Papers 2 & 3 style practice: Higher (calculator) — question paper (print)

Papers 2 & 3 style (Higher)

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QQ/GCSE-MATHS/P23HS

GCSE (9–1) · GCSE Mathematics (Higher)

Papers 2 & 3 style practice paper

Higher (calculator)

Duration: 90 minutes

Total: 80 marks

Name:

Class:

Date:

READ THESE INSTRUCTIONS FIRST

Answer all questions.
Section A contains ten multiple-choice questions. Choose one answer, A, B, C or D, for each.
Section B contains structured questions. Write your answers in the spaces provided.
A calculator may be used. Round non-exact answers to 3 significant figures unless told otherwise.
The number of marks is given in brackets [ ] at the end of each question or part question.
The total mark for this paper is 80.

Original practice material written for the MOE GCSE Mathematics syllabus. It is not an official MOE or school examination paper. Higher content with calculator work: standard form, bounds, algebraic solving including the quadratic formula, simultaneous equations with a quadratic, trigonometry, circle tangents, growth and decay, statistics and iteration.

For Examiner’s Use
Section A	/ 10
Section B	/ 70
Total	/ 80

Section A: Multiple-choice questions[10 marks]

Choose one answer for each question. Every correct answer scores one mark.

1
Work out 0.7 × 0.06.
1. A0.42
2. B4.2
3. C0.042
4. D0.0042
( )
2
(3.6 × 10⁴) ÷ (9 × 10²) equals
1. A4 × 10²
2. B4 × 10⁶
3. C4 × 10
4. D40 × 10³
( )
3
√40, correct to 3 significant figures, is
1. A6.3
2. B6.33
3. C6.324
4. D6.32
( )
4
£800 is invested at 4% compound interest for 2 years. The final amount is
1. A£864
2. B£865.28
3. C£866
4. D£832
( )
5
The interior angle of a regular 15-sided polygon is
1. A156°
2. B150°
3. C162°
4. D144°
( )
6
The gradient of the line through (2, 5) and (6, 13) is
1. A2
2. B½
3. C−2
4. D8
( )
7
For 0 < x < 90, sin x° = 0.5 gives x =
1. A60°
2. B30°
3. C45°
4. D26.6°
( )
8
The volume of a sphere of radius 3 cm, to 3 significant figures, is
1. A381.7 cm³
2. B108 cm³
3. C94.2 cm³
4. D113 cm³
( )
9
y is directly proportional to x². When x = 3, y = 18. When x = 5, y is
1. A30
2. B90
3. C50
4. D2
( )
10
The numbers 2, x and 7 have a mean of 5. x is
1. A6
2. B5
3. C4.5
4. D7
( )

Section B: Structured questions[70 marks]

Show all your working. A calculator may be used.

11
Standard form and bounds.
1. (a)
  Work out (2.4 × 10³) × (5 × 10⁵). Give your answer in standard form.
  [2]
2. (b)
  A length x is 4.5 correct to 1 decimal place. Write down the lower and upper bounds of x.
  [2]
3. (c)
  Find the upper bound of x².
  [2]
12
Solving quadratics.
1. (a)
  Factorise 6x² − x − 2.
  [2]
2. (b)
  Solve 4x² + 4x − 5 = 0 using the quadratic formula. Give your solutions to 3 significant figures.
  [4]
3. (c)
  Simplify fully (x + 2)⁄(x² − 4).
  [2]
13
Simultaneous equations.
1. (a)
  Solve 3x + 2y = 16 and x − y = 2.
  [4]
2. (b)
  Solve x² + y² = 20 and x − y = 2.
  [3]
14
Trigonometry.
1. (a)
  A right-angled triangle has a hypotenuse of 12 cm and an angle of 35°. Find the side opposite the 35° angle, to 3 significant figures.
  [3]
2. (b)
  A triangle has sides 7 cm and 10 cm with included angle 40°. Find its area, to 3 significant figures.
  [3]
3. (c)
  Write down the exact value of sin²30° + cos²60°.
  [2]
15
The circle x² + y² = 25.
1. (a)
  State the radius of the circle.
  [1]
2. (b)
  The point (4, 3) lies on the circle. Find the gradient of the radius from the origin to (4, 3).
  [2]
3. (c)
  Show that the gradient of the tangent at (4, 3) is −4⁄3.
  [2]
4. (d)
  Find the equation of the tangent at (4, 3). Give your answer in the form ax + by = c.
  [3]
16
Growth and decay.
1. (a)
  £2500 is invested at 3.5% compound interest per year for 6 years. Find the final amount, to the nearest penny.
  [3]
2. (b)
  A car worth £14 000 loses 15% of its value each year. Find its value after 4 years, to 3 significant figures.
  [3]
3. (c)
  Write down the single multiplier for a 15% yearly decrease.
  [1]
17
Cumulative frequency.
1. (a)
  The finishing times of 80 runners: 20 < t ≤ 30 min (8 runners), 30 < t ≤ 40 (18), 40 < t ≤ 50 (28), 50 < t ≤ 60 (16), 60 < t ≤ 70 (10). Write down the cumulative frequency at each upper boundary.
  [2]
2. (b)
  Estimate the median finishing time.
  [2]
3. (c)
  Estimate the interquartile range.
  [2]
4. (d)
  A second race’s box plot has median 49 and IQR 6. Compare the two races.
  [2]
18
Histogram.
1. (a)
  Parcels are weighed: 0 < w ≤ 4 kg (12 parcels), 4 < w ≤ 10 (24), 10 < w ≤ 16 (20), 16 < w ≤ 20 (16), 20 < w ≤ 30 (8). Explain why the bars have unequal widths.
  [1]
2. (b)
  Find the frequency density of the 4 < w ≤ 10 class.
  [2]
3. (c)
  Estimate how many parcels weigh 8 kg or less.
  [2]
4. (d)
  Estimate the proportion of parcels weighing more than 16 kg. Give your answer as a decimal.
  [2]
19
Iteration.
1. (a)
  Show that the equation x³ − 5x + 1 = 0 has a solution between x = 0 and x = 1.
  [2]
2. (b)
  Show that x³ − 5x + 1 = 0 can be rearranged as x = (x³ + 1)⁄5.
  [1]
3. (c)
  Use the iteration x₍n+1₎ = (xₙ³ + 1)⁄5 with x₀ = 0.2 to find a root of the equation correct to 4 decimal places.
  [4]
20
Capture–recapture and conditional probability.
1. (a)
  Ecologists catch and tag 40 fish. A week later they catch 60 fish and 8 of them are tagged. Estimate the total number of fish in the lake.
  [2]
2. (b)
  In a group, n(A ∩ B) = 18 and n(B) = 28. Find P(A | B) as a fraction in its simplest form.
  [2]

— End of paper —

Papers 2 & 3 style practice paper: Higher (calculator) — question paper