THE UNIVERSITY OF NEW SOUTH WALES
SCHOOL OF MATHEMATICS AND STATISTICS
Semester 1 2018
MATI-11131
MATI-IEMATICS IA
(1) TIME ALLOWED - 2 hours
(2) TOTAL NUMBER OF QUESTIONS - 4
(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
MAYBE USED
(8) A SHORT TABLE OF INTEGRALS IS APPENDED TO THE PAPER
All answers must be written in ink. Except where they are expressly required pencils
may only be used for drawing, sketching or graphical work.
Semester 1 2018 MATH1131 Page 2
USE A SEPARATE BOOK CLEARLY MARKED QUESTION 1
1. i) Calculate each of the following limits, or explain why it doesn't exist.
3x2 + sinx3 a) lim ----
x➔oo x2 + 2x + 1 b) lim ex - l
x➔O X ex -2c) lim --
x -�1n2 X
ii) Find the following integrals.
a) 11 = j xcosxdx
b) I2 = J x(l: lnx) dx
iii) Consider the curve defined implicitly by
dy a) Calculate the value of dx at the pomt ( 2, 1).b) Hence write down a Cartesian equation of the tangent line at thepoint (2, 1).c) Express your tangent line from part (b) in parametric vector form.
iv) Let a be a positive real number. Write the complex number -a+ ai inpolar form.
v) a) Solve z6 + 1 = 0.b) Factorise z6 + 1 into linear factors with complex coefficients.c) Factorise z6 + 1 into linear or irreducible quadratic factors with real coefficients. d) Factorise z6 + 1 into irreducible factors with rational coefficients.
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Semester 1 2018 MATH1131 Page 3
USE A SEPARATE BOOK CLEARLY MARKED QUESTION 2
2. i) Let p : [O, oo) -+ IR be the function defined by
p(x) = ln ( x + �)
a) Show that p has at least one root in the interval [O, 1].
b) Show that p has exactly one root in the interval [O, 1].
c) Explain how your arguments in parts (a) and (b) can be extended to
determine the number of roots of p in the interval [ 0, oo).
ii) a) Write down the formula for sinh(x) and cosh(x) in terms of exponen­
tials.
b) Hence, prove that for n E Z and x E IR
( cosh(x) + sinh(x) r = cosh(nx) + sinh(nx)
iii) Sketch the graph of the polar curve
r = l + cos(30)
for O ::::; 0 < 21r.
iv) A function f : [1, 5] -+ IR has the following properties
il!I f has a global maximum at 3,
il!I f is continuous everywhere except 4,
• f has no global minimum.
Draw a sketch of the graph of a possible f.
(You do not need to give a formula for your function.)
v) Let
You are given that the following vectors form an orthonormal set.
(You do not need to prove this.)
a) Calculate v1 · w.
b) Express w as a linear combination of v1, v2 and v3.
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Semester 1 2018 MATH1131
vi) Consider the lines £1 and £2 in JR3 defined below.
a) Show that the lines £1 and £2 intersect.
Page 4
b) Explain the Maple code below and how it is used to find the shortest
distance from the point P(l, 2, 3) to the plane containing the lines £1
and £2.
> with(LinearAlgebra):
> AP : = <2,0,1> - <1,2,3>;
> v1
> v2
> n
> 1
[ (:]
: = <1,2,-1>;
[ il ]
<0,2,3>
[:]
: = CrossProduct(v1,v2);
[ �3]
: = abs(n.AP)/sqrt(n.n);
l := 10
,Jft
77
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Semester 1 2018 MATH1131 Page 5
USE A SEPARATE BOOK CLEARLY MARKED QUESTION 3
3. i) Consider the points A, B, C, D and E with coordinate vectors
The plane containing A, B, C and D can be expressed in parametric
vector form as
a) Show that ABCD is a parallelogram.
b) Describe the values of ,\1 and ,\2 in the parametric form of the plane
given above that correspond to the edge CD of the parallelogram
ABCD.
c) Compute the area of the parallelogram ABCD.
d) Compute the volume of the parallelepiped formed by the vectors .AE,
AD and AB.
e) Does the point E lie in the plane containing the parallelogram ABC D?
Give reasons for your answer.
ii) Suppose that z and w are complex numbers.
a) Show that (z + w)(z + w) is real.
b) What is the geometric relationship between z + w and z + w?
iii) Consider the following system of equations.
x+2y - z - 3
2x- ,\y -2z 0
x+3y +.\z 5
a) For which values of ,\ does the system have no solution, a unique
solution or infinitely many solutions.
b) For the value or values of ,\ for which the system has infinitely many
solutions, write down the solutions in vector parametric form.
iv) Given that the invertible n x n matrix A satisfies
A2 = 2A+I,
express the inverse of A in terms of A and I.
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Semester 1 2018 MATH1131 Page 6
v) Using the Maple code below, or otherwise, find the determinant of
> with(LinearAlgebra):
> B : = <<1,0,2,3>1<2,2,4,0>1<1,1,2,1>1<4,-1,-2,0>>;
1 2 1 4
0 2 1 -1
2 4 2 -2
3 0 1 0
> Row□peration(B,[3,1],-2);
1 2 1 4
0 2 1 -1
0 0 0 -10
3 0 1 0
> Row□peration(%,[4,1],-3);
1 2 1 4
0 2 1 -1
0 0 0 -10
0 -6 -2 -12
> Row□peration(%,[4,2] ,3);
1 2 1 4
0 2 1 -1
0 0 0 -10
0 0 1 -15
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Semester 1 2018 MATH1131 Page 7
USE A SEPARATE BOOK CLEARLY MARKED QUESTION 4
4, i) Let f : JR ➔ JR be the function defined by
x4
f(x) = 1 sin(t3 + l)dt.
0
(Do NOT try to evaluate this integral.)
a) Explain why f is an even function.
b) Find f'(x).
ii) Find all real values of a and b such that the function defined by,
f(x) = {x
2 + bx
2x
is differentiable for all JR.
iii) a) State the Mean Value Theorem.
b) Prove that, for all x 2 0,
iv) The function f is defined by
if X < a
if X 2 a
f: (0, 2) ➔ JR where f(x) = e-x(l - x).
a) Explain why f has an inverse g = 1-1.
b) Find the domain and range of g.
c) Evaluate g'(0).
v) Do the following improper integrals converge or diverge? Give reasons
for your answer.100 2 a)
2 4 dx0 X +
b) [
oo x4
dx )1 lnx
vi) Use the E-definition of the limit to show that
lim (2 + e-x(sin(x)+2) sin(x) cos(x)) = 2.x-too
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Semester 1 2018 MATH1131 Page 8
BLANK PAGE
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Semester 1 2018 MATH1131
BASIC INTEGRALS
j ; dx = ln Ix I + C = ln I kx I,
J eax dx = ¾eax + C
Jax dx = -1-ax + C, Ina a=/ lJ sin ax dx =-¾cos ax+ C
j cos ax dx = ¾ sin ax + C
J sec2 ax dx = ¾ tan ax + C
j cosec2ax dx = -¾ cot ax+ C
j tanaxdx =} ln I secaxl + C
j cot ax dx = � ln I sin ax I + C
C=lnk
j sec ax dx =} ln I sec ax+ tanaxJ + C
j sinh ax dx = } cosh ax + C
J coshaxdx = ¾ sinhax + C
j sech2ax dx = 1 tanhax + C
j cosech2ax dx = -¾ coth ax+ C
J_d_x_ = !tan-1 �+c a2 + x2 a a J dx 1 _1 x
2 2 = - tanh - + C,a -x a a 1 l X = - coth- - + C,a a
= ]_ ln I a+ x I + C,2a a-x J dx . _1 x C---;::::::::::::=�=Slll -+ ,Ja2 _ x2 a J dx . h-1 x C---;::::::::::::=� = sm - +
-./x2 + a2 a J dx _1 x ---;::::::::::::=� = cosh - + C,
-./x2 - a2 a
Jxl < a
JxJ >a> 0
x�a>O
END OF EXAMINATION
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