The answer
(a) shown (\(124.1^{\circ}\))
(b) \(AD \approx 193\) m
(c) \(\approx 47.2^{\circ}\)
O-Level E-Math 2024 Paper 2 Question 5 · Verified worked solution by the Genius Plus Academy teaching team
What this question tests
This is Question 5 of the O-Level E-Math 2024 Paper 2. It tests bearings, in the Pythagoras & trigonometry area. It is worth 9 marks: 3 + 3 + 3. 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) Cosine rule in \(\triangle ABC\): \(\cos(\angle BAC) = \dfrac{AB^2 + AC^2 - BC^2}{2 \cdot AB \cdot AC} = \dfrac{175^2 + 280^2 - 405^2}{2(175)(280)} = \dfrac{-55000}{98000} = -0.56122\). So \(\angle BAC = \cos^{-1}(-0.56122) = 124.14^{\circ} = 124.1^{\circ}\) (1 d.p.). (shown)
(b) Bearing of \(AC\) from \(A\) \(= 048^{\circ} + 124.1^{\circ} = 172.1^{\circ}\). \(D\) is due west, bearing \(270^{\circ}\), so \(\angle DAC = 270^{\circ} - 172.1^{\circ} = 97.9^{\circ}\). In \(\triangle ACD\), \(\angle ADC = 180^{\circ} - 32^{\circ} - 97.9^{\circ} = 50.1^{\circ}\). Sine rule: \(\dfrac{AD}{\sin 32^{\circ}} = \dfrac{280}{\sin 50.1^{\circ}} \Rightarrow AD = \dfrac{280 \sin 32^{\circ}}{\sin 50.1^{\circ}} = 193.4 \approx 193\) m.
(c) Mast \(AT\) is vertical at \(A\). From \(C\): \(\tan 34^{\circ} = \dfrac{AT}{AC} \Rightarrow AT = 280 \tan 34^{\circ} = 188.86\) m. From \(B\): \(\tan(\text{elevation}) = \dfrac{AT}{AB} = \dfrac{188.86}{175} = 1.0792 \Rightarrow\) elevation \(= 47.2^{\circ}\) (1 d.p.).
Answer: (a) shown (\(124.1^{\circ}\))
(b) \(AD \approx 193\) m
(c) \(\approx 47.2^{\circ}\)
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 bearings question from Pythagoras & trigonometry, worth 9 marks: 3 + 3 + 3.
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