Question 3 - A-Level Maths

AQA Further AS Paper 1 2021 June — Question 3 1 marks

Exam Board	AQA
Module	Further AS Paper 1 (Further AS Paper 1)
Year	2021
Session	June
Marks	1
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Linear transformations
Type	3D transformation matrices
Difficulty	Moderate -0.3 This is a straightforward recall question about 3D rotation matrices. Students need to recognize that rotation matrices are orthogonal (columns are orthonormal), leading to a² + b² = 1 and the orthogonality condition ab + (√3/2)(-1/2) = 0. With the given values, this reduces to simple substitution into standard forms. Slightly easier than average as it's multiple choice and tests direct knowledge of rotation matrix properties rather than requiring problem-solving.
Spec	4.03f Linear transformations 3D: reflections and rotations about axes

3 The matrix $\mathbf { M }$ represents a rotation about the $x$-axis. $$\mathbf { M } = \left[ \begin{array} { c c c } 1 & 0 & 0 \\ 0 & a & \frac { \sqrt { 3 } } { 2 } \\ 0 & b & - \frac { 1 } { 2 } \end{array} \right]$$ Which of the following pairs of values is correct?
Tick ( $\checkmark$ ) one box. $$\begin{array} { l l } a = \frac { 1 } { 2 } \text { and } b = \frac { \sqrt { 3 } } { 2 } & \square \\ a = \frac { 1 } { 2 } \text { and } b = - \frac { \sqrt { 3 } } { 2 } & \square \\ a = - \frac { 1 } { 2 } \text { and } b = \frac { \sqrt { 3 } } { 2 } & \square \\ a = - \frac { 1 } { 2 } \text { and } b = - \frac { \sqrt { 3 } } { 2 } & \end{array}$$

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Question 3:

Answer	Marks	Guidance
Answer	Marks	Guidance
$a = -\dfrac{1}{2}$, $b = -\dfrac{\sqrt{3}}{2}$	B1 (AO1.1b)	Ticks correct box
Total: 1

## Question 3:
| Answer | Marks | Guidance |
|--------|-------|----------|
| $a = -\dfrac{1}{2}$, $b = -\dfrac{\sqrt{3}}{2}$ | B1 (AO1.1b) | Ticks correct box |
| **Total: 1** | | |

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3 The matrix $\mathbf { M }$ represents a rotation about the $x$-axis.

$$\mathbf { M } = \left[ \begin{array} { c c c } 
1 & 0 & 0 \\
0 & a & \frac { \sqrt { 3 } } { 2 } \\
0 & b & - \frac { 1 } { 2 }
\end{array} \right]$$

Which of the following pairs of values is correct?\\
Tick ( $\checkmark$ ) one box.

$$\begin{array} { l l } 
a = \frac { 1 } { 2 } \text { and } b = \frac { \sqrt { 3 } } { 2 } & \square \\
a = \frac { 1 } { 2 } \text { and } b = - \frac { \sqrt { 3 } } { 2 } & \square \\
a = - \frac { 1 } { 2 } \text { and } b = \frac { \sqrt { 3 } } { 2 } & \square \\
a = - \frac { 1 } { 2 } \text { and } b = - \frac { \sqrt { 3 } } { 2 } &
\end{array}$$

\hfill \mbox{\textit{AQA Further AS Paper 1 2021 Q3 [1]}}

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