Complex numbers, Differentiation, Integration by parts & Differential equations.
1
Given z1 = 5 − 12i and z2 = 3 + 4i , calculate
(a) z1 + z2
(c) z2 × z2
( b) z1 − z2
(d)
2
Express z = 5 + 12i in polar form.
3
Differentiate the following with respect to x :
(a) y = sin −1 5 x
y=
x
(b) y = e 2 cos −1 2 x
z1
z2
(e) z1
(f)
 1 + cos x 
(c) tan −1 

 sin x 
2x
, find the rate of change of y with respect to x, when x = 2 .
x+2
4
Given
5
Find the constraint equation of the curve defined parametrically by:
1
x = 2+ ,
t
6
z1
t2 +1
y= 2
t −1
1
2
A curve is defined parametrically by x = 2t − 3 − , y = t − 1 − .
t
t
(a) Find the equations of the tangents to the curve at the points where it crosses the
x-axis.
(b) Find the coordinates of the point of intersection.
dy
= 3( y + 2) .
dx
7
Find the general solution of the differential equation: 2( x + 3)
8
(a) Use the technique of integrating by parts to find the integral
∫ xe
−x
dx .
(b) Hence find the particular solution to the differential equation:
ex
9
dy
= xy 2 , given that y = 1 when x = 0 .
dx
Write down a mathematical model in the form of a differential equation for each of the
following statements :
(a) The rate of change of displacement (s) with time (t) is directly proportional to the
displacement.
Unit 1: Mathematics 2
Advanced Higher
(b) There are 500 football stickers to collect to fill an album. The rate at which the
number of stickers, N, in the album increases is directly proportional to the number of
stickers still to collect.
10
dP
= kP (100 − P )
dt
where P is the percentage of the field affected in t days. When t = 0, P = 1.
When t = 5, P = 60.
(a) Express P in terms of t.
Mildew hits a crop of corn in a field. Its spread can be modelled by
(b) Estimate the time it will take for 80% of the crop to be affected.
Unit 1: Mathematics 2
Advanced Higher