EMAT10100 Engineering Mathematics I 2022
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EMAT10100
Engineering Mathematics I
2022
SECTION A
A1. Find the polar form (modulus and argument) of the complex number 一16^2 一 16^2j .
Hence or otherwise find all complex roots to the equation
z5 + 16^2 + 16^2j = 0.
[You may leave your answers in polar form.]
Sketch the position of the roots in the Argand diagram (the complex plane).
(5 marks)
A2. (a) Find an equation in Cartesian form (that is, in terms of (x, y, z) coordinates) of the plane that passes through the point (x, y, z) = (1, 1, 1) and is normal to the vector v = 3i + 2j + k.
(1 mark)
(b) Find an equation in Cartesian form of the line that passes through the point (x, y, z) = (一1, 0, 1) and is in the direction of w = 2i + 3j 一 k.
(2 marks)
(c) Find the unique point of intersection between the plane in part (a) and the line in part (b).
(2 marks)
A3. Consider the following matrices
A = ╱ 一 、 B = .(╱) 丫(、) C = .(╱) 一 一 丫(、) D = ╱0(1) 0一2、
(a) State which of the following products are well defined (that is, the matrix dimen- sions are compatible)
A2 , ABT ACT , D2 BT , BT CT .
(2 marks)
(b) Compute each of the products that is well defined.
(3 marks)
A4. Solve the system of equations for the unknowns (x, y, z), written in matrix form as
一(一) 一 丫(、) 丫(、) = 丫(、)
leaving your answer in terms of the free parameter α .
(5 marks)
A5. (a) Use implicit differentiation to find in terms of x and y for the points (x, y) that lie on the curve C1 given by
(x2 + y2 一 1) 一 4x2y2 = 0 .
(2 marks)
(b) Use parametric differentiation to find an expression for in terms of t for points
that lie on a curve C2 described parametrically as
x(t) = cos3 (t), y(t) = sin3 (t) .
(2 marks)
(c) At which points t is the curve C2 not differentiable? What are the values of x and y at these points?
(1 mark)
A6. Use polynomial division and partial fractions to simplify
2x2 + 6x + 1
x2 一 x 一 6 .
Hence or otherwise calculate
x(6)x一+61 dx .
(5 marks)
A7. Given
u = ln ì and v = ^xy ,
use the chain rule for partial differentiation to find
∂F ∂x |
∂F and |
at the point (x, y) = (π/2, π/2) where
F (u, v) = u2 + v2 一 2u cos(v).
(5 marks)
A8. A biased coin is twice as likely to land ‘Heads’ as ‘Tails’ when tossed. A player is
invited to toss this biased coin three times. One point is awarded for each Head and three points for each Tail.
(a) Find the sample space and the probability distribution for the total points scored. (4 marks)
(b) Find the expected number of points scored.
(1 mark)
A9. Explain which of the standard probability distributions would best describe the output
of the following random process. Give the values of any parameters in the distribution.
(a) The number of false fire alarms to occur in Queen’s Building per year, if the prob- ability of false alarm on any given day is 0.01.
(2 marks)
(b) The number of left-handed students in a randomly-selected group of 5 students undertaking a group project, where the proportion of left-handed students in the entire student population is 0.25.
(1 mark)
(c) The total length of a student’s spotify playlist that contains 100 tracks, each of which is of average length 200 seconds with a standard deviation of 50 seconds.
(2 marks)
A10. The population p(t) of bacteria in a petri dish is estimated to obey the growth low
dp
where k > 0 and c > 0 are constants. Find a general expression in terms of k and c for the population p(t), given that the initial population satisfies
p(0) = 1
(5 marks)
A11. Find the solution x(t) to the differential equation
+ tx = 3t, x(0) = 1 .
dt
Sketch a graph of the solution for t > 0.
(5 marks)
A12. Consider the equation
cos2 (x) 一 x = 0.
(a) Show graphically that (1) has a unique solution x for 0 < x < 1.
(1)
(1 mark)
(b) Use the method of bisection to find an approximate solution x in the interval [0, 1] that is accurate to one decimal place.
(2 marks)
(c) Re-write (1) in the form x = f (x), and use the method of fixed point iteration to find a better solution that is accurate to three decimal places.
(2 marks)
SECTION B
B1. Consider the matrix
A = ╱ 1一2
(
1
一2
3
2一2、
2 丫 .
(a) Show carefully that the determinant of A is zero. Compute the rank of A.
(5 marks)
(b) Find all the eigenvalues of A.
(6 marks)
(c) Find eigenvectors corresponding to each of the two nonzero eigenvalues.
(6 marks)
(d) Describe carefully the effect of the linear transformation
x →| Ax
acting on an arbitrary position vector x.
(3 marks)
B2. Consider the function g : (0, &) →| (一&, &) defined by
g(x) = 3 + ln(x) 一 4x + x2 .
(a) (i) Find all the stationary points (also known as extrema or turning points) of g(x). Find the value of g(x) at each of these points, and classify each one as being either a maximum, minimum or inflection point.
(6 marks)
(ii) Show that x = 1 is a root of the function g .
(1 mark)
(iii) Find limá↓0 g(x) and limá↓太 g(x) .
(2 marks)
(iv) Using your answers to a(i)– (iii), sketch a graph of the function g(x), indicat-
ing any horizontal or vertical asymptotes, axis crossings and the location and value of stationary points.
(3 marks)
(v) How many roots does the function g(x) have?
(1 mark)
(vi) Use Newton’s method (also known as the Newton-Raphson method) to find
all the roots of g(x) other than x = 1, to two decimal places.
(4 marks)
(b) State whether the function g(x):
(i) is one-to-one (an injection) ;
(ii) is onto (a surjection) ;
(iii) has a well-defined derivative in the limit x → 0.
(3 marks)
B3. (a) The village of Sunnyridge has installed an array of 6 wind turbines. The probability of any day being insufficiently windy for the turbines to operate is found to be 0.3.
The probability on any day that any individual turbine needs to be shut down for maintenance is found to be 0.1.
[You may assume that all probabilities are mutually independent] A monitoring device is mounted upon one of the wind turbines.
(i) What is the probability that the monitored wind turbine operates on any given day?
(1 mark)
(ii) The monitored turbine is observed to be not operating. What is the proba-
bility that this turbine has been shut down for maintenance?
(2 marks)
(iii) What is the probability that more than 2 of the 6 turbines are shut down for
maintenance on any particular day?
(4 marks)
(iv) What is the probability that at least 4 turbines are operating on any particular
day?
(1 mark)
(b) Sunnyridge decide to additionally install a solar energy plant.
From prior weather records, it is estimated that the number of hours per day of sunshine at Sunnyridge S is a random variable that obeys a probability density
function
,. 0 ≤ s < 2
.
.
.
f母 (s) =... 1270(一) s 2 ≤ s < 12 .
.
.
.
.
(i) What is the probability of less than 4 hours of sunshine on any given day? (2 marks)
(ii) Find the mean and variance of the number of hours of daylight.
(6 marks) (iii) Find the probability of there being less than 1400 hours of sunshine in a year.
(4 marks)
B4. (a) Find the first, second and third derivative with respect to x of the function f (x) = 4x + x2 + arccos(x) defined for 0 ≤ x ≤ 1.
Hence show that the Maclaurin expansion of f (x) is
f (x) ≈ + 3x + x2 一 + h.o.t,
where h.o.t refers to higher-order terms of at least the fourth power of x.
(6 marks)
(b) Let Fγ (x) be the Maclaurin expansion of f (x) given in part (a) truncated after n terms [So that F1 = π/2, F2 = π/2 + 3x, etc.].
(i) Compute
Iγ = (0 1 Fγ (x)dx, for each n = 1, 2, 3, 4.
(4 marks) (ii) By comparing each integral Iγ with the true value of the integral
I = (0 1 f (x)dx = ,
find the ratio of the error between the (n + 1)st and the nth approximation to the integral for n = 1, 2, 3.
What evidence is there (if any) to suggest that the sequence Iγ converges linearly as n → &?
(4 marks)
(c) Another way to find an approximation for I is to compute the solution y(x) to the differential equation
dy
and let I = y(1).
(i) Use Euler’s method with stepsize h = 0.25 to compute an approximate solu- tion to this differential equation, in order to find y(1).
(4 marks)
(ii) What is the error in the value you obtain for y(1)? How could you obtain
a more accurate solution using a numerical method, without changing the stepsize?
(2 marks)
B5. (a) In a material testing device, the horizontal displacement x(t) of a particle of unit mass that has velocity one unit in the negative x direction, comes into into contact at time t = 0 with a resistive material. The resulting motion is found to satisfy the differential equation
+ 2 + 5x(t) = 0,
x(0) = 0, (0) = 一1.
(i) Find an expression for the displacement x(t) for t > 0.
(6 marks)
(ii) Use your answer to part (a)(i) to find the time t0 > 0 for the particle to
return to x = 0.
(1 mark)
(iii) What is the velocity at time t0 ?
(2 marks)
(b) Another material is placed into the testing device. The device is now shaken so that it undergoes a sinusoidal acceleration with period 2π . It is found that the subsequent motion of the particle is given by the differential equation
d2 x dx dx
dt2 dt dt
(i) Find an expression for the subsequent motion x(t).
(9 marks)
(ii) Use your answer to part (b)(i) to find the amplitude of the simple harmonic
motion that the particle settles to after a long time.
(2 marks)
2022-07-26