1.13 Exact calculations with fractions and π
9 exam questions worth 22 marks. Write your answer, open the mark scheme, then give yourself the marks you earned. Your answers are saved on this device.
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What is the exact area of a circle with radius 5 cm?
Choose one answer.
Correct Not quite: the answer is \( 25\pi \) cm²
Mark scheme
B: 25\pi cm²
Model answer
25\pi cm² (B). Area = πr² = π × 25.
A circle has diameter 14 cm. Work out its circumference. Give your answer in terms of π.
Mark scheme
M1: π × 14 or 2 × π × 7; A1: 14π cm
Model answer
Circumference = πd = 14π cm.
A circle has radius 6 cm. Work out its area. Give your answer (a) in terms of π (b) as a decimal to 1 decimal place.
Mark scheme
M1: π × 6²; A1: 36π; A1: 113.1
Model answer
(a) π × 6² = 36π cm². (b) 36π = 113.097... = 113.1 cm² to 1 d.p.
Work out \frac{2}{3} + \frac{1}{4}. Give your answer as an exact fraction.
Mark scheme
M1: common denominator 12 (8/12 and 3/12); A1: 11/12
Model answer
2/3 + 1/4 = 8/12 + 3/12 = 11/12.
A semicircle has a diameter of 10 cm. Work out its area in terms of π.
Mark scheme
B1: radius 5; M1: ½ × π × 5²; A1: 12.5π (or 25π/2) cm²
Model answer
The radius is 5 cm. Area = ½ × π × 25 = 12.5π cm².
A cylinder has radius 3 cm and height 10 cm. Work out its volume in terms of π.
Mark scheme
M1: π × 3² × 10; A1: 90π cm³
Model answer
Volume = πr²h = π × 9 × 10 = 90π cm³.
A circle has circumference 20\pi cm. Work out the radius of the circle.
Mark scheme
M1: 2πr = 20π (or πd = 20π); A1: 10 cm
Model answer
2πr = 20π, so 2r = 20 and r = 10 cm.
A circular path has an outer radius of 7 m and an inner radius of 4 m. Work out the area of the path. Give your answer in terms of π.
Mark scheme
M1: π × 7² or π × 4²; M1: 49π - 16π; A1: 33π m²
Model answer
Area = π × 49 - π × 16 = 49π - 16π = 33π m².
A circle of diameter 10 cm is drawn inside a square of side 10 cm. Work out the area of the square that is outside the circle. Give your answer (a) in terms of π (b) to 1 decimal place.
Mark scheme
B1: square area 100; B1: circle area 25π; B1: 100 - 25π; B1: 21.5
Model answer
Square: 10 × 10 = 100 cm². Circle: radius 5, area 25π cm². Shaded area = 100 - 25π cm² = 100 - 78.54 = 21.5 cm² (1 d.p.).
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