The answer
(a) \(77^{\circ}\)
(b) \(68^{\circ}\)
(c) \(57.6\) cm²
O-Level E-Math 2021 Paper 2 Question 6 · Verified worked solution by the Genius Plus Academy teaching team
What this question tests
This is Question 6 of the O-Level E-Math 2021 Paper 2. It tests angle in a semicircle, in the Circle properties / Mensuration 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) \(\angle AED = 90^{\circ}\) (angle in a semicircle, \(AD\) is a diameter). In \(\triangle AED\): \(\angle ADE = 180^{\circ} - 90^{\circ} - 35^{\circ} = 55^{\circ}\) (angle sum of a triangle). Since \(BE \parallel CD\), \(\angle BED + \angle EDC = 180^{\circ}\) (co-interior angles), and \(\angle EDC = \angle ADE + \angle ADC = 55^{\circ} + 48^{\circ} = 103^{\circ}\), so \(\angle BED = 180^{\circ} - 103^{\circ} = 77^{\circ}\).
(b) \(BCDE\) is a cyclic quadrilateral, so \(\angle BCD = 180^{\circ} - \angle BED = 180^{\circ} - 77^{\circ} = 103^{\circ}\) (opposite angles of a cyclic quadrilateral). \(\angle DCE = \angle DAE = 35^{\circ}\) (angles in the same segment, subtended by arc \(DE\)). So \(\angle BCE = \angle BCD - \angle DCE = 103^{\circ} - 35^{\circ} = 68^{\circ}\).
(c) The minor arc \(AC\) subtends \(\angle AOC = 2 \times \angle ADC = 96^{\circ}\) at the centre, so the reflex angle (the major sector \(OAEDC\)) is \(360^{\circ} - 96^{\circ} = 264^{\circ}\). Area \(= \dfrac{264}{360}\pi(5^2) = 57.6\) cm² (3 s.f.).
Answer: (a) \(77^{\circ}\)
(b) \(68^{\circ}\)
(c) \(57.6\) cm²
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 angle in a semicircle question from Circle properties / Mensuration, worth 9 marks: 3 + 3 + 3.
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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