Exam Structure Set A: Higher Paper 3

Question 1 [1 marks]

Indices and Standard Form

Write the number 45000 in standard form.

Question 2 [3 marks]

Quadratics

Factorise fully 3x^2 + 6x - 45

Question 3 [5 marks]

Graphs and Coordinates

The graph of y = x^3 is to be drawn for -2 <= x <= 2.

Complete the table of values for y = x^3. x : -2 , -1 , 0 , 1 , 2 y : -8 , _ , 0 , _ , 8

On the grid, plot the points from your table and join them with a smooth curve.

Write down the coordinates of the point where the curve crosses the y-axis.

Question 4 [1 marks]

Number and Calculation

Two offcuts of skirting board have lengths a = 14 cm and b = 9 cm, each measured correct to the nearest cm. Find the upper bound of a + b.

Question 5 [2 marks]

Statistics and Probability

24 pupils were asked how many siblings they have. One frequency, a, is missing from the table. Number of siblings (x): 0, 1, 2, 3 Frequency (f): 5, a, 6, 2 Total number of pupils = 24 Find the value of a.

Question 6 [2 marks]

Angles and Geometrical Reasoning

Fatima is making a scale drawing of a triangular field using a scale of 1 cm to 20 m. One side of the field is 130 m long.

Question 7 [2 marks]

Sequences

Work out an expression for the nth term of the sequence 1, 6, 15, 28.

Question 8 [2 marks]

Number and Calculation

Ffion uses her calculator to work out 4.5 x 3.2. She then uses the ANS key to subtract 2.6 from this answer. Work out her final result.

Question 9 [2 marks]

Angles and Geometrical Reasoning

A quadrilateral ABCD has angle A = 72 degrees, angle B = 99 degrees, angle C = 107 degrees and angle D = y.

Work out the size of angle y.

Give a reason for your answer to part (a).

Question 10 [3 marks]

Number and Calculation

A recipe for 8 people uses 3.85 kg of flour. By rounding suitable values to 1 significant figure, estimate the amount of flour needed for 31 people.

Question 11 [3 marks]

Area, Volume and Measures

A model car is made using a scale of 1 : 24. The real car has a length of 4.8 m.

Question 12 [3 marks]

Linear Algebra

n is an integer such that -2 <= n < 5.

List all the possible values of n.

Write down the smallest possible value of n.

Question 13 [3 marks]

Ratio and Proportion

A recipe uses flour and sugar in the ratio 5:2. Anna wants to make a batch using 800 g of flour. Work out how much sugar she must use.

Question 14 [3 marks]

Pythagoras and Trigonometry

Point A has coordinates (2, 3) and point B has coordinates (7, 11). Work out the distance AB. Give your answer correct to 3 significant figures.

Question 15 [3 marks]

Angles and Geometrical Reasoning

The plan of a solid, drawn on a centimetre grid, is an L-shape made from 5 unit squares, as shown. Every position in the plan has a stack of cubes exactly 3 cubes tall.

How many unit squares make up the plan?

Work out the total number of cubes in the solid.

Question 16 [3 marks]

Ratio and Proportion

A shop sells a jacket at 35% off the original price. The sale price is £78. Work out the original price of the jacket. Give your answer to the nearest 10p.

Question 17 [3 marks]

Angles and Geometrical Reasoning

OABC is a parallelogram, with vertices in order. OA = a and OC = c.

Find OB in terms of a and c.

Find CA in terms of a and c.

Question 18 [4 marks]

Fractions, Decimals and Percentages

A phone was sold for £306 after a 15% discount was applied. Lorraine says the original price was £360. Is she correct? Show your working.

Question 19 [4 marks]

Indices and Standard Form

Solve each of the following equations.

6^x = 216

7^(2x-1) = 343

Question 20 [7 marks]

Quadratics

The graph of y = x^2 + 2x - 3 is to be drawn for -4 <= x <= 2.

Complete the table of values for y = x^2 + 2x - 3. x : -4 , -3 , -2 , -1 , 0 , 1 , 2 y : _ , _ , _ , _ , _ , _ , _

Draw the graph of y = x^2 + 2x - 3 for -4 <= x <= 2.

Use the graph to write down the solutions of x^2 + 2x - 3 = 0.

Write down the coordinates of the minimum point of the graph.

Question 21 [4 marks]

Graphs and Coordinates

Find the equation of the straight line that is perpendicular to y = (1/2)x + 4 and passes through the point (3, -1). Give your answer in the form y = mx + c.

Question 22 [2 marks]

Angles and Geometrical Reasoning

Point S has coordinates (2, 2) and maps to the image point S'(-2, -2) under an enlargement with scale factor -1. Find the coordinates of the centre of enlargement.

Question 23 [4 marks]

Graphs and Coordinates

A curve y = r(x) passes through the point (8, 3).

After the transformation y = r(x) + c, the image of this point is (8, -5). Find the value of c.

The curve y = r(kx) has an image point (2, 3) that corresponds to (8, 3) on y = r(x). Find the value of k.

Question 24 [5 marks]

Indices and Standard Form

A student writes down the statement sqrt(16) + sqrt(9) = sqrt(16 + 9).

Show, by evaluating both sides, that the statement above is not correct.

By simplifying sqrt(20) + sqrt(5) and sqrt(20 + 5) separately, show that sqrt(a) + sqrt(b) is not equal to sqrt(a + b) in general.

Question 25 [6 marks]

Sequences

Here are the first four terms of a quadratic sequence. 23, 18, 11, 2

Show that the second difference of the sequence is -2.

Find an expression, in terms of n, for the nth term of the sequence.

Work out the 8th term of the sequence.