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Date May 2021 Marks available 3 Reference code 21M.2.AHL.TZ1.11
Level Additional Higher Level Paper Paper 2 Time zone Time zone 1
Command term Find Question number 11 Adapted from N/A

Question

The function f is defined by fx=3x+24x2-1, for xxpxq.

The graph of y=f(x) has exactly one point of inflexion.

The function g is defined by gx=4x2-13x+2, for x, x-23.

Find the value of p and the value of q.

[2]
a.

Find an expression for f'x.

[3]
b.

Find the x-coordinate of the point of inflexion.

[2]
c.

Sketch the graph of y=f(x) for -3x3, showing the values of any axes intercepts, the coordinates of any local maxima and local minima, and giving the equations of any asymptotes.

[5]
d.

Find the equations of all the asymptotes on the graph of y=g(x).

[4]
e.

By considering the graph of y=g(x)-f(x), or otherwise, solve f(x)<g(x) for x.

[4]
f.

Markscheme

attempt to solve 4x2-1=0 e.g. by factorising 4x2-1        (M1)

p=12, q=-12 or vice versa        A1

 

[2 marks]

a.

attempt to use quotient rule or product rule        (M1)

 

EITHER

f'x=34x2-1-8x3x+24x2-12=-12x2-16x-34x2-12        A1A1

 

Note: Award A1 for each term in the numerator with correct signs, provided correct denominator is seen.

 

OR

f'x=-8x3x+24x2-1-2+34x2-1-1        A1A1

 

Note: Award A1 for each term.

 

[3 marks]

b.

attempt to find the local min point on y=f'x OR solve f''x=0      (M1)

x=-1.60     A1

 

[2 marks]

c.

      A1A1A1A1A1

 

Note: Award A1 for both vertical asymptotes with their equations, award A1 for horizontal asymptote with equation, award A1 for each correct branch including asymptotic behaviour, coordinates of minimum and maximum points (may be seen next to the graph) and values of axes intercepts.
If vertical asymptotes are absent (or not vertical) and the branches overlap as a consequence, award maximum A0A1A0A1A1.

 

[5 marks]

d.

x=-23=-0.667         A1

(oblique asymptote has) gradient 43=1.33         (A1)

appropriate method to find complete equation of oblique asymptote         M1

    3x+24x2+0x-1                      43x-89

 4x2+83x-83x-1

  -83x-16979

y=43x-89=1.33x-0.889         A1

Note: Do not award the final A1 if the answer is not given as an equation.

 

[4 marks]

e.

attempting to find at least one critical value x=-0.568729, x=1.31872         (M1)

-23<x<-0.569  OR  -0.5<x<0.5  OR  x>1.32        A1A1A1

 

Note: Only penalize once for use of  rather than <.

 

[4 marks]

f.

Examiners report

[N/A]
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b.
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c.
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e.
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f.

Syllabus sections

Topic 5 —Calculus » SL 5.6—Differentiating polynomials n E Q. Chain, product and quotient rules
Show 127 related questions
Topic 2—Functions » AHL 2.13—Rational functions
Topic 2—Functions
Topic 5 —Calculus

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