Exam Structure Set B: Higher Paper 1

Question 1 [1 marks]

Ratio and Proportion

Simplify the ratio 35:14.

Question 2 [1 marks]

Linear Algebra

Solve 8 + x = 15

Question 3 [2 marks]

Sequences

Here are the first four terms of a sequence. 2, 6, 10, 14

Write down the next two terms of the sequence.

Describe, in words, the term-to-term rule for the sequence.

Question 4 [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 5 [1 marks]

Angles and Geometrical Reasoning

Point U has coordinates (-3, 5). Write down the coordinates of the image of U after a rotation of 180 degrees about the origin.

Question 6 [1 marks]

Indices and Standard Form

Simplify sqrt(50) fully.

Question 7 [2 marks]

Number and Calculation

Without using a calculator, estimate the value of sqrt(146). Show working to support your answer.

Question 8 [2 marks]

Indices and Standard Form

Simplify 20y^9 / 5y^3.

Question 9 [2 marks]

Area, Volume and Measures

Two similar triangles have corresponding sides in the ratio 5:8. A side of the smaller triangle is 15 cm. Work out the length of the corresponding side in the larger triangle.

Question 10 [2 marks]

Graphs and Coordinates

A line has equation 2y = 6x - 4.

Rearrange the equation into the form y = mx + c.

State the gradient of the line.

Question 11 [2 marks]

Statistics and Probability

A scatter graph shows the number of gym sessions attended per month (x) and the kilograms of weight lost that month (y), for data collected between 1 and 12 sessions per month. Aisha uses the line of best fit to estimate the weight lost by someone who attends 40 sessions in a month. Explain why this estimate would not be reliable.

Question 12 [3 marks]

Pythagoras and Trigonometry

Show that in any right-angled triangle the tangent of an acute angle is equal to the sine of that angle divided by the cosine of that angle. Give a short explanation using side lengths. This is a 'show that' question; all marks are for method and explanation.

Show that tan theta = sin theta / cos theta using opposite, adjacent and hypotenuse labels.

Question 13 [3 marks]

Angles and Geometrical Reasoning

Work out the value of x, then find the size of each angle.

Question 14 [3 marks]

Quadratics

Find the vertex (turning point) and minimum value of y = 2x^2 - 8x + 5.

Question 15 [3 marks]

Fractions, Decimals and Percentages

A class completed 5/8 of a test. Later, 3/10 of the pupils who had not completed it finished the test. What fraction of the class have now completed the test?

Question 16 [4 marks]

Quadratics

Write 6x^2 - 24x + 11 in the form a(x + p)^2 + q, stating the values of a, p and q.

Question 17 [4 marks]

Fractions, Decimals and Percentages

Show that increasing a number by 20% and then decreasing the result by 20% does not return to the original number. Use an algebraic approach with n as the original number and give the final expression in terms of n.

Question 18 [5 marks]

Angles and Geometrical Reasoning

On a diagram, A, B and C are points with position vectors relative to origin O: OA = (2, 1), OB = ( -1, 3), OC = (5, 4). (a) Find the vector AB. (b) Show that the three points are not collinear by using vectors.

Show that A, B and C are not collinear by using vectors.

Question 19 [5 marks]

Linear Algebra

A car park charges according to the formula C = 2.50 + 1.20h, where C is the total cost in £ and h is the number of hours parked.

Make h the subject of the formula.

Sasha pays £8.50 to park her car. Work out how many hours she parked for.

Question 20 [6 marks]

Statistics and Probability

A holiday company surveyed 120 travellers. 70 of the travellers were adults and the rest were children. 45 of the adults travelled by plane and the rest of the adults travelled by coach. Of the children, 28 travelled by coach and the rest travelled by plane.

Use this information to draw and complete a two-way table showing adults and children against plane and coach.

One traveller is picked at random from all 120 travellers. Find the probability that the traveller is a child who travelled by plane.

Question 21 [2 marks]

Indices and Standard Form

(4 x 10^n) x (2 x 10^3) = 8 x 10^8 Find the value of n.

Question 22 [2 marks]

Statistics and Probability

In a capture-recapture study, some of the marks put on animals in the first sample fall off before the second sample is taken.

Question 23 [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 24 [3 marks]

Linear Algebra

Solve (5x - 3) / 4 = (3x + 1) / 2 Show your working.

Question 25 [4 marks]

Statistics and Probability

The Venn diagram shows solar panels (S) and heat pumps (H) for the 48 houses on a street in Manchester. S only = 3y, H only = y, both S and H = 2y, and 12 houses have neither.

Form an equation in y and solve it.

A house is selected at random. Find P(S).

Question 26 [3 marks]

Graphs and Coordinates

The graph of y = cos(x - 90) is a transformation of y = cos x.

Describe fully the transformation from y = cos x to y = cos(x - 90).

By considering this transformation, or otherwise, write cos(x - 90) in terms of sin x.

Question 27 [4 marks]

Statistics and Probability

A cafe recorded the time, m minutes, that 26 + b customers spent queuing at the counter. One frequency, b, is unknown. Time, m (minutes): 0 < m <= 5, 5 < m <= 10, 10 < m <= 15, 15 < m <= 20 Frequency (f): 8, 12, b, 6 Given that the estimated mean queuing time is 10 minutes, find the value of b. You must show your working.

Question 28 [5 marks]

Angles and Geometrical Reasoning

Show that for three parallel lines cut by a single transversal, the angle between the transversal and the top line plus the angle between the transversal and the bottom line equals twice the angle between the transversal and the middle line. Give a short proof using algebra: let the acute angle at the middle intersection be a degrees, and express the other two in terms of a.