mathcad - cape - 2006 - math unit 1 paper 02

7/21/2019 Mathcad - CAPE - 2006 - Math Unit 1 Paper 02

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CAPE - 2006Pure Mathematics Unit

1Paper 02

Section A - Module 1

1 a( ) Solve the simultaneous equations x2

xy 6x2

xy

and x - 3y + 1 = 0 8 marks

b( ) The roots of the equation x2

4 x 1 0x2

4 x 1 are and . Without solving the

equation

i( ) state the values of + and 2 marks

ii( ) find the value of 2 2 3 marks

iii( ) find the equation whose roots are 11

and 1

1

7 marks

1



2 a( ) Prove by Mathematical Induction that

1

n

r

r

=

n

2n 1( )

1

n

r

r

=

nn 10 marks

b( ) Express in terms of n and in the SIMPLESTform

i( )

1

2 n

r

r

=

2 marks

ii( )

n 1

2 n

r

r

=

4 marks

c( ) Find n if

n 1

2 n

r

r

=

100

n 1

2 n

r

r

=

4 marks

2



Section B - Module 2

3 a( ) i( ) Find the coordinates of the centre and radius of the circle

x2

y2

2 x 4 y 4x2

y2

2 x 4 y 4 marks

ii( ) By writing x + 1 = 3 sin show that the parametric equations of this circleare

x = -1 + 3 sin and y = 2 + 3 sin 5 marks

iii( ) Show that the x-coordinates of the points of intersection of this circle withthe line x + y = 1 are

x 13

22 x 1

3

22 4 marks

b( ) Find the general solutions of the equation cos = 2 sin2 1 7 marks

3



4 a( ) Given that 4 sin x - cos x = R sin (x - ) R > 0 0 < < /2

i( ) find the values of R and correct to one decimal place 7 marks

ii( ) hence find ONE value of x between 0 and 360 for which the curve y = 4sin x - cos x has a stationary point

2 marks

b( ) Let z1

2 3i z2

3 4i

i( ) Find in the form a + bi a, b

R

a( ) z1

z2

1 mark

b( ) z1

z2

3 marks

c( ) z1

z2

5 marks

ii( ) Find the quadratic equation whose roots are z1

and z2

2 marks

4



6 a( ) In the diagram below not drawn to scale the area S under the line y = x for

0 x 1

is divided into a set of n rectangular strips each of width 1/n units

y = x

y

x

0 1/n 2/n 3/n 4/n (n-1)/n n/n

i( ) Show that the area S is approximately1

n2

2

n2

3

n2

...n 1

n2

6 marks

ii( ) Given that

1

n 1

r

r

=

n

2n 1( )

1

n 1

r

r

=

n n show that S = 1

21 1

n2 marks

b( ) i( ) Show that for f (x) =2 x

x2

4

d

dxf x( ) 8 2 x

2

x2

42

d

dxf x( ) 4 marks

ii( ) Hence evaluate

0

1

x24 6 x

2

x2

42

d 3 marks

c( ) Find the value of u > 0 if

u

2 u

x1

x4

d 7

192

u

2 u

x1

x4

d 5 marks

End of Test

mathcad - cape - 2006 - math unit 1 paper 02

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