Solutions to exam review

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


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Solution
Method 1

1 + sin q + 1 - sin q
LHS = 1 mark for common denominator
(1 - sin q )(1 + sin q )
2
= 1 mark for identity
cos 2 q
= 2sec 2 q 1 mark for identity

(
= 2 tan 2 q + 1 ) 1 mark for identity

= 2 tan 2 q + 2 = RHS 4 marks

Method 2

2sin 2 q 1 1
RHS = +2 ½ mark LHS = +
cos 2 q 1 - sin q 1 + sin q
2sin 2 q 2 cos 2 q ½ mark
= + 1 + sin q + 1 - sin q 1 mark for common
cos 2 q cos 2 q =
1 - sin q
2 denominator
2sin 2 q + 2 cos 2 q
=
cos 2 q
( )
2
2 sin 2 q + cos 2 q = 1 mark for identity
= ½ mark for cos 2 q
cos 2 q factoring LHS = RHS
2 ½ mark for
=
cos q 2
simplifying 4 marks

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


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3x + 4 = 7 2 x +1

Solution

log 3x + 4 = log 72 x +1 ½ mark for taking the log of both sides
( x + 4 ) log 3 = ( 2 x + 1) log 7 ½ mark for log theorem
x log 3 + 4 log 3 = 2 x log 7 + log 7 ½ mark for distributing
x log 3 - 2 x log 7 = log 7 - 4 log 3
½ mark for isolating and factoring x
x ( log 3 - 2 log 7 ) = log 7 - 4 log 3
log 7 - 4 log 3
x= ½ mark for solving for x
log 3 - 2 log 7
x = 0.877 ½ mark for consistent answer
3 marks

113.

Note(s):

· log 3 = 0.477 121 ln 3 = 1.098 612
log 7 = 0.845 098 ln 7 = 1.945 910
· 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


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Question No. 35
Outcome: D1, B3

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

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Answer

( 0, ¥ ) OR y > 0 1 mark


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Question No. 31
Outcome: B1

The graph of y = x 2 - 4 x - 5 crosses the x-axis at -1 and 5.
Where does the graph of y = ( x + 10 ) - 4 ( x + 10 ) - 5 cross the x-axis?
2

Answer
152.

-11, - 5 ½ mark for each correct answer

1 mark
153.

The graph of the function f ( x ) = 5sin x + p touches the x-axis once in the interval [ 0, 2p] .
Solutions possibleReview of p.
State a to Exam value Pre-Calculus 40s Old Exam Questions

Answer
154.

5 or –5 1 mark

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Marking Guide—Pre-Calculus Mathematics (40S)—January 2005 Page 43

Question No. 49
Outcome: B1, B5

Solution
156.

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

(-2, 3) (1, 3)

(1, 1) (4, 1)
(-4, 0)
x x

(-7, -2) (-2, -2)

1 mark 1 mark for absolute value 2 marks
1 mark for vertical shift
1 mark
2 marks

Note(s):

· in a) deduct ½ mark for one incorrect point
· in b) deduct 1 mark if they are done in the wrong order

Marking Guide—Pre-Calculus Mathematics (40S)—January 2005 Page 67

b) y = f ( 2x)
c) y = - f ( x) -3 3
x

Solutions to Exam )Review
d) y = f ( - x Pre-Calculus 40s Old Exam Questions

157.
Solution
a) y b) y
4

2

x x
-3 3 -1.5 1.5

c) d)
y y

2

x x
-3 3 -3 3

-2

4 marks


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