| Exam Board | OCR |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2006 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Proof by induction |
| Type | Suggest and prove formula |
| Difficulty | Standard +0.3 This is a straightforward induction question with a diagonal matrix where the pattern is immediately obvious from computing A² and A³. The inductive step requires only basic matrix multiplication of 2×2 matrices with mostly zero entries, making it more routine than typical induction proofs that involve algebraic manipulation or summation formulas. |
| Spec | 4.01a Mathematical induction: construct proofs4.03b Matrix operations: addition, multiplication, scalar |
| Answer | Marks | Guidance |
|---|---|---|
| Part (i) \(A^2 = \begin{pmatrix} 4 & 0 \\ 0 & 1 \end{pmatrix}\) | M1 | Attempt at matrix multiplication |
| A1, A1, 3 | Correct \(A^2\); Correct \(A^3\) | |
| \(A^3 = \begin{pmatrix} 8 & 0 \\ 0 & 1 \end{pmatrix}\) | ||
| Part (ii) \(A^n = \begin{pmatrix} 2^n & 0 \\ 0 & 1 \end{pmatrix}\) | B1 | Sensible conjecture made |
| B1, M1, A1, A1, 4 | State that conjecture is true for \(n = 1\) or 2; Attempt to multiply \(A^n\) and \(A\) or vice versa; Obtain correct matrix; Statement of induction conclusion |
**Part (i)** $A^2 = \begin{pmatrix} 4 & 0 \\ 0 & 1 \end{pmatrix}$ | M1 | Attempt at matrix multiplication
| A1, A1, 3 | Correct $A^2$; Correct $A^3$
$A^3 = \begin{pmatrix} 8 & 0 \\ 0 & 1 \end{pmatrix}$ |
**Part (ii)** $A^n = \begin{pmatrix} 2^n & 0 \\ 0 & 1 \end{pmatrix}$ | B1 | Sensible conjecture made
| B1, M1, A1, A1, 4 | State that conjecture is true for $n = 1$ or 2; Attempt to multiply $A^n$ and $A$ or vice versa; Obtain correct matrix; Statement of induction conclusion7 The matrix $\mathbf { A }$ is given by $\mathbf { A } = \left( \begin{array} { l l } 2 & 0 \\ 0 & 1 \end{array} \right)$.\\
(i) Find $\mathbf { A } ^ { 2 }$ and $\mathbf { A } ^ { 3 }$.\\
(ii) Hence suggest a suitable form for the matrix $\mathbf { A } ^ { n }$.\\
(iii) Use induction to prove that your answer to part (ii) is correct.
\hfill \mbox{\textit{OCR FP1 2006 Q7 [8]}}