Mathematics Revision Questions for the
University of Bristol School of Physics
Physics at University is more mathematical than at A level, which is why A
level maths, or the equivalent, is a requirement for our courses. Physics lecturers
will assume a familiarity with some basic mathematical tools and techniques. In
this document you will find a series of mathematical revision questions which
cover these techniques. Our hope is that you will work through this revision
material in the first few weeks of the course to ensure you are well prepared for
lectures.
If you find that you can’t do a question or if there is a particular area with
which you have difficulties, do not worry. We suggest you first look through your
A level notes but do not panic if you are completely stuck! Simply make a note
of the topic or question and ask your physics tutor in tutorial sessions. Another
possible source for more information may be the textbook ”Physics for Scientists
and Engineers” by Tipler and Mosca.
Students last year suggested it would be helpful to have these questions
before the start of term to give you an opportunity to refresh your memory before
lectures begin. Please note that there is no need to complete all the questions or
hand them in. They are for your benefit, so that you can understand what will
be expected of you, what you have forgotten and what you may need to learn.
1
Algebra
Revision Material
Quadratic equations of the form ax2 +bx+c = 0 where a, b and c are real valued
constants (in mathematical notation this written as a, b, c ∈ R) are solved by
the following equation,
√
−b ± b2 − 4ac
.
(1)
x=
2a
Polynomial long division allows for the solving of problems such as,
x3 + 3x2 + 3x + 1
= x2 + 2x + 1 ,
x+1
1
and can be solved in multiple ways. Two common schemes for solving such
expressions are,
1
or
x2 + 2x + 1
x+1
−1
x3 + 3x2 + 3x + 1
− x3 − x2
1
3
3
−1
−2
2
1
1
−1 .
0
2x2 + 3x
− 2x2 − 2x
x+1
−x−1
0
For partial fractions, there are a number of rules which are useful to know:
• For every linear factor such as (ax + b) in the denominator, there will be
a partial fraction of the form A/(ax + b).
• For every repeated factor such as (ax + b)2 in the denominator, there will
be two partial fractions: A/(ax + b) and B/(ax + b)2 . For higher powers
there will be correspondingly more terms.
• For quadratic factors in the denominator e.g.(ax2 + bx + c), there will be
a partial fraction of the form: (Ax + B)/(ax2 + bx + c).
The exponents of variables are combined as follows,
am an = am+n
am
= am−n .
an
Logarithms follow the below rules,
ax = n ⇔ x = loga n
n = aloga n ;
loga 1 = 0;
loga a = 1
m
loga (mn) = loga m + loga n;
loga
= loga m − loga n;
n
loga (mp ) = p loga m
2
Practice Problems
1.1. Solve the following quadratic equations by factorisation:
(a) x2 + 8x + 12 = 0
(g) 4x2 − 15x + 9 = 0
(b) x2 + x − 20 = 0
(h) 36x2 − 48x + 16 = 0
(c) 4x2 + 8x − 12 = 0
(i) 3(x2 + 2x) = 9
(d) x2 − x − 6 = 0
(j) x2 + 4x − 5 = 0
(e) 3x2 − 11x − 20 = 0
(k) 2x2 − 7x − 4 = 0
63
=2
(l) x −
x
(f) x + 3 = 2x2
1.2. Complete the square for the following quadratic equations i.e. rewrite them
in the form (x + a)2 + b = 0, and hence solve for x:
(a) x2 − 4x − 5 = 0
(e) 2x2 − 6x + 1 = 0
(b) x2 − 6x + 9 = 0
(f) x2 + 8x + 17 = 0
(c) x2 − 12x + 35 = 0
(g) 3x2 − 3x − 2 = 0
(d) x2 − 3x − 5 = 0
(h) 3x2 − 6x − 1 = 0
1.3. Solve the following quadratic equations using the quadratic formula given
in Equation (1):
(a) x2 = 8 − 3x
(e) 3x2 + 2x = 6
(b) (x − 2)(2x − 1) = 3
1 2 1
(f) x −
=
2
4
(g) 2x(x + 4) = 1
(c) (x + 2)2 − 3 = 0
4
(d) 2x + = 7
x
(h) 4x(x − 2) = −3
1.4. Solve the following cubic equations:
(a) x3 − 2x2 − 15x = 0.
(b) x3 − 4x2 + x + 6 = 0, given that x = −1 is a solution.
(c) x: x3 − 19x + 30 = 0, given x = 3 is a solution.
3
(d) x: x3 + x2 − 22x − 40 = 0, given x = −2 is a solution.
(e) x3 − 9x2 + 15x + 25 = 0, given x = 5 is a solution.
1.5. Solve the following equations:
(a) x1/3 − 3x−1/3 = 2.
(b) (2x2 − x)2 − 9(2x2 − x) + 18 = 0.
(c) 32x 2−x = (2.67)1/x .
(d) 32x − 3x+1 + 2 = 0.
(e) 52x = 7x+1 .
1.6. Express the following in partial fractions:
4
−4
2x + 1
(b)
(x − 1)(x + 2)2
5x3 + 2x2 + 5x
x4 − 1
7x + 2
(h)
(x + 2)2 (x − 2)
1
x2 + 5x + 6
x
(d)
(x + 1)(x − 1)2
10(x + 1)
(x + 3)(x2 + 1)
4x + 1
(j)
(x − 3)(x2 + x + 1)
(a)
(g)
x2
(i)
(c)
(e)
1
3
x −1
(k)
2x2 + 7
(x + 2)2 (x − 3)
x4 + 5x3 + 9x2 + 6x + 5
(l)
(x + 2)(x2 + 1)
x2 + 2x + 1
(f) 2
x +x−2
4
1.7. Simplify the following expressions
√
36x2 y 4
√
(e) 3 27x6 y 3
(x3 )2 y 4
(a)
x6 y
a2 b
(b)
abc4
(c) 5 × 43n+1 − 20 × 82n
(d)
1.8. Evaluate the following expressions:
(a)
103/2
10−1/2
(b) (161/4 )3
1.9. Solve the following expressions for x:
(a) 2x = 4
(c)
(b) 4x = 2
x
1
2
=
√
2
1.10. Questions on logarithms
(a) If loga q = 5 + loga b, and c = a4 , prove that q = abc.
(b) Simplify (i) aloga x ; (ii) a−2 loga x .
(c) Write down the values of log2 16 and log8 2.
(d) Given that logb a = c and logc b = a, prove that logc a = ac.
(e) If logx 10.24 = 2, find x.
(f) Without using a calculator, simplify and evaluate the following expression:
5
6
5
log2
+ log2
− log2
3
7
28
(g) Solve the following equation: loga (x2 + 3) − loga x = 2 loga 2.
5
2
Trigonometric Equations
Revision Material
Basic definitions:
tan θ =
sin θ
1
1
1
; cosec θ =
; sec θ =
; cot θ =
.
cos θ
sin θ
cos θ
tan θ
Pythagorean identities:
sin2 θ + cos2 θ = 1
sec2 θ = 1 + tan2 θ
cosec2 θ = 1 + cot2 θ
Addition formulae:
sin(A + B) = sin A cos B + cos A sin B
sin(A − B) = sin A cos B − cos A sin B
cos(A + B) = cos A cos B − sin A sin B
cos(A − B) = cos A cos B + sin A sin B
tan A + tan B
tan(A + B) =
1 − tan A tan B
tan A − tan B
tan(A − B) =
1 + tan A tan B
Double angle formulae:
sin 2A = 2 sin A cos A
cos 2A = cos2 A − sin2 A
cos 2A = 2 cos2 A − 1
cos 2A = 1 − 2 sin2 A
2 tan A
tan 2A =
1 − tan2 A
6
Sum and difference formulae:
A+B
A−B
= 2 sin
cos
2
2
A+B
A−B
= 2 cos
sin
2
2
A+B
A−B
= 2 cos
cos
2
2
A+B
A−B
= −2 sin
sin
2
2
sin A + sin B
sin A − sin B
cos A + cos B
cos A − cos B
Formulae using t = tan x2 :
2t
1 − t2
2t
sin x =
;
cos
x
=
;
tan
x
=
.
1 + t2
1 + t2
1 − t2
Transformation:
√
a cos x + b sin x = R cos(x − α), where R = a2 + b2 and tan α = b/a.
Practice Problems
2.1. Find all angles x in the range 0◦ ≤ x ≤ 360◦ which satisfy the following
equations:
(a) tan x = −0.4560
(c) cos 2x
= tan 155◦
3
= −0.5678
(b) sin 3x
2
= cot 108◦
(d) sin 3x
4
2.2. Find all values of x (0◦ ≤ x ≤ 360◦ ) which satisfy the following equations:
(a) 6 sin2 x − 5 cos x − 2 = 0
(b) 4 tan x − 2 cot x = 5cosec x
(d) cos 2x + 7 sin x + 3 = 0
(c) tan(45◦ +x)+cot(45◦ +x) = 4
(e) cos 3x − 3 cos x = cos 2x + 1
2.3. Using Trigonometric Formulae:
(a) Use addition formulae to show that cot(A − B) =
1 + cot A cot B
.
cot B − cot A
(b) Show that
sin(A+B+C) = cos A cos B cos C(tan A+tan B+tan C−tan A tan B tan C).
7
(c) Show that:
i. sin 3A = 3 sin A − 4 sin3 A;
ii. cos 3A = 4 cos3 A − 3 cos A.
(d) Show that sin x(cos 2x + cos 4x + cos 6x) = sin 3x cos 4x.
(e) Show that sin2 (A + B) − sin2 (A − B) = sin 2A sin 2B.
(f) Find the values of x between 0◦ and 180◦ that satisfy: cos x =
cos 2x + cos 4x.
(g) Without using a calculator, evaluate cos4 15◦ + sin4 15◦ .
2.4. Problems with tan x2 :
(a) If t = tan x2 , find the values of t that satisfy:
(a + 2) sin(x) + (2a − 1) cos x = 2a + 1
where a is a non-zero content.
Hence find two acute angles satisfying
√
the equation when a = 3.
√
(b) If tan x2 = cosec x − sin x, prove that tan2 x2 = −2 ± 5.
(c) If sec θ − tan θ = x, prove that tan 2θ = (1 − x)/(1 + x).
(d) Find
q
(1 + sin θ)(3 sin θ + 4 cos θ + 5) in terms of tan 2θ .
2.5. Transformations:
(a) Express cos x + sin x in the form R cos(x − α) given the values of R
and α. Hence find the maximum and minimum values of cos x+sin x.
(b) Find the values of x between 0◦ and 360◦ which satisfy the equations:
(i) 3 cos x + sin x = 1
(ii) 8 cos x + 9 sin x = 7.25
(iii) cos x + 7 sin x = 5 (iv) 7 cos x − 6 sin x = 2
(v) 5 sin x − 6 cos x = 4 (vi) 12 cos x − 5 sin x + 3 = 0
(c) Express y = W (sin α + µ cos α) in the form R cos(α −√
β), giving R
and tan β. Show that the maximum value of y is W 1 + µ2 and
that it occurs when tan α = 1/µ
8
2.6. Solve the following:
(a) Find a positive value of x satisfying the equation
tan−1 (2x) + tan−1 (3x) = 14 π.
√
(b) Solve the equation cos−1 (x 3) + 2 sin−1 x = 12 π.
3
Differentiation
Revision Material
For a general function y = f (x), the derivative of y with respect to (sometimes
abbreviated to w.r.t) x is denoted by:
dy
or y 0 or f 0 (x)
dx
and is defined by:
(
dy
f (x + δx) − f (x)
= lim
dx δx→0
δx
)
(2)
While this expression may generally be used to find any derivative from first
principles, it is more usual to rely on standard results, such as given in the table
below:
9
Function
Derivative
Comment
y = xn
dy
= nxn−1
dx
n may be positive,
negative or fractional.
y = ex
dy
= ex
dx
y = ln x
dy
1
=
dx
x
y = sin x
dy
= cos x
dx
y = cos x
dy
= − sin x
dx
y = tan x
dy
= sec2 x
dx
y = sin−1 x
dy
1
=√
dx
1 − x2
y = tan−1 x
1
dy
=
dx
1 + x2
y = f (x).g(x)
dy
= g(x)f 0 (x) + f (x)g 0 (x)
dx
The Product rule
dy
g(x)f 0 (x) − f (x)g 0 (x)
=
dx
g 2 (x)
The Quotient rule
dy
df
dg
=
×
dx
dg
dx
The Chain rule
y=
f (x)
g(x)
y = f [g(x)]
ln x ≡ loge x
(A function of a function)
10
Practice Questions
3.1. Differentiate w.r.t. x:
1
x2
1
(e) 4
x
(a) x3
(d)
(b) x2 + 4x + 5
(c) x4 − x2
3.2. Differentiate w.r.t. x:
1
(1 + x)
1
(b)
(1 − x)
(a)
(c)
1
(3x + 2)
3.3. Differentiate w.r.t. x:
(a) sin 3x
(d) sec x
(b) x + sin x
cos 5x
(c)
5
(e) tan 2x
3.4. Differentiate w.r.t. x:
(a) (1 − x)(2 − 3x + x2 )
5x3 − 3x
(b)
x2
(c)
(x + 1)2
x
3.5. Solve the following problems:
(a) If y + x = cos x, find
dy
.
dx
dy
= 0.
dx
w.r.t. x and evaluate the derivative when
(b) If y = 2x2 − 4x − 2, find the value of x for which
(c) Differentiate 3x2 +6+5x−2
x = 3.
dy
(d) If y(x + 1) = 4, find
.
dx
11
3.6. Differentiate w.r.t. x:
1
x2
(a) x5 + 5x4 + 10x3 + 8
(c) 3x2 −
(b) 3x5 − cos x + 2
(d) 2x 2 + 2x− 2 − 5
1
1
3.7. Differentiate w.r.t. x:
(a) (x + 2)(x2 + 4)
(e) (1 − x)3
(b) (1 + x2 )(1 − 3x2 )
(f) (3x + 4)3
(c) (x − 2)
(d) (x2 − 1)2 /x
2
2
(g) (3 + x)(4 − x)
3.8. Differentiate w.r.t. x:
(a) sin x cos x
(d) (x2 + 1) tan x
(e) sec x tan x
(b) x sin x + cos x
(f) cosec x cot x
1
(c) (x2 − 2) sin x + x cos x
2
(g) x sec x + cos x
3.9. Differentiate w.r.t. x:
2x
1 − x2
2 + x2
(b)
1−x
3x2
(c)
(x − 1)(x + 2)
(d) (3 − 2x2 )−1
(a)
(e)
x4
x4 + 4
3.10. Differentiate w.r.t. x:
sin x
2x
3 sin x
(b)
1 + cos x
5 + 3 sin x
(c)
3 + 5 sin x
2 + 7 cos x
3 + 5 cos x
sin x
(e)
1 + tan x
(a)
(d)
12
3.11. Differentiate w.r.t. x:
(h) cosec
(a) (2x − 3)4
√
(b) 2x − 3
√
x
1
(i) x + 1 −
x
2
1 + sin x
(j)
1 − sin2 x
(c) cos 5x
(d) sin(x2 )
(e) tan(2x + 5)
3
(k) sin3 x sin 3x
√
(l) 4x 1 + x2
(f) sec(x3 )
(g) sin 2x cos 2x
3.12. Differentiate w.r.t. x:
√
(c) x 1 − x2 + sin−1 x
(a) sin−1 2x
√
(b) 3 tan−1 x
(d) tan−1 (sin2 x)
3.13. Differentiate w.r.t. x:
sin x + cos x
sin x − cos x
(a) x(ln x − 1)
(e) ln
(b) ln(1/x)
(f) e−2x ln 3x
(c) ln(sec x)
(g) e2x ln(sec x)
2
(h) x2 ex tan−1 x
2x
(d)
ln x
3.14. Solve the following problems:
(a) If x = aθ2 and y = 2aθ, find dy/dx in terms of θ.
(b) If x = b(θ − sin θ) and y = b(1 − cos θ), show that
dy
dx
!2
=
!
cosec
2
θ
− 1.
2
(c) If x = 3t + t3 and y = 3 − t5/2 express dy/dx in terms of√t and prove
that, when d2 y/dx2 = 0, x has one of the values 0, ±6 3.
13
d2 y
d3 y
and
when (i) y = x5 + 2x3 + 4 and (ii) y = sin 4x.
dx2
dx3
(e) Find the maximum and minimum values of the function
2
(1 + 2x2 )e−x .
(d) Find
4
Integration
Revision Material
A brief list of standard integrals:
Z
Z
xn dx =
xn+1
+ C,
n+1
n 6= 1,
1
sin(ax + b) + C,
a
cos(ax + b) dx =
(ax + b)n+1
+ C,
(n + 1)a
Z
(ax + b)n dx =
Z
1
sin(ax + b) dx = − cos(ax + b) + C
a
Z
1
1
tan(ax + b) + C,
cosec2 (ax + b) dx = − cot(ax + b) + C
a
a
Z
sec2 (ax + b) dx =
Z
dx
1
−1 x
=
tan
+ C,
a2 + x 2
a
a
Z
dx
−1 x
√
=
sin
+C
a
a2 − x 2
Z
f 0 (x)
dx = ln f (x) + C,
f (x)
Z
eax dx =
eax
+C
a
Change of variable (integration by substitution):
Z
φ(x) dx =
Z
φ(x)
dx
du
du
Special cases:
Z
φ(ax + b) dx =
Z
n 6= 1
1Z
φ(u) du,
a
1Z
xφ(x ) dx =
φ(u) du,
2
2
u = ax + b
u = x2
Useful substitutions:
• If the integrand contains (a2 − x2 ), x = a sin u or x = a cos u may be
useful.
14
• If the integrand contains (a2 + x2 ), x = a tan u may be useful.
• If the integrand contains sinm x cosn x, where m and n are positive integers
and at least one of them is odd:
– If n is odd, sin x = u should be used;
– If m is odd, cos x = u should be used.
Integration by parts:
Zb
a
b
Z
dv
du
b
u
dx = [uv]a − v
dx
dx
dx
a
Practice Problems
4.1. Evaluate the following integrals:
(a)
(b)
Z
(4x2 − 2x + 9) dx
Z3 1
(c)
(d)
Z
Z1
3
2
+
−5
x3 x2
(f)
−1/2
dx
(g)
(2 sin x + 5cosec2 x) dx
(h)
√
(1 − x2 ) x dx
(e)
x4 + 1
dx
x2
(i)
3
4x + √
1 − x2
!
Z
3x4 + 5x2 + 2
dx
x2
Z3
x3 + 1
dx
x+1
0
0
Z
Z1/2
Z
x4 + x2 + 1
dx
x2 + x + 1
Z0
√
1 − 9x dx
dx
4.2. Evaluate the following integrals:
(a)
Z
sin(1 − x) dx
(c)
−1
(d)
(b)
π/2
Z
(e)
cos 5x dx
0
15
Z
(2x − 3)−3/2 dx
Z "
1
1
−
3
(x − 1)
(2 − x)3
#
dx
(f)
Z1
−1
(g)
Z1
3dx
dx
11 + 4x − 4x2
Z
2dx
(i)
dx
3x2 − 4x + 7
(h)
dx
dx
16x2 + 9
√
−1
dx
dx
16 − 9x2
Z
√
4.3. Integrate the following functions w.r.t. x using appropriate substitutions:
√
√
x
(a) x 1 − x2
(c)
x+1
i2
h
√
(b) x(1 + x2 )3/2
(d) x + 1 + x2
4.4. Integrate the following using the suggested substitutions:
(a) √
(b)
x2
, using a3 + x3 = u
3
3
a +x
x3
, using x4 = tan u
1 + x8
(c) tan x sec2 x, using tan x = u
x2
, using x3 = sin u
6
1−x
x
, using x2 = 3 tan u
(e) 4
x +9
x+1
(f) √ 2
,
x + 2x − 9
using x2 + 2x − 9 = u
(d) √
4.5. Integrate by substitution or otherwise:
(a) cos3 x
(f) sin3 2x
(b) sin5 x
(g)
(c) sin x cos x
2
2
(d) cos4 x sin3 x
sin3 x
cos2 x
cos x
(h) √
sin x
(e) sin3 x cos5 x
4.6. Evaluate the following integrals by parts, or otherwise:
(a)
Z
(b)
Z
cos 3x cos 2x dx
(c)
π/2
Z
sin 3x sin 2x dx
0
sin 3x cos x dx
16
(d)
(e)
Z
Zπ
cos 4x sin 2x dx
(j)
(f)
(g)
Z1
(k)
√
x 1 + x dx
(l)
√
x2 1 + x dx
(h)
(i)
Z
Z
Zπ
x2 sin x dx
(π − x) sin 3x dx
0
0
Z
x sin 2x dx
0
x cos x dx
0
Z
Zπ
sin−1 x dx
(m)
Z
(n)
Z
(o)
Z
x3 ln 5x dx
(e)
Z
x2 e2x dx
−1
x sin x
√
dx
1 − x2
x ln x dx
x ln(x + 4) dx
4.7. Evaluate the following integrals:
(a)
Z1
e−3x dx
0
(b)
Z1
x
(f)
−x 2
(e − e ) dx
Z1
2
x3 ex dx
0
0
(c)
Z2
(g)
(x − 1)ex dx
+ e−x
ex − e−x
Z " x
e
0
(d)
Z
x+2
e
(h)
dx
17
Z
#
dx
xe−1 + ex−1
dx
xe + e x