solutions to exam review

Solutions to Exam Review Pre-Calculus 40s Old Exam Questions


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Page 54 Marking Guide—Pre-Calculus Mathematics (40S)—January 2005

Question No. 43

Outcome: C1

Prove:

21 12 tan 2

1 sin 1 sinq

q q+ = +

- +

Solution

Method 1

( ) ( )

( )

2

2

2

2

1 sin 1 sinLHS

1 sin 1 sin

2

cos

2sec

2 tan 1

2 tan 2 RHS

q q

q q

q

q

q

q

+ + -=

- +

=

=

= +

= + =

Method 2

4 marks

1 mark for common denominator

1 mark for identity

1 mark for identity

1 mark for identity

½ mark

½ mark

½ mark for

factoring

½ mark for

simplifying

1 mark for common

denominator

1 mark for identity

2

2

1 1LHS

1 sin 1 sin

1 sin 1 sin

1 sin

2

cos

LHS RHS

q q

q q

q

q

= +- +

+ + -=

-

=

=

( )

2

2

2 2

2 2

2 2

2

2 2

2

2

2sinRHS 2

cos

2sin 2cos

cos cos

2sin 2cos

cos

2 sin cos

cos

2

cos

q

q

q q

q q

q q

q

q q

q

q

= +

= +

+=

+=

=4 marks


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Question No. 5

Outcome: D6

Solve for x algebraically.

Give your answer correct to 3 decimal places.

4 2 13 7x x+ +=

Solution

( ) ( )

( )

4 2 1log3 log 7

4 log3 2 1 log 7

log3 4log3 2 log 7 log 7

log3 2 log 7 log 7 4log3

log3 2log 7 log 7 4log3

log 7 4log3

log3 2log 7

0.877

x x

x x

x x

x x

x

x

x

+ +=

+ = +

+ = +

- = -

- = -

-=

-

=

Note(s):

·

· if brackets are omitted in the second step, but assumed to be there, deduct ½ mark

· the final answer must be correct to at least 3 decimal places

3 marks

½ mark for taking the log of both sides

½ mark for isolating and factoring x

½ mark for consistent answer

log3 0.477 121 ln 3 1.098 612

log 7 0.845 098 ln 7 1.945 910

= =

= =

½ mark for log theorem

½ mark for solving for x

½ mark for distributing


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Marking Guide—Pre-Calculus Mathematics (40S)—January 2005 Page 45

Question No. 35

Outcome: D1, B3

State the range of the function ( ) 2 xf x -= .

Answer

( )0, OR 0y¥ >

Note(s):

· give ½ mark for [ )0, or 0y¥ ³

1 mark


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Question No. 31

Outcome: B1

The graph of 2 4 5y x x= - - crosses the x-axis at 1 and 5.-

Where does the graph of ( ) ( )2

10 4 10 5y x x= + - + - cross the x-axis?

Answer

11, 5- -

1 mark

½ mark for each correct answer


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Question No. 33

Outcome: B7

The graph of the function ( ) 5sinf x x p= + touches the x-axis once in the interval [ ]0, 2p .

State a possible value of p.

Answer

5 or –5

1 mark


156.


Question No. 49

Outcome: B1, B5

Solution

a) ( )3y f x= + b) ( ) 2y f x= -

Note(s):

· in a) deduct ½ mark for one incorrect point

· in b) deduct 1 mark if they are done in the wrong order

1 mark 2 marks 1 mark for absolute value

1 mark for vertical shift

y

x

(1, 3) (-2, 3)

(-7, -2)

y

x(-4, 0)

(-2, -2)

(1, 1) (4, 1)

2 marks 1 mark


157.


Question No. 48

Outcome: B2, B3

Given the graph of ( )y f x= :

Sketch a clearly labelled graph of each of the following:

a) ( )2y f x=

b) ( )2y f x=

c) ( )y f x= -

d) ( )y f x= -

Solution

a) b)

c) d)

4 marks

y

x

2

-3 3

y

x

2

1.5 -1.5

y

x

4

3 -3

y

x

-2

-3 3

y

x -3 3

2

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