Exam Structure Set A: Higher Paper 1

Question 1 [1 marks]

Area, Volume and Measures

A cube has edge length 4 cm. Which of the following is the total surface area of the cube?

Question 2 [2 marks]

Graphs and Coordinates

A line has equation y = -2x + 7.

Write down the gradient of the line.

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

Question 3 [3 marks]

Angles and Geometrical Reasoning

Column vectors are written in the form (x, y), where x is the top (horizontal) number and y is the bottom (vertical) number. u = (4, 1) and v = (-2, 5).

Work out u + v as a column vector.

Work out u - v as a column vector.

Work out 3v as a column vector.

Question 4 [1 marks]

Pythagoras and Trigonometry

Write down the exact value of cos 45 degrees, giving your answer as a surd in its simplest form.

Question 5 [1 marks]

Statistics and Probability

Using the Venn diagram from Question 2, a student is chosen at random from the 30 students.

What is n(F union N)? A) 5 B) 13 C) 23 D) 30

Question 6 [1 marks]

Indices and Standard Form

Simplify sqrt(75) fully.

Question 7 [1 marks]

Ratio and Proportion

Given that m:n = 4:7, write n in terms of m as a linear function in the form n = km.

Question 8 [1 marks]

Sequences

Write down the second difference for the sequence 4, 9, 16, 25, 36.

Question 9 [1 marks]

Ratio and Proportion

A fruit drink is made from concentrate and water in the ratio 1:4. Write down what fraction of the drink is concentrate.

Question 10 [1 marks]

Statistics and Probability

Describe in words the region that represents A' and B'.

Question 11 [1 marks]

Linear Algebra

The diagram below shows a number line. There is an open circle at x = 3, with shading and an arrow extending to the left. Write down the inequality that is shown.

Question 12 [1 marks]

Graphs and Coordinates

Work out f(2) for f(x) = 3x + 1.

Question 13 [1 marks]

Number and Calculation

Ayesha has 4 different T-shirts and 3 different skirts. Work out the number of different outfits she can make by choosing one T-shirt and one skirt.

Question 14 [1 marks]

Graphs and Coordinates

Work out f(-2) for f(x) = 2x^2.

Question 15 [2 marks]

Number and Calculation

Write 18 as a product of its prime factors.

Question 16 [2 marks]

Quadratics

Consider the equation 3x^2 - 2x + 5 = 0.

Calculate the value of the discriminant, b^2 - 4ac.

Hence state, giving a reason, how many real solutions the equation has.

Question 17 [2 marks]

Linear Algebra

Solve 3(x - 2) < 15 Show your working.

Question 18 [2 marks]

Pythagoras and Trigonometry

Work out the exact value of sin 30 degrees + cos 60 degrees. Show your working.

Question 19 [2 marks]

Graphs and Coordinates

The graphs of y = 2x - 6 and y = -x + 6 are drawn on the grid. Write down the solution of the simultaneous equations y = 2x - 6 and y = -x + 6.

Question 20 [2 marks]

Indices and Standard Form

Three stars have these distances from Earth, in light-years. Star P: 3.6 x 10^2 Star Q: 9 x 10^1 Star R: 1.4 x 10^3 Put the stars in order of distance from Earth, starting with the nearest. Show how you have compared the distances.

Question 21 [2 marks]

Graphs and Coordinates

The graphs of y = 1 and y = 3x - 5 are drawn on the grid. Write down the solution of the simultaneous equations y = 1 and y = 3x - 5.

Question 22 [2 marks]

Linear Algebra

a = 6 and b = -3. Work out the value of 2a - b.

Question 23 [2 marks]

Indices and Standard Form

Simplify (3d^2)^3.

Question 24 [2 marks]

Graphs and Coordinates

The graphs of y = x - 1 and y = -2x - 7 are drawn on the grid. Write down the solution of the simultaneous equations y = x - 1 and y = -2x - 7.

Question 25 [2 marks]

Statistics and Probability

Priya, a gardener, recorded the height, in cm, of 15 tomato plants. The five-number summary is: Minimum = 30, Lower quartile = 45, Median = 55, Upper quartile = 68, Maximum = 140

Work out the interquartile range of the plant heights.

The maximum height of 140 cm is much larger than the rest of the data. Suggest one reason why this value might not be typical of the tomato plants.

Question 26 [3 marks]

Fractions, Decimals and Percentages

Convert 0.512512512... (512 recurring) to a fraction in its simplest form.

Question 27 [4 marks]

Ratio and Proportion

It takes 6 builders 15 days to build a wall, all working at the same constant rate. The number of days, D, needed is inversely proportional to the number of builders, n.

Find the value of k in the formula D = k/n.

Work out how many days it would take 9 builders, working at the same rate, to build the same wall.

Give one reason why this inverse proportion model might not give a realistic answer if a very large number of builders were used.

Question 28 [4 marks]

Quadratics

Solve x^2 + 14x + 9 = 0 by completing the square. Give your answers in surd form, simplified where possible.

Question 29 [4 marks]

Ratio and Proportion

y is directly proportional to sqrt(x). When x = 16, y = 12.

Find a formula for y in terms of x.

Work out the value of y when x = 25.

Question 30 [3 marks]

Angles and Geometrical Reasoning

For a certain regular polygon, the sum of its interior angles is 10 times the sum of its exterior angles. Work out the number of sides of the polygon.

Question 31 [3 marks]

Indices and Standard Form

Simplify sqrt(x^3) / x^(1/4). Give your answer as a single power of x.

Question 32 [3 marks]

Angles and Geometrical Reasoning

TA is a tangent to a circle at the point A. AB and AC are chords of the circle. The angle between the tangent and the chord AB is (3x + 8) degrees. The angle in the alternate segment, angle ACB, is (5x - 32) degrees. Diagram: circle, tangent TA at A, chords AB and AC drawn, tangent-chord angle (3x + 8) degrees marked at A, angle ACB = (5x - 32) degrees marked at C in the alternate segment.

Use the alternate segment theorem to form an equation in x, and solve it to find the value of x.

Work out the size of angle ACB.

Question 33 [4 marks]

Linear Algebra

The formula connecting m and n is m = (4n + 5)/(3 - n). Show that n = (3m - 5)/(m + 4)

Question 34 [4 marks]

Graphs and Coordinates

The graph of y = 2cos x - 1 is a transformation of y = cos x.

Write down the maximum and minimum values of y = 2cos x - 1.

Write down the coordinates of the y-intercept of the graph.

State the period of the graph.

Question 35 [4 marks]

Angles and Geometrical Reasoning

A, B and C are points on the circumference of a circle with centre O. B lies on the minor arc AC. The reflex angle AOC = 264 degrees. Diagram: circle, centre O, points A, B and C on the circumference, B on the minor arc AC, the reflex angle AOC = 264 degrees marked at the centre going the long way round, angle ABC marked at B.

Work out the size of the non-reflex angle AOC.

Work out the size of angle ABC.

Question 36 [5 marks]

Statistics and Probability

A gym in Glasgow, run by manager Callum, has 44 members in total. Of these, x members use the pool only, x^2 members use both the pool and the gym floor, and (2x + 1) members use the gym floor only. The remaining 3 members use neither facility.

Show that x^2 + 3x - 40 = 0.

Solve the equation to find the value of x, explaining why the other solution is not valid.

Hence find the probability that a randomly selected member uses the pool (including those who use both facilities).