How Can I Pass the Year 12 IB/A Level Maths Exam?

Year 12 is a big year for many. It can be filled with anxieties, uncertainty and those dreaded end-of-year exams. But there are things you can do along the way to help relieve some of the end-of-year pressures and ace Year 12.

In year 12 maths, students start learning new concepts like logarithms, binomial theorems, vectors, etc. Further, calculus is introduced, which is completely new to a year 12 maths student. Developing the skills in the following topics would help them to excel in their future years and courses.

These main areas of knowledge as crucial when studying Year 12 Maths Methods:

• Basic algebra – the ability to simplify and factorise a variety of families of expressions, to know techniques for solving equations. Most importantly, the need to set out solutions in a logical manner and follow conventions of the course in your solutions.

• The ability to simplify expressions and solve equations that involve polynomial functions, rational functions, exponential and logarithmic functions, and circular functions. Also, the ability to apply differentiation and anti-differentiation to many of these functions.

• Drawing clear graphs and presenting workings and answers logically. Always include details like showing coordinates of end-points and axes intercepts with exact values over a correct domain. Refer to examples in your textbook for guidance on how to set out your solution.

Developing the skills in the following topics would help them to excel in their future years and courses.

Logarithms and logarithmic functions
This is a completely new topic introduced in year 12. A logarithm is another way of writing exponent. It is used to solve equations involving exponents that cannot be solved even using the laws of exponents. There are some laws of logarithms as well, which students have to memorize in order to solve the questions. Further, year 12 also has logarithmic functions, where the students can understand the graphical representation of logarithms and their corresponding outputs.

• Example: If log 5 = a then what is the value of log (1/125) in terms of a?

  Solution:

  Using the properties of logarithms,

  log (1/125) = log 1 - log 125 (Quotient Property)
  = 0 - log 125 (as log 1 = 0)
  = - log 53
  = - 3 log 5 (Power Property)
  = - 3a (as log 5 = a)

  Answer: log (1/125) = -3a.

Binomial theorem
Binomial theorem primarily helps to find the expanded value of the algebraic expression of the form (x + y)n. Finding the value of (x + y)2, (x + y)3, (a + b + c)2 is easy and can be obtained by algebraically multiplying the number of times based on the exponent value. But finding the expanded form of (x + y)17 or other such expressions with higher exponential values involves too much calculation. It can be made easier with the help of the binomial theorem.

• Example: What is the binomial expansion of (x2 + 1)5 using the binomial theorem?

  Solution:

  The following formula derived from the Binomial Theorem is helpful to find the expansion.

  (x + y)n = (x+y)n = nC0 xny0 + nC1 xn-1y1 + nC2 xn-2 y2 + ... + nCk xn-kyk +....+ nCn x0yn

  (x2 + 1)5 = 5C0 (x2)510 + 5C1 (x2)5-111 + 5C2 (x2)5-2 12 + 5C3 (x2)5-3 13 + 5C4 (x2)5-4 14 + 5C5 (x2)5-5 15

  = x10 + 5 x8 + 10 x6 + 10 x4 + 5 x2 + 1

  Answer: (x2 + 1)5 = x10 + 5 x8 + 10 x6 + 10 x4 + 5 x2 + 1

Calculus
Calculus mainly deals with the rate of change and area between curves. It starts with limits and extends to derivatives and integrals. Year 12 mainly focuses on limits and derivatives. Further, as part of applications of derivatives, students learn the rate of change, slopes of tangents and normals, increasing and decreasing functions, concavity, etc.

• Example: Let f(y) = y2 and g(y) = ey. Use the chain rule in calculus to calculate h′(y) where h(y) = f(g(y)).

  Solution:

  Given, f(y) = y2 and g(y) = ey. First derivative of above functions are f'(y) = 2y and g'(y) = ey

  To find: h′(y)

  Now, h(y) = f(g(y))

  By chain rule of differentiation,

  h'(y) = f'(g(y)) g'(y)
  h'(y) = f'(ey) ey

  By substituting the values.

  h'(y) = 2ey × ey
  or h'(y) = 2e2y

  Answer: ∴ h'(y) = 2e2y

Passing your Year 12 Maths Exam requires thorough preparation and dedicated study. By understanding the syllabus, creating a study plan, practising questions and mental maths, and making flashcards and diagrams, you can increase your chances of succeeding in this crucial examination. Remember to stay motivated and focused throughout the preparation process, and best of luck in your studies!

Here are some tips to help you get the best exam results possible!