Standard +0.3 This is a straightforward error-spotting question on 3D transformations. Students need to recognize the standard rotation matrix form and identify that Anna used sin instead of cos in her angle calculation (should be cos θ = √3/2, giving θ = 150°). The matrix structure is given, requiring only pattern recognition and basic trigonometry rather than derivation or complex problem-solving.
Anna has been asked to describe the transformation given by the matrix
$$\begin{bmatrix} 1 & 0 & 0 \\ 0 & -\frac{\sqrt{3}}{2} & -\frac{1}{2} \\ 0 & \frac{1}{2} & -\frac{\sqrt{3}}{2} \end{bmatrix}$$
She writes her answer as follows:
The transformation is a rotation about the \(x\)-axis through an angle of \(\theta\), where
$$\sin \theta = \frac{1}{2} \quad \text{and} \quad -\sin \theta = -\frac{1}{2}$$
$$\theta = 30°$$
Identify and correct the error in Anna's work.
[2 marks]
Assesses the validity of Anna’s work by identifying her error,
e.g. that the angle is not 30
Answer
Marks
Guidance
or that has been ignor°ed.
2.3
B1
or
1
or
sin𝜃𝜃 = 2 ⇒ 𝜃𝜃 = 30° 𝜃𝜃 = 150°
√3
cos𝜃𝜃 = −2 ⇒ 𝜃𝜃 = 150° 𝜃𝜃 = −150°
∴ 𝜃𝜃 = 150°
cos𝜃𝜃
Answer
Marks
Guidance
States the correct angle 150 .
1.1b
B1
°
Answer
Marks
Guidance
Total
2
Q
Marking instructions
AO
Question 6:
6 | Assesses the validity of Anna’s work by identifying her error,
e.g. that the angle is not 30
or that has been ignor°ed. | 2.3 | B1 | Anna gave the wrong angle
or
1
or
sin𝜃𝜃 = 2 ⇒ 𝜃𝜃 = 30° 𝜃𝜃 = 150°
√3
cos𝜃𝜃 = −2 ⇒ 𝜃𝜃 = 150° 𝜃𝜃 = −150°
∴ 𝜃𝜃 = 150°
cos𝜃𝜃
States the correct angle 150 . | 1.1b | B1
°
Total | 2
Q | Marking instructions | AO | Marks | Typical solution
Anna has been asked to describe the transformation given by the matrix
$$\begin{bmatrix} 1 & 0 & 0 \\ 0 & -\frac{\sqrt{3}}{2} & -\frac{1}{2} \\ 0 & \frac{1}{2} & -\frac{\sqrt{3}}{2} \end{bmatrix}$$
She writes her answer as follows:
The transformation is a rotation about the $x$-axis through an angle of $\theta$, where
$$\sin \theta = \frac{1}{2} \quad \text{and} \quad -\sin \theta = -\frac{1}{2}$$
$$\theta = 30°$$
Identify and correct the error in Anna's work.
[2 marks]
\hfill \mbox{\textit{AQA Further AS Paper 1 2020 Q6 [2]}}