The answer
(a)(i)(a) \(\begin{pmatrix} 7 \\ -6 \end{pmatrix}\)
(b) \(\sqrt{85} \approx 9.22\)
(ii) \(k = 9\) or \(-3\)
(b)(i) \(\overrightarrow{CD} = \tfrac34\mathbf{p} - \mathbf{q}\)
(ii) \(\overrightarrow{AH} = \tfrac58\mathbf{p} - \tfrac12\mathbf{q}\)
O-Level E-Math 2021 Paper 2 Question 8 · Verified worked solution by the Genius Plus Academy teaching team
What this question tests
This is Question 8 of the O-Level E-Math 2021 Paper 2. It tests translation vector, in the Vectors area. It is worth 12 marks: (a)(i)(a) 1 + (b) 2, (ii) 3, (b)(i) 2 + (ii) 4. It is a worded / diagram-based question, so open your Ten-Year Series (TYS) or the official paper at this question, then follow our full worked solution below.
(a)(i)(a) \(\overrightarrow{RS} = S - R = \begin{pmatrix} 3 + 4 \\ -1 - 5 \end{pmatrix} = \begin{pmatrix} 7 \\ -6 \end{pmatrix}\).
(b) \(|\overrightarrow{RS}| = \sqrt{7^2 + (-6)^2} = \sqrt{85} = 9.22\).
(ii) \(\overrightarrow{ST} = \begin{pmatrix} k - 3 \\ 8 \end{pmatrix}\), \(|\overrightarrow{ST}|^2 = (k - 3)^2 + 64 = 100 \Rightarrow (k - 3)^2 = 36 \Rightarrow k = 9\) or \(k = -3\).
(b) In the parallelogram, \(\overrightarrow{BC} = \overrightarrow{AD} = \mathbf{p}\), so \(\overrightarrow{BG} = \tfrac34\mathbf{p}\) and \(\overrightarrow{AB} = \overrightarrow{AG} - \overrightarrow{BG} = \mathbf{q} - \tfrac34\mathbf{p}\).
(i) \(\overrightarrow{CD} = -\overrightarrow{AB} = \tfrac34\mathbf{p} - \mathbf{q}\) (since \(\overrightarrow{DC} = \overrightarrow{AB}\)).
(ii) \(\overrightarrow{AF} = \tfrac12\overrightarrow{AB} = \tfrac12\mathbf{q} - \tfrac38\mathbf{p}\). \(\overrightarrow{GF} = \overrightarrow{AF} - \overrightarrow{AG} = \left(\tfrac12\mathbf{q} - \tfrac38\mathbf{p}\right) - \mathbf{q} = -\tfrac38\mathbf{p} - \tfrac12\mathbf{q}\). Then \(\overrightarrow{AH} = \overrightarrow{AD} + \overrightarrow{DH} = \mathbf{p} + \overrightarrow{GF} = \mathbf{p} - \tfrac38\mathbf{p} - \tfrac12\mathbf{q} = \tfrac58\mathbf{p} - \tfrac12\mathbf{q}\).
Answer: (a)(i)(a) \(\begin{pmatrix} 7 \\ -6 \end{pmatrix}\)
(b) \(\sqrt{85} \approx 9.22\)
(ii) \(k = 9\) or \(-3\)
(b)(i) \(\overrightarrow{CD} = \tfrac34\mathbf{p} - \mathbf{q}\)
(ii) \(\overrightarrow{AH} = \tfrac58\mathbf{p} - \tfrac12\mathbf{q}\)
Same structure, different numbers
Swap the constants, dress a quadratic as a length, hide a derivative inside an integral, and a student sees a brand new problem. The structure underneath is the same, and so is the method. Once a student can name the structure, a whole row of questions that look different start to open the same way.
That is where marks really leak: in choosing the method, not in the algebra that follows. We call it Lock and Key, name the lock, then the key follows.
Want more questions like this, with worked solutions?
Join our mailing list and we will send practice sets and worked solutions. One email, no spam.
Genius Plus Academy · O-Level & IP Mathematics
Our O-Level E-Math tuition trains the same recognise-the-structure method these worked solutions show, taught by a team that has marked these papers for years. It runs within our weekly Secondary Math programme, Sec 1 to 4 and IP.
It is a translation vector question from Vectors, worth 12 marks: (a)(i)(a) 1 + (b) 2, (ii) 3, (b)(i) 2 + (ii) 4.
Yes. IP (Integrated Programme) schools teach the same O-Level Mathematics content; they just sequence it differently and set their own internal exams, so these worked solutions apply to IP students too.
Yes. Every worked solution here is free to read, with no sign-up wall.
Browse E-Math and A-Math by year in our worked-solutions library at /resources/solutions/o-level/.
Book a free trial and diagnostic. We will look at a real paper and show you exactly where the marks are going.