1. Crash Maths Practise Paper A
Helpful Hints

2. Helpful Questions to ask yourself at the start of each question
What is the main topic here? e.g. differentiation, trigonometry
What are the key words associated with this topic?
e.g. differentiation = gradient, stationary points, normal, tangents etc
These may be a good starting point. You my find it useful to write down the main topic and the key words on your question paper around the question.

3. General 1A 1B 1C Q1 How can you re-write √x?
Now re-write y so that each term is in terms of powers
Remember to show your un-simplified working out here. This will gain you marks
Write out your y = … again so you can see what you are doing. Remember, you are now differentiating
Write out your answer to part (b) again so you can see what you are dealing with
General 1A 1B 1C

4. General 2A 2B 2C Q2 What is the name for the form: a(x + b)² called?
Is this a straightforward complete the square? Remember, the 4 needs dealing with.
You may need to factorise 4x² - 5x by 4
Remember you are expected to use part (a) in part (b). You are told to make f(x) 0, so make your part (a) answer = 0 and solve
Compare 4(x – 1)² - 5(x - 1) = 6 to 4x² - 5x = 6
What do you notice? X has been replaced for x – 1.
Use your part (b) solutions here e.g. if x = 2, then x – 1 = 2 2A 2B 2C

5. Q3 3A
When dealing with powers, what must all of the bases be if we are going to deal with them?
Remember (2³)² = 2³ x 2³ = 26
Re-write all of the numbers in the same base
What happens with powers when you multiply them? Re-write as an equation without the bases
Make y the subject
Use your part (a) to help in part (b)
Sub your ‘y’ into 7x² + 42y = 130 3B

6. Q4 4
First principles is in your formula booklet. You just need to remember what each bit stands for
Since you haven’t been given an actual term to use but have been told for a constant function, you need to state algebraically a proof
E.g. Let f(x) = a, a ∈ R (simply writing this gained 1 mark)

7. Q5 5A
The question states use long division so you must do this to gain marks
Key words: One real solution.
What do you use to show how many solutions there are? What type of polynomial is this used in? (Discriminant)
Remember: Use part (a) in part (b). Did one of the answers in part a happen to be a factor?
Key words: Turning points. Normally we would complete the square here but you can’t complete the square on a cubic. What other areas/topics link to turning points?
Hint: Stationary points
You need to remember set notation here. The sign ∩ means the intersection. Essentially, where is ‘y’ less than your curve but also x ≤ 1? 5B 5C 5E

8. 6 Q6 What do you know about graph transformations?
What are the general tips to help you remember which way the graph transforms?
Inside the brackets = does the opposite of expected
Outside the brackets = does what is expected
Be careful with the asymptotes, these will affect part (ii)

9. Q7
General
What are the different proofs that you know? Which one suits this question best?
What would happen if you made 0 the subject?
Can the left hand side now be factorised?
Work forwards from here
Think carefully about which numbers would be good to sub in e.g 3x? 6-x?
Remember: (2x )² = 2x x 2x = 22x
Rearrange part (b) so that 0 is the subject
How else can 9-x be written? Now re-write your equation so it does not contain fractions
Can you substitute a value in so that a quadratic can be formed? E.g.
let y = 9x 7A 7B 7C

10. 8A 8B 8C Q8 What does a perpendicular bisector do?
What do you know about perpendicular gradients?
What does a tangent look like?
How can you use the tangent to find a line that will intersect with the perpendicular bisector you found in (a)? (HINT: Normal)
What will you find where the perpendicular bisector and the normal intersect? (Centre of circle)
If you were to draw the circle and plot the centre and point A or B, what could you draw so that the radius can be found?
Which famous theorem can be used to find the radius? 8B 8C

11. 9A 9B 9C Q9 Look for the identity here. How else can cos²θ be written?
From here follow the algebraic manipulation. You will need to look for elements you can cancel. Look for how you can match an element from the top with the bottom.
You have just expressed P in the form a + b sinθ. You now need to use this. Remember: part (a) is pretty much always needed in part (b)
You had a + bsinθ, you now have a + bsin(θ – 30)
Once you have the value of θ – 30, remember you are in the range of 0 ≤ θ ≤ 360. Think of the sine graph for this.
Need to be mindful of the range again (both in the x and y directions) 9B 9C

12. Q10
10A
Read the inequality here carefully: t ≥ 0. When Josh first releases his video has any time elapsed? T can be equal to 0.
You need to form an equation using the information given
What mathematical process has a relationship with exponentials?
You are using base e, what happens if you take a natural log of e?
What is special about the number 1024? Can it be written as a power?
Kloga = logak
Think of an exponential graph. What happens as the value of x continues to increase?
10B
10C

13. Q11 11
You have been given f’(x), but the curve C shows f(x). What process is needed to go from f’(x) to f(x)?
You have been told that the curve passes through (1,-2). You need to use this to find the constant ‘c’
Now you have f(x), look at the graph carefully. You need to find where the curve cuts the x-axis. What is the value of y when the curve cuts the x axis? Use this to find the points of intersection
Now you have where the curve cuts the x axis, you have your limits for definitive integration. Use this to find the area of R.

14. Your Turn
Write your own set of hints and tips for the question above. What are the main features? What questions should the person answering this question ask?