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This is the Exam of Applied Mathematics which includes Numerical Slips, Particles Collide, Distance, Projected Vertically, Downwards, Particle Falls, Initial Velocity, Distance, Value etc. Key important points are: Average Speed, Straight Line, Decelerating Uniformly, Speed Time Graph, Uniform Acceleration, Motion, Average Speed, Distance, Maximum Speed, Decelerating Uniformly
Typology: Exams
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General Guidelines
1 Penalties of three types are applied to candidates' work as follows:
Slips - numerical slips S(-1)
Blunders - mathematical errors B(-3)
Misreading - if not serious M(-1)
Serious blunder or omission or misreading which oversimplifies:
Attempt marks are awarded as follows: 5 (att 2), 10 (att 3).
2 Mark all answers, including excess answers and repeated answers whether cancelled
or not, and award the marks for the best answers.
3 Mark scripts in red unless candidate uses red. If a candidate uses red, mark the script
in blue or black.
4 Number the grid on each script 1 to 9 in numerical order, not the order of answering.
5 Scrutinise all pages of the answer book.
6 The marking scheme shows one correct solution to each question. In many cases there
are other equally valid methods.
At a certain instant athlete A is x metres from the finishing line and is running with a
constant speed of 8 m/s. At this instant athlete B is 6 metres behind A and is running
with a constant speed of 10 m/s. B catches up with A at the finishing line, so that the race ends in a dead heat.
(i) Find the velocity of B relative to A.
(ii) Find the value of x.
(b) A ferry F is travelling due east with a constant
speed of 12 km/hr.
A boat P is travelling in the direction α degrees
east of north with a constant speed of 20 km/hr.
At noon P is 1.6 km due south of F and t minutes later P intercepts F.
(i) Find the velocity of P relative to F,
in terms of i
r , j
r and α.
(ii) Find the value of α, correct to the nearest degree.
(iii) Find the value of t.
( ) ( )
24 m
3 s
(ii) time
2 i
10 i 8 i
(a) (i) V V V
2 2
1
BA B A
x
s ut at
v
v v
( ) ( )
20cos
(iii)
sin 0.
(ii) 20sin 12
20 sin i 20cos j 12 i
(b) (i) VPF VP VF
t
t α
α
α
α
α α
v r v
12 km/hr
1.6 km
20 km/hr α
moving with a speed of 10 m/s
collides directly with a smooth
sphere Q, of mass 3 kg, moving in the same direction with a
speed of 5 m/s on a smooth
horizontal table.
The coefficient of restitution for the collision is e.
After the collision, sphere Q continues to travel in the same direction but with a
speed of 8 m/s.
(i) Find the speed of P after the collision.
(ii) Find the value of e.
(iii) Find the fraction of kinetic energy lost due to the collision.
(iv) Find the magnitude of the impulse imparted to each sphere.
(iv) Impulse 3 8 3 5
FractionofKElost
KElost 1 37. 5 126.
KEaftercollision 2 5.5 3 8
(iii) KEbeforecollision 2 10 3 5
(ii) NEL v v e u u
(i) PCM 210 35 2 38
2 2
(^21) 2
1
2 2
(^21) 2
1
1 2 1 2
1
1
e
e
v
( ) v
10 m/s 5 m/s
P Q
2 kg 3 kg
(−x, −3), (2, y), (1, 3) and (x, y), respectively.
The centre of gravity of the four particles is at the origin.
Find the value of x and the value of y.
(b) A uniform lamina opqab consists
of a rectangle opqa and an isosceles
triangle oab.
|oa| = 18 cm and |ab| = |ob| = 15 cm.
The rectangular section has sides of
length 2ℓ cm and 18 cm as shown.
The centre of gravity of the lamina opqab
is at c, the midpoint of [oa].
Taking o as the origin, find the value of ℓ.
Give your answer in the form a b, where a b, ∈N.
c.g.of 9
areaof 18 2 36
c.g. of 9 4
areaof 18 12 108
(b) 12 cm
(a) 0
2
1
l
l l
l
l l
opqa ,
opqa
oab ,
oab
bc
y
y y
x
x x
2ℓ
18 cm
p q
o a
b
c
string 50 cm in length.
The mass describes a horizontal circle with constant speed 3 m/s on a smooth horizontal table.
The centre of the circle is also on the table.
(i) Show on a diagram all the forces acting on the particle.
(ii) Find the tension in the string.
(b) A smooth particle, of mass 4 kg, describes a horizontal
circle of radius r cm on a smooth horizontal table with constant speed 1.2 m/s.
The particle is connected by means of a light inelastic
string to a fixed point o which is 40 cm vertically above the centre of the circle.
The length of the string is 50 cm.
(i) Find the value of r.
(ii) Find the tension in the string. (iii) Find the normal reaction between the particle and the table.
(a)
R
T
4g
2
2
r
mv T
r cm
40 cm 50 cm
o
4 kg
α
α
30 cm
(b) (i) 50 40
2 2
r = −
(ii)
N
T
4g
( ) sin 4
c
2
2
iii T α N g
r
mv Tos α
