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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Quadratic Formula

quadratic formula

Discriminant and Nature of Roots

Discriminant, D = b2 – 4ac

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

no real roots

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

distinct roots

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

equal roots

Surd Rules

surd rules

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.

Sum and Difference of Cubes

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Partial Fractions

partial fractions

Indices Law

Laws for Same Base (For a > 0, and rational numbers m and n)

Laws for Same Index (For a, b > 0, and rational number m)

indices law for same index

Law for Zero Index (For a > 0)

indices law for zero power

Laws for Negative Index (For a, b > 0, and positive constants k, l, m and n)

indices law for negative index

Laws for Fractional Index (For a > 0, and positive constants m and n)

indices law for fraction index

Graphs of Exponential Functions

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

exponential graphs

 

Logarithm

Laws of Logarithms

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

Power Law

logarithm power law

Product Law:

logarithm product law

Quotient Law:

logarithm quotient law

Change−of−Base Law:

logarithm change base law

Graphs of Logarithmic Functions

math log graphs

 

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The Notation n!

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

Note that 0! = 1 (not 0).

n Choose r

Binomial expansion for (a + b)n

binomial expansion formula

Binomial expansion for (1 + b)n

The General Term (Tr+1)

general term for binomial expansion

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Length of Line Segment (Distance Between 2 Points)

Consider Figure 1 of the Cartesian plane shown below.

length between 2 points

Gradient and Equation of Line

Gradient, m

gradient formula

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.

ratio theorem

Midpoint of a Line Segment

midpoint formula

Gradients of Perpendicular Lines

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

 

Angle of Inclination

 

Area of Polygons (The “Shoelace” Method)

shoelace method formula for area of polygon

 

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.

bisector of 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.

linear law formula

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Trigonometric Ratios of Acute and Special Angles

The following table gives the trigonometric ratios of special angles.

special angles formula

Complementary Angles

Supplementary Angles

Trigonometric Ratios for Negative Angles

Radian Measure

Trigonometric Functions

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

Graphs of Trigonometric Functions

1) Graph of y = sin x

graph of sinx

2) Graph of y = cos x

graph of cosx

3) Graph of y = tan x

graph of tanx

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.

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  • Step-by-step explanations of all the must-know concepts.
  • Examples of top common exam questions!

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Cosecant, Secant and Cotangent Functions

Basic Trigonometric Identities

a-math trigo formulas

Addition Formulae

a-math addition formulas

Double Angle Formulae

a-math formula sheet double angle formula

R−Formulae

a-math formula sheet r-formula

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Rules and Techniques of Differentiation

1) Power Rule

2) Constant Multiple Rule

3) Sum and Difference Rule

4) Chain Rule

5) Product Rule

6) Quotient Rule

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.

 

Derivatives of Functions Involving ex and ln x