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Date May 2017 Marks available 2 Reference code 17M.2.sl.TZ2.6
Level SL only Paper 2 Time zone TZ2
Command term Show that Question number 6 Adapted from N/A

Question

Consider the function f(x)=x4+ax2+5, where a is a constant. Part of the graph of y=f(x) is shown below.

M17/5/MATSD/SP2/ENG/TZ2/06

It is known that at the point where x=2 the tangent to the graph of y=f(x) is horizontal.

There are two other points on the graph of y=f(x) at which the tangent is horizontal.

Write down the y-intercept of the graph.

[1]
a.

Find f(x).

[2]
b.

Show that a=8.

[2]
c.i.

Find f(2).

[2]
c.ii.

Write down the x-coordinates of these two points;

[2]
d.i.

Write down the intervals where the gradient of the graph of y=f(x) is positive.

[2]
d.ii.

Write down the range of f(x).

[2]
e.

Write down the number of possible solutions to the equation f(x)=5.

[1]
f.

The equation f(x)=m, where mR, has four solutions. Find the possible values of m.

[2]
g.

Markscheme

5     (A1)

 

Note:     Accept an answer of (0, 5).

 

[1 mark]

a.

(f(x)=)4x3+2ax     (A1)(A1)

 

Note:     Award (A1) for 4x3 and (A1) for +2ax. Award at most (A1)(A0) if extra terms are seen.

 

[2 marks]

b.

4×23+2a×2=0     (M1)(M1)

 

Note:     Award (M1) for substitution of x=2 into their derivative, (M1) for equating their derivative, written in terms of a, to 0 leading to a correct answer (note, the 8 does not need to be seen).

 

a=8     (AG)

[2 marks]

c.i.

(f(2)=)24+8×22+5     (M1)

 

Note:     Award (M1) for correct substitution of x=2 and  a=8 into the formula of the function.

 

21     (A1)(G2)

[2 marks]

c.ii.

(x=) 2, (x=) 0     (A1)(A1)

 

Note:     Award (A1) for each correct solution. Award at most (A0)(A1)(ft) if answers are given as (2 ,21) and (0, 5) or (2, 0) and (0, 0).

 

[2 marks]

d.i.

x<2, 0<x<2     (A1)(ft)(A1)(ft)

 

Note:     Award (A1)(ft) for x<2, follow through from part (d)(i) provided their value is negative.

Award (A1)(ft) for 0<x<2, follow through only from their 0 from part (d)(i); 2 must be the upper limit.

Accept interval notation.

 

[2 marks]

d.ii.

y     (A1)(ft)(A1)

 

Notes:     Award (A1)(ft) for 21 seen in an interval or an inequality, (A1) for “y \leqslant ”.

Accept interval notation.

Accept - \infty  < y \leqslant 21 or f(x) \leqslant 21.

Follow through from their answer to part (c)(ii). Award at most (A1)(ft)(A0) if x is seen instead of y. Do not award the second (A1) if a (finite) lower limit is seen.

 

[2 marks]

e.

3 (solutions)     (A1)

[1 mark]

f.

5 < m < 21 or equivalent     (A1)(ft)(A1)

 

Note:     Award (A1)(ft) for 5 and 21 seen in an interval or an inequality, (A1) for correct strict inequalities. Follow through from their answers to parts (a) and (c)(ii).

Accept interval notation.

 

[2 marks]

g.

Examiners report

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

Syllabus sections

Topic 6 - Mathematical models » 6.5 » Models using functions of the form f\left( x \right) = a{x^m} + b{x^n} + \ldots ; m,n \in \mathbb{Z} .
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