OCR MEI Further Extra Pure 2020 November — Question 1 5 marks

Exam BoardOCR MEI
ModuleFurther Extra Pure (Further Extra Pure)
Year2020
SessionNovember
Marks5
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Mark schemeDownload PDF ↗
TopicInvariant lines and eigenvalues and vectors
TypeFind eigenvalues of 2×2 matrix
DifficultyModerate -0.8 This is a straightforward application of the standard eigenvalue algorithm for a 2×2 matrix: solve det(A - λI) = 0 for a simple quadratic, then find eigenvectors by solving (A - λI)v = 0. The matrix has integer entries and the characteristic equation factors easily, making this a routine textbook exercise with no conceptual challenges beyond recalling the method.
Spec4.03c Matrix multiplication: properties (associative, not commutative)4.03h Determinant 2x2: calculation
1 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { r r } 0 & 2 \\ 3 & - 1 \end{array} \right)\).
Find
  • the eigenvalues of \(\mathbf { A }\),
  • an eigenvector associated with each eigenvalue.
Question 1:
AnswerMarks
1−λ 2
det(A−λI)= =λ(1+λ)−6
3 −1−λ
λ2 + λ – 6 [= 0 ]
So the eigenvalues are 2 and –3
0 2 x  2y  2x
e=2:   = = 
3 −1y 3x− y 2y
1
⇒x= y⇒ 
1
0 2 x  2y  −3x
e=−3:   = = 
3 −1y 3x− y −3y
−2
⇒3x=−2y⇒ 
AnswerMarks
 3 M1
A1
M1
A1
A1
AnswerMarks
[5]1.1a
1.1
1.1
1.1
AnswerMarks
1.1For ch eqn in any form
For both e-vals correct
Either equation correct in any
form FT
Or any non-zero multiple
AnswerMarks
Or any non-zero multipleCan be implied by correct e-
vals
Allow one sign error
If each e-vec is not paired
with its e-val (either
explicitly or in the working)
or if they are wrongly
assigned then SC1 if they
are both correct
Question 1:
1 | −λ 2
det(A−λI)= =λ(1+λ)−6
3 −1−λ
λ2 + λ – 6 [= 0 ]
So the eigenvalues are 2 and –3
0 2 x  2y  2x
e=2:   = = 
3 −1y 3x− y 2y
1
⇒x= y⇒ 
1
0 2 x  2y  −3x
e=−3:   = = 
3 −1y 3x− y −3y
−2
⇒3x=−2y⇒ 
 3  | M1
A1
M1
A1
A1
[5] | 1.1a
1.1
1.1
1.1
1.1 | For ch eqn in any form
For both e-vals correct
Either equation correct in any
form FT
Or any non-zero multiple
Or any non-zero multiple | Can be implied by correct e-
vals
Allow one sign error
If each e-vec is not paired
with its e-val (either
explicitly or in the working)
or if they are wrongly
assigned then SC1 if they
are both correct
1 The matrix $\mathbf { A }$ is given by $\mathbf { A } = \left( \begin{array} { r r } 0 & 2 \\ 3 & - 1 \end{array} \right)$.\\
Find

\begin{itemize}
  \item the eigenvalues of $\mathbf { A }$,
  \item an eigenvector associated with each eigenvalue.
\end{itemize}

\hfill \mbox{\textit{OCR MEI Further Extra Pure 2020 Q1 [5]}}
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Q1 5 Q2 5 Q3 12 Q4 13 Q5 8 Q6 17