Date | May 2022 | Marks available | 2 | Reference code | 22M.1.SL.TZ2.6 |
Level | Standard Level | Paper | Paper 1 (without calculator) | Time zone | Time zone 2 |
Command term | Show that | Question number | 6 | Adapted from | N/A |
Question
Consider the binomial expansion (x+1)7=x7+ax6+bx5+35x4+…+1 where x≠0 and a, b∈ℤ+.
Show that b=21.
The third term in the expansion is the mean of the second term and the fourth term in the expansion.
Find the possible values of x.
Markscheme
EITHER
recognises the required term (or coefficient) in the expansion (M1)
bx5=C27x512 OR b=C27 OR C57
b=7!2!5! (=7!2!(7-2)!)
correct working A1
7×6×5×4×3×2×12×1×5×4×3×2×1 OR 7×62! OR 422
OR
lists terms from row 7 of Pascal’s triangle (M1)
1, 7, 21,… A1
THEN
b=21 AG
[2 marks]
a=7 (A1)
correct equation A1
21x5=ax6+35x42 OR 21x5=7x6+35x42
correct quadratic equation A1
7x2-42x+35=0 OR x2-6x+5=0 (or equivalent)
valid attempt to solve their quadratic (M1)
(x-1)(x-5)=0 OR x=6±√(-6)2-4(1)(5)2(1)
x=1, x=5 A1
Note: Award final A0 for obtaining x=0, x=1, x=5.
[5 marks]
Examiners report
The majority of candidates answered part (a) correctly, either by using the Crn formula or Pascal's Triangle. In part (b) of the question, most candidates were able to correctly find the value of a=7 and set up a correct equation showing the mean of the second and fourth terms. While some struggled to complete the required algebra to solve the equation, the majority of candidates who found a correct quadratic equation were able to solve it correctly. A few candidates included x=0 in their final answer, thus not earning the final mark.