Edexcel Paper 2 2018 June — Question 3 5 marks

Exam BoardEdexcel
ModulePaper 2 (Paper 2)
Year2018
SessionJune
Marks5
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicProof
TypeRational and irrational number properties
DifficultyModerate -0.8 Part (a) requires finding a simple counterexample (e.g., √2 × √2 = 2), which is straightforward once you understand what's being asked. Part (b)(i) is routine graph sketching, and (b)(ii) requires basic reasoning about absolute values that can be verified by considering cases or geometric interpretation. This is easier than average A-level content, requiring mainly understanding of definitions rather than complex problem-solving.
Spec1.01c Disproof by counter example1.02l Modulus function: notation, relations, equations and inequalities1.02s Modulus graphs: sketch graph of |ax+b|
  1. (a) "If \(m\) and \(n\) are irrational numbers, where \(m \neq n\), then \(m n\) is also irrational."
Disprove this statement by means of a counter example.
(b) (i) Sketch the graph of \(y = | x | + 3\) (ii) Explain why \(| x | + 3 \geqslant | x + 3 |\) for all real values of \(x\).
Question 3:
Part (a):
AnswerMarks Guidance
Answer/WorkingMark Guidance
States or uses any pair of *different* irrational numbers, e.g. \(m = \sqrt{3}\), \(n = \sqrt{12}\)M1 1.1b
\(\{mn =\}\ (\sqrt{3})(\sqrt{12}) = 6\), \(\Rightarrow\) statement untrue or 6 is not irrational or 6 is rationalA1 2.4
Part (b)(i) and (ii) Way 1:
AnswerMarks Guidance
Answer/WorkingMark Guidance
V shaped graph reasonably symmetrical about \(y\)-axis with vertical intercept \((0,3)\) or 3 stated or marked on positive \(y\)-axisB1 1.1b
Superimposes the graph of \(y=\x+3\ \) on top of the graph of \(y=\
The graph of \(y=\x\ +3\) is either the same or above the graph of \(y=\
Part (b)(ii) Way 2:
AnswerMarks Guidance
Answer/WorkingMark Guidance
Reason 1: When \(x\geq 0\), \(\x\ +3=\
Reason 2: When \(x<0\), \(\x\ +3>\ 
# Question 3:

## Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| States or uses any pair of *different* irrational numbers, e.g. $m = \sqrt{3}$, $n = \sqrt{12}$ | M1 | 1.1b |
| $\{mn =\}\ (\sqrt{3})(\sqrt{12}) = 6$, $\Rightarrow$ statement untrue **or** 6 is not irrational or 6 is rational | A1 | 2.4 |

## Part (b)(i) and (ii) Way 1:
| Answer/Working | Mark | Guidance |
|---|---|---|
| V shaped graph reasonably symmetrical about $y$-axis with vertical intercept $(0,3)$ or 3 stated or marked on positive $y$-axis | B1 | 1.1b |
| Superimposes the graph of $y=\|x+3\|$ on top of the graph of $y=\|x\|+3$ | M1 | 3.1a |
| The graph of $y=\|x\|+3$ is either the same or above the graph of $y=\|x+3\|$ for corresponding values of $x$; **or** when $x\geq 0$, both graphs are equal; when $x<0$, graph of $y=\|x\|+3$ is above graph of $y=\|x+3\|$ | A1 | 2.4 |

## Part (b)(ii) Way 2:
| Answer/Working | Mark | Guidance |
|---|---|---|
| **Reason 1:** When $x\geq 0$, $\|x\|+3=\|x+3\|$ | M1 | 3.1a - Any one of Reason 1 or Reason 2 |
| **Reason 2:** When $x<0$, $\|x\|+3>\|x+3\|$ | A1 | 2.4 - Both Reason 1 and Reason 2 |

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\begin{enumerate}
  \item (a) "If $m$ and $n$ are irrational numbers, where $m \neq n$, then $m n$ is also irrational."
\end{enumerate}

Disprove this statement by means of a counter example.\\
(b) (i) Sketch the graph of $y = | x | + 3$\\
(ii) Explain why $| x | + 3 \geqslant | x + 3 |$ for all real values of $x$.

\hfill \mbox{\textit{Edexcel Paper 2 2018 Q3 [5]}}
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