Mathematics 1R Work - Week 10
Hand in Monday Week 11 :
Tutorial Exercises:
Further Exercises:
4.5.1; 4.5.2; 9.2.1 (i)
4.5.3; 4.5.7; 9.2.2; 9.3.2; 9.5.2; 9.5.3
Rest of 4.5, 5.1, 9.29.5
Question 4.5.1 Let A, B, X denote n n matrices, A being nonsingular. Wh
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 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
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 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 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
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
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
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 - 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
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
?, ?th ?, 2014
2 p.m. to 4 p.m.
EXAMINATION FOR THE DEGREES OF
M.A. AND B.Sc.
Mathematics 1S
An electronic calculator may be used provided that it does not have
a facility for either textual storage or display, or for graphical display.
Candidates must at
EXERCISES
1S
WEEK 10
Feedback Exercises
Algebra
FB 10.1 Prove by induction that the following statement is true.
#
#
"
"
1 2n
1 2
n
for all n N.
, then A =
If A =
0 1
0 1
#
1 2n
.
Solution Let Pn be the statement
=
0 1
#
#
"
"
1
2
1
2
.
and RHS =
Prove fo
1S
EXERCISES
WEEK 6
Feedback Exercises
Algebra
FB 6.1 A plane P has equation x 5y + 2z = 3. Find the equation
of the plane which is parallel to P and passes through the point
(1, 1, 0).
Solution The parallel plane is of the form x 5y + 2z = d, as it too
h
1S
EXERCISES
WEEK 9
Feedback Exercises
Algebra
FB 9.1 Prove1 the statement
for all odd x, y Z,
your proof should start Let x, y be
odd integers. Then.
1
4( x2 + y2 + 2)00 .
Solution Let x and y be odd integers. Then2 x = 2k + 1 and y =
2l + 1 for some k,
1S
EXERCISES
WEEK 7
Feedback Exercises
Algebra
FB 7.1 Let L be the line joining the points A(3, 9, 5) and B(6, 3, 2).
Find and clearly state1
a) a direction vector for L,
for example, write in sentences & ensure that any variables introduced are
properly
EXERCISES
1S
WEEK 5
Feedback Exercises
Algebra
FB 5.1 Use complex numbers to express sin6 in the form
a cos 6 + b cos 4 + c cos 2 + d
for some real numbers a, b, c and d.
Solution Let z = ei = cos + i sin . Then
and
z+
1
= 2 cos ,
z
z
1
= ei = cos i sin
EXERCISES
1S
WEEK 8
Feedback Exercises
Algebra
FB 8.1 Show that the line L with equations
x2
y+3
z1
=
=
1
2
1
lies on the plane P with equation 3x + y z = 2 and find
a) the equation of the plane perpendicular to P that contains L
b) equations for the line
Tuesday, 1st June, 2004
2.30 p.m. to 5.00 p.m.
EXAMINATION FOR THE DEGREES OF
M.A. AND B.Sc.
MATHEMATICS 1S
Candidates must attempt the WHOLE of Section A, TWO questions from
Section B and TWO questions from Section C.
Section A
Attempt BOTH questions fro
Monday, 9th June, 2003
2.30 p.m. to 5.00 p.m.
EXAMINATION FOR THE DEGREES OF
M.A. AND B.Sc.
MATHEMATICS 1S
Candidates must attempt the WHOLE of Section A, TWO questions from
Section B and TWO questions from Section C.
Section A
Attempt BOTH questions from
1S
EXERCISES
WEEK 6
Feedback Exercises
Algebra
FB 6.1 A plane P has equation x 5y + 2z = 3. Find the equation
of the plane which is parallel to P and passes through the point
(1, 1, 0).
Solution The parallel plane is of the form x 5y + 2z = d, as it too
h
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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