CAIE FP1 2018 November — Question 2 6 marks

Exam BoardCAIE
ModuleFP1 (Further Pure Mathematics 1)
Year2018
SessionNovember
Marks6
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Mark schemeDownload PDF ↗
TopicInvariant lines and eigenvalues and vectors
DifficultyStandard +0.3 This is a straightforward eigenvalue/eigenvector question on an upper triangular matrix where eigenvalues are immediately visible on the diagonal. Part (i) is direct computation (1 mark), part (ii) requires solving a simple system for the eigenvector corresponding to λ=-2, and part (iii) uses the property that eigenvectors are preserved under matrix powers. While it's Further Maths content, the triangular structure makes it easier than typical eigenvalue problems, placing it slightly above average difficulty.
Spec4.03h Determinant 2x2: calculation4.03i Determinant: area scale factor and orientation4.03n Inverse 2x2 matrix
It is given that $$\mathbf{A} = \begin{pmatrix} 2 & 3 & 1 \\ 0 & -2 & 1 \\ 0 & 0 & 1 \end{pmatrix}$$
  1. Find the eigenvalue of \(\mathbf{A}\) corresponding to the eigenvector \(\begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}\). [1]
  2. Write down the negative eigenvalue of \(\mathbf{A}\) and find a corresponding eigenvector. [3]
  3. Find an eigenvalue and a corresponding eigenvector of the matrix \(\mathbf{A} + \mathbf{A}^6\). [2]
Question 2:
AnswerMarks Guidance
2(i)2 B1
1
AnswerMarks Guidance
2(ii)Negative eigenvalue =−2 B1
4 3 1 i j k
 
A+2I= 0 0 1 4 3 1
 
 
AnswerMarks Guidance
0 0 3 0 0 1M1 Uses vector product (or equations) to find
corresponding eigenvector.
 3 
 
−4
 
 
AnswerMarks Guidance
 0 A1 Accept any non-zero scalar multiple of
 3 
 
−4 .
 
 
 0 
3
AnswerMarks Guidance
2(iii)An eigenvalue of A+ A6 is 2+26 =66, 62 or 2 B1
1  3  −6
     
Corresponding eigenvector is 0 −4 or 1 oe
     
     
AnswerMarks
0  0   3 B1
2
AnswerMarks Guidance
Quuestion AAnswer 
Question 2:
--- 2(i) ---
2(i) | 2 | B1 | Stated
1
--- 2(ii) ---
2(ii) | Negative eigenvalue =−2 | B1 | Stated
4 3 1 i j k
 
A+2I= 0 0 1 4 3 1
 
 
0 0 3 0 0 1 | M1 | Uses vector product (or equations) to find
corresponding eigenvector.
 3 
 
−4
 
 
 0  | A1 | Accept any non-zero scalar multiple of
 3 
 
−4 .
 
 
 0 
3
--- 2(iii) ---
2(iii) | An eigenvalue of A+ A6 is 2+26 =66, 62 or 2 | B1
1  3  −6
     
Corresponding eigenvector is 0 −4 or 1 oe
     
     
0  0   3  | B1
2
Quu | estion | AAnswer | Marks | Guii | dance
It is given that
$$\mathbf{A} = \begin{pmatrix} 2 & 3 & 1 \\ 0 & -2 & 1 \\ 0 & 0 & 1 \end{pmatrix}$$

\begin{enumerate}[label=(\roman*)]
\item Find the eigenvalue of $\mathbf{A}$ corresponding to the eigenvector $\begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}$. [1]

\item Write down the negative eigenvalue of $\mathbf{A}$ and find a corresponding eigenvector. [3]

\item Find an eigenvalue and a corresponding eigenvector of the matrix $\mathbf{A} + \mathbf{A}^6$. [2]
\end{enumerate}

\hfill \mbox{\textit{CAIE FP1 2018 Q2 [6]}}
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