Higher Secure B

Question 1 [1 marks]

Indices and Standard Form

Work out the value of 3^4.

Question 2 [2 marks]

Angles and Geometrical Reasoning

Draw a straight line segment AB of length 7 cm. Using a ruler and compasses only, construct the perpendicular bisector of AB. You must show all your construction lines.

Question 3 [2 marks]

Ratio and Proportion

P is directly proportional to Q. When Q = 5, P = 30. Find the value of P when Q = 9.

Question 4 [4 marks]

Area, Volume and Measures

The exchange rate is £1 sterling (GBP) = 1.16 euros (EUR).

Nadia changes £250 into euros. Work out how many euros she receives.

On the way home Nadia changes 87 euros back into £, using the same exchange rate. Work out how many £ she receives.

Question 5 [1 marks]

Graphs and Coordinates

The graph of y = g(x) is obtained from y = f(x) by first reflecting in the y-axis and then translating up by 6. Write g(x) in terms of f.

Question 6 [1 marks]

Statistics and Probability

A bag contains 3 blue and 9 red counters. One counter is chosen at random. Write down the probability it is red.

Question 7 [2 marks]

Graphs and Coordinates

Let f(x) = x^2 and g(x) = x + 1. Find (g o f)(3).

Question 8 [2 marks]

Quadratics

Factorise x^2 - 13x + 40

Question 9 [2 marks]

Number and Calculation

A plumber charges a call-out fee of £45 plus £30 per hour worked. Show that the total cost of a job lasting 2.5 hours is £120.

Question 10 [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 11 [2 marks]

Linear Algebra

Solve 2x + 5 = -3

Question 12 [2 marks]

Quadratics

Solve x^2 + 9x = 0

Question 13 [2 marks]

Linear Algebra

Anjali has p pounds. She has more than £15 but no more than £40. Write down an inequality, in terms of p, to show this information.

Question 14 [2 marks]

Area, Volume and Measures

Convert 0.003 km^2 into square metres (m^2).

Question 15 [3 marks]

Sequences

Here are the first four terms of a sequence. 1, 1.5, 2, 2.5

Write down the next term of the sequence.

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

Question 16 [3 marks]

Pythagoras and Trigonometry

In triangle LMN, angle L = 115 degrees, MN = 18 cm (opposite angle L), and LN = 9 cm (opposite angle M). Calculate the size of angle N. Give your answer correct to 1 decimal place.

Question 17 [3 marks]

Angles and Geometrical Reasoning

Enlarge triangle T with vertices (2, 1), (4, 1) and (2, 3) by a scale factor of -2 centre (1, 1). Find the coordinates of the image vertices.

Question 18 [5 marks]

Statistics and Probability

In a class of 30 students, 14 study French, 16 study Spanish and 5 study both French and Spanish. Every student studies French, Spanish, both, or neither.

Work out the number of students who study only French.

Work out the number of students who study neither French nor Spanish.

A student is picked at random from the class. Find the probability that the student studies both French and Spanish. Give your answer as a fraction in its simplest form.

Question 19 [5 marks]

Ratio and Proportion

A metal rod is stretched so that its length increases in direct proportion to the temperature increase. At 20 degrees C the rod is 1.25 m long. At 80 degrees C the rod is 1.253 m long. Assuming linear proportional change, find an expression for the length L (in metres) as a function of temperature t in degrees C, of the form L = m t + c. Give m and c to 6 decimal places.

Question 20 [6 marks]

Linear Algebra

A rectangle has length (2x + 5) cm and width (x + 3) cm. The perimeter of the rectangle is 46 cm.

Show that 6x + 16 = 46

Solve the equation to find the value of x.

Work out the length and the width of the rectangle.

Question 21 [7 marks]

Fractions, Decimals and Percentages

Deepa and her brother each invest £2000 for 5 years, in different types of account.

Account A pays compound interest at a rate of 2.5% per year. Deepa invests £2000 in Account A. Calculate the total amount in Account A after 5 years. Give your answer to the nearest penny.

Account B pays simple interest at a rate of 3% per year. Deepa's brother invests £2000 in Account B. Calculate the total amount in Account B after 5 years.

State which account gives the greater return after 5 years, and work out by how much.

Question 22 [1 marks]

Graphs and Coordinates

The graph of y = f(x) is transformed to give the graph of y = f(x) + 4. Which of the following correctly describes this transformation?

Question 23 [1 marks]

Quadratics

The equations y = x^2 - 1 and y = 2x + 2 are solved simultaneously. Which one of these points is a solution to both equations?

Question 24 [5 marks]

Ratio and Proportion

A metal alloy is made by mixing 300 g of metal A, which has a density of 8.4 g/cm^3, with 500 g of metal B, which has a density of 7.2 g/cm^3. Assuming there is no change in total volume when the metals are mixed, calculate the density of the alloy. Give your answer to 3 significant figures.

Question 25 [5 marks]

Graphs and Coordinates

Consider the family of cubics y = x^3 + kx^2 where k is a real constant. (a) Show that x = 0 is always a root. (b) For k = -3 find the other two roots and classify the nature of the turning points of the cubic (local max, local min or point of inflection).

For k = -3 find the other two roots. Then, by working out y at x = -1, 0, 1, 2, 3 and 4 and comparing neighbouring values, say whether the graph has a local maximum or a local minimum at x = 0 and at x = 2.

Question 26 [5 marks]

Number and Calculation

A padlock code has 4 digits. Each digit can be any number from 0 to 9.

The digits may be repeated. Work out the number of different codes possible.

The first digit of the code cannot be 0, although the other three digits may still be any digit from 0 to 9 and may repeat. Work out the number of different codes now possible.

Question 27 [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.