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GCSE Maths Higher: Geometry practice questions
Ten free GCSE Maths Higher tier questions on trigonometry, exact values, the cosine rule, circle theorems, spheres, arcs, vectors and the equation of a circle, each with a worked solution. Suitable for AQA, Edexcel and OCR.
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Timed mode
10 questions in 15 minutes, one question at a time. Your answers are marked when you submit or when the time runs out.
विषय: Trigonometry
1. A right-angled triangle has a hypotenuse of 10 cm and an angle of 30°. Find the side opposite the 30° angle.
हल
C. 5 cm
- sin 30° = opposite ÷ hypotenuse.
- Opposite = 10 × sin 30° = 10 × 0.5 = 5 cm.
विषय: Exact trig values
2. What is the exact value of cos 60°?
हल
B. 1/2
- Learn the exact values: cos 60° = 1/2 and sin 60° = √3/2.
विषय: Area of a triangle
3. Two sides of a triangle are 8 cm and 10 cm, and the angle between them is 30°. Find its area.
हल
A. 20 cm²
- Area = ½ab sin C.
- ½ × 8 × 10 × sin 30° = 40 × 0.5 = 20 cm².
विषय: Cosine rule
4. Two sides of a triangle are 5 cm and 7 cm, with 60° between them. Find the third side.
हल
D. √39 cm (about 6.24 cm)
- Cosine rule: c² = a² + b² − 2ab cos C.
- c² = 25 + 49 − 2 × 5 × 7 × 0.5 = 74 − 35 = 39.
- c = √39 ≈ 6.24 cm.
विषय: Circle theorems
5. The angle at the centre of a circle is ___ the angle at the circumference standing on the same arc.
हल
C. twice
- This is one of the circle theorems: the angle at the centre is twice the angle at the circumference.
विषय: Circle theorems
6. What is the angle in a semicircle?
हल
B. 90°
- An angle drawn from the ends of a diameter to the circumference is always a right angle.
विषय: Volume
7. Find the volume of a sphere of radius 3 cm. Give your answer in terms of π.
हल
A. 36π cm³
- Volume of a sphere = (4/3)πr³.
- (4/3) × π × 27 = 36π cm³.
विषय: Arc length
8. Find the arc length of a sector with radius 6 cm and angle 60°, in terms of π.
हल
D. 2π cm
- Arc length = (angle/360) × 2πr.
- (60/360) × 2π × 6 = (1/6) × 12π = 2π cm.
विषय: Vectors
9. a = (3, −2) and b = (1, 4) as column vectors. Find 2a − b.
हल
C. (5, −8)
- 2a = (6, −4).
- 2a − b = (6 − 1, −4 − 4) = (5, −8).
विषय: Equation of a circle
10. What is the equation of a circle with centre (0, 0) and radius 5?
हल
B. x² + y² = 25
- A circle with centre the origin and radius r is x² + y² = r².
- r = 5, so x² + y² = 25.