| Exam Board | WJEC |
|---|---|
| Module | Unit 1 (Unit 1) |
| Session | Specimen |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Proof |
| Type | Counter example to disprove statement |
| Difficulty | Standard +0.3 This question tests understanding of when equations imply equality, requiring students to recognize that squaring loses sign information (making A false with counterexample c=0, d=-1) while cubing preserves it (making B true). It's slightly easier than average as the concepts are straightforward, the counterexample is simple to find, and the proof requires only basic algebraic manipulation of cube roots. |
| Spec | 1.01a Proof: structure of mathematical proof and logical steps1.01c Disproof by counter example |
| Answer | Marks | Guidance |
|---|---|---|
| (a) For statement A: Choice of \(c = -\frac{1}{2}\) and \(d = -c – 1\) | M1 | AO2 |
| Correct verification that given equation is satisfied | A1 | AO2 |
| (b) For statement B: Use of the fact that any real number has a unique real cube root | M1 | AO2 |
| \((2c + 1)^3 = (2d + 1)^3 \Rightarrow 2c + 1 = 2d + 1\) | A1 | AO2 |
| \(2c + 1 = 2d + 1 \Rightarrow c = d\) | A1 | AO2 |
**(a)** For statement A: Choice of $c = -\frac{1}{2}$ and $d = -c – 1$ | M1 | AO2
Correct verification that given equation is satisfied | A1 | AO2
**(b)** For statement B: Use of the fact that any real number has a unique real cube root | M1 | AO2
$(2c + 1)^3 = (2d + 1)^3 \Rightarrow 2c + 1 = 2d + 1$ | A1 | AO2
$2c + 1 = 2d + 1 \Rightarrow c = d$ | A1 | AO2
**Total: [5]**
---In each of the two statements below, $c$ and $d$ are real numbers. One of the statements is true while the other is false.
A: Given that $(2c + 1)^2 = (2d + 1)^2$, then $c = d$.
B: Given that $(2c + 1)^3 = (2d + 1)^3$, then $c = d$.
\begin{enumerate}[label=(\alph*)]
\item Identify the statement which is false. Find a counter example to show that this statement is in fact false.
\item Identify the statement which is true. Give a proof to show that this statement is in fact true. [5]
\end{enumerate}
\hfill \mbox{\textit{WJEC Unit 1 Q6 [5]}}