The answer
(a) \(\approx 57.0^{\circ}\)
(b) \(319^{\circ}\)
O-Level E-Math 2018 Paper 1 Question 24 · Verified worked solution by the Genius Plus Academy teaching team
What this question tests
This is Question 24 of the O-Level E-Math 2018 Paper 1. It tests sine rule for an angle, in the Trigonometry & bearings area. It is worth 4 marks: 2 + 2. 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) By the sine rule in triangle \(SDM\): \(\dfrac{\sin(\angle DMS)}{SD} = \dfrac{\sin(\angle SDM)}{SM}\), so \(\sin(\angle DMS) = \dfrac{4140\sin29^{\circ}}{2392} = 0.8391\). The acute angle is \(\angle DMS = \sin^{-1}(0.8391) = 57.0^{\circ}\) (1 d.p.).
(b) The third angle is \(\angle DSM = 180^{\circ} - 29^{\circ} - 57.0^{\circ} = 94.0^{\circ}\). At \(M\) the bearing of \(D\) is \(290^{\circ}\). The point \(S\) lies on the southern side of \(MD\), so the bearing of \(S\) from \(M\) is \(290^{\circ} - \angle DMS = 290^{\circ} - 57.0^{\circ} = 233^{\circ}\). The back-bearing (bearing of \(M\) from \(S\)) is \(233^{\circ} - 180^{\circ} = 053^{\circ}\). At \(S\), turning from the direction \(SM\) towards \(SD\) through the interior angle \(\angle DSM = 94.0^{\circ}\) (anticlockwise, towards the west) gives the bearing of \(D\) from \(S\) as \(053^{\circ} - 94.0^{\circ} = -41^{\circ} \equiv 319^{\circ}\).
Answer: (a) \(\approx 57.0^{\circ}\)
(b) \(319^{\circ}\)
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 sine rule for an angle question from Trigonometry & bearings, worth 4 marks: 2 + 2.
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.
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