AQA Further Paper 2 2019 June — Question 9 13 marks

Exam BoardAQA
ModuleFurther Paper 2 (Further Paper 2)
Year2019
SessionJune
Marks13
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicInvariant lines and eigenvalues and vectors
TypeFind P and D for A = PDP⁻¹
DifficultyChallenging +1.8 This is a substantial Further Maths linear algebra question requiring eigenvalue/eigenvector computation for a non-square matrix (unusual and conceptually harder), diagonalization, limit behavior analysis, and geometric interpretation. The non-square matrix aspect and the limit convergence part require deeper understanding beyond routine procedures, though the individual techniques are standard for Further Maths students.
Spec4.03a Matrix language: terminology and notation4.03b Matrix operations: addition, multiplication, scalar4.03n Inverse 2x2 matrix4.03o Inverse 3x3 matrix
  1. Find the eigenvalues and corresponding eigenvectors of the matrix $$\mathbf{M} = \begin{bmatrix} 1 & 2 \\ 5 & 5 \\ -3 & 13 \\ 5 & 10 \end{bmatrix}$$ [5 marks]
  2. Find matrices \(\mathbf{U}\) and \(\mathbf{D}\) such that \(\mathbf{D}\) is a diagonal matrix and \(\mathbf{M} = \mathbf{U}\mathbf{D}\mathbf{U}^{-1}\) [2 marks]
  3. Given that \(\mathbf{M}^n \to \mathbf{L}\) as \(n \to \infty\), find the matrix \(\mathbf{L}\). [4 marks]
  4. The transformation represented by \(\mathbf{L}\) maps all points onto a line. Find the equation of this line. [2 marks]
Question 9:
AnswerMarks
9(a)Forms correct
characteristic
equation and solves.
(PI)
AnswerMarks Guidance
Condone one errorAO1.1a M1
−λ −λ + =0
  
5 10  25
λ=1 & λ=0.5 2
−4 2 0  .5 x −1.5𝜆𝜆+𝜆𝜆 = 0
0= 5 5
  
 −3 3y
5 10
or
−3 2x
0= 10 5
  
 −3 4y
5 5
1
λ=1 :  
2
4
λ=0.5:  
3
Obtains the correct
AnswerMarks Guidance
eigenvaluesAO1.1b A1
Uses correct equation
to find eigenvector for
either
AnswerMarks Guidance
λ=1 or λ=0.5 (PI)AO1.1a M1
Obtains correct
eigenvector for λ=1
Allow any scalar
AnswerMarks Guidance
multiple.AO1.1b A1
Obtains correct
eigenvector for
λ=0.5
Allow any scalar
AnswerMarks Guidance
multiple.AO1.1b A1
(b)Finds their correct U
with no zero column.AO1.1b B1F
U =   and D=  2 
3 2 0 1
or
1 4 1 0
U =   and D=  
2 3 0 1
2
Finds their correct D
(must be consistent
AnswerMarks Guidance
with their U)AO1.1b B1F
(c)U-1
Finds correct –
AnswerMarks Guidance
CAOAO1.1b B1
U-1 =
 
5−3 4 
14 1(1)n 0  2 −1
Mn =   2  
53 2 0 1n−3 4 
14 10 0 2 −1
L=
   
53 20 1−3 4 
−0.6 0.8
L=
 
−1.2 1.6
or
−1 3 −4
U-1 =
 
5 −2 1 
−11 4 1n 0   3 −4
Mn =  
   
5 2 30 (1)n −2 1 
 
2
−11 41 0 3 −4
L=
   
5 2 30 0−2 1 
−0.6 0.8
L=
 
−1.2 1.6
Multiplies their
matrices in correct
order with powers
Dn
AnswerMarks Guidance
taken insideAO1.1a M1
Correctly takes
n→∞ limit of their
Dn, (D must not
contain only ones and
AnswerMarks Guidance
zeros)AO2.2a M1
Obtains correct L
Correct answer
AnswerMarks Guidance
scores 4/4AO2.1 R1
(d)Obtains correct
equations for image
points from their L
AnswerMarks Guidance
PI by y =2xAO3.1a M1
x'= (−3x+4y)
5
2
y'= (−3x+4y)
5
y =2x
Obtains correct
equation for the line
AnswerMarks Guidance
Condone y'=2x'AO3.2a A1
Total13
QMarking Instructions AO 
Question 9:
--- 9(a) ---
9(a) | Forms correct
characteristic
equation and solves.
(PI)
Condone one error | AO1.1a | M1 | 1 13  6
−λ −λ + =0
  
5 10  25
λ=1 & λ=0.5 2
−4 2 0  .5 x −1.5𝜆𝜆+𝜆𝜆 = 0
0= 5 5
  
 −3 3y
5 10
or
−3 2x
0= 10 5
  
 −3 4y
5 5
1
λ=1 :  
2
4
λ=0.5:  
3
Obtains the correct
eigenvalues | AO1.1b | A1
Uses correct equation
to find eigenvector for
either
λ=1 or λ=0.5 (PI) | AO1.1a | M1
Obtains correct
eigenvector for λ=1
Allow any scalar
multiple. | AO1.1b | A1
Obtains correct
eigenvector for
λ=0.5
Allow any scalar
multiple. | AO1.1b | A1
(b) | Finds their correct U
with no zero column. | AO1.1b | B1F | 4 1 1 0
U =   and D=  2 
3 2 0 1
or
1 4 1 0
U =   and D=  
2 3 0 1
2
Finds their correct D
(must be consistent
with their U) | AO1.1b | B1F
(c) | U-1
Finds correct –
CAO | AO1.1b | B1 | 1 2 −1
U-1 =
 
5−3 4 
14 1(1)n 0  2 −1
Mn =   2  
53 2 0 1n−3 4 
14 10 0 2 −1
L=
   
53 20 1−3 4 
−0.6 0.8
L=
 
−1.2 1.6
or
−1 3 −4
U-1 =
 
5 −2 1 
−11 4 1n 0   3 −4
Mn =  
   
5 2 30 (1)n −2 1 
 
2
−11 41 0 3 −4
L=
   
5 2 30 0−2 1 
−0.6 0.8
L=
 
−1.2 1.6
Multiplies their
matrices in correct
order with powers
Dn
taken inside | AO1.1a | M1
Correctly takes
n→∞ limit of their
Dn, (D must not
contain only ones and
zeros) | AO2.2a | M1
Obtains correct L
Correct answer
scores 4/4 | AO2.1 | R1
(d) | Obtains correct
equations for image
points from their L
PI by y =2x | AO3.1a | M1 | 1
x'= (−3x+4y)
5
2
y'= (−3x+4y)
5
y =2x
Obtains correct
equation for the line
Condone y'=2x' | AO3.2a | A1
Total | 13
Q | Marking Instructions | AO | Marks | Typical Solution
\begin{enumerate}[label=(\alph*)]
\item Find the eigenvalues and corresponding eigenvectors of the matrix
$$\mathbf{M} = \begin{bmatrix} 1 & 2 \\ 5 & 5 \\ -3 & 13 \\ 5 & 10 \end{bmatrix}$$
[5 marks]

\item Find matrices $\mathbf{U}$ and $\mathbf{D}$ such that $\mathbf{D}$ is a diagonal matrix and $\mathbf{M} = \mathbf{U}\mathbf{D}\mathbf{U}^{-1}$
[2 marks]

\item Given that $\mathbf{M}^n \to \mathbf{L}$ as $n \to \infty$, find the matrix $\mathbf{L}$.
[4 marks]

\item The transformation represented by $\mathbf{L}$ maps all points onto a line.

Find the equation of this line.
[2 marks]
\end{enumerate}

\hfill \mbox{\textit{AQA Further Paper 2 2019 Q9 [13]}}
This paper (15 questions)
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