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Date November 2017 Marks available 3 Reference code 17N.2.SL.TZ0.T_5
Level Standard Level Paper Paper 2 Time zone Time zone 0
Command term Find Question number T_5 Adapted from N/A

Question

A function f is given by f ( x ) = ( 2 x + 2 ) ( 5 x 2 ) .

The graph of the function g ( x ) = 5 x + 6 x 6 intersects the graph of f .

Find the exact value of each of the zeros of f .

[3]
a.

Expand the expression for f ( x ) .

[1]
b.i.

Find f ( x ) .

[3]
b.ii.

Use your answer to part (b)(ii) to find the values of x for which f is increasing.

[3]
c.

Draw the graph of f for 3 x 3 and 40 y 20 . Use a scale of 2 cm to represent 1 unit on the x -axis and 1 cm to represent 5 units on the y -axis.

[4]
d.

Write down the coordinates of the point of intersection.

[2]
e.

Markscheme

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

1 ,   5 ,   5     (A1)(A1)(A1)

 

Note:     Award (A1) for –1 and each exact value seen. Award at most (A1)(A0)(A1) for use of 2.23606… instead of 5 .

 

[3 marks]

a.

10 x 2 x 3 + 10 2 x 2     (A1)

 

Notes:     The expansion may be seen in part (b)(ii).

 

[1 mark]

b.i.

10 6 x 2 4 x     (A1)(ft)(A1)(ft)(A1)(ft)

 

Notes:     Follow through from part (b)(i). Award (A1)(ft) for each correct term. Award at most (A1)(ft)(A1)(ft)(A0) if extra terms are seen.

 

[3 marks]

b.ii.

10 6 x 2 4 x > 0     (M1)

 

Notes:     Award (M1) for their f ( x ) > 0 . Accept equality or weak inequality.

 

1.67 < x < 1   ( 5 3 < x < 1 ,   1.66666 < x < 1 )     (A1)(ft)(A1)(ft)(G2)

 

Notes:     Award (A1)(ft) for correct endpoints, (A1)(ft) for correct weak or strict inequalities. Follow through from part (b)(ii). Do not award any marks if there is no answer in part (b)(ii).

 

[3 marks]

c.

N17/5/MATSD/SP2/ENG/TZ0/05.d/M     (A1)(A1)(ft)(A1)(ft)(A1)

 

Notes:     Award (A1) for correct scale; axes labelled and drawn with a ruler.

Award (A1)(ft) for their correct x -intercepts in approximately correct location.

Award (A1) for correct minimum and maximum points in approximately correct location.

Award (A1) for a smooth continuous curve with approximate correct shape. The curve should be in the given domain.

Follow through from part (a) for the x -intercepts.

 

[4 marks]

d.

( 1.49 ,   13.9 )   ( ( 1.48702 ,   13.8714 ) )     (G1)(ft)(G1)(ft)

 

Notes:     Award (G1) for 1.49 and (G1) for 13.9 written as a coordinate pair. Award at most (G0)(G1) if parentheses are missing. Accept x = 1.49 and y = 13.9 . Follow through from part (b)(i).

 

[2 marks]

e.

Examiners report

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

Syllabus sections

Topic 5—Calculus » SL 5.2—Increasing and decreasing functions
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Topic 5—Calculus » SL 5.3—Differentiating polynomials, n E Z
Topic 2—Functions » SL 2.3—Graphing
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Topic 2—Functions
Topic 5—Calculus

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