Higher Stretch A

Question 1 [1 marks]

Indices and Standard Form

Work out the value of 3^4.

Question 2 [1 marks]

Graphs and Coordinates

A line has gradient -3/4. What is the gradient of a line perpendicular to it? A) -3/4 B) 3/4 C) -4/3 D) 4/3

Question 3 [2 marks]

Fractions, Decimals and Percentages

Work out 7/9 - 4/9. Give your answer in its simplest form.

Question 4 [3 marks]

Sequences

Three sequences are shown below. Sequence A: 7, 11, 15, 19 Sequence B: 4, 12, 36, 108 Sequence C: 3, 6, 11, 18

State whether Sequence A is arithmetic, geometric, or neither of these. Give a reason for your answer.

State whether Sequence B is arithmetic, geometric, or neither of these. Give a reason for your answer.

State whether Sequence C is arithmetic, geometric, or neither of these. Give a reason for your answer.

Question 5 [4 marks]

Ratio and Proportion

A recipe for lemonade that makes 6 glasses needs 3 lemons and 250 ml of water. Nadia has 8 lemons and 900 ml of water. Work out the greatest number of glasses of lemonade Nadia can make, assuming she can only make whole batches of the recipe.

Question 6 [1 marks]

Number and Calculation

Write down the lower bound and upper bound for a number which is 12, correct to the nearest integer, using inequality notation.

Question 7 [1 marks]

Linear Algebra

Solve t - 8 = 2.

Question 8 [1 marks]

Number and Calculation

Round 7,246 to the nearest 100.

Question 9 [2 marks]

Area, Volume and Measures

Two mathematically similar solids have a linear scale factor of 4 from the smaller solid to the larger solid. The volume of the smaller solid is 3 cm^3. Work out the volume of the larger solid.

Question 10 [2 marks]

Quadratics

Solve the inequality 2x^2 - 8x >= 0. Give your answer in inequality form.

Question 11 [2 marks]

Pythagoras and Trigonometry

From a point on level ground 18 m from the base of a lighthouse, the angle of elevation to the top of the lighthouse is 52 degrees. Calculate the height of the lighthouse. Give your answer correct to 3 significant figures.

Question 12 [2 marks]

Angles and Geometrical Reasoning

The exterior angle of a regular polygon is 15 degrees. Work out the number of sides of the polygon.

Question 13 [2 marks]

Indices and Standard Form

A storage box for a school science kit in Bristol is a cube with volume 125 cm^3. Work out the length of one edge of the box.

Question 14 [3 marks]

Quadratics

Solve the equation 2x^2 - x - 6 = 0.

Question 15 [3 marks]

Angles and Geometrical Reasoning

Triangles ABC and DEF satisfy angle A = 50 degrees, angle C = 70 degrees and AC = DF. In triangle DEF, angle D = 50 degrees and angle F = 70 degrees. Show that the triangles are congruent and state two pairs of corresponding equal sides.

Question 16 [3 marks]

Sequences

The table shows the number of matchsticks used to make patterns of joined squares in a row. Pattern number (n): 1, 2, 3 Number of matchsticks: 4, 7, 10

Write down the number of matchsticks needed for Pattern number 4.

Find an expression, in terms of n, for the number of matchsticks in Pattern number n.

Question 17 [2 marks]

Angles and Geometrical Reasoning

w is the column vector (5, 11). Work out |w|, giving your answer correct to 1 decimal place.

Question 18 [4 marks]

Graphs and Coordinates

Show that the cubic y = x^3 - 6x^2 + 11x - 6 factorises as (x - 1)(x - 2)(x - 3). Hence, sketch the cubic and state the x-intercepts.

Show that the cubic factorises as stated.

Hence sketch the cubic and state the x-intercepts.

Question 19 [4 marks]

Quadratics

Solve 3x^2 = 13x - 4 by factorising. Show your working.

Question 20 [4 marks]

Angles and Geometrical Reasoning

O is the origin. The position vectors of points P, Q and R are OP = 2a - b, OQ = 6a + kb and OR = a + 2b, where a and b are non-parallel vectors and k is a constant. Given that P, Q and R lie on a straight line, find the value of k.

Question 21 [4 marks]

Area, Volume and Measures

A recipe for flapjacks that serves 6 people uses 750 g of oats.

Write 750 g in kilograms.

Bilal wants to make enough flapjacks to serve 15 people, using the recipe in the same proportions. Work out how many kilograms of oats he needs.

Question 22 [4 marks]

Statistics and Probability

A deck contains only cards numbered 1 to n, one of each. A card is chosen at random. The probability it is a multiple of 3 is 1/4. Find n, given n is a positive integer and the deck starts at 1.

Question 23 [4 marks]

Angles and Geometrical Reasoning

TP is a tangent to a circle with centre O, touching the circle at T. OT = 5 cm and OP = 13 cm. Since OT is a radius and TP is a tangent at T, angle OTP = 90 degrees. Diagram: circle, centre O, radius OT drawn to the point of contact T, tangent TP drawn perpendicular to OT, OP drawn from O to the external point P, right angle marked at T, OT = 5 cm, OP = 13 cm.

Work out the length TP.

Work out the size of angle TPO. Give your answer correct to 1 decimal place.

Question 24 [5 marks]

Statistics and Probability

The frequency table shows the results for Class B in the same test as Question 11 (also marked out of 10, also 20 pupils).

Calculate the mean score for Class B. Give your answer correct to 2 decimal places.

Find the range of scores for Class A and the range of scores for Class B.

Question 25 [9 marks]

Angles and Geometrical Reasoning

A surveyor produces a scale drawing of a triangular field. On the drawing the sides are 7.2 cm, 5.4 cm and 4.2 cm. The scale is 1:500 for linear measurements. (a) Work out the actual perimeter of the field in metres to 3 significant figures. (b) Hence, or otherwise, find the actual area of the field in square metres. Give your answer to 3 significant figures. (You may use Heron's formula.)

Question 26 [2 marks]

Statistics and Probability

A scientist plots the reaction time (in seconds) of runners against their age (in years), for ages 10 to 80. The points curve upwards steeply for ages under 20 and over 60, but are roughly flat for ages 20 to 60. Explain why it would not be appropriate to draw a single straight line of best fit for this data.

Question 27 [3 marks]

Linear Algebra

The kinetic energy of a moving object is given by E = (1/2)mv^2, where E is the kinetic energy in joules, m is the mass in kilograms and v is the speed in metres per second. Make v the subject of the formula.

Question 28 [3 marks]

Sequences

The nth term of a sequence is given by An + B, where A and B are numbers to be found. The 2nd term of the sequence is 11. The 5th term of the sequence is 23. Work out the values of A and B.

Question 29 [4 marks]

Quadratics

The function f is defined by f(x) = x^2 - 3x + 7.

Express f(x) in the form (x - a)^2 + b, where a and b are exact values.

Hence state the minimum value of f(x).

Question 30 [4 marks]

Statistics and Probability

A wildlife survey estimates the population of red squirrels in a forest in two consecutive years using capture-recapture. Year 1: 60 squirrels tagged in the first sample; second sample of 50, of which 12 are tagged. Year 2: 80 squirrels tagged in the first sample; second sample of 70, of which 14 are tagged.

Calculate the population estimate for Year 1 and for Year 2.

Work out the percentage change in the population estimate from Year 1 to Year 2.

Question 31 [5 marks]

Angles and Geometrical Reasoning

In triangle ABC, D is a point on BC. E is the foot of the perpendicular from D to AB, and F is the foot of the perpendicular from D to AC, with DE = DF. Prove that AD bisects angle BAC.

Question 32 [6 marks]

Pythagoras and Trigonometry

A vertical tree TB stands on horizontal ground, with its base at B. From a point C on the ground, in line with B, the angle of elevation of the top of the tree, T, is 42 degrees. From a point D, further from the tree than C, with C between B and D and CD = 15 m, the angle of elevation of T is 26 degrees.

By forming two equations for the height, h, of the tree (using triangles TBC and TBD), calculate h. Give your answer correct to 3 significant figures.

Hence calculate the distance BC.