1. (a) A lamina is defined by the regions R1 and R2:
Using a well proportioned and neatly drawn graph of these
regions, identify the area corresponding to the lamina.
(b) Using integral calculus, calculate the area defined in part a) of
(c) Using integral calculus determine the volume obtained if the
area identified in part a) of this question is swept 360° around
the positive x-axis.
2. (a) Using the complex numbers, 21 = 3 - 4i, and to=2+3i,calcu-
i. 21 + 22
ii. 21 - 22
(b) Using Euler's formula, 20 = cos () + isin(8), derive expressions
in terms of complex exponentials for;
iii. tan (0)
iv. Ecos ()
(c) The results of an experiment allowed the following equations to
i. Write the system of equations in the matrix form, A.X = B,
and calculate the determinant of the A matrix.
ii. Calculate the matrix of co-factors of the A matrix.
iii. Calculate the inverse matrix for the A matrix, and demon-
strate that the answer is the inverse of A.
iv. Using the inverse matrix, A-! find the values of the
unknown values, a, 3,
3. (a) On a dual carriageway, the front edge of a car overtaking a truck
draws level with the front edge of the truck. Exactly at this time,
70m. ahead, a large tree falls across both carriageways.
The truck is travelling at a constant 90k-ph, and the car, a con-
The driver of the truck reacts quickly, and the truck's brakes are
fully applied 0. 5s after the tree falls, thereafter, giving a constant
deceleration of the truck of 0.6g.
The driver of the car was distracted, and the car's brakes were
not fully applied until 0.7s after the tree fell, thereafter, giving a
constant deceleration of the car of 0.8g.
Assume that the acceleration due to gravity, g, is 9.81m/s².
(i) Calculate the distance travelled by each vehicle, assuming
constant velocity, before their brakes were applied. This is
the so-called "thinking distance".
(ii) Calculate the total stopping distance for the truck (thinking
distance + stopping distance).
(iii) Calculate the velocity with which the car hits the tree.
(b) The following data were obtained from an experimental test. Find
the linear regression line of the data and determine the linearity
coefficient of the regression line.
(c) Find the general solution for the following first order ordinary
dy =2y'sin(x) =
4. (a) A water tank is shown in figure 1. The tank has diameter,
D = 2.4 m and a hole of diameter, d = 4 cm in the centre of
the bottom of the tank. The initial depth of water in the tank is,
no = 2 m. Assume that the water velocity obeys Torricelli's Law,
U = V 2gh, where g = 9.81 m/s2 (acceleration due to gravity).
Figure 1: Water Tank
Calculate the time for the tank to empty to one quarter its initial
(b) Find the particular solution for the following ordinary differential
equation which has the boundary condition, I = 0, when y = 3.
5. (a) The volume of a solid, V, bounded by the curve
2 = 4.2 + 3y - 6 between the limits T = 0 and x = 4cm, and
y = 0 and y = 6cm is given by;
V = lo 0 4 / 6 4r² + 3y - 6 dy dr
Evaluate the volume.
(b) (i) Find the general solution for the following ordinary differen-
= - 5
3y2 + 5y + 4
(ii) Find the particular solution to the above ordinary differen- -
tial equation with the initial conditions, when T = 0, y = 3.
6. An inductor, capacitor, resistor series circuit is shown in figure 2.
L represents the inductance (H), C, the capacitance (F), R, the re-
sistance (), i, the current (A), and V, the potential difference (V).
Assume that each component is "ideal". For example, that the in-
ductor and capacitor have no dissipation.
Figure 2: Inductor, Capacitor, Resistor, Series Circuit
(a) Derive the differential equation for the circuit in terms of the
charge, q. Note, q = J idt.
(b) If the inductance, L, is 0.02H, the resistance, R, is 2209, the
capacitance, C, 3x10-GF, and the applied voltage, V, 5e-300 V.
Find the General Solution for the circuit.
(c) Determine the Particular Solution if g(0) = 0, and =
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