The answer
\(BC = (10 + 3\sqrt{3})\) cm
O-Level A-Math 2025 Paper 2 Question 1 · Verified worked solution by the Genius Plus Academy teaching team
What this question tests
This is Question 1 of the O-Level A-Math 2025 Paper 2. It tests rationalising the denominator, in the Surds area. It is worth 6 marks. 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.
Area of a trapezium \(= \tfrac12(\text{sum of parallel sides})\times\text{height}\): \[\tfrac12\big[(6 + \sqrt3) + l\big](4 - \sqrt3) = 26 \;\Rightarrow\; (6 + \sqrt3) + l = \frac{52}{4 - \sqrt3}.\] Rationalise: \(\dfrac{52}{4 - \sqrt3}\cdot\dfrac{4 + \sqrt3}{4 + \sqrt3} = \dfrac{52(4 + \sqrt3)}{16 - 3} = \dfrac{52(4 + \sqrt3)}{13} = 4(4 + \sqrt3) = 16 + 4\sqrt3.\) So \(l = (16 + 4\sqrt3) - (6 + \sqrt3) = 10 + 3\sqrt3\). Hence \(BC = (10 + 3\sqrt3)\) cm.
Answer: \(BC = (10 + 3\sqrt{3})\) cm
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 A-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 rationalising the denominator question from Surds, worth 6 marks.
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