The answer
(i) explained (no integer \(r\) gives \(x^0\))
(ii) \(-3024\)
O-Level A-Math 2020 Paper 2 Question 3 · Verified worked solution by the Genius Plus Academy teaching team
The question
(i) By considering the general term in the binomial expansion of \(\left(\dfrac{3}{x^2} + x\right)^8\), explain why every term is dependent on \(x\). [3]
(ii) Find the term independent of \(x\) in the expansion of \(\left(\dfrac{3}{x^2} + x\right)^8 (5 - 2x)\). [3]
(i) The general term is \[T_{r+1} = \binom{8}{r}\left(\dfrac{3}{x^2}\right)^{8-r}(x)^r = \binom{8}{r}\,3^{8-r}\,x^{-2(8-r)+r} = \binom{8}{r}\,3^{8-r}\,x^{3r-16}, \qquad r = 0, 1, \ldots, 8.\] A term independent of \(x\) needs \(3r - 16 = 0 \Rightarrow r = \dfrac{16}{3}\), which is not an integer in \(\{0, \ldots, 8\}\). So no term has \(x^0\), every term depends on \(x\).
(ii) In \(\left(\dfrac{3}{x^2} + x\right)^8 (5 - 2x)\), a constant term comes from \(5 \times (x^0\text{ term})\) and from \(-2x \times (x^{-1}\text{ term})\). As shown, there is no \(x^0\) term, so only the second contributes. For the \(x^{-1}\) term: \(3r - 16 = -1 \Rightarrow r = 5\), giving coefficient \(\binom{8}{5}3^{3} = 56 \times 27 = 1512\). Hence the term independent of \(x\) is \[(-2x)\big(1512\,x^{-1}\big) = -3024.\]
Answer: (i) explained (no integer \(r\) gives \(x^0\))
(ii) \(-3024\)
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 general term ⁿcr question from Binomial, worth 6 marks: 3 + 3.
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