UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS
General Certificate of Education Advanced Level
9709/33
MATHEMATICS
Paper 3 Pure Mathematics 3 (P3)
May/June 2013
1 hour 45 minutes
*9836805872*
Additional Materials:
Answer Booklet/Paper
Graph Paper
List
CAMBRIDGE INTERNATIONAL EXAMINATIONS
GCE Advanced Subsidiary Level and GCE Advanced Level
MARK SCHEME for the May/June 2013 series
9709 MATHEMATICS
9709/33
Paper 3, maximum raw mark 75
This mark scheme is published as an aid to teachers and candidates, to
Page 4
1
Syllabus
9709
EITHER: State or imply non-modular inequality (4x + 3)2 > x2, or corresponding equation
or pair of equations 4x + 3 = x
Obtain a critical value, e.g. 1
3
Obtain a second critical value, e.g.
5
3
State final answer x < 1, x >
5
OR:
2
1
Solve the inequality !4x + 3! > ! x !.
2
It is given that ln"y + 1# ln y = 1 + 3 ln x. Express y in terms of x, in a form not involving logarithms.
[4]
3
Solve the equation tan 2x = 5 cot x, for 0" < x < 180".
4
[5]
(i) Express "#3# cos x + sin x in t
2
1
Solve the inequality !4x + 3! > ! x !.
2
It is given that ln"y + 1# ln y = 1 + 3 ln x. Express y in terms of x, in a form not involving logarithms.
[4]
3
Solve the equation tan 2x = 5 cot x, for 0" < x < 180".
4
[5]
(i) Express "#3# cos x + sin x in t
Page 4
1
Syllabus
9709
EITHER: State or imply non-modular inequality (4x + 3)2 > x2, or corresponding equation
or pair of equations 4x + 3 = x
Obtain a critical value, e.g. 1
3
Obtain a second critical value, e.g.
5
3
State final answer x < 1, x >
5
OR:
2
1
Solve the inequality !4x + 3! > ! x !.
2
It is given that ln"y + 1# ln y = 1 + 3 ln x. Express y in terms of x, in a form not involving logarithms.
[4]
3
Solve the equation tan 2x = 5 cot x, for 0" < x < 180".
4
[5]
(i) Express "#3# cos x + sin x in t
Page 4
1
Syllabus
9709
EITHER: State or imply non-modular inequality (4x + 3)2 > x2, or corresponding equation
or pair of equations 4x + 3 = x
Obtain a critical value, e.g. 1
3
Obtain a second critical value, e.g.
5
3
State final answer x < 1, x >
5
OR:
2
1
Solve the inequality !4x + 3! > ! x !.
2
It is given that ln"y + 1# ln y = 1 + 3 ln x. Express y in terms of x, in a form not involving logarithms.
[4]
3
Solve the equation tan 2x = 5 cot x, for 0" < x < 180".
4
[5]
(i) Express "#3# cos x + sin x in t
Page 6
8
Mark Scheme
GCE AS/A LEVEL May/June 2013
Syllabus
9709
Paper
33
(i) Separate variables correctly and integrate at least one side
M1
Obtain term ln t, or equivalent
B1
Obtain term of the form a ln(k x3)
M1
2
A1
Obtain term ln(k x 3 ) , or equivale
Page 5
5
6
Mark Scheme
GCE AS/A LEVEL May/June 2013
Syllabus
9709
1
(i) Substitute x = , or divide by (2x + 1), and obtain a correct equation, e.g. a 2b + 8 = 0
2
1
Substitute x = and equate to 1, or divide by (2x 1) and equate constant remainder to 1
2
O
Page 5
5
6
Mark Scheme
GCE AS/A LEVEL May/June 2013
Syllabus
9709
1
(i) Substitute x = , or divide by (2x + 1), and obtain a correct equation, e.g. a 2b + 8 = 0
2
1
Substitute x = and equate to 1, or divide by (2x 1) and equate constant remainder to 1
2
O
3
7
The complex number ! is defined by ! = a + ib, where a and b are real. The complex conjugate of !
is denoted by !*.
(i) Show that ! ! !2 = !* and that "! ki#* = !* + ki, where k is real.
[2]
In an Argand diagram a set of points representing complex nu
3
7
The complex number ! is defined by ! = a + ib, where a and b are real. The complex conjugate of !
is denoted by !*.
(i) Show that ! ! !2 = !* and that "! ki#* = !* + ki, where k is real.
[2]
In an Argand diagram a set of points representing complex nu
Page 4
1
Syllabus
9709
EITHER: State or imply non-modular inequality (4x + 3)2 > x2, or corresponding equation
or pair of equations 4x + 3 = x
Obtain a critical value, e.g. 1
3
Obtain a second critical value, e.g.
5
3
State final answer x < 1, x >
5
OR:
2
1
Solve the inequality !4x + 3! > ! x !.
2
It is given that ln"y + 1# ln y = 1 + 3 ln x. Express y in terms of x, in a form not involving logarithms.
[4]
3
Solve the equation tan 2x = 5 cot x, for 0" < x < 180".
4
[5]
(i) Express "#3# cos x + sin x in t
2
1
Solve the inequality !4x + 3! > ! x !.
2
It is given that ln"y + 1# ln y = 1 + 3 ln x. Express y in terms of x, in a form not involving logarithms.
[4]
3
Solve the equation tan 2x = 5 cot x, for 0" < x < 180".
4
[5]
(i) Express "#3# cos x + sin x in t
Page 5
5
6
Mark Scheme
GCE AS/A LEVEL May/June 2013
Syllabus
9709
1
(i) Substitute x = , or divide by (2x + 1), and obtain a correct equation, e.g. a 2b + 8 = 0
2
1
Substitute x = and equate to 1, or divide by (2x 1) and equate constant remainder to 1
2
O