| Exam Board | Edexcel |
|---|---|
| Module | P4 (Pure Mathematics 4) |
| Year | 2022 |
| Session | October |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Proof |
| Type | Contradiction proof about integers |
| Difficulty | Standard +0.8 This requires systematic case analysis of factor pairs of 25, checking (1,25), (5,5), (25,1) and their negatives. While the method is straightforward once the factorization is seen, students must recognize all cases need checking and perform multiple algebraic calculations correctly—more demanding than routine proof questions but less conceptually deep than geometric or series proofs. |
| Spec | 1.01d Proof by contradiction |
| Answer | Marks |
|---|---|
| 8 | ( ) ( ) |
| Answer | Marks |
|---|---|
| 3 x 2 + 2 x y − y 2 = 2 5 '' | M1 |
Question 8:
8 | ( ) ( )
3 x − y = 2 5 a n d x + y = 1
Solves one of
( ) ( )
o r 3 x − y = 5 a n d x + y = 5
Correct solution of one.
3 x − y = 2 5 ( )
Either 4 x = 2 6 x = 6 . 5 , y = − 5 . 5
x + y = 1
3 x − y = 5 ( )
Or 4 x = 1 0 x = 2 .5 , y = 2 .5
x + y = 5
Solves both equations
Both solved correctly with a minimal reason given for the contradiction e.g
''not integers'' with conclusion ''hence there are no integers x and y such that
3 x 2 + 2 x y − y 2 = 2 5 '' | M1
A1
dM1
A1
(4)A student was asked to prove by contradiction that
"there are no positive integers $x$ and $y$ such that $3x^2 + 2xy - y^2 = 25$"
The start of the student's proof is shown in the box below.
\begin{center}
\fbox{\begin{minipage}{0.8\textwidth}
Assume that integers $x$ and $y$ exist such that $3x^2 + 2xy - y^2 = 25$
$\Rightarrow (3x - y)(x + y) = 25$
If $(3x - y) = 1$ and $(x + y) = 25$
$3x - y = 1\\
x + y = 25$ $\Rightarrow 4x = 26 \Rightarrow x = 6.5, y = 18.5$ Not integers
\end{minipage}}
\end{center}
Show the calculations and statements that are needed to complete the proof. [4]
\hfill \mbox{\textit{Edexcel P4 2022 Q8 [4]}}