| Exam Board | Edexcel |
|---|---|
| Module | AS Paper 1 (AS Paper 1) |
| Year | 2020 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Proof |
| Type | Contradiction proof of inequality |
| Difficulty | Moderate -0.3 Part (a) is a standard AM-GM inequality application requiring algebraic manipulation (multiply by ab, rearrange to perfect square form, or apply AM-GM directly), which is routine for AS level. Part (b) simply requires finding any negative values as a counterexample, which is trivial. Overall slightly easier than average due to the straightforward techniques involved. |
| Spec | 1.01a Proof: structure of mathematical proof and logical steps1.01c Disproof by counter example |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| States \((2a - b)^2 \geq 0\) | M1 | Condone \(>\) for first three marks |
| \(4a^2 + b^2 \geq 4ab\) | A1 | |
| Divide by \(ab\): \(\dfrac{4a^2}{ab} + \dfrac{b^2}{ab} \geq \dfrac{4ab}{ab}\) | M1 | Valid as \(a > 0, b > 0\) |
| Hence \(\dfrac{4a}{b} + \dfrac{b}{a} \geq 4\) | A1* | CSO; must state: squaring gives \(\geq 0\), and dividing by \(ab\) preserves inequality since \(a,b > 0\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(a = 5, b = -1 \Rightarrow \dfrac{4a}{b} + \dfrac{b}{a} = -20 - \dfrac{1}{5}\) which is less than 4 | B1 | Counter-example with values, calculation, and conclusion in words |
# Question 13:
## Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| States $(2a - b)^2 \geq 0$ | M1 | Condone $>$ for first three marks |
| $4a^2 + b^2 \geq 4ab$ | A1 | |
| Divide by $ab$: $\dfrac{4a^2}{ab} + \dfrac{b^2}{ab} \geq \dfrac{4ab}{ab}$ | M1 | Valid as $a > 0, b > 0$ |
| Hence $\dfrac{4a}{b} + \dfrac{b}{a} \geq 4$ | A1* | CSO; must state: squaring gives $\geq 0$, and dividing by $ab$ preserves inequality since $a,b > 0$ |
## Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $a = 5, b = -1 \Rightarrow \dfrac{4a}{b} + \dfrac{b}{a} = -20 - \dfrac{1}{5}$ which is less than 4 | B1 | Counter-example with values, calculation, and conclusion in words |
---\begin{enumerate}
\item (a) Prove that for all positive values of $a$ and $b$
\end{enumerate}
$$\frac { 4 a } { b } + \frac { b } { a } \geqslant 4$$
(b) Prove, by counter example, that this is not true for all values of $a$ and $b$.
\hfill \mbox{\textit{Edexcel AS Paper 1 2020 Q13 [5]}}