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.
You can click the headers below to go to the formulas:
Exponential and Logarithm Functions
Before you read on, you might want to download this entire A-Math formula sheet in PDF format to print it out, or to read it later.
This will be delivered to your email inbox.
Free Trial for A-Math Classes!
If you need more help in A-Math, click here to register for a free trial in our A-Math live online classes >> https://gradesolution.com.sg/online-zoom-classes-upper-secondary/
Quadratic Formula
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
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.
Before you forget, you might want to download this entire A-Math formula sheet in PDF format to print it out, or to read it later.
This will be delivered to your email inbox.
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
You may also like…
“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.
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)
Law for Zero Index (For a > 0)
Laws for Negative Index (For a, b > 0, and positive constants k, l, m and n)
Laws for Fractional Index (For a > 0, and positive constants m and n)
Graphs of Exponential Functions
The graphs of y = ax, 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
Before you forget, you might want to download this entire A-Math formula sheet in PDF format to print it out, or to read it later.
This will be delivered to your email inbox.
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)
You may also like…
“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
Gradient, m
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
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)
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 = 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.
Before you forget, you might want to download this entire A-Math formula sheet in PDF format to print it out, or to read it later.
This will be delivered to your email inbox.
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
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.
You may also like…
“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.
Cosecant, Secant and Cotangent Functions
Basic Trigonometric Identities
Addition Formulae
Double Angle Formulae
R−Formulae
Before you forget, you might want to download this entire A-Math formula sheet in PDF format to print it out, or to read it later.
This will be delivered to your email inbox.
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
Derivatives of Basic Trigonometric Functions
Integration
Integration of Trigonometric Functions
Integration of Exponential Functions and 1/(ax + b)
Definite Integrals
Area Between a Curve and the x-axis
Area Between a Curve and the y-axis
Applications of Differentiation and Integration in Kinematics
You may also like…
“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.
Chords of a Circle
The figure below shows a circle with centre O. AB and CD are chords within the circle.
(a) If PM ⊥ AB and AM = BM, then the centre O lies on PM. (⊥ bisector of chord)
(b) If OM ⊥ AB, then AM = BM. ( ⊥ from centre bisects chord)
(c) If AB = CD, then OM = ON. (equal chords are equidistant from centre)
(d) If AM = BM, then OM ⊥ AB. ( bisector of chord passes through centre)
Tangents to a Circle
(a) If TAS is a tangent to the circle at A, then OA ⊥ TS . (tangent ⊥ radius)
(b) If TA and TB are tangents to the circle at A and B respectively, then
(i) TA = TB
(ii) ∠ATO = ∠BTO (tangents from external point)
The diagram below shows a circle. PQ is a straight line.
If PQ is a tangent to the circle at A, then
(a) ∠x = ∠y,
(b) ∠j = ∠k (alternate segment theorem)
Midpoint Theorem
If AM = MB and AN = NC, then
(a) MN is parallel to BC and
(b) MN = half of BC.
For more A-Math revision tips, check out our free math eBooks and test papers for secondary school students!
Last Minute Revision for O-Level Math?
Check out our exam guide on other topics!
Free Trial for A-Math Classes!
If you need more help in A-Math, click here to register for a free trial in our A-Math live online classes >> https://gradesolution.com.sg/online-zoom-classes-upper-secondary/
Before you go, you might want to download this entire A-Math formula sheet in PDF format to print it out, or to read it later.
This will be delivered to your email inbox.
To Your Child’s Success,
Mr Tan Kok Heng
NIE Trained
Math Specialist
Jimmy Maths and Grade Solution Learning Centre
Does your child need help with Mathematics? Find out more about our Math Tuition Class Free Trial.
1) Live Zoom Lessons at Grade Solution Learning Centre
At Grade Solution Learning Centre, we are a team of dedicated educators whose mission is to guide your child to academic success. Here are the services we provide:
– Live Zoom lessons
– EdaptIQ™, a smart learning platform that tracks your child’s progress, strengths and weaknesses through personalised digital worksheets.
– 24/7 Homework Helper Service
We provide all these services above at a very affordable monthly fee to allow as many students as possible to access such learning opportunities. We also offer a free trial class and consultation with our tutors, to help make sure we are the best fit for your child.
We specialise in English, Math, and Science subjects.
You can see our fees and schedules here >>
2) Pre-recorded Online courses on Jimmymaths.com
If you are looking for something that fits your budget, or prefer your child learn at his or her own pace, you can join our pre-recorded online Math courses.
Your child can:
– Learn from recorded videos
– Get access to lots of common exam questions to ensure sufficient practice
– Get unlimited support and homework help
You can see the available courses here >>