MATH1231 MATHEMATICS AND STATISTICS
MATHEMATICS AND STATISTICS
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SCHOOL OF MATHEMATICS AND STATISTICS
MATH1231
MATHEMATICS 1B
(1) TIME ALLOWED – Two (2) hours
(2) TOTAL NUMBER OF QUESTIONS – 3
(3) ANSWER ALL QUESTIONS
(4) THE QUESTIONS ARE OF EQUAL VALUE
(5) ANSWER EACH QUESTION IN A SEPARATE BOOK
(6) THIS PAPER MAY BE RETAINED BY THE CANDIDATE
(7) ONLY CALCULATORS WITH AN AFFIXED “UNSW APPROVED” STICKER
MAY BE USED
(8) A SHORT TABLE OF INTEGRALS and A STANDARD NORMAL TABLE
ARE APPENDED ON THE LAST PAGES
(9) TO OBTAIN FULL MARKS, YOUR ANSWERS MUST NOT ONLY BE
CORRECT, BUT ALSO ADEQUATELY EXPLAINED, CLEARLY WRIT-
TEN AND LOGICALLY SET OUT.
All answers must be written in ink. Except where they are expressly required pencils
may only be used for drawing, sketching or graphical work.
2019 Term 1 MATH1231 Page 2
Use a separate book clearly marked Question 1
1. a) Suppose that f is a function of two variables, and z = f(x, y).
If x = u cosh v and y = u sinh v, find an expression for
∂z
∂v
in terms of
u, v and the partial derivatives of f .
b) Consider the surface x
1
2 + y
1
2 + z
1
2 = 6 at the point P (1, 4, 9).
i) Show that n =
63
2
is a vector normal to the surface at P .
ii) Hence write down the Cartesian equation of the tangent plane to the
surface at P .
c) Find ∫
1
x(x2 + 1)
dx.
d) Giving brief reasons, state whether the following is true or false?
∞∑
n=1
1
ns
diverges if s =
2
3
.
e) Let T : R2 → R3 be a linear transformation with the property that
T
(
1
0
)
=
14
−1
and T (2
3
)
=
5−1
1
.
Find a general formula for T
(
x
y
)
.
f) Consider the matrix
A =
(
8 1
−4 3
)
.
You are given that
v1 =
(
1
−1
)
and v2 =
(
1
−4
)
are eigenvectors of A.
i) Find the eigenvalues of A corresponding to v1 and v2.
ii) Prove that every b in R2 can be written as a linear combination of
eigenvectors of A.
Please see over . . .
2019 Term 1 MATH1231 Page 3
g) Using the following Maple output, answer the questions below. Give
reasons.
> with(LinearAlgebra):
> A := <<1,-1,2,-3,2>|<1,1,3,2,-1>|<1,-3,3,1,-2>|<2,0,5,-1,1>|
<2,4,7,9,-5>>;
A :=
1 1 1 2 2
−1 1 −3 0 4
2 3 3 5 7
−3 2 1 −1 9
2 −1 −2 1 −5
> LinearSolve(A,<0,0,0,0,0>);
t5 − t4
−3 t5 − t4
0
t4
t5
> B := GaussianElimination();
B :=
1 1 1 2 2 1 0 0 0 0
0 2 −2 2 6 1 1 0 0 0
0 0 2 0 0 −5
2
−1
2
1 0 0
0 0 0 0 0
47
4
−1
4
−9
2
1 0
0 0 0 0 0 0 −21
47
− 2
47
37
47
1
i) Find a basis for ker(A).
ii) Write down the value of rank(A).
iii) Find a basis for R5 containing as many columns of A as possible.
Please see over . . .
2019 Term 1 MATH1231 Page 4
Use a separate book clearly marked Question 2
2. a) i) Find
∫ pi
2
0
sin3 x cos2 x dx.
ii) By making a trigonometric substitution and using the result of (i),
or otherwise, find
∫ 2
0
x3
√
4− x2 dx.
b) Let x(t) denote the number of people in a population who have been
infected by a certain virus, with time t measured in days. Let y(t) denote
the number of susceptible people in the population, that is, those who are
presently uninfected but who are likely to become infected in the future.
Suppose that the rate of change of x(t) and y(t) with respect to time are
described by
dx
dt
= −αx, (1)
dy
dt
= −βxy, (2)
where α, β are positive constants.
i) Write down the solution of (1) given that x = x0 at t = 0, where x0
is a positive constant.
ii) Use the result from (i) to solve (2) for y(t), given that y = y0 at
t = 0. Here y0 is a positive constant.
iii) Find the limiting value of y(t) as t→ +∞.
c) Show that the following ordinary differential equation is exact, and hence
find the general solution.
(x2y + y3)
dy
dx
+ xy2 + sinh(2x) = 0
d) Find the general solution to the following ordinary differential equation
y′′(x) + 4y′(x) + 4y(x) = 8x.
e) i) Find the Maclaurin series for f(x) = cos(2x) and state where the
series is valid.
ii) Hence, or otherwise, show that
cos2 x = 1 +
∞∑
n=1
(−1)n2
2n−1x2n
(2n)!
.
f) Find the radius of convergence of
∞∑
n=1
n!
nn
xn.
Please see over . . .
2019 Term 1 MATH1231 Page 5
Use a separate book clearly marked Question 3
3. a) Let V be a vector space with dimension n, and let S be a set of m
vectors from V . Giving reasons, state the relationship (if any) between
the numbers m and n if
i) S is linearly independent;
ii) S is linearly dependent;
iii) S spans V .
b) Let X be a continuous random variable with probability density function
f(x) =
2
9
if 0 ≤ x < 1
c if 1 ≤ x < 3
0 otherwise
where c is a constant.
i) Find the value of c.
ii) Find P (X < 3
2
).
iii) Find the mean of X.
iv) Find the variance of X.
c) The Australian No–Hopers Party (ANHP) claims that 6% of voters sup-
port them. In a survey, 500 people are asked about their voting intentions.
i) Assume that the ANHP’s claim is true (so 6% of voters support
them), and let X be the random variable giving the number of voters
in the survey who support the ANHP. Then X will be modelled by a
binomial distribution. Find the mean and variance of this binomial
distribution.
ii) When the survey results are published, it is found that 22 of the 500
people supported the ANHP. Use the normal approximation to the
binomial to estimate the probability that X is at most 22.
iii) Does the result of (ii) provide clear evidence that actual support for
the ANHP is less than the 6% they claimed?
Please see over . . .
2019 Term 1 MATH1231 Page 6
d) At No–Name University there are three terms per year. All students have
to take three courses in Applied University Administration: ADMN1111
in term 1, ADMN2222 in term 2 and ADMN3333 in term 3. Students
attempt the term 2 and term 3 courses regardless of whether they pass
or fail previous courses. The following facts have been determined.
• 70% of students pass ADMN1111.
• Of the students who pass ADMN1111, 60% pass ADMN2222, while
of those who fail ADMN1111, 40% pass ADMN2222.
• Of those who pass ADMN2222, 30% pass ADMN3333, regardless of
their performance in ADMN1111. While of those who fail ADMN2222,
10% pass ADMN3333, again regardless of their performance in
ADMN1111.
i) What proportion pass ADMN3333?
ii) Of those who pass ADMN3333, what proportion passed ADMN1111?
iii) Are the events “passing ADMN1111” and “passing ADMN2222” in-
dependent? Give a reason for your answer.
Please see over . . .
2019 Term 1 MATH1231 Page 7
Standard normal probabilities P (Z ≤ z)