Exam Structure Set B: Higher Paper 3

Question 1 [2 marks]

Area, Volume and Measures

Rectangle A measures 5 cm by 8 cm.

Which of these rectangles is mathematically similar to rectangle A?

Write down the scale factor of enlargement from rectangle A to the rectangle you chose in part (a).

Question 2 [2 marks]

Fractions, Decimals and Percentages

Write 0.3333... (3 recurring) as a fraction in its simplest form.

Question 3 [5 marks]

Statistics and Probability

Priya Chandra manages Riverside Car Park. She recorded how many cars and vans were parked there in the morning and in the afternoon.

Question 4 [1 marks]

Angles and Geometrical Reasoning

A student is asked to enlarge a triangle by scale factor -2 from a given centre. The student draws the image on the same side of the centre as the original triangle, but twice the size. Explain the error in the student's diagram.

Question 5 [1 marks]

Graphs and Coordinates

Write down the equation of the graph obtained by reflecting y = f(x) in the y-axis.

Question 6 [1 marks]

Linear Algebra

Simplify 7y - 2y + 4y.

Question 7 [1 marks]

Indices and Standard Form

Work out the value of 16^(1/2).

Question 8 [2 marks]

Ratio and Proportion

A recipe for 6 portions uses 300 g of flour. Work out how much flour is needed for 15 portions, assuming the amount of flour is directly proportional to the number of portions.

Question 9 [2 marks]

Area, Volume and Measures

A sphere has radius 3 cm. Work out the volume of the sphere. Give your answer as a multiple of pi.

Question 10 [2 marks]

Linear Algebra

Make b the subject of the formula: a = (b + 11)/6

Question 11 [3 marks]

Number and Calculation

Find the highest common factor (HCF) of 24 and 36. You must show your method.

Question 12 [3 marks]

Quadratics

Consider the equation x^3 + 4x - 10 = 0 and the iteration x_{n+1} = (10 - x_n^3)/4. Starting with x_0 = 1.5, calculate x_1 and x_2. From these two iterations, state whether the sequence appears to be converging.

Question 13 [3 marks]

Pythagoras and Trigonometry

A cone has base radius 5 cm and perpendicular height 12 cm. Diagram: cone shown in cross-section, base radius marked 5 cm from the centre of the base to the edge, perpendicular height marked 12 cm from the centre of the base to the apex, and the slant height labelled l cm from the apex to the edge of the base. Calculate the slant height, l, of the cone.

Question 14 [4 marks]

Sequences

A sequence has nth term given by T_n = 5n - 3 for n odd, and T_n = 5n + 1 for n even. (a) Work out T_5 and T_6. (b) Explain briefly whether this sequence is arithmetic overall.

Question 15 [4 marks]

Indices and Standard Form

a = 3 x 10^5 b = 4 x 10^-2

Work out a x b. Give your answer in standard form.

Work out a / b. Give your answer in standard form.

Question 16 [4 marks]

Pythagoras and Trigonometry

An isosceles triangle has two equal sides of length 10 cm and a base of length 12 cm. A line is drawn from the apex of the triangle perpendicular to the base, meeting the base at its midpoint. Diagram: isosceles triangle with the two equal sides marked 10 cm, base marked 12 cm, and a dashed perpendicular line from the apex to the midpoint of the base labelled h cm.

Explain why the perpendicular line splits the base into two lengths of 6 cm each.

Calculate the perpendicular height, h, of the triangle.

Question 17 [4 marks]

Linear Algebra

Solve the simultaneous equations: y = 3x - 2 2x + y = 13

Question 18 [4 marks]

Number and Calculation

A rectangle has length 3 times its width. The perimeter of the rectangle is 96 cm. (a) Find the width and the length. (b) Work out the area of the rectangle.

Question 19 [4 marks]

Fractions, Decimals and Percentages

The table shows the number of apples picked at three orchards and the fraction of each orchard's apples that were sold at a farmers' market. Orchard | Apples picked | Fraction sold at market Ashcroft Farm | 96 | 3/8 Bramble Hill | 105 | 2/5 Cedar Grove | 90 | 7/9 Work out which orchard sold the most apples at the market. You must show your working to support your answer.

Question 20 [7 marks]

Statistics and Probability

Amelia recorded the number of text messages she sent on each of 13 days: 2, 5, 6, 6, 7, 8, 9, 9, 10, 12, 14, 15, 20

Find the median number of texts sent.

Find the lower quartile and the upper quartile of the number of texts sent.

Hence find the interquartile range of the number of texts sent.

Draw a box plot to represent Amelia's data on the grid provided.

Question 21 [1 marks]

Quadratics

An iteration formula is x_(n+1) = 3/(x_n - 2). Which of these values of x0 would make it impossible to calculate x1?

Question 22 [2 marks]

Statistics and Probability

In a capture-recapture study, n1 = 40 animals are marked and released. In the second sample, m = 5 of the recaptured animals are marked, and this gives a Lincoln-Petersen population estimate of N = 240. Find n2, the number of animals caught in the second sample.

Question 23 [3 marks]

Indices and Standard Form

Expand and simplify (3 + sqrt(5))(5 - sqrt(5)).

Question 24 [5 marks]

Area, Volume and Measures

A rectangular lawn ABCD has AB = 12 m and BC = 7 m. An isosceles triangular flower bed is attached to side AB, outside the rectangle, with the two equal slant sides of the triangle each measuring 7.5 m. Work out the total area of the lawn and the flower bed together.

Question 25 [3 marks]

Statistics and Probability

A histogram represents the finishing times, in minutes, of 200 athletes in a marathon, grouped into three classes: 180 <= t < 190 (width 10), 190 <= t < 210 (width 20), and 210 <= t < 240 (width 30). The heights of the three bars are in the ratio 3 : 2 : 1 respectively.

Question 26 [4 marks]

Graphs and Coordinates

A bus leaves a stop and accelerates uniformly to a speed of 12 m/s in 15 seconds. It then travels at 12 m/s for 40 seconds, before decelerating uniformly to rest in 10 seconds. The bus then waits at a second stop for 20 seconds, before accelerating uniformly back up to 12 m/s over the next 12 seconds.

Question 27 [3 marks]

Statistics and Probability

Using the cumulative frequency graph for the fun run times (60 runners, shown in Question 3), the organisers decide to award a medal to the fastest 20% of runners. Estimate the completion time below which a runner receives a medal.