Question 4 - A-Level Maths

OCR FP1 2013 June — Question 4 6 marks

Exam Board	OCR
Module	FP1 (Further Pure Mathematics 1)
Year	2013
Session	June
Marks	6
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Proof by induction
Type	Prove matrix power formula
Difficulty	Standard +0.3 This is a straightforward proof by induction on matrix powers with a simple 2×2 upper triangular matrix. The inductive step requires only basic matrix multiplication (4 entries to compute), and the algebra is clean with no complicated simplification. While it's a Further Maths topic, it follows the standard induction template exactly, making it slightly easier than an average A-level question overall.
Spec	4.01a Mathematical induction: construct proofs 4.03b Matrix operations: addition, multiplication, scalar

4 The matrix $\mathbf { M }$ is given by $\mathbf { M } = \left( \begin{array} { l l } 2 & 2 \\ 0 & 1 \end{array} \right)$. Prove by induction that, for $n \geqslant 1$, $$\mathbf { M } ^ { n } = \left( \begin{array} { c c } 2 ^ { n } & 2 ^ { n + 1 } - 2 \\ 0 & 1 \end{array} \right) .$$

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Question 4:

Answer	Marks	Guidance
Answer	Marks	Guidance
B1	Establish result true for $n=1$ or $n=2$
M1	Multiply M and $\mathbf{M}^k$, either order
$2(2^{k+1}-2)+2$ or $2^{k+1}+2^{k+1}-2$	A1	Obtain correct element
A1	Obtain other 3 correct elements
A1	Obtain $2^{k+2}-2$ convincingly
B1	Specific statement of induction conclusion, provided 5/5 earned so far and verified for $n=1$
[6]

## Question 4:

| Answer | Marks | Guidance |
|--------|-------|----------|
| | B1 | Establish result true for $n=1$ or $n=2$ |
| | M1 | Multiply **M** and $\mathbf{M}^k$, either order |
| $2(2^{k+1}-2)+2$ or $2^{k+1}+2^{k+1}-2$ | A1 | Obtain correct element |
| | A1 | Obtain other 3 correct elements |
| | A1 | Obtain $2^{k+2}-2$ convincingly |
| | B1 | Specific statement of induction conclusion, provided 5/5 earned so far and verified for $n=1$ |
| **[6]** | | |

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4 The matrix $\mathbf { M }$ is given by $\mathbf { M } = \left( \begin{array} { l l } 2 & 2 \\ 0 & 1 \end{array} \right)$. Prove by induction that, for $n \geqslant 1$,

$$\mathbf { M } ^ { n } = \left( \begin{array} { c c } 
2 ^ { n } & 2 ^ { n + 1 } - 2 \\
0 & 1
\end{array} \right) .$$

\hfill \mbox{\textit{OCR FP1 2013 Q4 [6]}}

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