N5 R1.1 Linear Algebra - Revision

Transcription

1 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 an exhaustive list of possible questions as the exam will have questions from new and unfamiliar contexts. Therefore, it is important not to learn different looking questions by rote, but to instead to learn not only how but when to apply certain skills. Topic Sub skills Questions Equation of a straight line Use of formula y b m x a and the gradient or equivalent to find the equation of a line given one point Identify gradient and y intercept from y mx c, 7a, 4d Identify gradient and y intercept from various forms of the equation of a straight line 7b - d, 8-6 Equations and inequations Simultaneous Equations 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 text in context 9 a-b, 4, 6a-b, 7, 8, 7 Solve a simultaneous equation graphically 9, 0 Solve a simultaneous equation algebraically 7 Rearranging formulae Function Notation Rearrange a linear equation 8 0 Rearrange an equation involving a square or square root (NOT IN THE LINEAR ALGEBRA NOVEMBER EXAM) - 5 Use of functional notation f x 6-8 When attempting a question, this key will give you additional important information. Key Note The question includes a reasoning element and can make a question more challenging. C Question should be completed without a calculator. Question should be completed with a calculator. Questions will be ordered by sub skill and typically will start of easier and then get more challenging. Some questions may also cover several sub skills from this outcome. Questions are gathered from multiple sources including ones we have created and from past papers. Extra challenge questions are for extension and are not essential for exam preparation.

2 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 6, 8 (b) Gradient of, passing through 4, 6 (c) Gradient of, passing through, 5 A straight line passes through the points, 7 and, 5 (a) Find the equation of this line (b) State the coordinates of the y-intercept of this line. A straight line cuts the as shown. x axis at the point 6, 0 and the y -axis at the point 0, 8 Find the equation of this line.

3 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 taxi fare consists of a call-out charge of 80 plus a fixed cost per kilometre. A journey of 4 kilometres costs 6 60 (a) Find the equation of the straight line. 4 (b) Calculate the fare for a journey of 7 kilometres. 6 Sketch a straight line with the equation y ax b satisfying the following two conditions: a 0 b 0

4 7 In each of the equations below, identify the gradient and state the coordinates of the y-intercept. (a) y x (b) y 4x 6 (c) x y 4 (d) 5x y 0 8 Four straight line graphs are shown below A B C D Which one of these above could represent the line with equation x y? Give two reasons to justify your answer. 9 Solve the following equations and inequations. (a) y (b) z (c) 5y 0 y 4 (d) x 0 Solve the equation x x 5 Solve the inequality x 5 4

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

6 6 There are two different options to pay entry into a local snooker club. The first option is to pay 7 each time you enter. Let x be the number of times that you visit this club. (a) Write an expression for the cost of visiting the snooker club x times using the first option. The second option is to buy a discount card with the following costs and benefits. A one off fee of 50 for the card A reduced cost 4 per visit The first 5 visits are free. (b) Write an expression for the cost of visiting the snooker club x times using the second option, where x 5. (c) Find algebraically the minimum number of times you would have to visit the club in order for the discount card to be the cheaper option. 4 7 C To hire a car costs 5 per day plus a mileage charge. The first 00 miles are free with each additional mile charged at pence. CAR HIRE 5 per day first 00 miles free each additional mile p (a) Calculate the cost of hiring a car for 4 days when the mileage is 640 miles. (b) A car is hired for d days 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 any 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 x photographs printed (where x is greater than ) plus a CD.

7 9 The graph opposite shows the line with equation y x (a) Copy the diagram and on the same diagram, sketch the line x y (b) Use your sketch to solve graphically the system of equations y x x y 0 The graph below shows the line x 6 y y 5 x + 6y = x (a) Copy the diagram and sketch the line x y 6 (b) Use your graph to solve the system of equations x 6 y x y 6 Solve the following system of equations algebraically. (a) x 4 y 5 x y 5 (b) 7 y x 8 y x

8 Brian, Molly 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 deluxe. A standard seat costs 4 and a deluxe seat costs 6. (a) Let x be the number of standard seats and y be the number of deluxe seats. Write down an equation to illustrate this information. (b) When all the seats are sold the ticket sales total 80. Write down an equation to illustrate this information. (c) How many standard seats and how many deluxe seats are in the cinema? 4 Aaron saves 50 pence and 0 pence coins in his piggy bank. Let x be the number of 50 pence coins in his bank. Let y be the number of 0 pence coins in his bank. (a) There are 60 coins in his bank. Write down an equation in x and y to illustrate this information. (b) The total value of the coins is 7 40 Write down another equation in x and y to illustrate this information. (c) Hence find algebraically the number of 50 pence coins Aaron has in his piggy bank.

9 5 A straight line has equation y mx c, where m and c are constants. (a) The point (, 7) lies on this line. 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 algebraically the values of m and c. (d) Write down the gradient of this line. 6 The graph below shows the two straight lines. y x x y 4 y y = x - P 0 x x + y = 4 The lines intersect at the point P. Find algebraically the coordinates of P. 4

10 7 In triangle PQR: PQ x centimetres PR 5x centimetres QR y centimetres. P 5x R x Q y (a) The perimeter of the triangle is 4 centimetres. Write down an equation in x and y to illustrate this information. (b) PR is centimetres longer than QR. Write down an equation in x and y to illustrate this information. (c) Hence calculate the values of x and y. 8 In each of the following equations, change the subject of the formula to k. (a) k x G y (b) R hk (c) v C m k 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 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 A 4 r to r. Change the subject of the formula s ut at to a.

11 6 A function is defined as f x x 5 (a) Evaluate f 6 (b) If t f, find the value of t. 7 8 A function is defined as f x 5 7x (a) Evaluate f (b) If a 5 f, find the value of a. A function is defined as g x 5 x (a) Evaluate g 8 (b) If p 0 g, find the value of p. (c) State the gradient and y intercept of the line g x y. (d) Extra challenge: A function is defined as g ( 7r) h( r )? h( x) 4 6x. What value of r satisfies the equation [END OF REVISION QUESTIONS] [Go to next page for the Marking Scheme]

12 Where suitable, you should always follow through an error as you may still gain partial credit. If you are unsure how to do this ask your teacher. Q Marking Scheme (a) Substitution y 8 5 x 6 Simplification y 5x (b) Substitution y 6 x 4 4 Simplification 4 y x 6 (c) 5 Substitution 5 y 5 x 6 Simplification 6 y x 8 or y x 6 Final answers must be in the form y ax b, cx dy e or fx gy h 0. Any other forms cannot gain,4,6. I.e. the known constants must be simplified. 5 7 (a) Find the gradient of the line m 4 Substitute into y b m x a 7 x 5 x y or y Simplify equation y x (b) 4 State coordinate 4 0, For 4 the answer must be given as a coordinate, do not accept c Evaluate gradient m Evidence of y mx c or y b m x a Eg y x c y 8 x 0 State equation y x 8 or 0 x 6 y or

13 4 5 6 (a) Evaluate gradient m Evidence of y b m x a y x 5 or y x 9 Simplify equation y x (b) 4 Substitute into equation 4 y 0 5 State the answer 5 4 (c) 6 Substitute into equation 6 5 x 7 State the answer 7 48 x 4 x (a) Evaluate gradient m Evidence of y mx c or y b m x a Simplify equation y x 8 4 Express equation with correct variables (b) 5 Substitute into equation and begin to evaluate 6 State the answer in context (units required) y x c y 8 x 0 or y 6 6 x 4 or 4 f d 8 5 f Evidence of negative gradient Line has a downward slope Evidence of negative y-intercept Line crosses y-axis below zero

14 7 8 (a) State gradient and y-intercept gradient, y-intercept 0, (b) Rearrange to the form y mx c y x State gradient and y-intercept gradient (c) 4 Rearrange to the form y mx c 4 x y 4 x 4 y x 4, y-intercept 0, 5 State gradient and y-intercept 5 4 gradient, y-intercept 0, (d) 6 Rearrange to the form y mx c 6 y 5x y 0 y 5x 5 y x 7 State gradient and y-intercept 7 5 gradient, y-intercept 0, Rearrange equation y x Interpret information Gradient is negative y-intercept is positive Select correct graph D for an answer with no working award 0/. the st mark can be awarded if the correct gradient and y-intercept are stated (if not rearranged)

15 9 0 (a) Expand brackets 8 0 y 4 or 5y 4 Collect like terms 6 0 y or 5y solve y (b) 4 Expand brackets 4 z Collect like terms 5 z 6 6 solve 6 z (c) 7 Collect variables 7 5y 8 Collect constants solve 9 y or y (d) 0 Expand brackets 0 8 6x 6 6 Collect like terms 0 6x solve 5 x or x 5 For correct answer without working award 0/ Deal with fraction 6x x 5 Collect like terms 5x 7 solve Collect like terms 7 x 5 x Solve x (a) Know that at B, y 0 0 x, Collect like terms Solve then state coordinate (coordinate must be stated) (b) 4 State values 4 x 6 x 6 x hence B 6, 0 This question involves an understanding of the straight line and the coordinate system.

16 (a) State expression x 8 (b) State expression (c) Collect facts to form equation x (or x 8 x 46 4 Collect like terms 4 5x 0 5 Solution for x 5 x x 0 5x or ) 6 State cycling speed (units required) 6 x kilometres per hour For answer to (c) without working award 0/4. This question involves the formula for distance: D ST 4 Strategy to find area of one triangle (there must be evidence of to gain this mark) Form equation x or x x x 4 Start to solve x 6 x 4 Solve 4 6 x 5 x or x This question involves the area of a triangle A lb and may involve multiplying fractions from Applications.. Other algebraic processes are acceptable but numerical methods are not. x (thinking of the area as A lb not A lb ) leading to same. For x solution award /4 5 C Valid strategy involving Form an equation Solution Eg () or x x 4 5x 8x x 49 9x 49 x Therefore Tom was paid 5 65 is for solving the equation from then finding an hourly rate. Follow through marks are available here.

17 6 7 C 8 C 9 (a) State expression 7 x (b) Start to form expression 50 4 Form expression x or 4x 0 (c) 4 Form inequality x 5 7x 5 Collect like terms 5 0 x 6 Solve for x 6 0 x or equivalent 7 State the minimum number 7 At least times. (a) calculation (b) Start formula 5 d Continue formula 0 m 4 Full formula 4 C 5d 0 m 00 (a) Start process Either or Calculation (units not required) (b) Start process Continue x 5 formula 5 C x C x C x ignore subsequent simplification. candidates may 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 or or (b) Solution x, y

18 0 (a) Evidence of at least one correct point Eg 0, 6 or 0 Completed line y 6, etc (b) Solution x 5, y 5 x + 6y = x (a) Multiply by appropriate factor Solve for x x 7 Solve for y y x 4 y 5 x 4 y 0 (b) 4 Multiply by appropriate factor 4 5 Solve for x 5 x 5 6 Solve for y 6 y 7 y x 8 6 y x 9 Equations must be scaled before first mark can be given. Substitution method may be used. Can get marks and if consistent with an error in mark 4. Ignore check and any check errors 5. Answers only 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 context 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 context of the question.

19 4 5 Form equation x y 00 Form equation 4x 6y 80 Evidence of scaling 4x 4 y 00 or 6x 6y Process for x 4 x 0 5 Process for y and answer in context 5 y 90 There are 0 standard seats There are 90 deluxe seats. For 5, the answer must be in the context of the question. (a) Equation x y 60 (b) Equation 50x 0 y 740 or 0 5x 0 y 7 40 (c) Evidence of scaling Eg 0x 0 y 00 4 Process 4 0x 540 or equivalent 5 State clearly the number of 50 pence coins (a) Form equation m c 7 (b) Form equation 4m c 7 5 There are 8 fifty pence coins. (c) Valid method Eg m 0 or 4m c 4 4 Value of m 4 m 5 5 Value of c 5 c (d) 6 State gradient 6 Gradient 5 and are only be awarded for equations in terms of m and c. 6 Evidence of valid strategy Eg x 4 Process valid strategy x or x y Eg 5x 6 4 or One value x 4 or y 5 4 Other value and state coordinate (answer must be in coordinate form) 4 4, 5 x y 4 4x y 6

20 7 (a) Start to form equation x 5x y Form equation 6x y 4 (b) Start to form equation An equation containing only the terms 5 x, y and 4 Form equation 4 Eg 5x y or 5x y (c) 5 Valid method 5 Eg x 44 6 Solve for one variable 6 x 4 or y 9 7 Solve for other variable 7 x 4 or y (a) Multiply by y yg k x add x yg x k yg x Divide by k (b) 4 Add hk 4 R hk (c) 5 Subtract R 5 hk R 6 Divide by h 6 7 subtract m 7 R k h C m 8 Multiply by k 8 k( C m) v 9 Divide by ( C m) 9 v k v k C m Multiply by t 7s 4 Subtract 4 t 4 7s t 4 Divide by 7 s 7

21 0 4 Multiply by P ( m 4) P Divide by m 4 P Add 4 4 m. Accept 8 m P for full marks Divide by B W H B Square root Multiply by k W B kl square kl m Multiply by 0 Divide by k m H 0 f kd 0 f d k 0 f Square root d k Divide by 4 A r 4 Square root A r 4

22 Subtract ut s ut at Multiply by Divide by s ut at t s ut s ut Accept a for full marks. t t a (a) Substitute 7 into formula f ( 6) 6 5 Evaluate f 6 (b) Substitute into formula t 5 4 Solve 4 (a) Substitute 7 t into formula f ( ) 5 7 Evaluate f 6 (b) Substitute into formula 5 5 7a 4 Solve 4. (a) Substitute 8 0 a 7 into formula g Evaluate g 8 (b) Substitute 0 into formula 0 5 p 4 Simplify equation p 5 Solve 5 p (c) 6 Simplify equation 6 y x 7 (d) 7 State gradient and coordinate of the y intercept Note, this question goes beyond the standard of previous National 5 exams. 7 m and ( 0, 7) 5 7r 4 6 r 5 r 8. The y intercept must be stated as a coordinate to receive 7 [END OF MARKING SCHEME]