Higher Test 5 Revision Question 1 Skill involved: K144a: Find the area of a full circle given its radius. The diagram shows a circle, centre π, with radius 15 cm. Work out the area of the circle. Give your answer in cm 2 correct to the nearest whole number. ………………………… cm 2 (2 marks) Question 2 Skill involved: E144: Find the area of a full circle. A circle, with centre π and radius 6 cm, contains another circle, with centre π and radius π₯ cm. Write down an expression, in terms of π and π₯, for the shaded area in cm 2 . ………………………… (2 marks) Question 3 Skill involved: E147: Find the area of composite shapes by adding or subtracting areas. The diagram shows the shape π΄π΅πΆπ·πΈ. The area of the shape is 91.8cm² Work out the value of π₯. π₯ = ………………………… (4 marks) Question 5 Skill involved: K203b: Plot a quadratic graph. Below is a table of values for π¦ = π₯2 − 4π₯ + 3 π₯ π¦ −2 15 −1 8 0 3 1 0 2 −1 3 0 4 3 On the grid, draw the graph of π¦ = π₯2 − 4π₯ + 3 for values of π₯ from −2 to 4 (2 marks) Question 6 Skill involved: K203a: Complete a table of values for a quadratic graph. π₯ Complete the table of values for π¦ = π₯2 − 2 − 3 π₯ π¦ −3 7.5 −2 −1 0 1 −2.5 2 3 4.5 (3 marks) Question 7 Skill involved: K203a: Complete a table of values for a quadratic graph. Complete the table of values for π¦ = 1 + 5π₯ − π₯2 π₯ π¦ −1 0 1 1 2 7 3 7 4 5 1 6 (2 marks) Question 8 Skill involved: K206f: Raise an algebraic term to a positive integer power. 3 Simplify (4π¦2 ) ………………………… (2 marks) Question 9 Skill involved: K206f: Raise an algebraic term to a positive integer power. 4 Simplify (3π2 ) ………………………… (2 marks) Question 10 Skill involved: K206e: Laws of indices for multiplying and dividing powers requiring two steps. Simplify 2 (4π₯5 π¦3 ) (3π₯ π¦2 ) ………………………… (2 marks) Question 11 Skill involved: K186b: Change the subject of a linear formula requiring two steps. Make π the subject of the formula π = 3π + 4 π = ………………………… (2 marks) Question 12 Skill involved: K186f: Change the subject of a formula with fractions. π Make π the subject of π΄ = π¦ − 5π§ π = ………………………… (2 marks) Question 13 Skill involved: K186b: Change the subject of a linear formula requiring two steps. Make π the subject of π¦ = ππ₯ − π π = ………………………… (2 marks) Question 14 Skill involved: K186d: Change the subject of a formula with a square root. Make π₯ the subject of the formula π¦ = π √π₯ π₯ = ………………………… (2 marks) Question 15 Skill involved: K186d: Change the subject of a formula with a square root. When you are β feet above sea level, you can see π miles to the horizon, where 3β π= √2 Make β the subject of the formula. β = ………………………… Question 16 Skill involved: E186: Change the subject of a formula where the subject appears once only. The diagram shows a frustum. The diameter of the base is 3π cm and the diameter of the top is π cm. The height of the frustum is β cm. The formula for the curved surface area, π ππ2 , of the frustum is π = 2π π√β2 +π2 Rearrange the formula to make β the subject. β = ………………………… (3 marks) Question 17 Skill involved: K58a: Find the mean of listed data. Kaz rolled a dice 10 times. Here are her scores. 2 6 5 4 4 2 1 3 4 3 Work out the mean. ………………………… (2 marks) Question 18 Skill involved: K257e: Use a cumulative frequency graph to estimate values. The table gives information about the times, in minutes, taken by 80 customers to do their shopping in a supermarket. This is the cumulative frequency graph for the table. f cf Time taken (π minutes) 0 < π‘ ≤ 10 7 7 10 < π‘ ≤ 20 26 33 20 < π‘ ≤ 30 24 57 30 < π‘ ≤ 40 14 71 40 < π‘ ≤ 50 7 78 50 < π‘ ≤ 60 2 80 One of the 80 customers is chosen at random. Use your graph to find an estimate for the probability that the time taken by this customer was more than 42 minutes. (2 marks) Question 19 Skill involved: E257: Cumulative Frequency Graphs The cumulative frequency graph gives information about the waiting times, in minutes, of people with appointments at Hospital A. The median of the waiting times at Hosptial A is 23 minutes. The interquartile range of the waiting times at Hospital A is 12 minutes. At a different hospital, Hospital B, the median waiting time is 28 minutes and the interquartile range of the waiting times is 19 minutes. Compare the waiting times at Hospital A with the waiting times at Hospital B by selecting two options from the list below. The median patient at Hospital A waits for less time. [ ] On average patients wait for longer at Hospital A. [ ] There is a smaller range of waiting times at Hospital B. [ ] The spread of how long patients wait at Hospital B is much wider. [ ] (2 marks) Question 20 Skill involved: K257c: Use a cumulative frequency graph to estimate the median. The manager of a call centre asked the 120 people, who rang the call centre last week, how long they each waited before their call was answered. This is the cumulative frequency The table gives information about their replies. graph: Time waited (t minutes) Cumulative frequency 0 < π‘≤5 8 0 < π‘ ≤ 10 23 0 < π‘ ≤ 15 40 0 < π‘ ≤ 20 68 0 < π‘ ≤ 25 101 0 < π‘ ≤ 30 120 Use the graph to find an estimate for the median of the times waited. ………………………… minutes (1 mark) Using the graph, find an estimate for the percentage of the 120 people who said that they waited longer than 23 minutes before their call was answered. ………………………… % (2 marks) Question 22 Skill involved: K256b: Find the interquartile range of listed data. Diyar recorded the distance, in kilometres, that he cycled each day for 11 days. Here are his results. 8 10 12 13 5 23 21 7 5 16 14 Find the interquartile range of his results. ………………………… km (3 marks) Question 23 Skill involved: E256: Quartiles from listed data. The temperature, in degrees Celsius, of a restaurant fridge is recorded each day. The temperatures over a 13 day period were: 3.2 4.2 3.3 3.6 3.7 3.2 4.2 3.3 3.4 3.6 3.2 4.1 2.9 Calculate the interquartile range for this set of data. ………………………… (1 mark) Question 24 Skill involved: K114a: Write a number as the product of its prime factors. Write 800 as a product of its prime factors. ………………………… (2 marks) Question 25 Skill involved: K114a: Write a number as the product of its prime factors. Write 880 as a product of powers of its prime factors. ………………………… (3 marks) Question 26 Skill involved: K315b: Add or subtract surds that require simplifying. Express √40 + 4√10 + √90 as a surd in its simplest form. ………………………… (3 marks) Question 27 Skill involved: K315b: Add or subtract surds that require simplifying. Write √20 + √45 in the form π√5. ………………………… (3 marks) Question 28 Skill involved: E315: Add and subtract surds. Expand and simplify (√7 + 2)(√7 − 2) ………………………… Question 29 Skill involved: E315: Add and subtract surds. Express (5 − √8 )(1 + √2) in the form π + π√2, where π and π are integers. ………………………… (3 marks) Question 30 Skill involved: K316a: Rationalise the denominator of a fraction where the denominator is a single surd. 28 Rationalise the denominator of √7 ………………………… (2 marks) Question 31 Skill involved: K316a: Rationalise the denominator of a fraction where the denominator is a single surd. Given that π is a prime number, simplify (Could not display math) Give your answer in the form π₯ + π¦√π, where π₯ and π¦ are fractions. π₯ = ...........,π¦ = ........... (2 marks) Question 32 Skill involved: K317c: Rationalise the denominator of a fraction in the form π + √π π + √π . √12 Show that √3+2 can be written in the form π + √π where π and π are integers. π = ...........,π = ........... (3 marks) Question 33 Skill involved: K317c: Rationalise the denominator of a fraction in the form π + √π π + √π . 8 Express √5−1 in the form √π + π where π and π are integers. Show each stage of your working clearly. ………………………… (3 marks) Question 34 Skill involved: K203f: Plot a cubic graph. The table below shows some values for 1 π¦ = 2 π₯3 − 2π₯ + 3 π₯ −3 −2 −1 0 1 2 3 π¦ −4.5 3 4.5 3 1.5 3 10.5 1 On the grid, draw the graph of π¦ = 2 π₯3 − 2π₯ + 3 for −3 ≤ π₯ ≤ 3 (2 marks) Question 35 Skill involved: E299: Find the volume of more complex compound 3D shapes. A traffic bollard is in the shape of a cylinder with a hemisphere on top. The bollard has diameter 24 centimetres height 70 centimetres. Calculate the volume of the bollard. Give your answer correct to 3 significant figures. ………………………… cm 3 (5 marks) Question 36 Skill involved: E299: Find the volume of more complex compound 3D shapes. The diagram shows a storage tank. The storage tank consists of a hemisphere on top of a cylinder. The height of the cylinder is 30 metres. The radius of the cylinder is 3 metres. The radius of the hemisphere is 3 metres. Calculate the total volume of the storage tank. Give your answer correct to 3 significant figures. ………………………… m 3 (3 marks) Mark scheme Question 5 Question 1 707cm 2 Question 2 36π − π π₯2 Question 3 π₯ =4.8 Question 6 π₯ −3 −2 −1 0 π¦ 7.5 2 1 2 1.5 3 −2.5 0 4.5 Question 4 288π₯ − 36π₯2 Question 7 π₯ −1 0 1 2 3 4 5 6 π¦ -5 1 5 7 7 5 1 5 Question 8 64π¦6 3 Question 14 Question 9 π2 π¦2 81π8 Question 10 12π₯7 π¦5 Question 11 π−4 3 Question 15 2π2 3 Question 16 √π 2 −4π2 π4 4π2 π2 Question 12 π¦(π΄ + 5π§) Question 17 3.4 Question 13 π¦+π π₯ Question 18 8 80 Question 25 24 × 5 × 11 or 2 × 2 × 2 × 2 × 5 × 11 Question 19 The median patient at Hospital A waits for less time. and The spread of how long patients wait at Hospital B is much wider. Question 20 minutes Question 21 Question 26 % 9√10 Question 22 9km Question 27 5√5 Question 23 0.7 Question 28 3 Question 29 Question 24 2 × 2 × 2 × 2 × 2 × 5 × 5 or 25 × 52 1 + 3√2 Question 30 Question 34 4√7 Question 31 π₯ = 1 ,π¦ 2 = 1 10 Question 32 π = −6, π = 48 Question 35 29900cm 3 Question 33 √20 + 2 Note that the following marking instruction was also given for this question: Question 36 m3
