The answer
(a)(i) \(t(t + 18)\)
(ii) difference \(= 72\) for all \(t\)
(iii) largest \(= 99\)
(b)(i) \(8n + 12\)
(ii) \(8n + 12 = 4(2n + 3)\), a multiple of 4
O-Level E-Math 2019 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 2019 Paper 2. It tests algebraic expression for grid positions, in the Sequences & number patterns area. It is worth 9 marks: 1 + 2 + 3 + 2 + 1. 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) Each cell in a column is 6 more than the one above (the grid has 6 columns), so the four numbers are \(t\), \(t + 6\), \(t + 12\), \(t + 18\).
(i) Product of top and bottom \(= t(t + 18)\).
(ii) Product of the middle two \(= (t + 6)(t + 12) = t^2 + 18t + 72\). Product of top and bottom \(= t(t + 18) = t^2 + 18t\). Their difference is \[(t^2 + 18t + 72) - (t^2 + 18t) = 72,\] which has no \(t\) in it, so it is 72 whatever the column position.
(iii) Sum of the four \(= t + (t + 6) + (t + 12) + (t + 18) = 4t + 36 = 360\), so \(4t = 324\) and \(t = 81\). The largest (bottom) number is \(t + 18 = 99\).
(b) The sequence is arithmetic. From the 3rd term (36) to the 6th term (60) is 3 steps, so the common difference \(d = \dfrac{60 - 36}{3} = 8\). The first term is \(36 - 2(8) = 20\).
(i) \(n\)th term \(= 20 + 8(n - 1) = 8n + 12\). (Check: \(n = 3 \to 36\), \(n = 6 \to 60\).)
(ii) \(8n + 12 = 4(2n + 3)\). Since \(2n + 3\) is a whole number, every term is \(4 \times\) a whole number, i.e. a multiple of 4.
Answer: (a)(i) \(t(t + 18)\)
(ii) difference \(= 72\) for all \(t\)
(iii) largest \(= 99\)
(b)(i) \(8n + 12\)
(ii) \(8n + 12 = 4(2n + 3)\), a multiple of 4
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 algebraic expression for grid positions question from Sequences & number patterns, worth 9 marks: 1 + 2 + 3 + 2 + 1.
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