Date | November 2021 | Marks available | 6 | Reference code | 21N.2.AHL.TZ0.6 |
Level | Additional Higher Level | Paper | Paper 2 | Time zone | Time zone 0 |
Command term | Determine | Question number | 6 | Adapted from | N/A |
Question
Prove the identity (p+q)3-3pq(p+q)≡p3+q3.
The equation 2x2-5x+1=0 has two real roots, α and β.
Consider the equation x2+mx+n=0, where m, n∈ℤ and which has roots 1α3 and 1β3.
Without solving 2x2-5x+1=0, determine the values of m and n.
Markscheme
METHOD 1
(p+q)3-3pq(p+q)≡p3+q3
attempts to expand (p+q)3 M1
p3+3p2q+3pq2+q3
(p+q)3-3pq(p+q)≡p3+3p2q+3pq2+q3-3pq(p+q)
≡p3+3p2q+3pq2+q3-3p2q-3pq2 A1
≡p3+q3 AG
Note: Condone the use of equals signs throughout.
METHOD 2
(p+q)3-3pq(p+q)≡p3+q3
attempts to factorise (p+q)3-3pq(p+q) M1
≡(p+q)((p+q)2-3pq) (≡(p+q)(p2-pq+q2))
≡p3-p2q+pq2+p2q-pq2+q3 A1
≡p3+q3 AG
Note: Condone the use of equals signs throughout.
METHOD 3
p3+q3≡(p+q)3-3pq(p+q)
attempts to factorise p3+q3 M1
≡(p+q)(p2-pq+q2)
≡(p+q)((p+q)2-3pq) A1
≡(p+q)3-3pq(p+q) AG
Note: Condone the use of equals signs throughout.
[2 marks]
Note: Award a maximum of A1M0A0A1M0A0 for m=-95 and n=8 found by using α, .
Condone, as appropriate, solutions that state but clearly do not use the values of and .
Special case: Award a maximum of A1M1A0A1M0A0 for and obtained by solving simultaneously for and from product of roots and sum of roots equations.
product of roots of
(seen anywhere) A1
considers by stating M1
Note: Award M1 for attempting to substitute their value of into .
A1
sum of roots of
(seen anywhere) A1
considers and by stating M1
Note: Award M1 for attempting to substitute their values of and into their expression. Award M0 for use of only.
A1
[6 marks]