**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

## Discriminant and Nature of Roots

### Discriminant, D = b^{2} – 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

## 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 (a*x* + *b*) is a known factor of f(*x*), then f(- b/a) = 0,

and conversely

(ii) if f(- b/a) = 0, then a*x* + *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

## Indices Law

__Laws for Same Base (For __*a* > 0, and rational numbers *m *and *n*)

*a*> 0, and rational numbers

*m*and

*n*)

__Laws for Same Index (For __*a*, *b* > 0, and rational number *m*)

*a*,

*b*> 0, and rational number

*m*)

__Law for Zero Index (For ____a____ > 0)__

__a__

__Laws for Negative Index ____(For __*a*, *b *> 0, and positive constants *k*, *l*, *m *and *n*)

*a*,

*b*> 0, and positive constants

*k*,

*l*,

*m*and

*n*)

__Laws for Fractional Index ____(For __*a* > 0, and positive constants *m *and *n*)

*a*> 0, and positive constants

*m*and

*n*)

## Graphs of Exponential Functions

The graphs of *y = a ^{x}*, where

*a*> 0 and

*a*≠ 1, are shown below.

## Logarithm

## Laws of Logarithms

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

### Power Law

### Product Law:

### Quotient Law:

### Change−of−Base Law:

## 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).

## n Choose r

## Binomial expansion for (a + b)^{n}

## Binomial expansion for (1 + b)^{n}

## The General Term (Tr+1)

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## “Your Ultimate A-Math Revision Notes”

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

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

Consider Figure 1 of the Cartesian plane shown below.

## Gradient and Equation of Line

### Gradient, m

For a line with gradient *m* and passing through the point (x_{1}, y_{1}) , the equation of the line is given by:

y – y_{1 }= m(x – x_{1}).

**You may still use *y* =* mx* + *c* and substitute (x_{1}, y_{1}) 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

## Gradients of Perpendicular Lines

If two perpendicular lines L_{1} and L2 have gradients m_{1} and m2 respectively, then m_{1} × m2 = -1.

## Angle of Inclination

## Area of Polygons (The “Shoelace” Method)

## Equation of a Circle

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

(*x* – *a*)^{2} + (*y* – *b*)^{2} = *r*^{2}.

In **general form**, the equation of the circle is *x*^{2} + *y*^{2} + 2*gx* + 2*fy* + *c* = 0 with centre *C*(–*g*, –*f*) and radius, r = √(f^{2} + g^{2 }– 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.**

## 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*

*2) Graph of y = cos x*

*3) Graph of y = tan x*

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

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

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## “Your Ultimate A-Math Revision Notes”

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- 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.

## Cosecant, Secant and Cotangent Functions

## Basic Trigonometric Identities

## Addition Formulae

## Double Angle Formulae

## R−Formulae

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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.