Question 8 - A-Level Maths

OCR MEI Further Pure Core AS 2018 June — Question 8 6 marks

Exam Board	OCR MEI
Module	Further Pure Core AS (Further Pure Core AS)
Year	2018
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 with matrices, following a standard template. The base case is trivial (n=1), and the inductive step requires only matrix multiplication of 2×2 matrices with simple entries, plus basic algebraic manipulation of powers of 2. While it's a Further Maths topic, the execution is mechanical with no conceptual obstacles or novel insights required.
Spec	4.01a Mathematical induction: construct proofs 4.03b Matrix operations: addition, multiplication, scalar

Prove by induction that \(\begin{pmatrix} 1 & 1 \\ 0 & 2 \end{pmatrix}^n = \begin{pmatrix} 1 & 2^n - 1 \\ 0 & 2^n \end{pmatrix}\) for all positive integers \(n\). [6]

Show mark scheme Show mark scheme source

Question 8:

Answer	Marks
8	1 11 1 211

When n = 1, 

0 2 0 21

1 1k 1 2k 1

[Assume] n = k: 

0 2 0 2k

1 1k1 1 2k1 1 1

Then 

0 2 0 2k 0 2

1 12k12 1 2k11

 

0 2k1 0 2k1

so if true for n = k then true for n = k + 1

Answer	Marks
[as true for n = 1] therefore true for all n.	B1

M1

A1*

M1dep*

A1dep*

Answer	Marks
[6]	2.1

2.2a

1.1 2.2a

Answer	Marks
2.4	1 1 1 2k1

or

0 2 0 2k

1 2k 12k

or

0 2k1

must clearly ‘assume’ n = k

must have established truth for

Answer	Marks
n = 1 for final mark	‘true for n = k and k + 1’ is

M0

but need not re-state it at

the end

Question 8:
8 | 1 11 1 211
When n = 1, 
0 2 0 21
1 1k 1 2k 1
[Assume] n = k: 
0 2 0 2k
1 1k1 1 2k1 1 1
Then 
0 2 0 2k 0 2
1 12k12 1 2k11
 
0 2k1 0 2k1
so if true for n = k then true for n = k + 1
[as true for n = 1] therefore true for all n. | B1
M1
M1
A1*
M1dep*
A1dep*
[6] | 2.1
2.2a
1.1
1.1
2.2a
2.4 | 1 1 1 2k1
or
0 2 0 2k
1 2k 12k
or
0 2k1
must clearly ‘assume’ n = k
must have established truth for
n = 1 for final mark | ‘true for n = k and k + 1’ is
M0
but need not re-state it at
the end

Show LaTeX source

Prove by induction that $\begin{pmatrix} 1 & 1 \\ 0 & 2 \end{pmatrix}^n = \begin{pmatrix} 1 & 2^n - 1 \\ 0 & 2^n \end{pmatrix}$ for all positive integers $n$. [6]

\hfill \mbox{\textit{OCR MEI Further Pure Core AS 2018 Q8 [6]}}

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Q1 4 Q2 3 Q3 5 Q4 5 Q5 7 Q6 4 Q7 9 Q8 6 Q9 9 Q10 8