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Date May Example question Marks available 2 Reference code EXM.2.AHL.TZ0.8
Level Additional Higher Level Paper Paper 2 Time zone Time zone 0
Command term Show that Question number 8 Adapted from N/A

Question

The function f is given by f(x)=mx3+nx2+px+q, where m, n, p, q are integers.

The graph of f passes through the point (0, 0).

The graph of f also passes through the point (3, 18).

The graph of f also passes through the points (1, 0) and (–1, –10).

Write down the value of q.

[1]
a.

Show that 27m+9n+3p=18.

[2]
b.

Write down the other two linear equations in m, n and p.

[2]
c.

Write down these three equations as a matrix equation.

[3]
d.i.

Solve this matrix equation.

[3]
d.ii.

The function f can also be written f(x)=x(x1)(rxs) where r and s are integers. Find r and s.

[3]
e.

Markscheme

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

q = 0     A1  N1

[1 mark]

a.

Attempting to substitute (3, 18)          (M1)

m33+n32+p3=18       A1

27m+9n+3p=18        AG  N0

[2 marks]

b.

m + n + p = 0      A1    N1

m + np = −10      A1    N1

[2 marks]

c.

Evidence of attempting to set up a matrix equation                (M1)

Correct matrix equation representing the given equations          A2   N3

eg (2793111111)(mnp)=(18010)

[3 marks]

d.i.

(253)               A1A1A1    N3

[3 marks]

d.ii.

Factorizing       (M1)

eg f(x)=x(2x25x+3),  f(x)=(x2x)(rxs) 

r=2  s=3      (accept f(x)=x(x1)(2x3))              A1A1    N3

[3 marks]

e.

Examiners report

[N/A]
a.
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b.
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c.
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d.i.
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d.ii.
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e.

Syllabus sections

Topic 1—Number and algebra » AHL 1.14—Introduction to matrices
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Topic 1—Number and algebra

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