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Mathematical Methods for Physics and Engineering: A Comprehensive Guide

K. F. Riley, M. P. Hobson, S. J. Bence

Chapter 1

Preliminary algebra - all with Video Answers

Educators

Chapter Questions

07:52

Problem 1

Continue the investigation of equation (1.7), namely
$$
g(x)=4 x^{3}+3 x^{2}-6 x-1
$$
as follows.
(a) Make a table of values of $g(x)$ for integer values of $x$ between $-2$ and $2 .$ Use it and the information derived in the text to draw a graph and so determine the roots of $g(x)=0$ as accurately as possible.
(b) Find one accurate root of $g(x)=0$ by inspection and hence determine precise values for the other two roots.
(c) Show that $f(x)=4 x^{3}+3 x^{2}-6 x-k=0$ has only one real root unless $-5 \leq k \leq \frac{7}{4}$
any real roots it has.

Jeff Vermeire
Jeff Vermeire
Numerade Educator
07:52

Problem 1

Continue the investigation of equation (1.7), namely
$$
g(x)=4 x^{3}+3 x^{2}-6 x-1
$$
as follows.
(a) Make a table of values of $g(x)$ for integer values of $x$ between $-2$ and 2 . Use it and the information derived in the text to draw a graph and so determine the roots of $g(x)=0$ as accurately as possible.
(b) Find one accurate root of $g(x)=0$ by inspection and hence determine precise values for the other two roots.
(c) Show that $f(x)=4 x^{3}+3 x^{2}-6 x-k=0$ has only one real root unless $-5 \leq k \leq \frac{7}{4}$

Jeff Vermeire
Jeff Vermeire
Numerade Educator
04:27

Problem 2

Determine how the number of real roots of the equation
$$
g(x)=4 x^{3}-17 x^{2}+10 x+k=0
$$
depends upon $k$. Are there any cases for which the equation has exactly two distinct real roots?
has.

Jeff Vermeire
Jeff Vermeire
Numerade Educator
02:00

Problem 3

Continue the analysis of the polynomial equation
$$
f(x)=x^{7}+5 x^{6}+x^{4}-x^{3}+x^{2}-2=0
$$
investigated in subsection 1.1.1, as follows.
(a) By writing the fifth-degree polynomial appearing in the expression for $f^{\prime}(x)$ in the form $7 x^{5}+30 x^{4}+a(x-b)^{2}+c$, show that there is in fact only one positive root of $f(x)=0$
(b) By evaluating $f(1), f(0)$ and $f(-1)$, and by inspecting the form of $f(x)$ for negative values of $x$, determine what you can about the positions of the real roots of $f(x)=0$

Khanh Ha
Khanh Ha
Numerade Educator
02:40

Problem 4

Given that $x=2$ is one root of
$$
g(x)=2 x^{4}+4 x^{3}-9 x^{2}-11 x-6=0
$$
use factorisation to determine how many real roots it has.

James Kiss
James Kiss
Numerade Educator
06:22

Problem 5

Construct the quadratic equations that have the following pairs of roots: (a) $-6,-3 ;$ (b) 0,$4 ;$ (c) 2,$2 ;$ (d) $3+2 i, 3-2 i$, where $i^{2}=-1 .$

Shelby Mohamed
Shelby Mohamed
Numerade Educator
06:12

Problem 6

Use the results of (i) equation (1.13), (ii) equation (1.12) and (iii) equation (1.14) to prove that if the roots of $3 x^{3}-x^{2}-10 x+8=0$ are $\alpha_{1}, \alpha_{2}$ and $\alpha_{3}$ then
(a) $\alpha_{1}^{-1}+\alpha_{2}^{-1}+\alpha_{3}^{-1}=5 / 4$,
(b) $\alpha_{1}^{2}+\alpha_{2}^{2}+\alpha_{3}^{2}=61 / 9$,
(c) $x_{1}^{3}+x_{2}^{3}+x_{3}^{3}=-125 / 27$
(d) Convince yourself that eliminating (say) $\alpha_{2}$ and $\alpha_{3}$ from (i), (ii) and (iii) does not give a simple explicit way of finding $\alpha_{1}$.
Trigonometric identities

Uma Kumari
Uma Kumari
Numerade Educator
02:11

Problem 7

Prove that
$$
\cos \frac{\pi}{12}=\frac{\sqrt{3}+1}{2 \sqrt{2}}
$$
by considering.
(a) the sum of the sines of $\pi / 3$ and $\pi / 6$,
(b) the sine of the sum of $\pi / 3$ and $\pi / 4$

Himanshu Kushwaha
Himanshu Kushwaha
Numerade Educator
02:11

Problem 8

(a) Use the fact that $\sin (\pi / 6)=1 / 2$ to prove that $\tan (\pi / 12)=2-\sqrt{3}$
(b) Use the result of (a) to show further that $\tan (\pi / 24)=q(2-q)$ where $q^{2}=2+\sqrt{3}$

Himanshu Kushwaha
Himanshu Kushwaha
Numerade Educator
05:43

Problem 9

Find the real solutions of
(a) $3 \sin \theta-4 \cos \theta=2$,
(b) $4 \sin \theta+3 \cos \theta=6$,
(c) $12 \sin \theta-5 \cos \theta=-6$

Jeff Vermeire
Jeff Vermeire
Numerade Educator
05:27

Problem 10

If $s=\sin (\pi / 8)$, prove that
$$
8 s^{4}-8 s^{2}+1=0
$$
and hence show that $s=[(2-\sqrt{2}) / 4]^{1 / 2}$

Jarrett Lancaster
Jarrett Lancaster
Numerade Educator
06:59

Problem 11

Find all the solutions of
$$
\sin \theta+\sin 4 \theta=\sin 2 \theta+\sin 3 \theta
$$
that lie in the range $-\pi<\theta \leq \pi .$ What is the multiplicity of the solution $\theta=0$ ?
Coordinate geometry

Jeff Vermeire
Jeff Vermeire
Numerade Educator
03:10

Problem 12

Obtain in the form $(1.38)$ the equations that describe the following:
(a) a circle of radius 5 with its centre at $(1,-1)$;
(b) the line $2 x+3 y+4=0$ and the line orthogonal to it which passes through $(1,1)$
(c) an ellipse of eccentricity $0.6$ with centre $(1,1)$ and its major axis of length 10 parallel to the $y$-axis.

Allison Knapp
Allison Knapp
Numerade Educator
04:36

Problem 13

Determine the forms of the conic sections described by the following equations:
(a) $x^{2}+y^{2}+6 x+8 y=0$;
(b) $9 x^{2}-4 y^{2}-54 x-16 y+29=0$
(c) $2 x^{2}+2 y^{2}+5 x y-4 x+y-6=0 ;$
(d) $x^{2}+y^{2}+2 x y-8 x+8 y=0$

Tamara Worner
Tamara Worner
Numerade Educator
16:10

Problem 14

For the ellipse
$$
\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1
$$
with eccentricity $e$, the two points $(-a e, 0)$ and $(a e, 0)$ are known as its foci. Show that the sum of the distances from any point on the ellipse to the foci is $2 a$. (The constancy of the sum of the distances from two fixed points can be used as an alternative defining property of an ellipse.).
Partial fractions

James Schroeder
James Schroeder
Numerade Educator
04:50

Problem 15

Resolve the following into partial fractions using the three methods given in section 1.4, verifying that the same decomposition is obtained by each method:
(a) $\frac{2 x+1}{x^{2}+3 x-10}$
(b) $\frac{4}{x^{2}-3 x}$
$1.16$ Express the following in partial fraction form:
(a) $\frac{2 x^{3}-5 x+1}{x^{2}-2 x-8}$
(b) $\frac{x^{2}+x-1}{x^{2}+x-2}$

Trinity Steen
Trinity Steen
Numerade Educator
05:24

Problem 16

Express the following in partial fraction form:
(a) $\frac{2 x^{3}-5 x+1}{x^{2}-2 x-8}$,
(b) $\frac{x^{2}+x-1}{x^{2}+x-2}$.

Trinity Steen
Trinity Steen
Numerade Educator
04:43

Problem 17

Rearrange the following functions in partial fraction form:
(a) $\frac{x-6}{x^{3}-x^{2}+4 x-4}$
(b) $\frac{x^{3}+3 x^{2}+x+19}{x^{4}+10 x^{2}+9}$

Sirat Shah
Sirat Shah
Numerade Educator
04:31

Problem 18

Resolve the following into partial fractions in such a way that $x$ does not appear in any numerator:
(a) $\frac{2 x^{2}+x+1}{(x-1)^{2}(x+3)}$
(b) $\frac{x^{2}-2}{x^{3}+8 x^{2}+16 x}$
(c) $\frac{x^{3}-x-1}{(x+3)^{3}(x+1)}$
Binomial expansion

Trinity Steen
Trinity Steen
Numerade Educator
03:22

Problem 19

Evaluate those of the following that are defined: (a) ${ }^{5} C_{3}$, (b) ${ }^{3} C_{5}$, (c) ${ }^{-5} C_{3}$, (d) ${ }^{-3} \mathrm{C}_{5-}$
$1.20$ Use a binomial expansion to evaluate $1 / \sqrt{4.2}$ to five places of decimals, and compare it with the accurate answer obtained using a calculator.

Jacob Shpiece
Jacob Shpiece
Numerade Educator
05:52

Problem 20

Use a binomial expansion to evaluate $1 / \sqrt{4.2}$ to five places of decimals, and compare it with the accurate answer obtained using a calculator.

Jacob Shpiece
Jacob Shpiece
Numerade Educator
07:24

Problem 21

Prove by induction that
$$
\sum_{r=1}^{n} r=\frac{1}{2} n(n+1) \quad \text { and } \quad \sum_{r=1}^{n} r^{3}=\frac{1}{4} n^{2}(n+1)^{2}
$$

Jeff Vermeire
Jeff Vermeire
Numerade Educator
07:24

Problem 22

Prove by induction that
$$
1+r+r^{2}+\cdots+r^{k}+\cdots+r^{n}=\frac{1-r^{n+1}}{1-r}
$$

Jeff Vermeire
Jeff Vermeire
Numerade Educator
03:47

Problem 23

Prove that $3^{2 n}+7$, where $n$ is a non-negative integer, is divisible by $8 .$

Jacob Shpiece
Jacob Shpiece
Numerade Educator
01:39

Problem 24

If a sequence of terms $u_{n}$ satisfies the recurrence relation $u_{n+1}=(1-x) u_{n}+n x$ with $u_{1}=0$ then show, using induction, that for $n \geq 1$
$$
u_{n}=\frac{1}{x}\left[n x-1+(1-x)^{n}\right]
$$

Angela Guo
Angela Guo
Numerade Educator
07:24

Problem 25

Prove by induction that
$$
\sum_{r=1}^{n} \frac{1}{2^{r}} \tan \left(\frac{\theta}{2^{r}}\right)=\frac{1}{2^{n}} \cot \left(\frac{\theta}{2^{n}}\right)-\cot \theta
$$

Jeff Vermeire
Jeff Vermeire
Numerade Educator
07:14

Problem 26

The quantities $a_{1}$ in this exercise are all positive real numbers.
(a) Show that
$$
a_{1} a_{2} \leq\left(\frac{a_{1}+a_{2}}{2}\right)^{2}
$$
(b) Hence prove by induction on $m$ that
$$
a_{1} a_{2} \cdots a_{p} \leq\left(\frac{a_{1}+a_{2}+\cdots+a_{p}}{p}\right)^{P}
$$
where $p=2^{\prime \prime \prime}$ with $m$ a positive integer. Note that each increase of $m$ by unity doubles the number of factors in the product.

Sriparna Bhattacharjee
Sriparna Bhattacharjee
Numerade Educator
08:14

Problem 27

Establish the values of $k$ for which the binomial coefficient ${ }^{P} C_{k}$ is divisible by $p$ when $p$ is a prime number. Use your result and the method of induction to prove that $n^{p}-n$ is divisible by $p$ for all integers $n$ and all prime numbers $p$. Deduce that $n^{5}-n$ is divisible by 30 for any integer $n$.

Heather Zimmers
Heather Zimmers
Numerade Educator
03:08

Problem 28

An arithmetic progression of integers $a_{n}$ is one in which $a_{n}=a_{0}+n d$, where $a_{0}$ and $d$ are integers and $n$ takes successive values $0,1,2, \ldots .$
(a) Show that if any one term of the progression is the cube of an integer then so are infinitely many others.
(b) Show that no cube of an integer can be expressed as $7 n+5$ for some positive integer $n$

AG
Ankit Gupta
Numerade Educator
04:11

Problem 29

Prove, by the method of contradiction, that the equation
$$
x^{n}+a_{w-1} x^{N-1}+\cdots+a_{1} x+a_{0}=0
$$
in which all the coefficients $a_{i}$ are integers, cannot have a rational root, unless that root is an integer. Deduce that any integral root must be a divisor of $a_{0}$ and hence find all rational roots of
(a) $x^{4}+6 x^{3}+4 x^{2}+5 x+4=0$
(b) $x^{4}+5 x^{3}+2 x^{2}-10 x+6=0$

Gregory Higby
Gregory Higby
Numerade Educator
05:07

Problem 30

Prove that the equation $a x^{2}+b x+c=0$, in which $a, b$ and $c$ are real and $a>0$, has two real distinct solutions IFF $b^{2}>4 a c$.

Shelby Mohamed
Shelby Mohamed
Numerade Educator
04:56

Problem 31

For the real variable $x$, show that a sufficient, but not necessary, condition for $f(x)=x(x+1)(2 x+1)$ to be divisible by 6 is that $x$ is an integer.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:45

Problem 32

Given that at least one of $a$ and $b$, and at least one of $c$ and $d$, are non-zero, show that $a d=b c$ is both a necessary and sufficient condition for the equations
$$
\begin{aligned}
&a x+b y=0 \\
&c x+d y=0
\end{aligned}
$$
to have a solution in which at least one of $x$ and $y$ is non-zero.

Shelby Mohamed
Shelby Mohamed
Numerade Educator
02:11

Problem 33

The coefficients $a_{l}$ in the polynomial $Q(x)=a_{4} x^{4}+a_{3} x^{3}+a_{2} x^{2}+a_{1} x$ are all integers. Show that $Q(n)$ is divisible by 24 for all integers $n \geq 0$ if and only if all the following conditions are satisfied:
(i) $2 a_{4}+a_{3}$ is divisible by 4 ;
(ii) $a_{4}+a_{2}$ is divisible by 12 ;
(iii) $a_{4}+a_{3}+a_{2}+a_{1}$ is divisible by 24 .

Andrija Isakov
Andrija Isakov
Numerade Educator