The answer
(a) \(132\) cm²
(b) \(19.8\) cm
(c) \(\approx 126\) cm²
O-Level E-Math 2022 Paper 2 Question 7 · Verified worked solution by the Genius Plus Academy teaching team
What this question tests
This is Question 7 of the O-Level E-Math 2022 Paper 2. It tests area of a trapezium on a face, in the Pythagoras & trigonometry / Mensuration area. It is worth 10 marks: 3 + 2 + 5. 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) \(EX = \tfrac38 \times EF = \tfrac38(24) = 9\) cm. \(ABXE\) lies in the front face, a trapezium with parallel sides \(AB = 24\) and \(EX = 9\), height \(AE = 8\). Area \(= \tfrac12(24 + 9)(8) = 132\) cm².
(b) \(X\) and \(G\) are both on the top face. With \(XF = 24 - 9 = 15\) and \(FG = BC = 13\), \(XG = \sqrt{15^2 + 13^2} = \sqrt{394} = 19.8\) cm.
(c) Place \(A\) at the origin: \(B(24, 0, 0)\), \(G(24, 13, 8)\), \(X(9, 0, 8)\). Vectors from \(B\): \(\overrightarrow{BX} = (-15, 0, 8)\), \(\overrightarrow{BG} = (0, 13, 8)\). Cross product \(\overrightarrow{BX} \times \overrightarrow{BG} = (-104, 120, -195)\), of magnitude \(\sqrt{104^2 + 120^2 + 195^2} = 251.5\). Area \(= \tfrac12(251.5) = 126\) cm². *(Equivalently by Heron's rule on \(BG = \sqrt{233}\), \(BX = 17\), \(GX = \sqrt{394}\).)*
Answer: (a) \(132\) cm²
(b) \(19.8\) cm
(c) \(\approx 126\) 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 area of a trapezium on a face question from Pythagoras & trigonometry / Mensuration, worth 10 marks: 3 + 2 + 5.
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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