Edexcel IALMathematicsGeneral Applications11 Marks

Find the first four terms in ascending powers of x of the binomial expansion of (1 - x/2)¹⁰.

(a)Find the first four terms in ascending powers of x of the binomial expansion of (1 - x/2)¹⁰.[4]
(b)Hence use your expansion to find an approximation for 0.995¹⁰, giving your answer to 5 decimal places.[2]
(c)Given that (2 + ax)(1 - x/2)¹⁰ = 2 - 9x + bx² + ..., find the value of a and the value of b.[5]

Mark scheme

Marking points from the official mark scheme. The mark or mark type is shown in brackets.

(a) B1 for 1 (first term).
(a) M1 for attempting the general term for nCr(1)^(n-r)(-x/2)^r for at least two terms with r>0, using correct binomial coefficient and power of (-x/2).
(a) A1 for the next two terms: -5x and + (10 * 9 / 2) * (-x/2)² = 45/4 x² (or 11.25x²).
(a) A1 for the fourth term: (10 * 9 * 8 / (3 * 2 * 1)) * (-x/2)³ = -15/2 x³ (or -7.5x³). (All four terms must be correct and simplified).
(b) M1 for setting 1 - x/2 = 0.995 => x = 0.01 and substituting this x value into their expansion (at least the first 3 terms).
(b) A1 for 0.95112 (must be 5 decimal places, following through their expansion).
(c) M1 for multiplying (2 + ax) by their expansion from (a) for terms up to x².
(c) A1 for correctly expanding terms involving x and equating to -9x: (2)(-5x) + (ax)(1) = -10x + ax = -9x.
(c) M1 for equating the coefficient of x to -9 and solving for a: -10 + a = -9.
(c) A1 for a = 1.
(c) M1 for correctly expanding terms involving x²: (2)(45/4 x²) + (ax)(-5x).
(c) A1 for b = 2(45/4) + (1)(-5) = 45/2 - 5 = 35/2 (or 17.5).

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