Higher Stretch B

Question 1 [1 marks]

Pythagoras and Trigonometry

The diagram shows a right-angled triangle with the right angle marked. The sides are labelled p, q and r, where r is the side opposite the right angle. Diagram: right-angled triangle, right angle shown at the bottom-left corner, side p along the base, side q up the vertical side, side r as the slanted side joining the two ends.

Question 2 [1 marks]

Linear Algebra

Solve 36 = 4x

Question 3 [1 marks]

Statistics and Probability

A market trader sells sun hats. State the type of correlation you would expect between the number of hours of sunshine in a day and the number of sun hats sold that day.

Question 4 [2 marks]

Number and Calculation

A value v = 3.80, correct to 3 significant figures. Write down the error interval for v.

Question 5 [3 marks]

Ratio and Proportion

Mrs Okafor is comparing two bags of flour. Bag A: 750 g for £1.05. Bag B: 1.2 kg for £1.56.

Question 6 [3 marks]

Quadratics

Solve x^2 + 3x - 5 = 0. Give your solutions correct to 2 decimal places.

Question 7 [1 marks]

Indices and Standard Form

720000 is to be written in standard form. Put a ring around the number that is correctly written in standard form.

Question 8 [1 marks]

Linear Algebra

Solve y - 5 = 9.

Question 9 [2 marks]

Sequences

The nth term of a sequence is 2n + 7. Work out which term of the sequence is equal to 35.

Question 10 [2 marks]

Fractions, Decimals and Percentages

Work out 2 2/5 + 1 4/5. Give your answer as a mixed number.

Question 11 [2 marks]

Area, Volume and Measures

A parallelogram has an area of 108 cm^2 and a base of 9 cm. Work out the perpendicular height of the parallelogram.

Question 12 [2 marks]

Ratio and Proportion

Work out 150 US dollars in pounds. Use the exchange rate 1 GBP = 1.35 USD. Give your answer to 2 decimal places.

Question 13 [2 marks]

Linear Algebra

Solve 12 - 2x >= 4 Show your working.

Question 14 [2 marks]

Ratio and Proportion

The ratio of the length of Path A to the length of Path B is 15:40.

Simplify the ratio 15:40 fully.

Using your simplified ratio, write the length of Path A as a fraction of the length of Path B.

Question 15 [3 marks]

Graphs and Coordinates

Given points A(-1, 2), B(3, 5) and C(6, 1) are three consecutive vertices of a parallelogram ABCD in that order. Find the coordinates of D.

Question 16 [3 marks]

Area, Volume and Measures

A large rectangle is 18 cm by 10 cm. A central rectangle of 8 cm by 6 cm is not shaded. Find the area of the shaded part of the large rectangle.

Question 17 [3 marks]

Pythagoras and Trigonometry

For each right-angled triangle described below, write down which trigonometric ratio (sine, cosine or tangent) should be used.

The hypotenuse and angle theta are known. The side opposite theta is unknown. Which ratio should be used to find it?

The hypotenuse and angle theta are known. The side adjacent to theta is unknown. Which ratio should be used to find it?

The side opposite theta and the side adjacent to theta are known. Angle theta is unknown. Which ratio should be used to find it?

Question 18 [3 marks]

Area, Volume and Measures

A cube has volume 343 cm^3. Work out the total surface area of the cube.

Question 19 [4 marks]

Angles and Geometrical Reasoning

ABCD is a cyclic quadrilateral. Side AB is extended to a point E, so that angle CBE is the exterior angle of the quadrilateral at B. Angle CBE = 115 degrees. Diagram: circle with cyclic quadrilateral ABCD drawn on the circumference in order A, B, C, D, side AB extended beyond B to point E, angle CBE = 115 degrees marked at B.

Work out the size of angle ADC. Give a reason for your answer.

Work out the size of angle ABC.

Question 20 [4 marks]

Pythagoras and Trigonometry

A right-angled triangle PQR has right angle at Q. Side PQ = 8.0 cm and PR = 13.0 cm. Find QR and then find angle R to the nearest degree.

Find QR to 3 significant figures.

Find angle R to the nearest degree.

Question 21 [4 marks]

Statistics and Probability

The box plot below shows the number of goals scored by a football team in each of its last 32 matches.

Write down the median number of goals scored.

Work out the interquartile range of the number of goals scored.

The team is said to have had an "attacking game" if it scored more goals than the upper quartile. Estimate the number of matches, out of the 32, in which the team had an attacking game.

Question 22 [4 marks]

Linear Algebra

The sum of three consecutive odd integers is 51. Let the integers be n, n + 2 and n + 4. Form an equation and solve for n. State the three integers.

Question 23 [5 marks]

Statistics and Probability

Tom plays a game twice. In each round, the probability that Tom wins is 0.4, independently of any other round.

Complete the probability tree diagram for the two rounds by writing the missing probability on each of the four second-stage branches.

Work out the probability that Tom wins exactly one of the two rounds.

Question 24 [5 marks]

Area, Volume and Measures

A company sells two mathematically similar spherical balloons. The small balloon has diameter 20 cm and costs £3.50. The large balloon has diameter 35 cm and costs £9.50. The company intends the cost of each balloon to be proportional to the amount of material used to make it (its surface area).

Work out the surface area scale factor from the small balloon to the large balloon.

Use your answer to part (a) to work out what the large balloon should cost if the cost is exactly proportional to surface area.

The large balloon actually costs £9.50. Comment on whether the large balloon is good value for money compared with the small balloon, giving a reason.

Question 25 [5 marks]

Fractions, Decimals and Percentages

Grace's salary was £28000. It increased by 4% in year 1, by 3% in year 2, and by 2.5% in year 3.

Work out Grace's salary after year 3. Give your answer to the nearest penny.

Work out the overall percentage increase in Grace's salary over the three years. Give your answer correct to 1 decimal place.

Question 26 [5 marks]

Ratio and Proportion

It takes 6 builders 15 days to build a wall. Assume all builders work at the same constant rate, and the time taken is inversely proportional to the number of builders.

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

The wall needs to be completed in 5 days. Work out the minimum number of builders needed.

Question 27 [2 marks]

Number and Calculation

Explain why rounding 148 to 1 significant figure gives 100, and not 150.

Question 28 [3 marks]

Quadratics

n is a positive integer. Two consecutive odd numbers can be written as (2n + 1) and (2n - 1). Prove that the difference of their squares is always a multiple of 8.

Question 29 [4 marks]

Linear Algebra

Simplify fully: [1/x + 1/y] divided by [1/x - 1/y]. Give your answer as a single fraction in terms of x and y.

Question 30 [5 marks]

Graphs and Coordinates

The table shows values of f(x) = x^3 - 3x + 1 for integer values of x from -2 to 2. x : -2 , -1 , 0 , 1 , 2 f(x) : -1 , 3 , 1 , -1 , 3

Show that f(-1) = 3.

Using the table, write down the three intervals (each of width 1) in which the equation x^3 - 3x + 1 = 0 has a solution.

State the total number of real solutions to the equation x^3 - 3x + 1 = 0.

Question 31 [6 marks]

Ratio and Proportion

A shop sells fabric by the metre. The price per metre is p pounds. There is a special offer: buy at least 2 metres and get 15% off the total price; buy 5 or more metres and get 25% off the total price. (a) For a customer buying 4 metres, show in terms of p the total cost after discount. (b) For what values of p (p > 0) will the cost for 4 metres after that discount be less than £40? Give your answer as an inequality and solve.

(a) Show the total cost after discount for 4 metres in terms of p.

(b) For what values of p will the cost for 4 metres after discount be less than £40? Solve the inequality.

Question 32 [7 marks]

Graphs and Coordinates

A circle with centre O, the origin, has equation x^2 + y^2 = 25 P is the point (-3, 4).

Find the equation of the tangent to the circle at P.

This tangent intersects the line with equation y = 2x + 1 at a single point. Find the coordinates of this point.