User interface language: English | Español

Date May 2017 Marks available 3 Reference code 17M.2.AHL.TZ2.H_11
Level Additional Higher Level Paper Paper 2 Time zone Time zone 2
Command term State Question number H_11 Adapted from N/A

Question

It is given that f ( x ) = 3 x 4 + a x 3 + b x 2 7 x 4 where a and b are positive integers.

Given that x 2 1 is a factor of f ( x ) find the value of a and the value of b .

[4]
a.

Factorize f ( x ) into a product of linear factors.

[3]
b.

Using your graph state the range of values of c for which f ( x ) = c has exactly two distinct real roots.

[3]
d.

Markscheme

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

g ( x ) = 3 x 4 + a x 3 + b x 2 7 x 4

g ( 1 ) = 0 a + b = 8      M1A1

g ( 1 ) = 0 a + b = 6      A1

a = 7 ,   b = 1      A1

[4 marks]

a.

3 x 4 + 7 x 3 + x 2 7 x 4 = ( x 2 1 ) ( p x 2 + q x + r )

attempt to equate coefficients     (M1)

p = 3 ,   q = 7 ,   r = 4      (A1)

3 x 4 + 7 x 3 + x 2 7 x 4 = ( x 2 1 ) ( 3 x 2 + 7 x + 4 )

= ( x 1 ) ( x + 1 ) 2 ( 3 x + 4 )      A1

 

Note:     Accept any equivalent valid method.

 

[3 marks]

b.

c > 0      A1

6.20 < c < 0.0366      A1A1

 

Note:     Award A1 for correct end points and A1 for correct inequalities.

 

Note:     If the candidate has misdrawn the graph and omitted the first minimum point, the maximum mark that may be awarded is A1FTA0A0 for c > 6.20 seen.

 

[3 marks]

d.

Examiners report

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

Syllabus sections

Topic 2—Functions » AHL 2.12—Factor and remainder theorems, sum and product of roots
Show 28 related questions
Topic 2—Functions

View options