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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+q3-3pqp+qp3+q3.

[2]
a.

The equation 2x2-5x+1=0 has two real roots, α and β.

Consider the equation x2+mx+n=0, where m, n and which has roots 1α3 and 1β3.
Without solving 2x2-5x+1=0, determine the values of m and n.

[6]
b.

Markscheme

METHOD 1

p+q3-3pqp+qp3+q3

attempts to expand p+q3                 M1

p3+3p2q+3pq2+q3

p+q3-3pqp+qp3+3p2q+3pq2+q3-3pqp+q

p3+3p2q+3pq2+q3-3p2q-3pq2                 A1

p3+q3                 AG


Note: Condone the use of equals signs throughout.

 

METHOD 2

p+q3-3pqp+qp3+q3

attempts to factorise p+q3-3pqp+q                 M1

p+qp+q2-3pq p+qp2-pq+q2

p3-p2q+pq2+p2q-pq2+q3                 A1

p3+q3                 AG


Note: Condone the use of equals signs throughout.

 

METHOD 3

p3+q3p+q3-3pqp+q

attempts to factorise p3+q3                 M1

p+qp2-pq+q2

p+qp+q2-3pq                 A1

p+q3-3pqp+q                 AG


Note: 
Condone the use of equals signs throughout.


[2 marks]

a.

Note: Award a maximum of A1M0A0A1M0A0 for m=-95 and n=8 found by using α,β=5±174 α,β=0.219, 2.28.
Condone, as appropriate, solutions that state but clearly do not use the values of α and β.
Special case: Award a maximum of A1M1A0A1M0A0 for m=-95 and n=8 obtained by solving simultaneously for α and β from product of roots and sum of roots equations.


product of roots of x2-52x+12=0

αβ=12 (seen anywhere)                      A1

considers 1α31β3 by stating 1αβ3=n                      M1


Note: Award M1 for attempting to substitute their value of αβ into 1αβ3.

1αβ3=1123

n=8                      A1

sum of roots of x2-52x+12=0

α+β=52 (seen anywhere)                A1

considers 1α3 and 1β3 by stating α+β3-3αβα+βαβ3 α+βαβ3-3α+βαβ2=-m                      M1


Note: Award M1 for attempting to substitute their values of α+b and αβ into their expression. Award M0 for use of α+β3-3αβα+β only.


=523-325218 =125-30=95

m=-95                A1

x2-95x+8=0


[6 marks]

b.

Examiners report

[N/A]
a.
[N/A]
b.

Syllabus sections

Topic 1—Number and algebra » SL 1.6—Simple proof
Show 41 related questions
Topic 2—Functions » AHL 2.12—Factor and remainder theorems, sum and product of roots
Topic 1—Number and algebra
Topic 2—Functions

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