Skip to main content

EXAM TIPS

O levels

A levels:

Pure (P1):

It is recommended to have taken Add Maths in O levels for Pure Mathematics 1 in A levels  as there is a content overlap between the syllabi of both subjects, but it is not a requirement.

  • Quadratics: Remember the use of the discriminant and what it says about the function/which region of the graph of the pick. Know the difference between an equation and an inequality.
    • Use the finding the square method to find the coordinates of the vertex of a quadratic function (this allows for the most ease in sketching)
    • When simultaneous equations are given where one is quadratic and one is linear, skip over the elimination method and use substitution.
  • Functions: 
    • Know if a function is one-one by using the horizontal line test and seeing if the graph crosses the line at more than one point. 
    • Have a good understanding of the domain and range, especially for composite functions (eg: the range of the inner function should be equal or less than the domain of the outer function in order for it to exist)
    • Practicing transformations and their applications are also important, as well as finding inverses for various types of functions.
  • Coordinate Geometry (& Circles): 
    • Knowledge of the basic formulas (eg: length of a line, gradient) are extremely important. 
    • Recall the properties of circles learned in circle theorems in O level Math as they are often used in questions. 
    • Simultaneous equations pop up very frequently in questions where a diagram is presented to you (where different lines meet), so become better at solving them.
  • Circular Measure: 
    • This is one of the easier chapters as it has a heavy reliance on formulas (made easier by the fact that everything is in radians). The only strategy involved is when areas get involved, which is where it’s best to plan what to calculate and which areas to subtract before you start solving. 
  • Trigonometry: This topic can be difficult to some, as it includes many identities. Instead of having to refer to the MF-19 each time, it’s best to memorize them. This comes naturally with practice. 
    • Make sure that the correct quadrants are selected when doing ASTC (eg: sin(x) in the first and second quadrant, tan(x) in the first and third, and cos (x) in the first and fourth). 
    • Use identities to bring the entire question down to one trigonometric function so that it can be exposed as a disguised quadratic (or solved directly).
    • Be conscious of when the question involves an altered domain (eg: sin(2x+A)) and find the value of x by reversing the steps taken to turn x into 2x+A on your angle values). 
    • Also look out for a domain that spans from negative to positive (eg: -180<x<90) and go clockwise instead of counterclockwise when calculating ASTC values if answers are to be found in the negative domain.
  • Series:
    • The most difficult questions in this topic are ones that are based on real life examples. Always identify what goes where in the formula before plugging anything in, and whether the case is arithmetic or geometric). 
    • Otherwise heavily formula based. Although given in MF-19, recall the sum and nth term formulas beforehand.
  • Binomial Theorem: follows a strict base formula (given in MF-19) that should be followed correctly: (a+bx)n= ∑nr=0nCr an-r(bx)r
    • The x term and constant can be swapped when finding an expansion with the power of x in descending order. 
    • In questions where finding the term independent of x (or a coefficient of x in any power) is required, double check that the variable is to the correct power (i.e correct r value was used) before extracting the coefficient for the final answer
  • Differentiation
  • Integration

Statistics (S1):

  • Representation of Data
  • Permutations and Combinations
    • Always make cases no matter what.
    • Use permutations if order matters, (eg. people arranged in a line, arrangement of cars in a parking lot)
    • Use combination if order doesn’t matter (eg. people being selected for a committee, books to be picked for reading)
    • If you have trouble differentiating between when perms or combs are to be used, stick to permutations and use factorials in the denominator to eliminate the repeats/shuffling order (giving the same values as combinations)
  • Geometric Distribution
    • This distribution is to be used when the final value is unknown (eg. first 3 days rather than first 3 days of 8) – The number of trials aren’t fixed
    • The formulae for this distribution are given in the MF-19.
    • There is only 2 possible outcomes (eg. Eats breakfast/doesn’t eat breakfast) – This is considered success (denoted: p) and failure (denoted: q)
    • Example: X~Geo(0.25) – For P(X=3) → 0.75^2*0.25For P(X≤r) → 1-q^r (1-P(No success in ‘r’ trials))For P(X>r) → q^r  (1-P(X≤r) → 1-(1-q^r))E(X) is 1/p and Var(X) is q/(p^2)
  • Binomial Distribution: This distribution follows a set of conditions that are important to remember. 1. The number of trials (‘n’ value) must be fixed (eg: 10 calls made), unlike the geometric distribution. 2. There must be exactly 2 outcomes, typically a success and a failure (eg: call answered, call not answered) 
    • The trials must be independent of one another 4. The probability of success (p) must remain the same for each trial
    • Noted by B~(n,p) and the set formula (provided in MF-19) must be used but can be used several times to fit the requirements of the case in the question (eg: r from 0 to 3 must be plugged in separately for X<4) B~(x:n,p) = nCx p^x (q)^n-x where q= 1-p (probability of failure)
    • The mean is found by calculating np, and variance by npq (hence standard deviation= (npq)^0.5
  • Normal Distribution
    • It’s not necessary to refer from the table, you can use the calculator but you should use the table to check your answer.
    • Denoted by N~(μ,σ^2) where μ=the mean (np from Binomial) and σ for standard deviation (npq)^0.5 
    • When going from binomial to normal distribution, ensure that correction of continuation is used.
  • Probability
    • Probability is denoted as a fraction or decimal and must never be greater than 1.
    • Try to make a tree diagram if you have trouble understanding the probability question.
    • Probability diagram is made if there are two events (eg. two unbiased dice thrown)
    • For conditional probability, memorize the formula P(A|B) =P(A∩B)/P(B) where P(A∩B)  is the probability of both events (A & B) occurring without condition and P(B) is the given condition. (eg. probability of raining tomorrow given it will rain day after tomorrow; A=raining tomorrow, B=raining day after tomorrow)
  • Discrete Random variables
    • Always make a table and plug in values within the table.
    • Stay within the range when creating the table.
    • Sum of probabilities in the table is equal to 1. E(X) is provided in MF-19.

Mechanics (M1):

  • Forces & Equilibrium
  • Kinematics of motion in a straight line
  • Momentum
  • Newton’s Laws of motion
  • Energy, work & power

Further Math:

Knowledge of P1, P3 and M1 is assumed to be known.

It is recommended to take AS Math accelerated in First Year since the majority of the knowledge is O level Add math in AS Math.

Pure 1:

For this component, knowing Integration & Differentiation is most important. Do a lot of practice for it beforehand to be prepared.

  • Roots of Polynomials
    • There are two formulae which are not provided in MF-19 which are crucial to memorize. (Formulae for alpha squared and alpha cubed).
  • Rational Function Graphs
    • Remember to find all required information before sketching (intercepts, turning points, asymptotes, asymptote intersections).
    • If asked to differentiate, divide the function to form a new function and differentiate the new function.
    • If asked to sketch a modulo function, modulus of function is a reflection on the x-axis and modulus of x is a reflection on the y-axis.
  • Summation of Series
    • Although given in MF-19, you should memorize the three summation formulae.
    • Knowledge from O level Maths’ Series is optimal here when formulating new equations.
    • Remember you can not use the formulae if the initial term is not 1.
  • Matrices
    • There is a set way of finding the invariant line, and it is recommended you use that way only.
    • Often times Matrices questions aren’t difficult so you shouldn’t spend too much time on this chapter
  • Polar Coordinates
    • This is the hardest chapter in this component, and knowledge of integration and differentiation is optimal.
    • Use sine and cosine to compare cartesian to polar.
    • When (x,y) in cartesian, (sqrt(x^2+y^2), sin^-1(x/y)) in polar.
    • If the question is too difficult, do not ponder on it and come back to it during the exam.
    • Do a lot of A2 Math integration practice (P3).
  • Vectors
    • This chapter is heavily based on memorization. It is recommended you memorize everything in this chapter (eg. when to use cross/dot product)
    • It is unexpected for the exam to ask something outside the memorized.
    • Memorize all the formulae and just know when to apply each.
  • Proof by Induction
    • Every question requires you to attempt it the same way.
    • Always find the base case (n=0 or start of the range)
    • Always find the ‘k’th case (n=k)
    • Always find the (k+1)th case (n=k+1) using the ‘k’th case. This can be either by multiplying or differentiating depending on the question.
    • Once proven, write a final inductive statement (Since n=k proves n=k+1, through proof by induction, the case is proved). Be specific in your inductive statement.

Mechanics:

  • Projectile Motion
    • This is identical to projectile motion in Physics (if taken)
    • The SUVAT formulae will mostly be used.
    • Always resolve components of velocity from the beginning.
      Vertical velocity is 0 at the highest point.
      Memorize Range and Height formulae since they are not provided in
      MF-19.
      ALWAYS take g=10 NOT 9.81.
  • Forces & Equilibrium
    • This is the most difficult chapter of this component.
    • Always pick one side to use as your basis when identifying the center of mass.
    • For hollow surfaces, DO NOT find volumes for the center of mass, find Surface Areas.
    • Identify all the forces in the beginning and RESOLVE them all in ONE direction.
    • For moments, take ALL forces in a SINGLE direction into account.
    • Make two equations, one for moments and one for forces along the surface of the body (usually horizontal). Equate positive and negative in this case if the body is in equilibrium.
    • In cases of friction F<(Mew)R.
  • Circular Motion
    • There are 4 main cases for horizontal:
    • Inside a bead – Here, there will be a Normal Resistance Force in both directions.
    • On a table: Normal Resistance Force will only be upwards
    • Attached to a string through a ring onto a table: There will be tension upwards and if connected to another particle in circular motion, tensions are equal. There is also a Normal Resistance Force upwards at an acute angle to the string.
    • Attached to a string: There is only tension in the string.
    • For vertical circular motion, Weight will also play a role, so another force would be weight acting on the particle.
    • If attached to a string, there will also be tension.
    • If falling from a sphere, there will be a Normal Resistance Force.
    • Remember to equate Centripetal force (mv^2/r or mrw^2) in all cases.
    • If it goes into projectile motion, the SUVAT or projectile motion equation will be used with velocity as final velocity in circular motion.
  • Hooke’s Law
    • Always start with the Tension or Elastic Potential Energy formulae.
    • Identify which unknowns need to be found and which variables are present.
    • Vertical forces are always equal IF the body is in equilibrium.
    • If on a slope, RESOLVE forces and equate them accordingly.
  • Linear Motion under Variable Force
    • Always start with the F=ma equation being F=mdv/dt OR F=mvdv/dx depending on the scenario.
    • Knowledge of partial differentiation is very important.
    • Try to integrate with limits accordingly rather than trying to find the +c.
  • Momentum
    • For a perfectly elastic collision, equate momentums to each other.
    • For collision with a wall, use eusintheta against the wall and use the same rebound costheta along the wall where theta is the angle made by the particle and the wall when collided.
    • For oblique collisions, make diagrams for before and after collisions and RESOLVE velocities.
    • Vertical velocities are always the same, horizontal velocities will change.
    • Horizontal velocities can be found through simultaneous equations with equation of resolution of momentum (m1u1+m2u2=m1v1+m2v2) and the equation of coefficient of resolution (VA-VB=-e(UA-UB))