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CAPE - Mathematics - Unit 2 - 2004Paper 01
Section A (Module 2.1)
1 In the diagram below (not drawn to scale) the line y = 3 cuts the curve y e2 xx
at the
point (a, b)
Calculate the values of a and b 4 marks
e2 x
3e2 x
x1
2ln 3.
1
2ln a
1
2ln 3.
1
2ln b = 3
2 Differentiate with respect to x
a( ) x ln 3 x2. x x 0 2 marks
b( ) y sin2
x cos. x.x x 3 marks
a( )1
3 x2
6 x( )2
x
1
3 x2
6 x( )x
b( ) 2 sin x. cos. x. cos x.( ). cos x. sin2 x. 2 sin x. cos2. x sin3 x2 sin x. cos. x. cos x.( ). cos x. sin2 x. x x x
1
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3 a( ) Find the gradient at the point (1, 1) on the curve 2 xy y2
3 02 xy y2
3 5 marks
b( ) Solve for x the equation e2 x
3 ex
4 0e2 x
3 ex
4 3 marks
a( ) 2 xdy
dx. 2 y 2 y
dy
dx. 02 x
dy
dx. 2 y 2 y
dy
dx.
dy
dx
2 y
2 x 2 y
dy
dx
y
x y
dy
dx
y
x y
dy
dx
y
x y
dy
dx1 1,
1
2
dy
dx
b( ) ex
1 ex
4 0ex
1 ex
4 x 2 ln 2.2 ln
4 a( ) Express in partial fractionsx 1
x x 2( ).5 marks
b( ) Hence find xx 1
x x 2( ).d x > 0 3 marks
a( ) x 1 A x 2( ). B x( ).x 2( ) x( )x B x A1
2B
1
2
1
2 x
1
2 x 2( )
b( ) x1
2 xd x
1
2 x 2( ).d
1
2ln x x 2( ).( ). Cx x 2( ).( ) Cx x
1
2ln x.
1
2ln x 2( ). ln K.
5 Find xx2
ln x. d 5 marks
Total 30 marks
1
3x
3ln x. x
1
3x
3 1
xd
1
3x
3ln x.
1
9x
3C
1
9x
3Cx x x
1
9x
33 ln x. 1( ) C
2
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Section B (Module 2.2)
6 Find the term independent of x in the expansion of 3 x1
2 x2
9
5 marks
9
r3 x( )
9 r 1
2 x2
r 9 r 2 r9 r r r = 39
33( )
6 1
2
315309
2
9
33( )
6 1
2
3
(7654.5)
7 The first four terms of an AP are 2, 5, (2x + y + 7) and (2x - 3y) respectively where
x and y are constants. Find the value of x and the value of y 8 marks
2x + y + 7 = 8 2x - 3y = 11
2 x 3 1 2 x( ) 11 02 x 3 1 2 x( ) 11 has solution(s) x7
4y
5
2
8 a( ) Find the sum to n terms of the geometric series
4 2 11
2... 5 marks
b( ) Deduce the sum to infinity of the series 2 marks
a( ) Sn
4 11
2
n
11
2
n
nS
n8 1
1
2nnn
b( ) S∞
∞n
8 11
2n
lim∞
yields S∞
8∞
3
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9 If 1 ax( )n
1 6 x 16 x2
...... 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
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Section C (Module 2.3)
11 A bag contains two red balls R1
and R2
one green ball G and two black balls B1
and B2
Randomly two balls are drawn together from the bag
a( ) Describe the sample space 2 marks
b( ) Determine the probability that
i( ) Both balls are the same colour 2 marks
ii( ) at least one ball is black 3 marks
a( ) S R1 R2 B1 B2,
R1 G,
R2 G,
B1 G,
B2 G,
R1 B1,
R1 R2 B1 B2,
R1 G,
R2 G,
B1 G,
B2 G,
R1 B1,
R2
B1
R1
B2
, R2
B2
,
b( ) i( ) P R1
R2
. or P. B1
B2
.1
5
1
5
1
5
1
5P R
1R
2. or P. B
1B
2.
2
25
ii( ) P(one black ball or two black balls)
B1 G or B2 G.
or B1 R1.
or B1 R2.
or B2 R1.
or B2 R2.
or B1 B2.
71
5
1
5
7
257
1
5
1
5
5
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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
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14 a( ) Which one of the following situations describes a mutually exclusive event?
i( ) Selecting either an even or a prime number from the set of realnumbers
ii( ) Selecting either a negative integer or a perfect square from the set of integers
iii( ) Selecting either a perfect square or an odd number from the set of numbers 1 to 100
1 mark
b( ) The probability that A hits a target is1
4and the probability that B hits the
same target is2
5The event that A hits the target is independent of the
event that B hits the target. What is the probability that both A and B hit the target
5 marks
a( ) ii( )
b( ) P ( A ^ B ) =1
4
2
5
1
10
1
4
2
5
15 A marksman shoots at a target and he either hits the target (H) or misses it (M). Theprobability of H is 0.4. He shoots at the target 3 times
Determine
a( ) the elements of the event A associated with the marksman hitting the targetexactly twice
2 marks
b( ) the probability that the marksman hits the target at least once 3 marks
Total 30 marks
a( ) HHM( ) HMH( ) MHH( )
b( ) 1 - P (MMM) = 1 0.6( )3
0.7841 0.6( )3
7