| Exam Board | OCR |
|---|---|
| Module | H240/02 (Pure Mathematics and Statistics) |
| Year | 2022 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Topic | Proof |
| Type | Proof by exhaustion with cases |
| Difficulty | Moderate -0.3 Part (a) is a standard proof by exhaustion requiring students to square three cases (3m, 3m+1, 3m+2) and show none yield 3n+2—this is a textbook exercise in modular arithmetic. Part (b) requires counting favorable outcomes when selecting three integers with replacement, considering residue classes mod 3, which involves some combinatorial reasoning but follows directly from the setup. Overall slightly easier than average due to the heavily scaffolded structure and routine nature of both parts. |
| Spec | 1.01a Proof: structure of mathematical proof and logical steps2.03a Mutually exclusive and independent events |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \((3m+0)^2 = 9m^2\) | B1 | \(9m^2\) alone, not as part of longer expression |
| \((3m+1)^2 = 9m^2+6m+1\) and \((3m+2)^2 = 9m^2+12m+4\) | M1 | At least one expansion attempted using \(r=1\) or \(2\). Must include three (or four) terms, allow one error |
| \(= 3(3m^2+2m)+1\) or \(= 3(3m^2+4m+1)+1\) or \(3(3m^2+4m)+4\) | A1 | At least one of these seen explicitly |
| None of these is of the form \(3n+2\); allow "\(\neq 3n+2\)" | A1 | Must see the statement oe. Can be seen once at end or with each separate case. Dep complete method, with all three cases seen |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \((3m+r)^2 = 9m^2+6mr+r^2 = 3(3m^2+2mr)+r^2 = 3n+r^2\) | M1, A1 | Attempted. Must include 3 (or 4) terms, allow one error. Explicit |
| But \(r^2 = 0, 1\) or \(4\) | B1 | |
| Hence not in the form \(3n+2\) for any \(r\) | A1 | Must see the statement oe. Dep complete method |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Let \((3m+r)^2 = 3n+2\) | M1 | |
| \(3(3m^2+2mr-n) = 2-r^2\) | A1 | |
| Hence \(2-r^2\) is divisible by 3 | B1 | |
| But \(2-0^2=2\), \(2-1^2=1\), \(2-2^2=-2\); none of these is divisible by 3 | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \((3m)^2=(9m^2-2)+2\); \((3m+1)^2=(9m^2+6m-1)+2\); \((3m+2)^2=(9m^2+12m+2)+2\) | B1, M1, A1 | Allow one arithmetical error. Both correct |
| \((9m^2-2)=3(3m^2)-2\); \((9m^2+6m-1)=3(3m^2+2m)-1\); \((9m^2+12m+2)=3(3m^2+4m)+2\) | or \(3(3m^2-\frac{2}{3})+2\); or \(3(3m^2+2m-\frac{1}{3})+2\); or \(3(3m^2+4m+\frac{2}{3})+2\) | |
| Hence none is divisible by 3 | A1 | None of the brackets is an integer |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Either imply three digits all of the same type or imply three digits all of different types | M1* | Could be numerical or algebraic or in words. If listed, must be clear which ones are selected |
| \(P(0,0,0)\) or \(P(1,1,1)\) or \(P(2,2,2)\): \(\left(\frac{1}{3}\right)^3\) | M1 dep | M1 for \(\left(\frac{1}{3}\right)^3\) associated with at least one of these |
| \(P(0,1,2)\): \(\left(\frac{1}{3}\right)^3 \times 6\) or \(1\times\frac{2}{3}\times\frac{1}{3}\) oe | M1 dep | M1 for \(\left(\frac{1}{3}\right)^3 \times k\) where \(k=4,5\) or \(6\), associated with \((0,1,2)\) |
| Alternative: No. of cases \(= 3^3=27\); No. divisible by \(3 = (3+6=)\ 9\) | M1, M1 | Allow 7 or 8 |
| \(\frac{9}{27}\) or \(\frac{1}{3}\) or \(0.333\) (3 sf) | A1 | Correct answer with no working: M0M0M0A0 |
## Question 7(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| $(3m+0)^2 = 9m^2$ | B1 | $9m^2$ alone, not as part of longer expression |
| $(3m+1)^2 = 9m^2+6m+1$ and $(3m+2)^2 = 9m^2+12m+4$ | M1 | At least one expansion attempted using $r=1$ or $2$. Must include three (or four) terms, allow one error |
| $= 3(3m^2+2m)+1$ or $= 3(3m^2+4m+1)+1$ or $3(3m^2+4m)+4$ | A1 | At least one of these seen explicitly |
| None of these is of the form $3n+2$; allow "$\neq 3n+2$" | A1 | Must see the statement oe. Can be seen once at end or with each separate case. Dep complete method, with all three cases seen |
**Alternative method 1:**
| Answer | Mark | Guidance |
|--------|------|----------|
| $(3m+r)^2 = 9m^2+6mr+r^2 = 3(3m^2+2mr)+r^2 = 3n+r^2$ | M1, A1 | Attempted. Must include 3 (or 4) terms, allow one error. Explicit |
| But $r^2 = 0, 1$ or $4$ | B1 | |
| Hence not in the form $3n+2$ for any $r$ | A1 | Must see the statement oe. Dep complete method |
**Alternative method 2:**
| Answer | Mark | Guidance |
|--------|------|----------|
| Let $(3m+r)^2 = 3n+2$ | M1 | |
| $3(3m^2+2mr-n) = 2-r^2$ | A1 | |
| Hence $2-r^2$ is divisible by 3 | B1 | |
| But $2-0^2=2$, $2-1^2=1$, $2-2^2=-2$; none of these is divisible by 3 | A1 | |
**Alternative method 3:**
| Answer | Mark | Guidance |
|--------|------|----------|
| $(3m)^2=(9m^2-2)+2$; $(3m+1)^2=(9m^2+6m-1)+2$; $(3m+2)^2=(9m^2+12m+2)+2$ | B1, M1, A1 | Allow one arithmetical error. Both correct |
| $(9m^2-2)=3(3m^2)-2$; $(9m^2+6m-1)=3(3m^2+2m)-1$; $(9m^2+12m+2)=3(3m^2+4m)+2$ | | or $3(3m^2-\frac{2}{3})+2$; or $3(3m^2+2m-\frac{1}{3})+2$; or $3(3m^2+4m+\frac{2}{3})+2$ |
| Hence none is divisible by 3 | A1 | None of the brackets is an integer |
---
## Question 7(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Either imply three digits all of the same type or imply three digits all of different types | M1* | Could be numerical or algebraic or in words. If listed, must be clear which ones are selected |
| $P(0,0,0)$ or $P(1,1,1)$ or $P(2,2,2)$: $\left(\frac{1}{3}\right)^3$ | M1 dep | M1 for $\left(\frac{1}{3}\right)^3$ associated with at least one of these |
| $P(0,1,2)$: $\left(\frac{1}{3}\right)^3 \times 6$ or $1\times\frac{2}{3}\times\frac{1}{3}$ oe | M1 dep | M1 for $\left(\frac{1}{3}\right)^3 \times k$ where $k=4,5$ or $6$, associated with $(0,1,2)$ |
| **Alternative:** No. of cases $= 3^3=27$; No. divisible by $3 = (3+6=)\ 9$ | M1, M1 | Allow 7 or 8 |
| $\frac{9}{27}$ or $\frac{1}{3}$ or $0.333$ (3 sf) | A1 | Correct answer with no working: M0M0M0A0 |
---7 It is given that any integer can be expressed in the form $3 m + r$, where $m$ is an integer and $r$ is 0,1 or 2 .
Use this fact to answer the following.
\begin{enumerate}[label=(\alph*)]
\item By considering the different values of $r$, prove that the square of any integer cannot be expressed in the form $3 n + 2$, where $n$ is an integer.
\item Three integers are chosen at random from the integers 1 to 99 inclusive. The three integers are not necessarily different.
By considering the different values of $r$, determine the probability that the sum of these three integers is divisible by 3 .
\end{enumerate}
\hfill \mbox{\textit{OCR H240/02 2022 Q7 [8]}}