|----------|
| After first pass: $(3x-5) \leq (x+1)$, giving $x \leq 3$ — **contradiction**, so $3x-5 \geq x+1$ | | |
| $(3x-5) \leq (x+1) \Rightarrow 2x \leq 6 \Rightarrow x \leq 3$ | B1 | |
| $(x+1) \leq (4x-13) \Rightarrow 14 \leq 3x \Rightarrow x \geq \frac{14}{3}$, so $x \geq 5$ | B1 | |
| $(4x-13) \leq (2x+3) \Rightarrow 2x \leq 16 \Rightarrow x \leq 8$ | B1 | Three inequalities required |
**(b)(ii)**
| Answer | Mark | Guidance |
|--------|------|----------|
| $(x+1) \leq (4x-13) \Rightarrow x \geq \frac{14}{3}$, i.e. $x \geq 5$ | B1 | |
| $(4x-13) \leq (3x-5) \Rightarrow x \leq 8$ | B1 | Two inequalities required |
**(b)(iii)**
| Answer | Mark | Guidance |
|--------|------|----------|
| $(3x-5) \leq (2x+3) \Rightarrow x \leq 8$ | B1 | One inequality required |
**(c)**
| Answer | Mark | Guidance |
|--------|------|----------|
| Combining inequalities: $x \geq 5$ and $x \leq 8$, with $x \geq 4$ from part (a) | M1 | Must use results to find a specific value |
| Need all four integers **distinct**; testing values shows $x = 7$ gives distinct integers | A1 | $x = 7$: values are $16, 17, 8, 15$ — all distinct |8 Four distinct positive integers are $( 3 x - 5 ) , ( 2 x + 3 ) , ( x + 1 )$ and $( 4 x - 13 )$.
\begin{enumerate}[label=(\alph*)]
\item Explain why $x \geqslant 4$.
\item The four integers are to be sorted into ascending order using a bubble sort.
The original list is\\
$\begin{array} { c c c c } ( 3 x - 5 ) & ( 2 x + 3 ) & ( x + 1 ) & ( 4 x - 13 ) \end{array}$\\
After the first pass, the list is\\
$( 3 x - 5 ) \quad ( x + 1 ) \quad ( 4 x - 13 ) \quad ( 2 x + 3 )$\\
After the second pass, the list is\\
$( x + 1 )$\\
$( 4 x - 13 )$\\
$( 3 x - 5 )$\\
$( 2 x + 3 )$\\
After the third pass, the list is\\
$( 4 x - 13 ) \quad ( x + 1 )$\\
$( 3 x - 5 )$\\
( $2 x + 3$ )
\begin{enumerate}[label=(\roman*)]
\item By considering the list after the first pass, write down three inequalities in terms of $x$.
\item By considering the list after the second pass, write down two further inequalities in terms of $x$.
\item By considering the list after the third pass, write down one further inequality in terms of $x$.
\end{enumerate}\item Hence, by considering the results above, find the value of $x$.
\begin{center}
\includegraphics[max width=\textwidth, alt={}]{5a414265-6273-41c5-ad5f-f6316bd774d0-19_2486_1714_221_153}
\end{center}
\end{enumerate}
\hfill \mbox{\textit{AQA D1 2012 Q8 [10]}}