Introduction and Overview

Question 1

Angle of depression

Answer 1

The angle between the horizontal and a direction below the horizontal.

Question 2

Angle of elevation

Answer 2

The angle between the horizontal and a direction above the horizontal.

Question 3

Arbitrary constant

Answer 3

Arbitrary constant
When finding an indefinite integral, you should add a constant (called the arbitrary constant or the constant of integration). This is needed as when differentiating, the derivative of the constant would be zero.

For example, ∫2xdx=x^2+c
.

Question 4

Area of a triangle

Answer 4

The area of any triangle ABC is given by the formula:

Area=1/2absinC

where a
is the length of the side opposite to angle A
, etc.

So to find the area of a triangle, you need to know the lengths of two of the sides, and the angle between them. (If you don’t have this information, you may be able to use the sine rule or the cosine rule first so that you have the necessary information to find the area).

Question 5

Area under a curve

Answer 5

The definite integral ∫ba f(x)dx
calculates the area between the curve y=f(x)
and the x
-axis.

If the curve is above the x
-axis, so that the value of y
is positive, the definite integral works out to be positive.

However, if the curve is below the x
-axis, so that y
is negative, the integral works out to be negative.

Question 6

Asymptote

Answer 6

A straight line which is approached by a curve, but the curve never reaches the line. Asymptotes are usually marked on graphs as dotted lines.

For example, the graph of y=1/x
has asymptotes which are the x
-axis and the y
-axis.

Question 7

Base

Answer 7

In an expression involving indices, the base is the number that is being raised to a power.

So, for example, in 84, 8 is the base and 4 is the index.

Question 8

Base of logarithms

Answer 8

The base of a logarithm is the same as the base of an index.

The statement a^x=b
can be written in logarithmic form as log(a)b=x
a
can be described as the base of the index or as the base of the logarithm.

Question 9

Bearing

Answer 9

The three figure bearing of point B from point A is the angle, measured clockwise from North, that the line AB makes with a line drawn in the North direction from A.

Question 10

Binomial coefficients

Answer 10

Binomial coefficients are the coefficients in the expansion of an expression of the form (a+b)^n
.

The binomial coefficient for the term in a^rb^(n−r)
is given by

nCr=n!/(n−r)!r!=n(n−1)(n−2)…(n−r+1)/1×2×…×r

Question 11

Binomial expansion

Answer 11

The expansion of an expression of the form (a+b)^n
. It is called binomial because it involves two variables (“bi” means two).

Question 12

Chord

Answer 12

A straight line joining two points on a curve or on the circumference of a circle.

Question 13

Circle properties

Answer 13

The angle in a semicircle is a right angle
The perpendicular from the centre of a circle to a chord bisects the chord
The tangent to a circle at a point is perpendicular to the radius through that point.

Question 14

Coefficient

Answer 14

For example, in the expression 2x^3−3x^2−x+4
, the coefficient of x^3
is 2
, the coefficient of x^2
is −3
, and the coefficient of x
is −1
. (The final 4
is referred to as the constant term).

Question 15

Completing the square

Answer 15

Any quadratic expression can be written in the form A(x+B)^2+C
, where A
, B
and C
are constants. This process is called completing the square, and it is particularly useful for finding the vertex of a quadratic graph. This method is also used to derive the quadratic formula, and it can be used to solve quadratic equations (it is usually easier to use the quadratic formula, but if you already have an expression in the completed square form, it can be easy to use this to solve a related quadratic equation).

Question 16

Component

Answer 16

Vectors can be written in terms of Cartesian components.

Vectors in two dimensions may be written

in component form, e.g. 3i−2j
, where i
and j
are unit vectors in the x
and y
directions respectively
in column vector form, e.g. (3 )
(−2)

Question 17

Conjecture

Answer 17

The dictionary definition of conjecture is “the formation of opinion on incomplete grounds”. A mathematical conjecture is a statement for which there is some evidence (there should certainly be no known counter-examples) but which has yet to be proved.

One of the most famous conjectures is Goldbach’s conjecture, which states that every even number is the sum of two prime numbers (including 1 as a prime in this context).

Question 18

Continuous graph

Answer 18

A graph which has no breaks is said to be continuous. A graph which has a vertical asymptote, such as y=1/x
, is not continuous.

Question 19

Converse

Answer 19

If a theorem states that A ⇒
B, then the converse of the theorem is that A ⇐
B (or B ⇒
A). The converse of a theorem may or may not be true. If it is, then you can write A ⇔
B.

For example: The angles of a triangle are equal ⇒ the triangle is equilateral.
The converse of this statement is: A triangle is equilateral ⇒ the angles of a triangle are equal.
The converse is also true, so you can write: The angles of a triangle are equal ⇔ the triangle is equilateral.
However, although the statement The shape ABCD is a square ⇒ the shape ABCD has four equal sides
is true, its converse:
is not true, since ABCD could be a rhombus. The shape ABCD has four equal sides ⇒ the shape ABCD is a square

Question 20

Coordinates

Answer 20

Coordinates
A means of describing a position relative to some fixed point (the origin).

In Cartesian coordinates, position is given in terms of two perpendicular directions, x
and y

Question 21

Cosine function

Answer 21

Cosine rule
For any triangle ABC

the cosine rule is: a^2=b^2+c^2−2bccosA
This can be written as cosA=b^2+c^2−a^2/2bc
The first form is used to find the third side given the lengths of two sides and the angle between them, and the second form is used for finding an angle given the lengths of all three sides.

Question 22

Counter-example

Answer 22

A counter-example is a particular example where a conjecture can be shown to be false.

For example, if the statement is made that “All odd numbers are prime”, then the statement can be disproved by giving just one example of an odd number which is not prime, such as 9.

Question 23

Cubic equation

Answer 23

A cubic equation has the form ax^3+bx^2+cx+d=0
, where a
, b
and c
are constants, and a≠0
.

If a cubic equation has a root which is an integer, this root can be found by trial, and the factor theorem gives a linear factor of the equation. The cubic equation can then be factorised into the linear factor and a quadratic factor. The quadratic factor can then be used to find the other two roots of the equation (if they exist).

Question 24

Cubic expression

Answer 24

A polynomial expression in which the highest term is a term in x^3
. A cubic expression may also contain terms in x^2
, x
and a constant term.