The answer
(a) \(a = \dfrac{5}{2}\) or \(-\dfrac{5}{2}\)
(b)(i) \(\overrightarrow{AB} = \begin{pmatrix} 48 \\ 37 \end{pmatrix}\)
(ii) \(C\) does not lie on the line
O-Level E-Math 2018 Paper 1 Question 18 · Verified worked solution by the Genius Plus Academy teaching team
The question
(a) \(\overrightarrow{PQ} = \begin{pmatrix} a \\ 3a \end{pmatrix}\) and \(\left|\overrightarrow{PQ}\right| = \dfrac{5\sqrt{10}}{2}\). Find the two possible values of \(a\). [2]
(b) A line joins \(A(-5, 1)\) and \(B(43, 38)\). (i) Find \(\overrightarrow{AB}\). [1] (ii) Use vectors to show whether or not \(C(11, 13)\) lies on this line. [2]
(a) \(\left|\overrightarrow{PQ}\right|^2 = a^2 + (3a)^2 = 10a^2\). Set \(10a^2 = \left(\dfrac{5\sqrt{10}}{2}\right)^2 = \dfrac{25 \times 10}{4} = \dfrac{250}{4}\), so \(a^2 = \dfrac{25}{4}\) and \(a = \pm\dfrac{5}{2}\).
(b)(i) \(\overrightarrow{AB} = \begin{pmatrix} 43 - (-5) \\ 38 - 1 \end{pmatrix} = \begin{pmatrix} 48 \\ 37 \end{pmatrix}\).
(ii) \(\overrightarrow{AC} = \begin{pmatrix} 11 - (-5) \\ 13 - 1 \end{pmatrix} = \begin{pmatrix} 16 \\ 12 \end{pmatrix}\). If \(C\) lay on the line \(AB\), then \(\overrightarrow{AC}\) would be a scalar multiple of \(\overrightarrow{AB}\). Comparing components: \(\dfrac{16}{48} = \dfrac{1}{3}\) but \(\dfrac{12}{37} \neq \dfrac{1}{3}\). The ratios differ, so \(\overrightarrow{AC}\) is not parallel to \(\overrightarrow{AB}\) and \(C\) does not lie on the line.
Answer: (a) \(a = \dfrac{5}{2}\) or \(-\dfrac{5}{2}\)
(b)(i) \(\overrightarrow{AB} = \begin{pmatrix} 48 \\ 37 \end{pmatrix}\)
(ii) \(C\) does not lie on the line
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.
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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 magnitude of a column vector question from Vectors in two dimensions, worth 5 marks: 2 + 1 + 2.
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