N5 R1.1 Linear Equations - Revision

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1 N5 R Linear Equations - Revision This revision pack covers the skills at Unit Assessment and eam level for Linear Equations so ou can evaluate our learning of this outcome. It is important that ou prepare for Unit Assessments but ou should also remember that the final eam is considerabl more challenging, thus practice of eam content throughout the course is essential for success. The SQA does not currentl allow for the creation of practice assessments that mirror the real assessments so ou should make sure our knowledge covers the sub skills listed below in order to achieve success in assessments as these revision packs will not cover ever possible question that could arise in an assessment. Topic Unit Sub skills Questions Equation of a straight line R Use of formula b m a or equivalent to find the equation of a line given one point and the gradient Use of functional notation f - 6 This will be covered in S4 during R: Quadratics Identif gradient and intercept from m c, 7a, 4d Identif gradient and intercept from various forms of the equation of a straight line 7b - d, 8 Equations and inequations Simultaneous Equations Rearranging formulae R R R Solve an equation or inequation where the coefficients are a member of Z and solutions are a member of Q Solve an equation or inequation where the coefficients are a member of Q 0-6 Construct equations from tet in contet 9 a-b, 4, 6a-b, 7, 8, 7 Solve a simultaneous equation graphicall 9, 0 Solve a simultaneous equation algebraicall 7 Rearrange a linear equation 8 0 Rearrange an equation involving a square or square root - 5 When attempting a question, this ke will give ou additional important information. Ke Note Question is at unit assessment level, a similar question could appear in a unit assessment or an eam. Question is at eam level, a question of similar difficult will onl appear in an eam. * C The question includes a reasoning element and tpicall makes a question more challenging. Both the Unit Assessment and eam will have reasoning questions. If a star is placed beside one of the above smbols that indicates the question involves sub skills from previousl learnt topics. If ou struggle with this question ou should go back and review that topic, reference to the topic will be in the marking scheme. Question should be completed without a calculator. Question should be completed with a calculator. Questions will be ordered b sub skill and tpicall will start of easier and then get more challenging. Some questions ma also cover several sub skills from this outcome or even include sub skills from previousl learnt topics (denoted with a *). Questions are gathered from multiple sources including ones we have created and from past papers. Etra challenge questions are for etension and are not essential for either Unit Assessment or eam preparation.

2 FORMULAE LIST The roots of a b c 0 are b b 4ac a Sine rule: a sin A b sin B c sin C Cosine rule: a b c bccos A or b cos A c a bc Area of a triangle: A absin C Volume of a sphere: V 4 r Volume of a cone: V r h Volume of a pramid: V Ah Standard deviation: s n n n, where n is the sample size.

3 Q Questions Marks Determine the equation of the following lines given the gradient and the point on the line. (a) Gradient of 5, passing through (b) Gradient of 6, 8 4, 6, passing through (c) Gradient of, 5, passing through, 7 A straight line passes through the points and, 5 (a) Find the equation of this line (b) State the coordinates of the -intercept of this line. A straight line cuts the shown. 9, 0 -ais at the point and the 0, 8 -ais at the point as Find the equation of this line.

4 4 Teams taking part in a quiz answer questions on film and sport. This scatter graph shows the scores of some of the teams. A line of best fit is drawn as shown above. (a) Find the equation of this best-fitting straight line. (b) Use this equation to estimate the sport score for a team with a film score of 0. (c) Use this equation to estimate the film score of a different team with a sports score of 5 5 A tai fare consists of a call-out charge of 80 plus a fied cost per kilometre. A journe of 4 kilometres costs 6 60 (a) Find the equation of the straight line. 4 (b) Calculate the fare for a journe of 7 kilometres. 6 Sketch a straight line with the equation a b satisfing the following two conditions: a 0 b 0

5 7 In each of the equations below, identif the gradient and state the coordinates of the -intercept. (a) (b) 4 6 (c) 4 (d) Four straight line graphs are shown below A B C D Which one of these above could represent the line with equation? Give two reasons to justif our answer. 9 Solve the following equations and inequations. (a) (b) z (c) (d) Solve the equation 5 Solve the inequalit 5 4

6 * Part of the graph of the straight line with equation, is shown below. B 0 (a) Find the coordinates of the point B. (b) For what values of is 0 * Jane enters a two-part race. (a) She ccles for hours at a speed of Write down an epression in for the distance ccled. 8 kilometres per hour. (b) She then runs for 0 minutes at a speed of kilometres per hour. Write down an epression in for the distance run. (c) The total distance of the race is 46 kilometres. Calculate Jane s ccling speed. 4 4 * Two triangles have dimensions as shown. All units are in centimetres. The triangles are equal in area. Calculate algebraicall the value of. 4 5 C Tom and Samia are paid the same hourl rate. Harr is paid more per hour than Tom. Tom worked 5 hours, Samia worked 8 hours and Harr worked hours. The were paid a total of 49. How much was Tom Paid?

7 6 There are two different options to pa entr into a local snooker club. The first option is to pa Let (a) 7 each time ou enter. be the number of times that ou visit this club. Write an epression for the cost of visiting the snooker club times using the first option. The second option is to bu a discount card with the following costs and benefits. A one off fee of 50 for the card A reduced cost per visit The first 5 visits are free. 4 (b) Write an epression for the cost of visiting the snooker club times using the second option, where. 5 (c) Find algebraicall the minimum number of times ou would have to visit the club in order for the discount card to be the cheaper option. 4 7 C 5 To hire a car costs per da plus a mileage charge. The first 00 miles are free with each additional mile charged at CAR HIRE 5 per da pence. first 00 miles free each additional mile p (a) Calculate the cost of hiring a car for 4 das when the mileage is 640 miles. (b) A car is hired for d das and the mileage is m miles where m 00. Write down a formula for the cost C of hiring the car. 8 C Quick-Smile Photographers charge the following rates: 50 pence per photograph for the first photographs printed 5p per photograph for an further photographs printed 4 5 for a CD of the photographs. (a) How much does it cost to have 6 photographs printed plus a CD? (b) Find a formula for C, the cost in pounds, of having photographs printed (where is greater than ) plus a CD.

8 9 The graph opposite shows the line with equation. (a) (b) Cop the diagram and on the same diagram, sketch the line Use our sketch to solve graphicall the sstem of equations 0 The graph below shows the line = (a) Cop the diagram and sketch the line 6 (b) Use our graph to solve the sstem of equations 6 6 Solve the following sstem of equations algebraicall. (a) (b) 7 8

Brian, Moll and their four children visit Waterworld. The total cost of their tickets is 56 (a) Let a pounds be the cost of an adult s ticket and c pounds the cost of a child s ticket.

9 Brian, Moll and their four children visit Waterworld. The total cost of their tickets is 56 (a) Let a pounds be the cost of an adult s ticket and c pounds the cost of a child s ticket. Write down an equation in terms of a and c to illustrate this information. Sarah and her three children visit Waterworld. The total cost of their tickets is 6 (b) Write down an equation in terms of a and c to illustrate this information. (c) (i) Find the cost of one adult ticket and one child ticket (ii) James and his four children also visit Waterworld. Find the cost of their visit. A Cinema has 00 seats which are either standard or delue. A standard seat costs 4 and a delue seat costs. (a) Let 6 be the number of standard seats and be the number of delue seats. Write down an equation to illustrate this information. (b) 80 When all the seats are sold the ticket sales total. Write down an equation to illustrate this information. (c) How man standard seats and how man delue seats are in the cinema? 4 Aaron saves 50 pence and 0 pence coins in his pigg bank. Let be the number of 50 pence coins in his bank. Let be the number of 0 pence coins in his bank. (a) There are 60 coins in his bank. Write down an equation in and to illustrate this information. (b) The total value of the coins is 7 40 Write down another equation in and to illustrate this information. (c) Hence find algebraicall the number of 50 pence coins Aaron has in his pigg bank.

10 5 A straight line has equation (a) The point (, 7) m c lies on this line., where m and c are constants. Write down an equation in m and c to illustrate this information. (b) A second point ( 4, 7) lies on this line. Write down an equation in m and c to illustrate this information. (c) Hence calculate algebraicall the values of m and c. (d) Write down the gradient of this line. 6 The graph below shows the two straight lines. 4 = - P 0 + = 4 The lines intersect at the point P. Find algebraicall the coordinates of P. 4

11 7 In triangle PQR: PQ PR 5 QR centimetres centimetres centimetres. P 5 R Q (a) The perimeter of the triangle is 4 centimetres. Write down an equation in and to illustrate this information. (b) PR is centimetres linger than QR. Write down an equation in and to illustrate this information. (c) Hence calculate the values of and. 8 In each of the following equations, change the subject of the formula to k. k (a) G (b) R hk (c) C v m k 9 0 Change the subject of the formula Change the subject of the formula Change the subject of the formula Change the subject of the formula Change the subject of the formula 7 4 t s to s. ( m 4) P to m. W BH to H. m L to m. k kd f to d. 0

12 4 Change the subject of the formula A 4 r to r. 5 Change the subject of the formula s ut at to a. [END OF REVISION QUESTIONS] [Go to net page for the Marking Scheme]

13 Where suitable, ou should alwas follow through an error as ou ma still gain partial credit. If ou are unsure how to do this ask our teacher. Q Marking Scheme (a) Substitution Simplification 5 (b) Substitution Simplification 4 6 (c) 5 Substitution 5 6 Simplification 6 Final answers must be in the form other forms cannot gain,4,6 a b 5 8 or 8, c d e or f g h 0. An 5 7 (a) Find the gradient of the line m 4 Substitute into b m a 7 5 or Simplif equation (b) 4 State coordinate 4 0, For 4 the answer must be given as a coordinate, do not accept c Evaluate gradient Evidence of m c or b m a m Eg c 8 0 State equation 8 or 0 6 or

14 (a) Evaluate gradient m Evidence of b m a 5 or 9 Simplif equation (b) 4 Substitute into equation and begin to evaluate (a) 5 State the answer 5 4 Evaluate gradient 4 0 m Evidence of b m a 5 or 9 Simplif equation (b) 4 Substitute into equation State the answer 5 4 (c) 6 Substitute into equation State the answer (a) Evaluate gradient m Evidence of m c or b m a Simplif equation 8 4 Epress equation with correct variables (b) 5 Substitute into equation and begin to evaluate 6 State the answer in contet (units required) c 8 0 or 66 4 or 4 f d 8 5 f Evidence of negative gradient Line has a downward slope Evidence of negative -intercept Line crosses -ais below zero

15 7 8 (a) State gradient and -intercept gradient (b) Rearrange to the form m c State gradient and -intercept gradient (c) 4 Rearrange to the form m c State gradient and -intercept 5 gradient (d) 6 Rearrange to the form m c 6 4, -intercept, -intercept State gradient and -intercept 7 gradient 0, 0, 4, -intercept 0, 5 5 Rearrange equation Interpret information Gradient is negative -intercept is positive Select correct graph D 5, -intercept 0, for an answer with no working award 0/. the st mark can be awarded if the correct gradient and -intercept are stated (if not rearranged)

16 9 0 * (a) Epand brackets or 5 4 Collect like terms solve 6 0 or (b) 4 Epand brackets 4 z Collect like terms 5 z 6 6 solve 6 (c) 7 Collect variables 7 z 5 8 Collect constants solve 9 5 or 5 (d) 0 Epand brackets Collect like terms solve or 4 For correct answer without working award 0/ Deal with fraction 6 5 Collect like terms solve Collect like terms Solve (a) Know that at B, 0 0, Collect like terms Solve then state coordinate (coordinate must be stated) (b) 4 State values hence B 6, 0 This question involves an understanding of the straight line and the coordinate sstem.

17 (a) State epression (b) State epression (c) Collect facts to form equation 8 (or Collect like terms Solution for 5 6 State ccling speed (units required) kilometres per hour For answer to (c) without working award 0/4. This question involves the formula for distance: D ST or ) 4 * Strateg to find area of one triangle (there must be evidence of this mark) to gain Form equation 4 Start to solve 6 4 Solve or or This question involves the area of a triangle A lb and ma involve multipling fractions from Applications.. Other algebraic processes are acceptable but numerical methods are not.. For (thinking of the area as A lb not A lb ) leading to same solution award /4 5 C Valid strateg involving Form an equation Solution Eg () or Therefore Tom was paid 5 65 is for solving the equation from then finding an hourl rate. Follow through marks are available here.

18 6 7 C 8 C 9 (a) State epression (b) Start to form epression 50 4 Form epression or (c) 4 Form inequalit Collect like terms 5 6 Solve for or equivalent State the minimum number 7 At least times. (a) calculation (b) Start formula Continue formula 5d 0 m 4 Full formula 4 C 5d 0m 00 (a) Start process Either or Calculation (units not required (b) Start process Continue formula 5 C ignore subsequent simplification. candidates ma work in pence, but final answer must be in pounds (a) Evidence of at least one correct point Eg 0, or 0 Completed line, etc 0 5 (b) Solution,

19 0 (a) Evidence of at least one correct point Eg Completed line 0, 6 6, 0 or etc = (b) Solution (a) Multipl b appropriate factor Solve for Solve for (b) 4 Multipl b appropriate factor 4 5 Solve for 6 Solve for 5, Equations must be scaled before first mark can be given. Substitution method ma be used. Can get marks and if consistent with an error in mark 4. Ignore check and an check errors 5. Answers onl or Guess and check method 0 mark (a) Form equation a 4c 56 (b) Form equation a c 6 (c) Evidence of scaling 4 Process for a 4 a Eg a 6c 7 or 5 Process for c and answer in contet 5 c 8 One adult ticket costs One child ticket costs 8 6a c 68 4a c 44 (ii) 6 Find cost (units required) 6 Eg a 4 c 4( 8) 44 For 5, the answer must be in the contet of the question.

20 4 5 Form equation 00 Form equation Evidence of scaling or Process for 5 Process for 4 0 and answer in contet 5 90 There are 0 standard seats There are 90 delue seats. For 5, the answer must be in the contet of the question. (a) Equation 60 (b) Equation or (c) Evidence of scaling Eg Process 4 5 State clearl the number of 50 pence coins (a) Form equation m c 7 (b) Form equation 4m c 7 or equivalent There are 8 fift pence coins. (c) Valid method Eg m 0 or 4m c 4 4 Value of m 5 Value of c 4 m 5 5 c (d) 6 State gradient 6 Gradient and are onl be awarded for equations in terms of m and c. 5 6 or Evidence of valid strateg Eg 4 Process valid strateg Eg or One value 4 or 5 4 Other value and state coordinate (answer must be in coordinate form) 4 4,

21 7 8 9 (a) Start to form equation 5 Form equation 6 4 (b) Start to form equation An equation containing onl the terms, and 4 Form equation 4 Eg 5 or 5 (c) 5 Valid method 5 Eg 44 6 Solve for one variable 6 4 or 7 Solve for other variable 7 (a) Multipl b add 5 4 or G k G k G Divide b k (b) 4 Add hk (c) 5 Subtract R 6 Divide b h 7 subtract m 8 Multipl b k 9 Divide b ( C m) R hk hk R R k h C m v k k( C m) v v k C m Multipl b t 7s 4 Subtract 4 t 4 7s t 4 Divide b 7 s 7

22 0 4 Multipl b P ( m 4) P Divide b m 4 P Add 4 4 m. Accept 8 m P for full marks Divide b B Square root Multipl b k W B H W B kl square Multipl b 0 Divide b k H m kl m 0 f kd 0 f d k 0 f Square root d k Divide b 4 Square root A r 4 A r 4

23 5 Subtract ut s ut at Multipl b Divide b t s ut Accept a for full marks. t s ut at s ut t a [END OF MARKING SCHEME]

N5 R1.1 Linear Algebra - Revision

N5 R1.1 Linear Algebra - Revision N5 R Linear Algebra - Revision This revision pack covers the core linear algebra skills and provides opportunities to apply these skills to standard and challenging exam level questions. This pack is not