Moderate -0.3 Part (a) requires finding a simple counterexample (e.g., p=2√2, q=√2 gives ratio 2), which is straightforward once you understand the concept. Part (b)(i) is routine graph sketching. Part (b)(ii) requires understanding the triangle inequality but can be verified by cases or geometric reasoning. Overall slightly easier than average due to the accessible counterexample and standard inequality verification.
3. a. "If \(p\) and \(q\) are irrational numbers, where \(p \neq q , q \neq 0\), then \(\frac { p } { q }\) is also irrational."
Disprove this statement by means of a counter example.
b. (i) Sketch the graph of \(y = | x | - 2\).
(ii) Explain why \(| x - 2 | \geq | x | - 2\) for all real values of \(x\).
States or uses any valid pair of different irrational numbers e.g. \(p=\sqrt{2},\ q=\sqrt{8}\): \(\dfrac{p}{q}=\dfrac{\sqrt{2}}{\sqrt{8}}=\dfrac{1}{2}\)
M1
States or uses any pair of different numbers that will disprove the statement; e.g. \(\sqrt{2},\sqrt{8}\); \(\sqrt{3},\sqrt{12}\); \(\pi, 2\pi\); \(3e, 7e\)
\(\dfrac{1}{2}\) is rational, therefore statement is disproved
A1
Uses correct reasoning to disprove with correct conclusion
(2 marks)
Answer
Marks
Guidance
### Part (b)(i): Sketch \(y =
x
- 2\)
Answer
Mark
Guidance
V-shaped graph symmetrical about \(y\)-axis with intercepts \((2,0)\), \((0,-2)\) and \((-2,0)\)
B1
Must show correct intercepts
### Part (b)(ii): Explain why \(
x-2
\geq
Answer
Mark
Guidance
Draw graph of \(y =
x-2
\) on same axes
Graph of \(y=
x-2
\) is either the same as or above graph of \(y=
(3 marks)
## Question 3:
### Part (a): Disprove by counter example
| Answer | Mark | Guidance |
|--------|------|----------|
| States or uses any valid pair of different irrational numbers e.g. $p=\sqrt{2},\ q=\sqrt{8}$: $\dfrac{p}{q}=\dfrac{\sqrt{2}}{\sqrt{8}}=\dfrac{1}{2}$ | M1 | States or uses any pair of different numbers that will disprove the statement; e.g. $\sqrt{2},\sqrt{8}$; $\sqrt{3},\sqrt{12}$; $\pi, 2\pi$; $3e, 7e$ |
| $\dfrac{1}{2}$ is rational, therefore statement is disproved | A1 | Uses correct reasoning to disprove with correct conclusion |
**(2 marks)**
### Part (b)(i): Sketch $y = |x| - 2$
| Answer | Mark | Guidance |
|--------|------|----------|
| V-shaped graph symmetrical about $y$-axis with intercepts $(2,0)$, $(0,-2)$ and $(-2,0)$ | B1 | Must show correct intercepts |
### Part (b)(ii): Explain why $|x-2| \geq |x| - 2$
| Answer | Mark | Guidance |
|--------|------|----------|
| Draw graph of $y = |x-2|$ on same axes | M1 | Draws $y=|x-2|$ on top of $y=|x|-2$ |
| Graph of $y=|x-2|$ is either the same as or above graph of $y=|x|-2$; or when $x \geq 0$, $y=|x-2|$ is above $y=|x|-2$ | A1 | Correct conclusion from graphical comparison |
**(3 marks)**
3. a. "If $p$ and $q$ are irrational numbers, where $p \neq q , q \neq 0$, then $\frac { p } { q }$ is also irrational."
Disprove this statement by means of a counter example.\\
b. (i) Sketch the graph of $y = | x | - 2$.\\
(ii) Explain why $| x - 2 | \geq | x | - 2$ for all real values of $x$.\\
\hfill \mbox{\textit{Edexcel PMT Mocks Q3 [5]}}