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Date May 2015 Marks available 2 Reference code 15M.1.hl.TZ1.5
Level HL only Paper 1 Time zone TZ1
Command term Show that Question number 5 Adapted from N/A

Question

The functions f and g are defined by f(x)=ax2+bx+c, xR and g(x)=psinx+qx+r, xR where a, b, c, p, q, r are real constants.

Given that f is an even function, show that b=0.

[2]
a.

Given that g is an odd function, find the value of r.

[2]
b.

The function h is both odd and even, with domain R.

Find h(x).

[2]
c.

Markscheme

EITHER

f(x)=f(x)     M1

ax2bx+c=ax2+bx+c2bx=0, (xR)     A1

OR

y-axis is eqn of symmetry     M1

so b2a=0     A1

THEN

b=0     AG

[2 marks]

a.

g(x)=g(x)psin(x)qx+r=psinxqxr

psinxqx+r=psinxqxr     M1

 

Note:     M1 is for knowing properties of sin.

 

2r=0r=0     A1

 

Note:     In (a) and (b) allow substitution of a particular value of x

[2 marks]

b.

h(x)=h(x)=h(x)2h(x)=0h(x)=0, (x)     M1A1

 

Note:     Accept geometrical explanations.

[2 marks]

Total [6 marks]

c.

Examiners report

Sometimes backwards working but many correct approaches.

a.

Some candidates did not know what odd and even functions were. Correct solutions from those who applied the definition.

b.

Some realised: just apply the definitions. Some did very strange things involving f and g.

c.

Syllabus sections

Topic 2 - Core: Functions and equations » 2.1 » Odd and even functions.

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