O-LevelAdditional MathematicsSeries4037/22 · May/June 2019 · Q88 Marks

In the binomial expansion of `(a - x/2)^6`, the coefficient of x³ is 120 times the coefficient of x⁵. Find the possible values of the constant a.

(a)In the binomial expansion of `(a - x/2)^6`, the coefficient of x³ is 120 times the coefficient of x⁵. Find the possible values of the constant a.[4]
(b)(i)Expand `(1+2x)^20` in ascending powers of x, as far as the term in x³. Simplify each term.[2]
(b)(ii)Use your expansion to show that the value of `0.98^20` is 0.67 to 2 decimal places.[2]

Mark scheme

Marking points from the official mark scheme (May/June 2019). The mark or mark type is shown in brackets.

[B2] (a) For `(6 C 3) a^3 (-x/2)^3` and `(6 C 5) a^1 (-x/2)^5` oe soi (B1 for either `(6 C 3) a^3 (-x/2)^3` or `- (20/8)a^3 [x^3]` oe OR `(6 C 5) a^1 (-x/2)^5` or `(6/32)a[x^5]` oe, OR for `(20/8)a^3 [x^3]` and `(6/32)a[x^5]` oe, oe where k > 0)
[M1] (20/8)a³ = 120 * (6/32)a oe soi
[A1] [a=] ±3
[B2] (b)(i) 1+40x + 760x² +9120x³ (B1 for three out of the four terms correct. If B0 then SC1 for 1,40x, 760x²,9120x³ seen but not summed)
[B2] (b)(ii) 1+40(-0.01) + 760(-0.01)² + 9120(-0.01)³ OR 1-0.4+0.076-0.00912 oe leading to 0.66688 cao (or M1 for use of x = −0.01 oe in their expansion seen or implied by e.g. 0.66688 without working or 1-0.4+0.076 – 0.00912)

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