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.
Show that 27m+9n+3p=18.
Write down the other two linear equations in m, n and p.
Write down these three equations as a matrix equation.
Solve this matrix equation.
The function f can also be written f(x)=x(x−1)(rx−s) where r and s are integers. Find r and s.
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]
Attempting to substitute (3, 18) (M1)
m33+n32+p3=18 A1
27m+9n+3p=18 AG N0
[2 marks]
m + n + p = 0 A1 N1
−m + n − p = −10 A1 N1
[2 marks]
Evidence of attempting to set up a matrix equation (M1)
Correct matrix equation representing the given equations A2 N3
eg (2793111−11−1)(mnp)=(180−10)
[3 marks]
(2−53) A1A1A1 N3
[3 marks]
Factorizing (M1)
eg f(x)=x(2x2−5x+3), f(x)=(x2−x)(rx−s)
r=2 s=3 (accept f(x)=x(x−1)(2x−3)) A1A1 N3
[3 marks]