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
Solution
Method 1
( )( )
( )
2
2
2
2
1 sin 1 sin
LHS
1 sin 1 sin
2
cos
2sec
2 tan 1
2tan 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 1
LHS
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
2sin
RHS 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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4 2 1
3 7x x+ +
=
Solution
( ) ( )
( )
4 2 1
log3 log7
4 log3 2 1 log7
log3 4log3 2 log7 log7
log3 2 log7 log7 4log3
log3 2log7 log7 4log3
log7 4log3
log3 2log7
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 ln3 1.098 612
log7 0.845 098 ln 7 1.945 910
= =
= =
½ mark for log theorem
½ mark for solving for x
½ mark for distributing

115.
116.
Question No. 35
Outcome: D1, B3
State the range of the function ( ) 2 x
f x -
= .
Answer
( )0, OR 0y¥ > 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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Marking Guide—Pre-Calculus Mathematics (40S)—January 2005 Page 43
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.
Marking Guide—Pre-Calculus Mathematics (40S)—January 2005 Page 67
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 marks1 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.
b) ( )2y f x=
c) ( )y f x= -
d) ( )y f x= -
Solution
a) b)
c) d)
4 marks
x
-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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