Date | November 2021 | Marks available | 1 | Reference code | 21N.1.AHL.TZ0.9 |
Level | Additional Higher Level | Paper | Paper 1 (without calculator) | Time zone | Time zone 0 |
Command term | State | Question number | 9 | Adapted from | N/A |
Question
Consider the expression 1√1+ax-√1-x1√1+ax−√1−x where a∈ℚ, a≠0.
The binomial expansion of this expression, in ascending powers of x, as far as the term in x2 is 4bx+bx2, where b∈ℚ.
Find the value of a and the value of b.
State the restriction which must be placed on x for this expansion to be valid.
Markscheme
attempt to expand binomial with negative fractional power (M1)
1√1+ax=(1+ax)-12=1-ax2+3a2x28+… A1
√1-x=(1-x)12=1-x2-x28+… A1
1√1+ax-√1-x=(1-a)2x+(3a2+18)x2+…
attempt to equate coefficients of x or x2 (M1)
x : 1-a2=4b; x2 : 3a2+18=b
attempt to solve simultaneously (M1)
a=-13, b=16 A1
[6 marks]
|x|<1 A1
[1 mark]