National 5 Mathematics Revision Homework with Worked Solutions. Alexander Forrest

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1 National 5 Mathematics Revision Homework with Worked Solutions Alexander Forrest

2 Contents Mathematics (National 5) Expressions and Formulae... Mathematics (National 5) Relationships...3 Mathematics (National 5) Applications...4 Arcs & Sectors...5 Brackets...6 Completing the Square...7 Equations and Inequalities...8 Factorisation Factorisation...10 Formulae...11 Fractions Fractions...13 Line of Best Fit...14 Money & Finance Money & Finance...16 Properties of Shapes...17 Pythagoras Theorem...18 Quadratic Check-up...19 Scale Factor and Area...0 Scale Factor and Volume...1 Similar Triangles... Simultaneous Equations...3 Standard Deviation & Boxplots...4 The Straight Line The Straight Line...6 Surds & Indices...7 Trigonometry...8 Trig Equations...9 Trig Area of triangle and exact values...30 Trig graphs...31 Vectors...34 Volume...36 Solutions...37 An attempt has been made to match these homework sheets to the relevant outcomes of the new Scottish National 4 and National 5 units, but it is stressed that this is the author s interpretation only.

3 Mathematics (National 5) Expressions and Formulae Topic Homework Sheet E&F 1.1 Working with surds Surds & Indices Pythagoras' Theorem E&F 1.1 Simplifying expressions using the laws of indices Surds & Indices E&F 1. Working with algebraic expressions involving expansion of brackets Brackets E&F 1. E&F 1. E&F 1.3 E&F 1.3 Factorising an algebraic expression Completing the square in a quadratic expression with unitary x coefficient Reducing an algebraic fraction to its simplest form Applying one of the four operations to algebraic fractions Factorisation 1 Factorisation Quadratic Check-up Completing the square Quadratic Check-up Fractions 1 Fractions Fractions 1 Fractions Formulae E&F 1.4 E&F 1.4 E&F 1.4 E&F 1.4 E&F.1 E&F. Determining the gradient of a straight line, given two points Calculating the length of arc or the area of a sector of a circle Calculating the volume of a standard solid (Rounding to a given number of significant figures) Interpreting a situation where mathematics can be used and identifying a strategy Explaining a solution and/or relating it to context The Straight Line Arcs & Sectors Properties of shapes Volume Formulae Volume Fractions Fractions E&F 1.1 E&F 1.1 E&F 1.1 E&F 1.1 Mathematics (National 4) Expressions and Formulae Using the distributive law in an expression with a numerical common Brackets factor to produce a sum of terms Simplifying an expression which has Brackets more than one variable Formulae Evaluating an expression or a formulae which has more than one variable Brackets Calculating the gradient of a straight line from horizontal and vertical distances The Straight Line 1 Page National 5 Mathematics Revision Homework A Forrest 013

4 Mathematics (National 5) Relationships Topic Homework Sheet Rel 1.1 Rel 1.1 Determining the equation of a straight line, given the gradient Working with linear equations and inequations The Straight Line 1 The Straight Line Equations and Inequalities Rel 1.1 Working with simultaneous equations Simultaneous Equations Rel 1.1 Changing the subject of a formula Formulae Recognise and determine the equation of Rel 1. a quadratic function from its graph Completing the square Rel 1. Sketching a quadratic function Completing the square Rel 1. Identifying features of a quadratic function Quadratic Check-up Rel 1.3 Working with quadratic equations Quadratic Check-up Rel 1.4 Applying the Pythagoras theorem Pythagoras' Theorem Rel 1.4 Rel 1.4 Rel 1.5 Rel 1.5 Rel.1 Rel. Applying the properties of shapes to determine an angle involving at least two steps Using similarity Working with the graphs of trigonometric functions Working with trigonometric relationships in degrees Interpreting a situation where mathematics can be used and identifying a strategy Explaining a solution and relating it to context Properties of shapes Scale Factor and Area Scale Factor and Volume Similar Triangles Trig Equations Trig Equations Trig Area of triangle and exact values Trig graphs Properties of shapes Properties of shapes Rel 1.1 Mathematics (National 4) Relationships Drawing and recognising a graph of a linear equation. The Straight Line 1 Rel 1.1 Solving linear equations. Equations and Inequalities Rel 1.1 Changing the subject of a formula. Formulae Rel 1. Using Pythagoras theorem Pythagoras' Theorem Rel 1.4 Drawing and applying a best-fitting straight line Line of Best Fit National 5 Mathematics Revision Homework A Forrest 013 Page 3

5 Mathematics (National 5) Applications Topic Homework Sheet Calculating the area of a triangle using Trigonometry Apps 1.1 trigonometry Trig Area of triangle and exact values Using the sine and cosine rules to find a Apps 1.1 Trigonometry side or angle Apps 1.1 Using bearings with trigonometry Trigonometry Apps 1. Working with D vectors Vectors Apps 1. Working with 3D coordinates Vectors Apps 1. Using vector components Vectors Apps 1.3 Working with percentages Money & Finance - Worksheet 1 Apps 1.3 Working with fractions Fractions 1 Formulae Apps 1.4 Comparing data sets using statistics Standard Deviation & Boxplots Forming a linear model from a given set Apps 1.4 Line of Best Fit of data Interpreting a situation where Apps.1 mathematics can be used and identifying Money & Finance - Worksheet a strategy Explaining a solution and/or relating it to Money & Finance - Worksheet Apps. context Vectors Page 4 National 5 Mathematics Revision Homework A Forrest 013

6 Arcs & Sectors 1) Calculate the area of sector OAB. A ) Calculate the length of arc AB. (Give your answers to dp.) O 6cm 60 B 3) Find angles A ĈB and A ÔB. (Give your answers to 1 dp.) B 6 cm A 7cm O C 4) What is the length of chord AB? (Give your answer to dp.) 5) What is the area of ΔABC? (Give your answer to dp.) E&F 1.4 National 5 Mathematics Revision Homework A Forrest 013 Page 5

7 Nat 4 E&F 1.1 Brackets 1) Simplify the following : a) 5x 3x b) y 6y c) - z - z d) k - k e) 5(m n) 3( m + n) f) 3( a + b) = ( a 5b) ) Find the sum of : a) x + 3y and -6x + 1y b) p 3q r and 5p + 3q + r c) 4c 6d + 5e and c 3d 7e d) x - x - 4 and 3x x + 5 3) Simplify the following: a) 4x 4x x x b) k( k 1) k( k + ) c) x(x 3) x(x +) + (x + x) d) (x -x) +3(x + x) (x + x) 4) Simplify the following: a) (a + )(b + 5) b) (x 1)( y 4) c) (x + 3)(x x 3) d) (a + b) ( a b) e) (x + 3) (3x + ) f) 1 1 x x x x 5) Writing.01 as , show that.01 = 4.04 to two decimal places. 6) Prove that ( a + b)(a ab + b ) = a 3 + b 3. Page 6 National 5 Mathematics Revision Homework A Forrest 013

8 Completing the Square 1 ) Write each expression in the form (x + p ) + q a x x y y z z ) 6 10 b) 3 c) 8 10 d a a b b c c ) 10 5 e) f) 40 1 ) Write each expression in the form p - (x + q ) a) 4 x- x b) 5 4x- x c) x- x d) 6 3x- x e) x- x- x f) 4x- 6 - x 3) Write each expression in completed square form a x x y y z z ) 4 1 b) c) d a a b b c c ) 1 e) f) 3 6 4) (i) (ii) (iii) For each function : Write down the equation of the axis of symmetry; Write down the coordinates of the turning point; and Sketch the graph. a) y x 3 b) y x c) y x 1 d) y 10 x 3 E&F 1. National 5 Mathematics Revision Homework A Forrest 013 Page 7

Solve 1) 6x 8 6 ) 5x -15 30 3) 7-4x 8 7x- 9 4) 3(x 6) 4( x 8) 5) 3(-3y y) 5(1-y 8)-6y 6) x 8 6 7) 3x -15 30 8) 7 4x 8 7x 7 9) 3(x- 6) 4( x- 8) 10) 3( x) 5(1 x- 8) Equations and Inequalities 11) The

9 Solve 1) 6x 8 6 ) 5x ) 7-4x 8 7x- 9 4) 3(x 6) 4( x 8) 5) 3(-3y y) 5(1-y 8)-6y 6) x 8 6 7) 3x ) 7 4x 8 7x 7 9) 3(x- 6) 4( x- 8) 10) 3( x) 5(1 x- 8) Equations and Inequalities 11) The graph below shows the equation 3x +5 < x + 1. Which of the two points A( -6,) or B( -3,1) lie within the solution set? E&F 1. Nat 4 Rel 1.1 Rel 1.1 Rel 1. Page 8 National 5 Mathematics Revision Homework A Forrest 013

10 Factorisation 1 1) Remove the brackets a) ( x 4)( x 5) b) (x 5)( x 1) c) ( a 5)( a 5) d) ( x 4)(3x 15) ) Copy and complete 3) Factorise ) 7 1 ( 4)( ) a x x x x ) 6 5 ( )( ) b x x x x ) 8 15 ( 3)( ) c x x x x ) 1 0 ( )( ) d x x x a x x ) 3 b x x ) c b b ) 10 5 d x x ) 5 36 e c c ) 9 8 f r r ) 6 7 g x x ) 4 45 h z z ) 71 7 i) 4 11x x j) 5 10a a 4) Factorise a x x ) 5 3 b a a ) 6 7 c c c ) d x x ) 19 9 e c c ) f r r ) g x x ) 1 h z z ) i) 1 3x 18x j) 15 7p p 5) Factorise (Difference of two squares) a) b) c) x t x 36 4 y d) 16x 9y e) 4c 100 National 5 Mathematics Revision Homework A Forrest 013 Page 9

11 Factorisation Factorise Fully a) 8x 18y b) a 1 b c) 3a ab b d) a b 4c e) 5x 9xy y f x x ) g x x ) h x x ) 3 1 i x x ) j x 4 ) 16 k x x ) l x x 3 ) 4 m) 5x 0y n) ab ac Expand a) x 3y 3x 4y b) 3x y x y c) x 5y 3x y d) x 3y e) 5x 3y Simplify a) 3x x 5 x 6x 9 b) 4 a 3c 5d 3 a 5c 4d c) 3x y x 3y d) 75 5 e) x x 3 1 6x 3x E&F 1. Page 10 National 5 Mathematics Revision Homework A Forrest 013

12 Formulae 1) The cost of tuition, c, is 0 per hour, plus a booking fee of 5. a) Write a formula to express the cost of tuition for x hours. b) Use the formula to calculate the cost of 1 hours tuition. ) The formula E k = ½mv is used to calculate the kinetic energy (E k ) of an object with mass (m) kilograms travelling at velocity (v) metres per second. Use the formula to calculate the energy of an object of mass 10 kg travelling at velocity 15m/s. 3) Use Newton s Law of Gravitation, Mm F G R to calculate the gravitational force between Earth and the Moon. Give your answer correct to 3 significant figures. M = 5.97x 10 4 Kg (Mass of the Earth.) m = x 10 Kg ( Mass of the Moon.) G = 6.67 x N(m/kg) (Gravitational Constant.) R = km (Distance between Earth and the Moon.) 4) Rearrange the formula v = u + at to make t the subject. Mm 5) Rearrange the formula F G to make R the subject. R 6) A square and a triangle both have the same base and an area of 81 cm. Calculate the dimensions of the triangle. 81cm 81cm E&F 1. National 5 Mathematics Revision Homework A Forrest 013 Page 11

13 Fractions 1 E&F 1.3 E&F 1.1 Simplify the following : 1 0 3a 1x 7y 1) ) 3) 4) 5) ab 48 81y Change to mixed numbers: ) 7) 8) 9) 10) Change to improper (top heavy) fractions: ) 1 1) 13) 5 14) 3 15) Simplify the following : ) 17) 18) ) 4 3 0) 9 3 x y Nat 4 Rel 1.1 Nat 4 E&F 1.1 E&F 1.3 E&F 1.4 Apps 1.3 Apps 1.3 Rel 1.1 Page 1 National 5 Mathematics Revision Homework A Forrest 013

14 Fractions Simplify the following : 4x 6 1a 8 14a 10b 1) ) 3) 4 3 8x 16y 18a 1a 3x 4) 5) 6) 6a xy xz 3p 3 a 4 x y 7) 8) 9) p p 1 a x y Solve the equations: x 6 a 4 3 a 3 3a 10 10) =10 11) = 1) = ) A fox broke into a henhouse and killed half of the chickens. It then took another one and left. A mink then slunk in and killed half of the remaining flock, leaving 6 terrified chickens. How many chickens were there before the break in? E&F National 5 Mathematics Revision Homework A Forrest 013 Page 13

15 Line of Best Fit Test scores for an S4 maths and physics test are shown below: Pupil Maths Physics They are plotted on a scatter graph. Maths and Physics scores Physics Score Maths Score a) Is there a correlation between scoring well in maths and physics? Explain your answer. b) Draw a line of best fit on the scatter graph and use it to find an equation linking the physics and maths test results for this data set. c) Use your equation to predict the physics test score for a pupil who scored 55 in the maths test. Apps 1.3 Apps 1.4 Page 14 National 5 Mathematics Revision Homework A Forrest 013

16 Money & Finance 1 1) Glenfox Lodge was valued at 365,000 on 30 th April 001. If appreciation is 4% per annum, what is the value of Glenfox Lodge on 30 th April 011? ) A car was bought for 8000 in Each year, it depreciated in value by 0%. What was the car worth 4 years later? 3) Calculate the compound interest earned on: a) 500 for 3 years at 6% per annum. b) 875 for years at 4.5% per annum c) 550 for 7 months at 6.% 4) Greg Gregson earns 96 per week. He pays 6% superannuation. a) How much superannuation does he pay per week? b) How much superannuation does he pay per year? National Insurance is payable at % on the first 56 earned and 9% on the rest. c) How much National Insurance does he pay per week? d) How much National Insurance does he pay per year? He also pays tax and.80 union dues each week. e) What is his weekly net pay? 5) HMRC Income tax rates Basic rate ,400 at 0 per cent ,000 at 0 per cent. Higher rate Additional rate 37, ,000 at 40 per cent. Over 150,000 at 50 per cent. 35, ,000 at 40 per cent. Over 150,000 at 50 per cent. Charlie Charleson has a personal allowance of 7,475.Her salary is 44,350. a) Calculate her annual income tax for the tax years 010/11 and 011/1. b) How much more tax does she have to pay in 011/01? c) Express this increase as a percentage of her annual income tax for 010/11. d) Calculate her monthly income tax for 011/1. Nat 4 Rel 1.4 Apps 1.3 National 5 Mathematics Revision Homework A Forrest 013 Page 15

17 Money & Finance 1) You wish to buy a new car which has a cash price of 13,335. The following options are available: Finance Package Deposit Plus 35 monthly payments of 199 and walk away. Final GFV payment to keep the vehicle / future trade in. Bank of Bod Loan at fixed rate of 3.% pa for 36 months. Which is the best option? Give reasons for your answer. ) A high street electrical retailer offers a television set for 700. Hire purchase is available on the product for a 10% deposit and years of monthly payments. The total amount payable is (i) (ii) Calculate the monthly repayment due. Express the cost of the hp as a percentage of the cost of the television set. Page 16 National 5 Mathematics Revision Homework A Forrest 013

18 Properties of Shapes 1) Find the area of quadrilateral ACOB in terms of r and x by considering it as two rightangled triangles. ) Hence show that an expression in terms of r and x for the shaded area is: Shaded area = r x tan x 180 B r O x x A C Remember to justify all the steps of your working and write your solution so that it is clearly understood by anyone who reads it. Does the solution read well? Is there sufficient explanation? The reader should be led, line by line, through your argument. Rel Apps.1. E&F 1.4 National 5 Mathematics Revision Homework A Forrest 013 Page 17

19 Pythagoras Theorem 1) Use Pythagoras Theorem to find x for each triangle. Round any decimals answers to 1 decimal place. Give the answers to h, i and j as exact values in their simplest form. 3 a) b) c) 1 6 x 15 x 5 1 x d) e) f) 3 9 x 8 11 x 7 x 13 g) h) i) 1 x 4 j) 7 x x x ) Is it possible to have a right-angled triangle with sides 5, 7 and 11 cm? Give reasons for your answer. Page 18 National 5 Mathematics Revision Homework A Forrest 013

20 Quadratic Check-up 1) Solve each of the following quadratic equations by factorisation: ) 3 0 a x x ) 0 b x x ) c x x ) 1 0 d x x ) e x x ) f x x ) g x x ) h x x ) Solve the following by using the quadratic formula Give answers, where they exist, to dp. ) a x x ) 5 0 b x x ) c x x ) 1 0 d x x ) 0 e x x ) f x x ) g x x ) 0 h x x b x b 4ac a 3) Sketch each of the following quadratics on separate diagrams. Clearly mark the roots and y-intercept on each diagram. a y x x ) 1 b y x x ) 1 c y x x ) 3 8 d y x x ) 8 e y x ) Nat 4 Rel 1. E&F 1.1 Rel 1.4 National 5 Mathematics Revision Homework A Forrest 013 Page 19

21 1) A photograph is enlarged in the ratio 4:1. What is the area of the enlargement? Scale Factor and Area 8.5 cm x cm ) What is the area of the larger circle? 8.5 cm x cm cm 4 cm 3) What is the area of the smaller similar triangle? x cm 5 cm cm 4) A photograph is reduced in the ratio :3. What is the area of the smaller photograph? 4 cm 4 cm x cm Rel1. Rel 1.3 E&F 1. Page 0 National 5 Mathematics Revision Homework A Forrest 013

22 1) What is the volume of the enlargement? Scale Factor and Volume 15 cm 3 x cm 3 3 cm 9 cm ) What is the volume of the larger can? 150 x cm 3 cm 3 cm 4 cm 3) Orangi is a new drink sold in similar triangular cartons. What is the volume of the smaller carton? 640 cm 3 x cm 3 5cm 0 cm Rel 1.4 National 5 Mathematics Revision Homework A Forrest 013 Page 1

23 Similar Triangles For each of the following isosceles triangles: Sketch the two similar triangles; and Find the value of the missing length x. 1) 0 cm 4 cm 5 cm x cm ) 1 cm x cm cm 6 cm Rel 1.4 Page National 5 Mathematics Revision Homework A Forrest 013

24 Simultaneous Equations 1) Use the graph to write down the solution to the simultaneous equations y= 3x +5 and y = -x. ) Solve the equations: a) b) c) x 3y 13 5x 3y 7 x 5y 39 x 3y 3 3x 4y 7 4x 3y 4 3) The total cost of an order for three hamburger meals and two chicken meals is The total cost of an order for two hamburger meals and three chicken meals is Find the individual costs for each meal. Rel 1.1 Rel 1.4 National 5 Mathematics Revision Homework A Forrest 013 Page 3

25 Standard Deviation & Boxplots 1) The number of punishment exercises handed out to pupils in various schools on a certain day were counted in a survey: a) Calculate the standard deviation. b) Calculate a 5 figure summary of the data. c) Calculate the range. d) Calculate the interquartile range. e) Calculate the semi interquartile range. f) Draw a boxplot of the data. ) A second survey was carried out one week later a) Calculate the standard deviation. b) Calculate a 5 figure summary of the data. c) Calculate the range. d) Calculate the interquartile range. e) Calculate the semi interquartile range. f) Draw a boxplot of the data on the same axis as the one drawn in question (1)(f) above. 3) What do you notice when you compare the two sets of data? s / x x n n-1 Page 4 National 5 Mathematics Revision Homework A Forrest 013

26 The Straight Line 1 1) Find the gradient, y-intercept and equation of each of the following lines : y a) b) y x x ) Write down the equation of the line which passes through the point (0, 5) with gradient 3. 3) For each of the following, copy and complete the table and sketch each line. a) y = 4x 3 x y b) y - 6x 4 = 0 x y Apps 1.4 National 5 Mathematics Revision Homework A Forrest 013 Page 5

27 1) Find the gradient of the lines joining: a) P(-1, 7) to Q( 5,10) b) A(-6,-) to B(, -4) c) V( 1,-) to W( 5, 0) The Straight Line What can you say about the lines PQ and VW? ) Write down the equation of the line which passes through (0, 5) with gradient 3. 3) For each of the following, plot 3 or more points and sketch each line: a) y = 4x 3 b) 6y + 3x 18 = 0 4) Write down the equation of the graph below: Nat 4 Rel 1.1 Nat 4 E&F 1.1 Rel 1.1 Page 6 National 5 Mathematics Revision Homework A Forrest 013

28 Surds & Indices 1) Simplify the following : a) b) 40 c) 00 d) 189 e) 8181 ) Rationalise the denominator, expressing your answer in its simplest form: a) b) c) d) e) ) Calculate each of the following, expressing your answer with a rationalised denominator in its simplest form: a) b) c) d) ) Calculate: a) 3 4 b) 5 3 c) 7 5 x 3 5 d) -3 e) 4-5 f) 3 4 x 3 4 g) (5 3 ) 4 h) ) Write out the following as roots: a) 3 1/ b) 5-1/3 c) 7 /3 d) -3/4 e) 4 5/8 Rel 1.1 E&F 1.4 National 5 Mathematics Revision Homework A Forrest 013 Page 7

29 Trigonometry Apps 1.1 1) Calculate the missing values a) 7 cm x cm b) c) x cm x cm 1 cm 7 cm 70 8 cm ) Calculate the bearing of the robot from the jogger. 150 m 168 m 5 m E&F 1.1 Page 8 National 5 Mathematics Revision Homework A Forrest 013

30 Trig Equations Solve 1) sin x x 360 ) cosx 1 0 x 70 3) cosx 1 0 x 360 4) 3cosx 1 0 x 70 5) 8cosx x 360 6) 1sinx x 360 7) sin x x 360 8) cosx x 70 9) cos x 1 0 x ) tanx x 70 11) Write down the equation of the following graph and state the values of x that give the solution y =1 for 0 x 450. Rel 1.5 National 5 Mathematics Revision Homework A Forrest 013 Page 9

31 1) Calculate the area of the triangle. Trig Area of triangle and exact values 7 cm 50 8 cm ) Given that the area of the triangle below is 74 cm, calculate the angle θ. Give your answer to 1 dp. 10 cm 15 cm θ 3) Find the exact values of a) sin 135 b) cos 5 c) tan40 d) sin10 e) cos 300 f) tan315 Rel 1.5 Page 30 National 5 Mathematics Revision Homework A Forrest 013

32 Trig graphs For each of the following six graphs, write down its equation, amplitude and period. 1) ) Rel 1.5 National 5 Mathematics Revision Homework A Forrest 013 Page 31

33 3) 4) Page 3 National 5 Mathematics Revision Homework A Forrest 013

34 5) 6. National 5 Mathematics Revision Homework A Forrest 013 Page 33

Vectors 1) Write out each of the following vectors in component form. ) In the following picture, a Helicopter (H) must take on some fuel at (F) before rescuing the canoeist (C).

35 Vectors 1) Write out each of the following vectors in component form. ) In the following picture, a Helicopter (H) must take on some fuel at (F) before rescuing the canoeist (C). North is in the direction of the y axis. Distance is in units. Given the points C( -3, 1) F(0,0) and H( 3,3) :- a) Work out the individual components of the helicopter flight and sum them to find the resultant vector HC. b) Use the answer to a to state the direction and how many units the helicopter has travelled from its original starting place. c) Sketch the vector diagram of the helicopter flight, clearly showing the resultant vector. Page 34 National 5 Mathematics Revision Homework A Forrest 013

36 3) Write down the co-ordinates of corners B, C, D, E, F and G. Apps ) The vector b is applied to the point Z (3,-1, 5) to make 1 ZP. What are the co-ordinates of P? 5) 5 8 a b 4 c d Calculate a + b + c - d. National 5 Mathematics Revision Homework A Forrest 013 Page 35

37 Volume Calculate the volume of the following, giving your answers correct to significant figures : 1) ) 7cm 7cm 16cm 6 cm 3) V V V sphere cone cylinder 4 r r h 3 r h cm 4) The volume of this globe is 430 cm 3. What is its radius? Page 36 National 5 Mathematics Revision Homework A Forrest 013

Sample : 6 worksheets without solutions

Sample : 6 worksheets without solutions Contents Mathematics (National 5) Expressions and Formulae... Mathematics (National 5) Relationships...3 Mathematics (National 5) Applications...4 Arcs & Sectors...5 Brackets...6 Completing the Square...7