| Exam Board | AQA |
|---|---|
| Module | Further Paper 2 (Further Paper 2) |
| Year | 2020 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | 3x3 Matrices |
| Type | Volume/area scale factors |
| Difficulty | Hard +2.3 This is a challenging Further Maths question requiring recognition that the determinant in (a) has a special structure amenable to factorization (likely involving Vandermonde-like patterns), then applying this abstract result to find det(M) and solve a cubic equation. The factorization requires significant algebraic insight beyond routine determinant expansion, and connecting parts (a) and (b) requires pattern recognition under parameter substitution. |
| Spec | 4.03j Determinant 3x3: calculation4.03k Determinant 3x3: volume scale factor |
| Answer | Marks |
|---|---|
| 8(a) | Shows understanding that |
| Answer | Marks | Guidance |
|---|---|---|
| or row operation is used. | 1.1a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| determinant. | 1.1a | M1 |
| Expands the determinant | 1.1a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| factor | 1.1b | A1 |
| Correctly finds two factors | 1.1b | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| 6 marks | 1.1b | A1 |
| Answer | Marks |
|---|---|
| 8(b) | Forms an expression for the |
| Answer | Marks | Guidance |
|---|---|---|
| 300 | 3.1a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| volume scale factor = det M | 1.1a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| π₯π₯ β1Β±2β7 | 2.2a | A1 |
| Total | 9 | π₯π₯ = β1+2β7 , π₯π₯ = β1β2β7 |
| Q | Marking Instructions | AO |
Question 8:
--- 8(a) ---
8(a) | Shows understanding that
value of the determinant is
unchanged when a column
or row operation is used. | 1.1a | M1 | 2 2
2 π₯π₯+ππ π₯π₯ +ππ
2
πποΏ½β1 ππ βππ οΏ½
2
1 ππ ππ
2 2
0 π₯π₯βππ π₯π₯ βππ
2
πποΏ½β1 ππ βππ οΏ½
2
1 ππ ππ
0 1 π₯π₯+ππ
2
ππ(π₯π₯βππ)οΏ½β1 ππ βππ οΏ½
2
1 ππ ππ
0 1 π₯π₯+ππ
ππ(π₯π₯βππ)(ππ+ππ)οΏ½0 1 ππβπποΏ½
2
1 ππ ππ
ππ(π₯π₯βππ)(ππ+ππ){(ππβππ)β(π₯π₯+ ππ)}
Demonstrates
understanding that a factor
can be extracted from the
determinant. | 1.1a | M1
Expands the determinant | 1.1a | M1
Correctly extracts one
factor | 1.1b | A1
Correctly finds two factors | 1.1b | A1
Obtains the determinant in
fully factorised form
OE
βCoππr(rππec+t ππa)n(sππw+erπ₯π₯ s)e(π₯π₯enβ: ππ)
6 marks | 1.1b | A1
--- 8(b) ---
8(b) | Forms an expression for the
volume scale factor
300 | 3.1a | M1 | βππ(π₯π₯βππ)(ππ+ππ)(π₯π₯+ππ)
ππ = 5,ππ = 3 and ππππ = Β±480
β5Γ8Γ(π₯π₯β3)(π₯π₯+5) = Β± 480
(π₯π₯β3)(π₯π₯+5) = Β± 12
π₯π₯ = 1, π₯π₯ = β3
Forms an equation f=or 0th.6e25ir
volume scale factor = det M | 1.1a | M1
Deduces all four correct
values of : -3, 1,
π₯π₯ β1Β±2β7 | 2.2a | A1
Total | 9 | π₯π₯ = β1+2β7 , π₯π₯ = β1β2β7
Q | Marking Instructions | AO | Marks | Typical Solution\begin{enumerate}[label=(\alph*)]
\item Factorise $\begin{vmatrix} 2u + h + x & x + h & x^2 + h^2 \\ 0 & a & -a^2 \\ a + b & b & b^2 \end{vmatrix}$ as fully as possible.
[6 marks]
\item The matrix $\mathbf{M}$ is defined by
$$\mathbf{M} = \begin{bmatrix} 13 + x & x + 3 & x^2 + 9 \\ 0 & 5 & 25 \\ 8 & 3 & 9 \end{bmatrix}$$
Under the transformation represented by $\mathbf{M}$, a solid of volume $0.625 \text{m}^3$ becomes a solid of volume $300 \text{m}^3$
Use your answer to part (a) to find the possible values of $x$.
[3 marks]
\end{enumerate}
\hfill \mbox{\textit{AQA Further Paper 2 2020 Q8 [9]}}