2024 IGCSE Additional Mathematics May/June 0606/23 Q4

2024 IGCSE Additional Mathematics May/June 0606/23 Q4

8 marks
  1. Find and simplify the term independent of \(x\) in the expansion of \(\left(x^2-\dfrac1{2x^3}\right)^{10}\).[2]
  2. DO NOT USE A CALCULATOR IN THIS PART OF THE QUESTION.
    1. Use the binomial theorem to show that \((1+2\sqrt2)^4-(1-2\sqrt2)^4=k\sqrt2\), where \(k\) is an integer to be found.[4]
    2. Hence write \(\dfrac{(1+2\sqrt2)^4-(1-2\sqrt2)^4}{1+\sqrt2}\) in the form \(a+b\sqrt2\), where \(a\) and \(b\) are integers.[2]

Solution:

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Answer:(a) \(\dfrac{105}8\). (b)(i) \(k=144\). (ii) \(288-144\sqrt2\).

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