# Your Complete A-Math Formula Sheet

We have compiled the complete list of must-know formulas inside this A-Math Formula Sheet so that you will be well prepared to tackle O-Level A-Math exams.

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## Discriminant and Nature of Roots

### Discriminant, D = b2 – 4ac

Case 1: When D < 0, there are no real roots.

Case 2: When D > 0, there will be two distinct roots for x.

Case 3: When D = 0, there will be equal roots

## Rationalisation of Surds

Case 1: Where the denominator is in the form √k, multiply the numerator and denominator by √k.

Case 2: Where the denominator is in the form a + b√k, multiply the numerator and denominator by a – b√k.

Note: a + b√k is the conjugate of a – b√k and vice versa.

Case 3: Where the denominator is in the form a√h + b√k , multiply the numerator and denominator by a√h – b√k .

Note: a√h + b√k is the conjugate of a√h – b√k and vice versa.

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## Remainder Theorem

When a polynomial f(x) is divided by a linear divisor (x + b), f(x) can be expressed in the following manner:

f(x) = (x + b) × Q(x) + R, where Q(x) is a polynomial of x and R is a constant remainder.

When x = −b,

f(-b) = (-b+b) × Q(-b) + R

= R

## Factor Theorem

(i)         if (ax + b) is a known factor of f(x), then f(- b/a) = 0,

and conversely

(ii)        if f(- b/a) = 0, then ax + b must be a factor of f(x).

**Note that both Remainder and Factor Theorems only work for linear divisors.

## “Your Ultimate A-Math Revision Notes”

Contains:

• More than 180 pages of content, carefully curated by our team of subject experts.
• Step-by-step explanations of all the must-know concepts.
• Examples of top common exam questions!

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## Graphs of Exponential Functions

The graphs of y = ax, where a > 0 and a ≠ 1, are shown below.

## Laws of Logarithms

If a, x, y are positive numbers and a ≠ 1, then

## Graphs of Logarithmic Functions

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## The Notation n!

n! = n × (n − 1) × (n – 2) × (n – 3) ×  × 3 × 2 × 1.

Note that 0! = 1 (not 0).

## “Your Ultimate A-Math Revision Notes”

Contains:

• More than 180 pages of content, carefully curated by our team of subject experts.
• Step-by-step explanations of all the must-know concepts.
• Examples of top common exam questions!

Click the button below to find out more.

## Length of Line Segment (Distance Between 2 Points)

Consider Figure 1 of the Cartesian plane shown below.

## Gradient and Equation of Line

For a line with gradient m and passing through the point (x1, y1) , the equation of the line is given by:

y – y1 = m(x – x1).

**You may still use y = mx + c and substitute (x1, y1) into the equation to find the value of c. But it may involve more steps.

## Parallel Lines and Collinear Points

When two lines are parallel, then the two lines must have the same gradient.

Conversely, when two lines have the same gradient, then the two lines must be parallel.

If two line segments have the same gradient and there is a common point between the two line segments, then the line segments must be collinear.

## Ratio Theorem

Ratio Theorem allows us to determine a point on a line segment that is divided in the ratio m : n.

## Midpoint of a Line Segment

If two perpendicular lines L1 and L2 have gradients m1 and m2  respectively, then m1 × m2 = -1.

## Equation of a Circle

In standard form, the equation of the circle with centre C(a, b) and radius r units is

(xa)2 + (yb)2 = r2.

In general form, the equation of the circle is x2 + y2 + 2gx + 2fy + c = 0 with centre C(–g, –f) and radius, r = √(f2 + g2 – c).

## Perpendicular Bisector of a Chord

Consider the circle with centre C and chord XZ in the diagram shown below. The centre of the circle must lie on the perpendicular bisector of the chord.

## Transforming Equations to Linear Form

A non−linear equation can be transformed into a linear equation of the form Y = mX + c, where X and Y are each functions of x and/or y, and m and c are constants.

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## Trigonometric Ratios of Acute and Special Angles

The following table gives the trigonometric ratios of special angles.

## Trigonometric Functions

**Tip: To remember which trigonometric ratios are positive, consider the following acronym: ASTC (Add Sugar To Coffee).

## Graphs of Trigonometric Functions

### 3) Graph of y = tan x

1Period of the trigonometric graph refers to the interval for 1 complete cycle or wave.

2The amplitude is the distance between the maximum value and the equilibrium.

## “Your Ultimate A-Math Revision Notes”

Contains:

• More than 180 pages of content, carefully curated by our team of subject experts.
• Step-by-step explanations of all the must-know concepts.
• Examples of top common exam questions!

Click the button below to find out more.

## R−Formulae

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## Connected Rates of Change

If two variables x and y are connected by the equation y = f(x), then

## Stationary Points and Their Nature

Consider the graph of y = f(x) as shown in the figure below.

Note that:

(i)          dy/dx = 0 at A, B and C.

(ii)        We call points A, B and C stationary points. Points A and B are also turning points.

(iii)       Point A is a maximum point.

(iv)       Point B is a minimum point.

(v)        Point C is a stationary point of inflexion since it is neither a maximum or minimum point.

## First Derivative Test

Use the table below to help organise the investigative facts.

## Second Derivative Test

**Note that:
The Second Derivative Test is inconclusive should the value of f″(x) becomes 0 or undefined. In such cases, we need to use the First Derivative Test to determine the nature of the stationary point.