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A level math formula

Saturday 17 June 2023

                  Formulas Of  A level Math

In this article we are covering all the basic formulas of A level math.



  •              Mensuration
  •              Binomial series
  •              Logarithms and exponentials
  •              Trigonometric Identities
  •              Integration
  •              Differentiation  
  •              Arithmetic series
  •              Geometric Series                                                    


  •              Probability
  •              Standard deviation
  •              Statistical tables


  •              Kinematics
  •              Centre s of mass
  •               Motion in a circle


     1. Mensuration
                surface area of a sphere =4πr2

               Curved surface area of a cone =πrl 

     2.  Binomial Series

              The binomial series is   the power function f(x) = (1 + x)m. You can find the series expansion with a formula:
             (a + b)n = an + (nC1)an-1b + (nC2)an-2b2 + … + (nCn-1)abn-1 + bn


   3.   Differentiation
                    Differentiation allows us to find rates of change. It also allows us to find the rate of 

change of x with respect to y, which on a graph of y against x is the gradient of the curve.:
             a.  If y = xn, dy/dx = nxn-1


             b. If y = kxn, dy/dx = nkxn-1(where k is a constant- in other words a number)


  4. Logarithms and exponentials:
           Logarithms are another way of writing indices.
            If a = bc then c = logba

     The properties of indices can be used to show that the following rules for logarithms hold:


   a.      logax +   logay = loga(xy)

    b.      logax – logay = loga(x/y)

    c.      logaxn = nlogax

             e x ln a = ax
5. Integration:
        Integration is the reverse of differentiation.


              If y = 2x + 3, dy/dx = 2

              If y = 2x + 5, dy/dx = 2

                If y = 2x, dy/dx = 2      

6.    Arithmetic series:
         An arithmetic progression is a sequence where each term is a certain number larger than the 

previous term.  The terms in the sequence are said to increase by a common difference, d.

For example: 3, 5, 7, 9, 11, is an arithmetic progression where d = 2. The nth term of this sequence 

is 2n + 1 .

        In general, the nth term of an arithmetic progression, with first term a and common difference d, 

is: a + (n - 1)d . So for the sequence 3, 5, 7, 9, ... Un = 3 + 2(n - 1) = 2n + 1, which we already knew

      Arithmetic progression is given by
              un = a +(n-1) d
     sum to n terms of an arithmetic progression  is given by
             Sn = ½ n [ 2a + (n - 1)d ]



 7.  Geometric series:


             A geometric progression is a sequence where each term is r times larger than the previous 

term. r is known as the common ratio of the sequence. The nth term of a geometric progression, 

where a is the first term and r is the common ratio, is:



    For example, in the following geometric progression, the first term is 1, and the common ratio is 2:

                      1, 2, 4, 8, 16, ...

      The nth term is therefore 2n-1  


      The sum of the first n terms of a geometric progression is:


                                     a(1 - r)

                                        1 – r


        1. Probability:


           The probability of an event occurring is the chance or likelihood of it occurring.

The probability of an event A, written P(A), can be between zero and one, with P(A) = 1


Probability  =

   the number of successful outcomes of an experiment


the number of possible outcomes


 The probability of a certain event occurring, for example, can be represented by P(A). The 

probability of a different event occurring can be written P(B). Clearly, therefore, for two events A and B,

                              P(A) + P(B) - P(AÇB) = P(AÈB)


                             P(A ∪ B) = P(A) + P(B) − P(A ∩ B) 


                            P(A ∩ B) = P(A) P(B | A)


                            P(A | B) = P( )P( ) P( )P( ) P( )P( )

                      Bayes’ Theorem: P(Aj | B) = P( )P( ) P( )P( ) j j i i A BA Σ A BA

         2. Standard deviation:

            The formulae for the variance and standard deviation are given below. m means the mean 

of the data.

 Variance=s2=S (xr - m)2


              The standard deviation, s, is the square root of the variance.

        What the formula means:

          1.   xr - m  means take each value in turn and subtract the mean from each value.

          2.   (xr - m)2  means square each of the results obtained from step (1)

          3.  S(xr - m)2  means add up all of the results obtained from step (2).


          4.   Divide step (3) by n, which is the number of numbers

          5.  For the standard deviation, square root the answer to step (4)


  1. Kinematics:


       Motion in a circle Transverse velocity: v r = θ  

       Transverse acceleration: v r = θ  

        Radial acceleration: 2 2 v r r


    2. Centre of Mass:

          The centre of mass (/gravity) of a body or a system of particles is the resultant of the weights 

        of the individual particles making up the body or system.   

          Formula to calculate the position of the center of mass with respect to a reference         point is given as:

  • { X }_{ cm }\quad =\quad \frac { { m }_{ 1 }{ x }_{ 1 }\quad +\quad { m }_{ 2 }{ x }_{ 2 } }{ { m }_{ 1 }{ \quad +\quad m }_{ 2 } }


   3. Motion in a circle

        If a body is moving around a circle, even if it is moving at a constant speed it is accelerating. This is because it is changing direction (it isn't moving in a straight line).

The direction of this acceleration is towards the centre of the circle and the magnitude is given by:

  • a = v2/r

where v is the speed and r is the radius of the circle.

Using our formula above, this can also be written as:

  • a = r w2

Which of these you use will depend on whether you are dealing with speed or angular speed.

The acceleration occurs because there is a force acting:

Imagine that you are in a car going fast round a bend to the left. You will feel a force pulling you to one side (the left hand side). This is the force causing the acceleration. The force acts towards the centre of the circle.

Essential A-Level Math Formulas for Success


A-Level Mathematics is a challenging subject that requires a solid understanding of various mathematical concepts and the ability to apply them effectively. One key aspect of mastering A-Level Math is being familiar with essential formulas that are commonly used in different topics. In this article, we will explore a selection of fundamental formulas that every A-Level Math student should know. These formulas will not only assist you in solving problems efficiently but also serve as a foundation for more advanced mathematical concepts.

1. Quadratic Formula:

The quadratic formula is a fundamental tool for solving quadratic equations of the form ax^2 + bx + c = 0. It states that for any quadratic equation in the standard form, x = (-b ± √(b^2 - 4ac))/(2a). This formula enables us to find the roots of a quadratic equation accurately and efficiently.

2. Binomial Expansion Formula:

The binomial expansion formula allows us to expand expressions of the form (a + b)^n, where 'a' and 'b' are constants, and 'n' is a positive integer. The formula is given by (a + b)^n = C(n, 0)a^n b^0 + C(n, 1)a^(n-1) b^1 + C(n, 2)a^(n-2) b^2 + ... + C(n, r)a^(n-r) b^r + ... + C(n, n)a^0 b^n. Here, C(n, r) represents the binomial coefficient, which can be calculated using combinations.

3. Differentiation Formulas:

Differentiation is a crucial concept in calculus. Memorizing differentiation formulas can greatly simplify the process of finding derivatives. Some key differentiation formulas include:

d/dx (k) = 0 (derivative of a constant)

d/dx (x^n) = nx^(n-1) (power rule)

d/dx (sin(x)) = cos(x) (derivative of sine function)

d/dx (cos(x)) = -sin(x) (derivative of cosine function)

d/dx (e^x) = e^x (derivative of exponential function)

4.Integration Formulas:

Integration is the reverse process of differentiation and is also an essential concept in calculus. Here are a few integral formulas that are frequently used:

∫k dx = kx + C (integral of a constant)

∫x^n dx = (x^(n+1))/(n+1) + C (power rule for integration)

∫sin(x) dx = -cos(x) + C (integral of sine function)

∫cos(x) dx = sin(x) + C (integral of cosine function)

∫e^x dx = e^x + C (integral of exponential function)

5. Trigonometric Identities:

Trigonometry plays a significant role in A-Level Mathematics. Understanding and applying trigonometric identities can help simplify equations and solve problems effectively. Some key trigonometric identities include:

sin^2(x) + cos^2(x) = 1 (Pythagorean identity)

sin(2x) = 2sin(x)cos(x) (double-angle formula)

cos(2x) = cos^2(x) - sin^2(x) (double-angle formula)

sin(A ± B) = sin(A)cos(B) ± cos(A)sin(B) (sum and difference formulas)


Mastering A-Level Mathematics requires a thorough understanding of concepts and the ability to apply formulas effectively. The formulas mentioned in this article are


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