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Grade 10 — Practice Paper 2

Calculator Allowed  |  60 Marks  |  75 Minutes  |  Cambridge IGCSE (0580) Extended Style

✅ Calculator 📋 60 marks  |  ⏱ 75 minutes
75:00
Section A — Number & Proportion (14 marks)
Questions 1–5
Q1 3 marks
y is directly proportional to x². When x = 4, y = 48.
(a) Find the value of k in y = kx².
(b) Find y when x = 5.
(c) Find x when y = 75. Give the positive value.
Q2 3 marks
y is inversely proportional to x. When x = 3, y = 20.
(a) Find y when x = 12.
(b) Find x when y = 4.
(c) A different quantity p is inversely proportional to x². When x = 2, p = 9. Find p when x = 6.
Q3 3 marks
A map has scale 1 : 40 000.
(a) Two towns are 6.5 cm apart on the map. Find the actual distance in km.
(b) A lake has an area of 4.5 km² on the ground. Find the area on the map in cm².
(c) A road is 12 km long. Find its length on the map in cm.
km
cm²
cm
Q4 3 marks
A car is bought for $18 000. It depreciates at 15% per year.
(a) Find the value after 3 years. Give your answer to the nearest dollar.
(b) After how many complete years will the value first fall below $9 000?
years
Q5 2 marks
The mass of a sphere is measured as 4.7 kg correct to 1 decimal place. Write down the error interval for the mass m.
(a) Lower bound of m.
(b) Upper bound of m.
kg
kg
Section B — Algebra & Graphs (16 marks)
Questions 6–10
Q6 3 marks
Solve the simultaneous equations: y = 3x − 2 and y = x² + x − 4.
(a) Enter the smaller x-value (to 2 d.p. if not exact).
(b) Enter the larger x-value.
Q7 3 marks
A geometric sequence has terms: 2, 6, 18, 54, ...
(a) Find the 10th term.
(b) Find the sum of the first 8 terms.
(c) A different sequence has first term 100 and sum to infinity 250. Find the common ratio.
Q8 3 marks
A velocity-time graph shows:
  • From t=0 to t=4s: velocity increases uniformly from 0 to 12 m/s
  • From t=4 to t=10s: constant velocity 12 m/s
  • From t=10 to t=14s: velocity decreases uniformly from 12 m/s to 0
(a) Find the total distance travelled.
(b) Find the acceleration during the first 4 seconds (m/s²).
(c) Find the deceleration during the last 4 seconds (m/s²). Give as positive value.
m
m/s²
m/s²
Q9 4 marks
The curve y = x³ − 6x² + 9x − 2 has derivative dy/dx = 3x² − 12x + 9.
(a) Find the x-coordinates of the two turning points.
(b) Find the gradient of the tangent at x = 4.
(c) Find the equation of the tangent at x = 4 in the form y = mx + c. Enter c.
Q10 3 marks
The graph of y = f(x) passes through (−3, 4), (0, 2), and (2, −1).
(a) Write down the y-coordinate of the point (0, ?) on y = f(x) + 3.
(b) Write down the x-coordinate of the point (?, 4) on y = f(x − 2). (What x maps to the same y-value as (−3,4) in the original?)
(c) Write down the y-coordinate of the point (2, ?) on y = −f(x).
Section C — Geometry & Trigonometry (18 marks)
Questions 11–15
Q11 4 marks
Triangle ABC has AB = 9 cm, BC = 7 cm, and angle ABC = 64°.
(a) Find the length AC using the cosine rule. Give answer to 3 s.f.
(b) Find the area of triangle ABC. Give answer to 3 s.f.
cm
cm²
Q12 3 marks
In triangle PQR, PQ = 11 cm, QR = 8 cm, PR = 7 cm.
(a) Find the angle QPR using the cosine rule. Give to 1 d.p.
(b) Using the sine rule, find angle PQR to 1 d.p.
°
°
Q13 4 marks
A cuboid has length 8 cm, width 6 cm, and height 5 cm.
(a) Find the length of the space diagonal. Give exact answer as √k — enter k.
(b) Find the angle the space diagonal makes with the base ABCD (the base is 8×6). Give to 1 d.p.
(c) A pyramid has a square base of side 10 cm and vertical height 12 cm. Find the length of a slant edge (corner to apex). Give to 3 s.f.
°
cm
Q14 4 marks
Matrix M = [2, 1; 0, 3] (rows: [2,1] and [0,3]).
(a) Find the image of the point (3, 2) under M.
(b) Find the determinant of M.
(c) Find the inverse matrix M⁻¹. Apply M⁻¹ to the image you found in (a) and verify it returns (3, 2). Enter the x-coordinate after applying M⁻¹.
(d) The matrix N = [0, -1; 1, 0] represents a rotation. State the angle of rotation (degrees, anticlockwise).
, )
°
Q15 3 marks
A ship sails from port P on a bearing of 052° for 60 km to reach point A. It then sails on a bearing of 126° for 45 km to reach point B.
(a) Find the angle PAB inside the triangle. (Hint: use co-interior/supplementary bearing angles.)
(b) Find the distance PB using the cosine rule. Give to 3 s.f.
°
km
Section D — Statistics & Probability (12 marks)
Questions 16–19
Q16 3 marks
The table shows the frequency density for a histogram of heights (cm) of plants:

ClassFreq. DensityFrequency
0 – 101.2?
10 – 202.4?
20 – 401.8?
40 – 800.6?
(a) Find the total frequency.
(b) State the modal class.
(c) Estimate the mean height. (Use midpoints 5, 15, 30, 60.)
Q17 3 marks
In a survey of 50 people: 30 like coffee (C), 25 like tea (T), 12 like both.
(a) Draw a Venn diagram (in your notes) and find n(neither C nor T).
(b) A person is selected at random. Find P(C ∩ T).
(c) Given the person likes tea, find P(also likes coffee). P(C | T) =
Q18 3 marks
A test for a disease has:
  • P(positive | has disease) = 0.95
  • P(positive | no disease) = 0.08
  • P(has disease) = 0.02
(a) Find P(positive test) using the law of total probability.
(b) Find P(has disease | positive test). Give to 3 s.f.
Q19 3 marks
The scatter graph of 8 students shows hours of study (x) and test score (y).
Mean hours = 4.5, Mean score = 62.
The line of best fit passes through (4.5, 62) and (8, 76).
(a) Find the gradient of the line of best fit.
(b) Write the equation of the line of best fit in the form y = mx + c. Enter c.
(c) Predict the score for a student who studies 10 hours. Is this interpolation or extrapolation? Enter 1 for interpolation, 2 for extrapolation.

Final Score

Complete all sections, then calculate your grade.