FP1 June 2009 Q5

EdexcelOld spec8 marksMatrices

5. \(\mathbf{R} = \begin{pmatrix} a & 2 \\ a & b \end{pmatrix}\), where \(a\) and \(b\) are constants and \(a > 0\).

(a) Find \(\mathbf{R}^2\) in terms of \(a\) and \(b\). (3)

Given that \(\mathbf{R}^2\) represents an enlargement with centre \((0,\ 0)\) and scale factor 15,

(b) find the value of \(a\) and the value of \(b\). (5)

Scheme	Marks
\(\mathbf{R}^2 = \begin{pmatrix} a^2 + 2a & 2a + 2b \\ a^2 + ab & 2a + b^2 \end{pmatrix}\)	M1 A1 A1
	(3)

Mark scheme (b)

Scheme	Marks
Puts their \(a^2 + 2a = 15\) or their \(2a + b^2 = 15\) or their \((a^2 + 2a)(2a + b^2) - (a^2 + ab)(2a + 2b) = 225\) (or to 15),	M1,
Puts their \(a^2 + ab = 0\) or their \(2a + 2b = 0\)	M1
Solve to find either \(a\) or \(b\)	M1
\(a = 3, \quad b = -3\)	A1, A1
	(5)
	[8]

Alternative for (b)

Scheme	Marks
Uses \(\mathbf{R}^2 \times\) column vector \(= 15 \times\) column vector, and equates rows to give two equations in \(a\) and \(b\) only	M1, M1
Solves to find either \(a\) or \(b\) as above method	M1 A1 A1

Notes

(b) M1 M1 as described in scheme (In the alternative scheme column vector can be general or specific for first M1 but must be specific for 2^nd M1)

M1 requires solving equations to find \(a\) and/or \(b\) (though checking that correct answer satisfies the equations will earn this mark) This mark can be given independently of the first two method marks.

So solving \(\mathbf{M}^2 = 15\mathbf{M}\) for example gives M0M0M1A0A0 in part (b)

Also putting leading diagonal = 0 and other diagonal = 15 is M0M0M1A0A0 (No possible solutions as \(a > 0\))

A1 A1 for correct answers only

Any Extra answers given, e.g. \(a = -5\) and \(b = 5\) or wrong answers – deduct last A1 awarded

So the two sets of answers would be A1 A0

Just the answer. \(a = -5\) and \(b = 5\) is A0 A0

Stopping at two values for \(a\) or for \(b\) – no attempt at other is A0A0

Answer with no working at all is 0 marks