GCSE Maths 2026: Paper 3 - challenging questions

 

1) 

1. Simplify: ∛(27x⁶y³)
2. Solve: 2³ˣ⁻¹ = 1 / √128
3. Which one of the following is the bigger? Show your working with all necessary steps:

        3⁵² or 4³⁹

2) 

ABC is a right-angled triangle. AB = BC = 1 cm. The side, BC, is extended in such a way that AC = CD.

1. Find the value of CD and leave the answer in surd form.
2. Hence find the value of tan 22.5° without using the calculator and leave the answer in surd form.
3. Leave the answer to part 2) in rationalized form.

3) 

1. Using the formula, area of a triangle = 1/2 ab sin C, show that the area of a right-angled triangle is 1/2 height x base.
2. Find the area of the triangle ACD.

4) 

A clear glass jar contains x milk chocolates wrapped in gold foil and 5 white chocolates wrapped in silver foil. A chocolate is taken at random from the jar and eaten. Then, a second chocolate is taken at random and eaten. The probability that both chocolates eaten are milk chocolates (gold foil) is 3/7.

Prove that: 2x² – 17x – 30 = 0

Hence, find the value of x.

5) 

1) Sketch the graph of y = cos x for 0° ≤ x ≤ 360°. 

2) Use the graph to show that cos 120° = - cos 60°.

3) PQR is an isosceles triangle with PQ = PR. QS = c, SR = d and PS = e.

Show that cos Θ = (c - d)/2e.

6)