The answer
(a) \(q = \dfrac{A - 2\pi p^2}{\pi p}\)
(b) \(l = 3r\)
O-Level E-Math 2015 Paper 1 Question 21 · Verified worked solution by the Genius Plus Academy teaching team
What this question tests
This is Question 21 of the O-Level E-Math 2015 Paper 1. It tests rearrange a formula for a chosen variable, in the Change of subject / Mensuration (surface area) area. It is worth 5 marks: 2 + 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) Expand: \(A = 2\pi p^2 + \pi p q\). Then \(\pi p q = A - 2\pi p^2\), so \(q = \dfrac{A - 2\pi p^2}{\pi p}\) (equivalently \(q = \dfrac{A}{\pi p} - 2p\)).
(b) The solid is a cylinder (curved surface only, since both flat ends are capped by hemispheres) plus two hemispheres. The two hemispheres together form one whole sphere. - Curved surface of the cylinder \(= 2\pi r \times (\text{length}) = 2\pi r \times 2r = 4\pi r^2\). - Two hemispheres \(=\) one sphere \(= 4\pi r^2\). - Total surface area of the solid \(= 4\pi r^2 + 4\pi r^2 = 8\pi r^2\).
The total surface area of a cone of radius \(r\) and slant height \(l\) is \(\pi r^2 + \pi r l = \pi r(r + l)\). Twice this is \(2\pi r(r + l)\). Setting the two equal: \[8\pi r^2 = 2\pi r(r + l) \Rightarrow 4r = r + l \Rightarrow l = 3r.\]
Answer: (a) \(q = \dfrac{A - 2\pi p^2}{\pi p}\)
(b) \(l = 3r\)
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 rearrange a formula for a chosen variable question from Change of subject / Mensuration (surface area), worth 5 marks: 2 + 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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