CAPE - Mathematics - Unit 2 - 2004 Paper 01 Section A (Module 2.1) 1 In the diagram below (not drawn to scale) the line y = 3 cuts the curve y e 2 x at the point (a, b) Calcul ate the values of a and b 4 marks e 2 x 3 e 2 x x 1 2 ln 3 . 1 2 ln a 1 2 ln 3 . 1 2 ln b = 3 2 Differentiate with respect to x a ( ) x ln 3 x 2 . x x 0 2 marks b ( ) y sin 2 x cos . x . x x 3 marks a ( ) 1 3 x 2 6 x ( ) 2 x 1 3 x 2 6 x ( ) x b ( ) 2 sin x . cos . x . cos x . ( ) . cos x . sin 2 x . 2 sin x . cos 2 . x sin 3 x 2 sin x . cos . x . cos x . ( ) . cos x . sin 2 x . x x x 1
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7/29/2019 Mathcad - CAPE - 2004 - Math Unit 2 - Paper 01
...... find the value of the constant a and the value of the
constant n 6 marks
1 n ax( ).n n 1( ).
2a
2x
2... 1 6 x 16 x
21 n ax( ).
n n 1( ).
2a
2x
2...
na = 6n n 1( ).
2a
216
n n 1( ).
2a
2 n n 1( )
2
6
n
2
16 0n n 1( )
2
6
n
2
16 n = 9 a2
3
10 A craftsman estimated the side of a square tile to be 16 cm but the actual measurement was16.14 cm. Calculate to two decimal places the percentage error in the actual measurementof the area of the tile
4 marks
Total 30 marks
15.5 side 16.5< areamin
15.52
minarea
min240.25
min
areamax
16.52
maxarea
max272.25
max
maximum error. 272.25 240.25maximum error.
oercentage error =272.25 240.25
16.142
100( ) 12.284=
4
7/29/2019 Mathcad - CAPE - 2004 - Math Unit 2 - Paper 01
12 Only three horses A, B and C are in a race. The probability that A wins the race is twice theprobability that B wins. The probability that B wins is twice the probability that C wins.Find the probability of winning for each of the horses.
5 marks
let x be the random variable "horse wins"
x A B C
P X x( ). 4 k 2 k k k 1
7
4
7
2
7
1
7
13 Let A and B be the events such that P ( A U B ) =3
4P ( A' ) =
2
3P ( A ^ B ) =
1
4
Find
a( ) P ( A ) 1 mark
b( ) P ( B ) 3 marks
c( ) P ( A ^ B' ) 3 marks
a( ) P A( ).1
3P A( ).
b( ) P ( B ) =3
4
1
3
1
4
2
3
3
4
1
3
1
4
c( ) P ( A ) - P ( A ^ B ) =1
3
1
4
1
12
1
3
1
4
6
7/29/2019 Mathcad - CAPE - 2004 - Math Unit 2 - Paper 01