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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 ) = m x 3 + n x 2 + p x + 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 27 m + 9 n + 3 p = 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 ( x 1 ) ( r x s ) 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)

m 3 3 + n 3 2 + p 3 = 18        A1

27 m + 9 n + 3 p = 18        AG  N0

[2 marks]

b.

m  + n  + p = 0      A1    N1

m  + n p = −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  ( 27 9 3 1 1 1 1 1 1 ) ( m n p ) = ( 18 0 10 )

[3 marks]

d.i.

( 2 5 3 )                A1A1A1    N3

[3 marks]

d.ii.

Factorizing       (M1)

eg  f ( x ) = x ( 2 x 2 5 x + 3 ) ,   f ( x ) = ( x 2 x ) ( r x s )  

r = 2   s = 3       (accept f ( x ) = x ( x 1 ) ( 2 x 3 ) )              A1A1    N3

[3 marks]

e.

Examiners report

[N/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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