# Why Is A Level Maths Hard?

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# Why is A Level Maths So Hard?

Having now graduated from the top 20% of the O Level cohort, the syllabus is now made much tougher to further differentiate among all of you.

Here are some reasons why A Level Mathematics is so hard:

## Tedious Workings

As we progress from O Level to A Level, the workings will become longer and longer.

You will also find some expansions to be tedious, especially when cubic equations are given.

Evaluating the cross product of two vectors are cumbersome with much room for careless mistakes.

For interest-rate questions involve geometric progression, it can be extremely difficult to formulate the general expression.

With no indication as to you have obtained the correct expression, you might end up losing marks further down.

Answers used to be round numbers in O Level to make solving the questions easier.

In A Level, the final answers tend to be “more realistic” and you will see decimals, fractions appearing.

In reality, solutions are unlikely to look nice.

## Numerous Solving Techniques To Master

### Integration

Anyone who has done integration would tell you that there are so many techniques to approach the questions.

Depending on whether you have algebra, logarithms, exponential, trigonometry, and inverse trigonometry, the approach you adopt will be vastly different.

At times you will need to use partial fractions to divide and conquer the function.

Or you might need to use integration by parts.

And identifying which one to integrate, which one to differentiate is tough.

Not one size fits all

### Permutation and combinations

In the case of permutation and combinations, there are so many different scenarios.

Sometimes you will use the slotting method, sometimes complementary.

Unlike integration, the problem is all these approaches seem logical.

This means you might not get stuck while solving them and there is no way of knowing if you have attained the right answer.

## Curve Sketching

As A Level students, you are also expected to know the shapes of basic functions, such as polynomials logarithms, exponential, trigonometry.

Given a function, you are also expected to know how to draw the inverse function, f(-|x|), -f(-x) with asymptotes in mind

Sometimes without the aid of the graphing calculator.

## Need To Learn Manipulation

To derive the final answer, students are now exposed to the new technique of manipulation.

To manipulate in Maths is to create the desired term you wish to see for your solution.

For example, they can be used to replace long division, factorize functions, complete the square, or to prove identities.

## Abstract Concepts

### 3D vectors

The introduction of lines (and skew lines!) and planes in vectors have stumbled many.

For most students, 3D vectors are hard to visualize.

It is even harder to solve questions revolving intersections of planes, lines, or a combination of both.

Most students also do not appreciate the purpose of doing the cross product.

And questions like finding the foot of perpendicular tend to trip students

### Complex numbers

The idea that numbers can be imaginary is mindblowing in the first place.

### Volume finding

The fact that you can find the volume of a 3D object by integrating a function by pen and paper is fascinating.

But if you are the one doing it I doubt you will feel engrossing.

How much to rotate, by π or 2π also depends on whether your graph spans one half or both of the y-axis.

### Functions

For functions, it is difficult to determine the range of a function.

Students often learn through the hard way that they cannot simply substitute extreme x values to find the range of a function.

Also, determining whether a composite function exists or not requires an understanding of the connectivity of the substituent functions.

To determine a one-to-one function, do you use the horizontal line test or the vertical line test? Comment below.

## Not Knowing Where To Start

### Proving trigo identities

A common grievance students have is not knowing where and how to start.

Topics like these include proving trigonometric identities, where no formula seems appropriate to transform from one function to another, such as double-angle formula and R-formula.
Formula list such as this may only be helpful to a certain extent.

### Binomial within binomial

Questions such as drawing samples behaving a binomial distribution and concurrences within the samples have binomial probability density functions are complicated.

## Complex Terminology

Many terminologies appear in the big topic of statistics, such as sample mean, variance, probability density functions, cumulative functions, discrete random variables, continuous random variables, hypothesis, regression lines, etc.

Since different known conditions can affect which of the distributions can be employed, students have to appreciate the terminologies and remember (and probably understand) nuances like why sample variance formula not the same as population variance

### Venn diagrams

Venn diagrams can be annoying, especially with the deliberate incomplete values and you are asked to determine the maximum and the minimum values of a segment.

Especially where a subset can occur.

### Transformation

Graphs transformation is annoying.

To shift y=x graph towards the right by 1 unit you use y=x-1 yet to shift the same graph up by 1 unit you use y = x + 1.

Darkness descends upon you when you are asked to determine the series of transformations to attain the final form.

### P. S.

If you do want your parents to know your pain in understanding Maths, make sure they read this.