2014-15 Mathematics 1R Workshop 3 - Solutions
Rearrange to get cos(4x) =
3/2.
One solution is given by 4x = /6. Hence the general solution is
4x = /6 + 2k,
(k Z),
Hence
x = /24 + k/2.
More simply
x=
(1 + 12k)
.
24
The solutions x (0, ) are
11 13 23
,
,
,
Mathematics 1R
Sample Workshop Questions 4
1. Suppose that x and y satisfy the inequalities 1 x < 3 and 2 < y < 4. Find the
corresponding inequality satised by 2y 3x.
2. Find the reduced echelon form of the matrix
0 2 3 1
2 3 1 0 .
1 0 2 3
3. By consideri
Mathematics 1R
Sample Workshop Questions 2
1. Simplify the expressions
(a) 2 log(x/y) log(x3 y),
2. Find x such that
(b) log(xey2 log x ).
exp( 1 log x) = 2 x.
3
3. The increase in the biomass M with time of a colony of bacteria is modelled by
M = M0 ekt
1R
EXERCISES
WEEK 1
Hand-in Exercises
Algebra
H 1.1 By adaptation of the proof in the lecture notes that
irrational, prove that 3 is also an irrational number.
Solution Assume
2 is
3 Q with representation
3=
m
n
where m and n are natural numbers with no c
Chapter 4
Exercises on the properties of
matrices
4.1
Addition of matrices
Answer 4.1.1
Answer 4.1.2
5 5
A + B 2C = 0 10
7 2
= + = 2 + 2 (1) = 0.
Answer 4.1.3 The trace of A is a11 + a22 + a33 and the trace of B is b11 + b22 + b33 . The
diagonal entries
University of Glasgow
Department of Mathematics
Course 1R, Answers - Dierentiation
These are just answers and/or hints to proofs, not model solutions.
8.1 First principles In each case consider the dierence quotient
=
f (x + h) f (x)
,
h
simplify, and det
University of Glasgow
Department of Mathematics
Course 1R, Answers - Trigonometric functions
These are just answers and/or hints to proofs, not model solutions.
7.1 Radians and values of trigonometric funtions
1. (i) , (ii)
4
3
,
4
(iii) , (iv)
3
2
,
3
(v
Mathematics 1R Work - Week 5
Hand in Monday Week 6 :
Tutorial Exercises:
Further Exercises:
Question 2.3.7
2.3.7; 7.3.1; 7.3.2
2.3.5; 2.3.8; 2.3.11; 7.3.3; 7.3.4
Rest of 2.3 and 7.3
Show that for R,
Re
ei
i + ei
is independent of .
Answer
Calculation give
Mathematics 1R
Sample Workshop Questions 1
1. Find the greatest common divisor of (a) 27 and 126 and of (b) 839 and 345.
2. Find integers m and n satisfying the equality 39m + 182n = 13.
3. Find the quotient and the remainder polynomials when x4 + 2x3 + x
Mathematics 1R
Sample Workshop Questions 3
1. Find the general solution of the equation 2 cos 3x + 1 = 0 and identify those solutions
satisfying 0 < x < /2.
2. Use the identity cos 2 = 1 2 sin2 to prove that
sin
1
=
8
2
2
2.
3. Prove the identity
(cosec c
Chapter 1
Answers to Exercises on the
properties of numbers
1.1
Euclidean algorithm and gcd
Answer 1.1.1
m = 3.
First divide the entire equation by 2 to obtain 4m + 13n = 1. Take n = 1 and
Answer 1.1.2
m = 8 and n = 1.
Answer 1.1.3
Thee gcd is 19.
Answer
Chapter 2
Answers to Exercises on Complex
Numbers
2.1
Complex arithmetic
Answer 2.1.1
x = 2 and y = 4.
Answer 2.1.2
z + w = 1 + 5i,
z w = 5 + 3i,
zw = 10 5i,
2 11i
z
=
,
w
5
iz + w = 6 + 2i .
Answer 2.1.3
(i)
3 4i
5
1 3i
2
(ii)
Answer 2.1.4
(i) 1
Answer 2
1R
EXERCISES
WEEK 1
Hand-in Exercises
Algebra
H 1.1 By adaptation of the proof in the lecture notes that
irrational, prove that 3 is also an irrational number.
2 is
Calculus
H 1.2 Make two properly labeled sketches,
a) of a function and
b) of a non-functi
University of Glasgow
Department of Mathematics
Course 1R, Answers - Curve sketching
These are just answers and/or hints to proofs, not model solutions.
1.
(ii)
(i)
(iii)
(iv)
2. (i) Horizontal asymptote y = 0. y 0+ as x , y 0 as x . (ii)
Horizontal asymp
EXERCISES
1R
WEEK 2
Hand-in Exercises
Algebra
H 2.1 Given two polynomials f ( x ) and g( x ) of degrees m and n
respectively with m > n, describe how to nd a monic polynomial of
highest degree dividing f ( x ) and g( x ) quoting any relevant results
from
Mathematics 1R Work - Week 4
Hand in Monday Week 5 :
Tutorial Exercises:
Further Exercises:
Question 2.2.6
2.2.6 (i); 7.2.1; 7.2.3 (ii)
2.2.6 (ii); 2.2.9; 2.3.1; 2.3.2 (i); rest of 7.2.3; 7.2.4
Rest of 2.2 and 7.2
(i) By putting z = x + iy (x, y R), solve
Mathematics 1R Work - Week 3
Hand in Monday Week 4 :
Tutorial Exercises:
Further Exercises:
Question 2.1.3
2.1.3 (iii), (v); 6.4.4 (iv); 6.4.5
2.2.1; 6.4.8; 6.4.12
Rest of 2.1, 2.2.2, 2.2.3, Rest of 6.4 and 7.1
Express each of the following in the form x
University of Glasgow
Department of Mathematics
Course 1R, Answers - Applications of Dierentiation
These are just answers and/or hints to proofs, not model solutions.
9.1 Equations of tangents and normals
1. (i) f (x) = 4x 1 = 3 where x = 1.
The tangent y
University of Glasgow
Department of Mathematics
Course 1R, Answers - Elementary matters
These are just answers and/or hints to proofs, not model solutions.
6.1 Functions and graphs
1. (i) f (3) = 4,
(ii) f (x + 2) = 1 4x x2 .
2. g
2
x
g(x) =
2
x.
x
3. x
Mathematics 1R Work - Week 7
Hand in Monday Week 8 :
Tutorial Exercises:
Further Exercises:
3.3.1 (viii); 3.4.1 (iv); 8.1.1 (ii), (iv)
3.3.1 (vi); 3.5.1; 8.3.1(i)-(vi); 8.3.3 (i)-(iii)
Rest of 3.3. 3.4, 3.5 and 8.3
Question 3.3.1
(viii) Draw up a table of
Mathematics 1R Work - Week 9
Hand in Monday Week 10 :
Tutorial Exercises:
Further Exercises:
Question 4.3.4
4.3.4, 9.1.1 (ii)(iii)
4.4.4 ; 8.5.3
Rest of 4.3, 4.4, 8.5 and 9.1
It is given that
x1
3x1
x1
for some constant k, the system of equations
+ 3x2 +
Wednesday, 10th January, 2007
2.30 p.m. to 5.00p.m.
EXAMINATION FOR THE DEGREES OF
M.A. AND B.Sc.
MATHEMATICS 1R
Candidates must attempt the WHOLE of Section A, TWO questions from
Section B and TWO questions from Section C.
Section A Compulsory Questions
Wednesday, 11th January, 2006
2.30 p.m. to 5 p.m.
EXAMINATION FOR THE DEGREES OF
M.A. AND B.Sc.
MATHEMATICS 1R
Candidates must attempt the WHOLE of Section A, TWO questions from
Section B and TWO questions from Section C.
Section A Compulsory Questions
At
Wednesday, 22nd August, 2007
2.30 p.m. to 5.00 p.m.
EXAMINATION FOR THE DEGREES OF
M.A. AND B.Sc.
MATHEMATICS 1R
Candidates must attempt the WHOLE of Section A, TWO questions from
Section B and TWO questions from Section C.
Section A Compulsory Questions