Question 11 OR - A-Level Maths

CAIE FP1 2011 November — Question 11 OR

Exam Board	CAIE
Module	FP1 (Further Pure Mathematics 1)
Year	2011
Session	November
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Invariant lines and eigenvalues and vectors
Type	Find eigenvectors given eigenvalue
Difficulty	Challenging +1.2 This is a standard Further Maths eigenvalue/eigenvector question with a conceptual extension about invariant planes. The first part (finding eigenvalues and eigenvectors of a 3×3 matrix) is routine FP1 material. The second part requires understanding that planes spanned by eigenvectors are invariant under the transformation, which is a direct application of the definition. The final part simply asks for three such planes (spanned by pairs of eigenvectors), requiring no novel insight beyond the worked example. More challenging than typical A-level but standard for Further Maths.
Spec	4.03n Inverse 2x2 matrix 4.03o Inverse 3x3 matrix 4.04b Plane equations: cartesian and vector forms

Find the eigenvalues and corresponding eigenvectors of the matrix $\mathbf { A }$, where $$\mathbf { A } = \left( \begin{array} { r r r } 1 & 1 & 2 \\ 0 & 2 & 2 \\ - 1 & 1 & 3 \end{array} \right)$$ The linear transformation $\mathrm { T } : \mathbb { R } ^ { 3 } \rightarrow \mathbb { R } ^ { 3 }$ is defined by $\mathbf { x } \mapsto \mathbf { A x }$. Let $\mathbf { e } , \mathbf { f }$ be two linearly independent eigenvectors of $\mathbf { A }$, with corresponding eigenvalues $\lambda$ and $\mu$ respectively, and let $\Pi$ be the plane, through the origin, containing $\mathbf { e }$ and $\mathbf { f }$. By considering the parametric equation of $\Pi$, show that all points of $\Pi$ are mapped by T onto points of $\Pi$. Find cartesian equations of three planes, each with the property that all points of the plane are mapped by T onto points of the same plane.

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Question 11 (EITHER):

Mean Value, Arc Length, Surface Area

Answer	Marks	Guidance
Answer/Working	Marks	Guidance
Mean value $= \frac{\int_0^3 y\,dx}{3-0} = \left[\frac{2}{3}x^{\frac{3}{2}}-\frac{2}{15}x^{\frac{5}{2}}\right]_0^3 \div 3$	M1M1, A1	Uses formula and integrates $y$ wrt $x$
$= \sqrt{3}\left[\frac{2}{3}\times3-\frac{2}{15}\times9\right]\div3 = \frac{4\sqrt{3}}{15}\ (=0.462)$	A1	4 marks
$y' = \frac{1}{2\sqrt{x}}-\frac{1}{2}\sqrt{x} \Rightarrow \frac{ds}{dx}=\sqrt{1+\frac{1}{4}\left(\frac{1}{x}-2+x\right)}$	B1	Uses $\sqrt{1+(y')^2}$
$\Rightarrow \frac{ds}{dx} = \sqrt{\frac{1}{4}\left(\frac{1}{x}+2+x\right)} = \left(\frac{1}{2\sqrt{x}}+\frac{1}{2}\sqrt{x}\right)$	B1 (AG)
$s = \frac{1}{2}\int_0^3\left(x^{-\frac{1}{2}}+x^{\frac{1}{2}}\right)dx = \frac{1}{2}\left[2\sqrt{x}+\frac{2}{3}x^{\frac{3}{2}}\right]_0^3 = 2\sqrt{3}\ (=3.46)$	M1A1, M1A1	6 marks
$S = 2\pi\int_0^3\left(x^{\frac{1}{2}}-\frac{1}{3}x^{\frac{3}{2}}\right)\left(\frac{1}{2}x^{-\frac{1}{2}}+\frac{1}{2}x^{\frac{1}{2}}\right)dx$	M1	Surface area formula
$= \pi\int_0^3\left(1+\frac{2}{3}x-\frac{1}{3}x^2\right)dx = \left[x+\frac{1}{3}x^2-\frac{1}{9}x^3\right]_0^3 = 3\pi$	M1A1, A1	4 marks, total [14]

Question 11 (OR):

Eigenvalues and Eigenvectors

Answer	Marks	Guidance
Answer/Working	Marks	Guidance
$\det(\mathbf{A}-\lambda\mathbf{I})=0 \Rightarrow \lambda^3-6\lambda^2+11\lambda-6=0$	M1A1	Forms characteristic equation
$\Rightarrow \lambda=1,2,3$	A1A1	Solves
$\mathbf{e}_1=-2\mathbf{j}+\mathbf{k},\quad \mathbf{e}_2=\mathbf{i}+\mathbf{j},\quad \mathbf{e}_3=2\mathbf{i}+2\mathbf{j}+\mathbf{k}$	M1A1, A1	7 marks, eigenvectors
$\mathbf{r}=s\mathbf{e}+t\mathbf{f}$; $\mathbf{A}(s\mathbf{e}+t\mathbf{f})=s\mathbf{A}\mathbf{e}+t\mathbf{A}\mathbf{f}=(s\lambda)\mathbf{e}+(t\mu)\mathbf{f}$	B1, M1A1	States equation of plane, 3 marks
$\mathbf{e}_1\times\mathbf{e}_2=-\mathbf{i}+\mathbf{j}+2\mathbf{k}\Rightarrow x-y-2z=0$	M1A1
$\mathbf{e}_1\times\mathbf{e}_3=2\mathbf{i}-\mathbf{j}-2\mathbf{k}\Rightarrow 2x-y-2z=0$	A1
$\mathbf{e}_2\times\mathbf{e}_3=\mathbf{i}-\mathbf{j}\Rightarrow x-y=0$	A1	4 marks, total [14]

# Question 11 (EITHER):

## Mean Value, Arc Length, Surface Area

| Answer/Working | Marks | Guidance |
|---|---|---|
| Mean value $= \frac{\int_0^3 y\,dx}{3-0} = \left[\frac{2}{3}x^{\frac{3}{2}}-\frac{2}{15}x^{\frac{5}{2}}\right]_0^3 \div 3$ | M1M1, A1 | Uses formula and integrates $y$ wrt $x$ |
| $= \sqrt{3}\left[\frac{2}{3}\times3-\frac{2}{15}\times9\right]\div3 = \frac{4\sqrt{3}}{15}\ (=0.462)$ | A1 | 4 marks |
| $y' = \frac{1}{2\sqrt{x}}-\frac{1}{2}\sqrt{x} \Rightarrow \frac{ds}{dx}=\sqrt{1+\frac{1}{4}\left(\frac{1}{x}-2+x\right)}$ | B1 | Uses $\sqrt{1+(y')^2}$ |
| $\Rightarrow \frac{ds}{dx} = \sqrt{\frac{1}{4}\left(\frac{1}{x}+2+x\right)} = \left(\frac{1}{2\sqrt{x}}+\frac{1}{2}\sqrt{x}\right)$ | B1 (AG) | |
| $s = \frac{1}{2}\int_0^3\left(x^{-\frac{1}{2}}+x^{\frac{1}{2}}\right)dx = \frac{1}{2}\left[2\sqrt{x}+\frac{2}{3}x^{\frac{3}{2}}\right]_0^3 = 2\sqrt{3}\ (=3.46)$ | M1A1, M1A1 | 6 marks |
| $S = 2\pi\int_0^3\left(x^{\frac{1}{2}}-\frac{1}{3}x^{\frac{3}{2}}\right)\left(\frac{1}{2}x^{-\frac{1}{2}}+\frac{1}{2}x^{\frac{1}{2}}\right)dx$ | M1 | Surface area formula |
| $= \pi\int_0^3\left(1+\frac{2}{3}x-\frac{1}{3}x^2\right)dx = \left[x+\frac{1}{3}x^2-\frac{1}{9}x^3\right]_0^3 = 3\pi$ | M1A1, A1 | 4 marks, total [14] |

---

# Question 11 (OR):

## Eigenvalues and Eigenvectors

| Answer/Working | Marks | Guidance |
|---|---|---|
| $\det(\mathbf{A}-\lambda\mathbf{I})=0 \Rightarrow \lambda^3-6\lambda^2+11\lambda-6=0$ | M1A1 | Forms characteristic equation |
| $\Rightarrow \lambda=1,2,3$ | A1A1 | Solves |
| $\mathbf{e}_1=-2\mathbf{j}+\mathbf{k},\quad \mathbf{e}_2=\mathbf{i}+\mathbf{j},\quad \mathbf{e}_3=2\mathbf{i}+2\mathbf{j}+\mathbf{k}$ | M1A1, A1 | 7 marks, eigenvectors |
| $\mathbf{r}=s\mathbf{e}+t\mathbf{f}$; $\mathbf{A}(s\mathbf{e}+t\mathbf{f})=s\mathbf{A}\mathbf{e}+t\mathbf{A}\mathbf{f}=(s\lambda)\mathbf{e}+(t\mu)\mathbf{f}$ | B1, M1A1 | States equation of plane, 3 marks |
| $\mathbf{e}_1\times\mathbf{e}_2=-\mathbf{i}+\mathbf{j}+2\mathbf{k}\Rightarrow x-y-2z=0$ | M1A1 | |
| $\mathbf{e}_1\times\mathbf{e}_3=2\mathbf{i}-\mathbf{j}-2\mathbf{k}\Rightarrow 2x-y-2z=0$ | A1 | |
| $\mathbf{e}_2\times\mathbf{e}_3=\mathbf{i}-\mathbf{j}\Rightarrow x-y=0$ | A1 | 4 marks, total [14] |

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Find the eigenvalues and corresponding eigenvectors of the matrix $\mathbf { A }$, where

$$\mathbf { A } = \left( \begin{array} { r r r } 
1 & 1 & 2 \\
0 & 2 & 2 \\
- 1 & 1 & 3
\end{array} \right)$$

The linear transformation $\mathrm { T } : \mathbb { R } ^ { 3 } \rightarrow \mathbb { R } ^ { 3 }$ is defined by $\mathbf { x } \mapsto \mathbf { A x }$. Let $\mathbf { e } , \mathbf { f }$ be two linearly independent eigenvectors of $\mathbf { A }$, with corresponding eigenvalues $\lambda$ and $\mu$ respectively, and let $\Pi$ be the plane, through the origin, containing $\mathbf { e }$ and $\mathbf { f }$. By considering the parametric equation of $\Pi$, show that all points of $\Pi$ are mapped by T onto points of $\Pi$.

Find cartesian equations of three planes, each with the property that all points of the plane are mapped by T onto points of the same plane.

\hfill \mbox{\textit{CAIE FP1 2011 Q11 OR}}

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Q1 6 Q2 6 Q3 7 Q4 7 Q5 7 Q6 8 Q7 9 Q8 10 Q9 13 Q10 13 Q11 EITHER Q11 OR