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INTERMEDIATE 2 – ADDITIONAL QUESTION BANK

INTERMEDIATE 2 – ADDITIONAL QUESTION BANK. You have chosen to study:. Simultaneous Equations. UNIT 2 :. Please choose a question to attempt from the following:. 1. 2. 3. 4. Back to Unit 2 Menu. EXIT. Simultaneous Equations : Question 1. The diagram shows the graph of

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INTERMEDIATE 2 – ADDITIONAL QUESTION BANK

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  1. INTERMEDIATE 2 – ADDITIONAL QUESTION BANK You have chosen to study: Simultaneous Equations UNIT 2 : Please choose a question to attempt from the following: 1 2 3 4 Back to Unit 2 Menu EXIT

  2. Simultaneous Equations : Question 1 The diagram shows the graph of 3x + 2y = 17. Copy the diagram and on your diagram draw the graph of y = 2x – 2 , hence solve 3x + 2y = 17 y = 2x – 2 3x + 2y = 17. Reveal answer only Go to Comments Get hint EXIT Go to full solution Go to Sim Eq Menu

  3. Simultaneous Equations : Question 1 The diagram shows the graph of 3x + 2y = 17. Copy the diagram and on your diagram draw the graph of y = 2x – 2 , hence solve 3x + 2y = 17 y = 2x – 2 To draw graph:Construct a table of values with at least 2 x-coordinates. Plot and join points. Solution is where lines cross 3x + 2y = 17. What would you like to do now? Reveal answer only Go to Comments EXIT Go to full solution Go to Sim Eq Menu

  4. Simultaneous Equations : Question 1 The diagram shows the graph of 3x + 2y = 17. Copy the diagram and on your diagram draw the graph of y = 2x – 2 , hence solve 3x + 2y = 17 y = 2x – 2 Solution is x = 3 & y = 4 3x + 2y = 17. What would you like to do now? Try another like this Go to Comments EXIT Go to full solution Go to Sim Eq Menu

  5. Question 1 1. Construct a table of values with at least 2 x coordinates. The diagram shows the graph of 3x + 2y = 17. y = 2x - 2 x 0 5 Copy the diagram and on your diagram draw the graph of y = 2x – 2 , hence solve 3x + 2y = 17 y = 2x – 2 y -2 8 Begin Solution Continue Solution Comments Sim Eq Menu Back to Home

  6. Question 1 2. Plot and join points. Solution is where lines cross. The diagram shows the graph of 3x + 2y = 17. Copy the diagram and on your diagram draw the graph of y = 2x – 2 , hence solve 3x + 2y = 17 y = 2x – 2 Begin Solution Try another like this Comments Solution is x = 3 & y = 4 Sim Eq Menu What would you like to do now? Back to Home

  7. Markers Comments There are two ways of drawing the line y = 2x - 1 Method 1 Finding two points on the line: x = 0 y = 2 x 0 – 1 = -1 x = 2 y = 2 x 2 - 1 = 3 First Point (0,-1) Second Point (-1,3) Plot and join (0,-1), and (-1,3). 1. Construct a table of values with at least 2 x coordinates. y = 2x - 2 x 0 5 y -2 8 Next Comment Sim Eqs Menu Back to Home

  8. Markers Comments Method 2 Using y = mx + c form: y = mx + c y = 2x - 1 gradient m = 2 y - intercept c = -1 y - intercept gradient 2. Plot and join points. Solution is where lines cross. Plot C(0, -1) and draw line with m = 2 Next Comment Sim Eqs Menu Solution is x = 3 & y = 4 Back to Home

  9. Simultaneous Equations : Question 1B The diagram shows the graph of 2x + y = 8. Copy the diagram and on your diagram draw the graph of y = 1/2x + 3 , hence solve 2x + y = 8 y = 1/2x + 3 Reveal answer only Go to Comments Get hint EXIT Go to full solution Go to Sim Eq Menu

  10. Simultaneous Equations : Question 1B The diagram shows the graph of 2x + y = 8. Copy the diagram and on your diagram draw the graph of y = 1/2x + 3 , hence solve 2x + y = 8 y = 1/2x + 3 To draw graph:Construct a table of values with at least 2 x-coordinates. Plot and join points. Solution is where lines cross What would you like to do now? Reveal answer only Go to Comments EXIT Go to full solution Go to Sim Eq Menu

  11. Simultaneous Equations : Question 1B The diagram shows the graph of 2x + y = 8. Copy the diagram and on your diagram draw the graph of y = 1/2x + 3 , hence solve 2x + y = 8 y = 1/2x + 3 What would you like to do now? Solution is x = 2 & y = 4 Go to Comments EXIT Go to full solution Go to Sim Eq Menu

  12. Question 1B 1. Construct a table of values with at least 2 x coordinates. The diagram shows the graph of 2x + y = 8. y = 1/2x + 3 x 0 6 y 3 6 Copy the diagram and on your diagram draw the graph of y = 1/2x + 3 , hence solve 2x + y = 8 y = 1/2x + 3 Begin Solution Continue Solution Comments Sim Eq Menu Back to Home

  13. Question 1B 2. Plot and join points. Solution is where lines cross. The diagram shows the graph of 2x + y = 8. y = 1/2x + 3 x 0 6 y 3 6 Copy the diagram and on your diagram draw the graph of y = 1/2x + 3 , hence solve 2x + y = 8 y = 1/2x + 3 What would you like to do now? Begin Solution Continue Solution Comments Sim Eq Menu Solution is x = 2 & y = 4 Back to Home

  14. Markers Comments There are two ways of drawing the line y = ½ x + 3 Method 1 Finding two points on the line: x = 0 y = ½ x 0 + 3 = 3 x = 2 y = ½ x 2 +3 = 4 First Point (0, 3) Second Point (2, 4) Plot and join (0, 3), and (2, 4). 1. Construct a table of values with at least 2 x coordinates. y = 1/2x + 3 x 0 6 y 3 6 Next Comment Sim Eqs Menu Back to Home

  15. Markers Comments Method 2 Using y = mx + c form: y = mx + c y = ½ x + 3 gradient m = ½ y - intercept c = +3 y - intercept gradient 2. Plot and join points. Solution is where lines cross. y = 1/2x + 3 x 0 6 y 3 6 Plot C(0, 3) and draw line with m = ½ Next Comment Sim Eqs Menu Solution is x = 2 & y = 4 Back to Home

  16. Simultaneous Equations : Question 2 Solve3u - 2v = 4 2u + 5v = 9 Get hint Reveal answer only Go to full solution Go to Comments Go to Sim Eq Menu EXIT

  17. Simultaneous Equations : Question 2 Eliminate either variable by making coefficient same. Solve3u - 2v = 4 2u + 5v = 9 What would you like to do now? Substitute found value into either of original equations. Reveal answer only Go to full solution Go to Comments Go to Sim Eq Menu EXIT

  18. Simultaneous Equations : Question 2 Solve3u - 2v = 4 2u + 5v = 9 Solution is u = 2 & v = 1 What would you like to do now? Try another like this Go to full solution Go to Comments Go to Sim Eq Menu EXIT

  19. Question 2 (x5) (x2) 3u - 2v = 4 2u + 5v = 9 1. Eliminate either u’s or v’s by making coefficient same. Solve3u - 2v = 4 2u + 5v = 9 1 2 Now get: (x5) 1 = 3 (x2) 2 4 = 3 + 4 Begin Solution Continue Solution Comments Sim Eq Menu Back to Home

  20. Question 2 (x5) (x2) 3u - 2v = 4 2u + 5v = 9 2. Substitute found value into either of original equations. Solve3u - 2v = 4 2u + 5v = 9 1 2 Substitute 2 for u in equation 2 4 + 5v = 9 5v = 5 v = 1 Solution is u = 2 & v = 1 Begin Solution Try another like this What would you like to do now? Comments Sim Eq Menu Back to Home

  21. Markers Comments (x5) (x2) 3u - 2v = 4 2u + 5v = 9 e.g. 2x + 3y = 4 6x + 3y = 4 Subtract the equations Note: When the “signs” are the same subtract to eliminate. When the “signs” are different add to eliminate. 1. Eliminate either u’s or v’s by making coefficient same. 1 2 Now get: (x5) 1 = 3 (x2) 2 4 = 3 + 4 Next Comment Sim Eqs Menu Back to Home

  22. Markers Comments (x5) (x2) 3u - 2v = 4 2u + 5v = 9 e.g. 2x + 3y = 4 6x - 3y = 4 Add the equations Note: When the “signs” are the same subtract to eliminate. When the “signs” are different add to eliminate. 1. Eliminate either u’s or v’s by making coefficient same. 1 2 Now get: (x5) = 3 1 (x2) 2 4 = 3 + 4 Next Comment Sim Eqs Menu Back to Home

  23. Simultaneous Equations : Question 2B Solve5p + 3q = 0 4p + 5q = -2.6 Get hint Reveal answer only Go to full solution Go to Comments Go to Sim Eq Menu EXIT

  24. Simultaneous Equations : Question 2B Eliminate either variable by making coefficient same. Solve5p + 3q = 0 4p + 5q = -2.6 What would you like to do now? Substitute found value into either of original equations. Reveal answer only Go to full solution Go to Comments Go to Sim Eq Menu EXIT

  25. Simultaneous Equations : Question 2B Solve5p + 3q = 0 4p + 5q = -2.6 Solution is q = -1 & q = 0.6 What would you like to do now? Go to full solution Go to Comments Go to Sim Eq Menu EXIT

  26. Question 2B (x4) (x5) 5p + 3q = 0 4p + 5q = -2.6 1. Eliminate either p’s or q’s by making coefficient same. Solve5p + 3q = 0 4p + 5q = -2.6 1 2 Now get: (x4) 1 = 3 (x5) 2 4 = 4 - 3 Begin Solution Continue Solution Comments Sim Eq Menu Back to Home

  27. Question 2B (x4) (x5) 5p + 3q = 0 4p + 5q = -2.6 2. Substitute found value into either of original equations. Solve5p + 3q = 0 4p + 5q = -2.6 1 2 Substitute -1 for q in equation 1 5p + (- 3) = 0 5p = 3 p =3/5 = 0.6 Solution is q = -1 & p = 0.6 Begin Solution Continue Solution What would you like to do now? Comments Sim Eq Menu Back to Home

  28. Markers Comments (x4) (x5) 5p + 3q = 0 4p + 5q = -2.6 e.g. 2x + 3y = 4 6x + 3y = 4 Subtract the equations Note: When the “signs” are the same subtract to eliminate. When the “signs” are different add to eliminate. 1. Eliminate either p’s or q’s by making coefficient same. 1 2 Now get: (x4) 1 = 3 (x5) 2 4 = 4 - 3 Next Comment Sim Eqs Menu Back to Home

  29. Markers Comments (x4) (x5) 5p + 3q = 0 4p + 5q = -2.6 e.g. 2x + 3y = 4 6x - 3y = 4 Add the equations Note: When the “signs” are the same subtract to eliminate. When the “signs” are different add to eliminate. 1. Eliminate either p’s or q’s by making coefficient same. 1 2 Now get: (x4) 1 = 3 (x5) 2 4 = 4 - 3 Next Comment Sim Eqs Menu Back to Home

  30. Simultaneous Equations : Question 3 If two coffees & three doughnuts cost £2.90 while three coffees & one doughnut cost £2.60 then find the cost of two coffees & five doughnuts. Get hint Reveal answer only Go to full solution Go to Comments Go to Sim Eq Menu EXIT

  31. Simultaneous Equations : Question 3 If two coffees & three doughnuts cost £2.90 while three coffees & one doughnut cost £2.60 then find the cost of two coffees & five doughnuts. Form two equations, keeping costs in pence to avoid decimals. Eliminate either c’s or d’s by making coefficient same. What would you like to do now? Substitute found value into either of original equations. Reveal answer only Remember to answer the question!!! Go to full solution Go to Comments Go to Sim Eq Menu EXIT

  32. Simultaneous Equations : Question 3 If two coffees & three doughnuts cost £2.90 while three coffees & one doughnut cost £2.60 then find the cost of two coffees & five doughnuts. Two coffees & five doughnuts = £3.90 Try another like this Go to full solution Go to Comments Go to Sim Eq Menu EXIT

  33. Question 3 (x1) (x3) 2c + 3d = 290 3c + 1d = 260 1. Form two equations, keeping costs in pence to avoid decimals. If two coffees & three doughnuts cost £2.90 while three coffees & one doughnut cost £2.60 then find the cost of two coffees & five doughnuts. Let coffees cost c pence & doughnuts d pence then we have 1 2 2. Eliminate either c’s or d’s by making coefficient same. (x1) 1 = 3 (x3) 2 4 = Begin Solution Try another like this 4 - 3 Comments Sim Eq Menu Back to Home

  34. Question 3 (x1) (x3) 2c + 3d = 290 3c + 1d = 260 3. Substitute found value into either of original equations. If two coffees & three doughnuts cost £2.90 while three coffees & one doughnut cost £2.60 then find the cost of two coffees & five doughnuts. Let coffees cost c pence & doughnuts d pence then we have 1 2 Substitute 70 for c in equation 2 What would you like to do now? 210 + d = 260 d = 50 Two coffees & five doughnuts Begin Solution = (2 x 70p) + (5 x 50p) Try another like this = £1.40 + £2.50 Comments = £3.90 Sim Eq Menu Back to Home

  35. Comments (x1) (x3) 2c + 3d = 290 3c + 1d = 260 Step 1 Form the two simultaneous equations first by introducing a letter to represent the cost of a coffee ( c) and a different letter to represent the cost of a doughnut (d). 1. Form two equations, keeping costs in pence to avoid decimals. Let coffees cost c pence & doughnuts d pence then we have 1 2 i.e. 2c + 3d = 290 1d + 2c = 260 2. Eliminate either c’s or d’s by making coefficient same. (x1) 1 = 3 Note change to pence eases working. (x3) 2 4 = Next Comment 4 - 3 Sim Eqs Menu Back to Home

  36. Comments (x1) (x3) 2c + 3d = 290 3c + 1d = 260 Step 2 Solve by elimination. Choose whichever variable it is easier to make have the same coefficient in both equations. 1. Form two equations, keeping costs in pence to avoid decimals. Let coffees cost c pence & doughnuts d pence then we have 1 2 2. Eliminate either c’s or d’s by making coefficient same. (x1) 1 = 3 (x3) 2 4 = Next Comment 4 - 3 Sim Eqs Menu Back to Home

  37. Comments 2c + 3d = 290 3c + 1d = 260 Step 3 Once you have a value for one variable you can substitute this value into any of the equations to find the value of the other variable. It is usually best to choose an equation that you were given in question. 3. Substitute found value into either of original equations. Let coffees cost c pence & doughnuts d pence then we have Substitute 70 for c in equation Step 4 Remember to answer the question!!! 210 + d = 260 d = 50 Two coffees & five doughnuts = (2 x 70p) + (5 x 50p) Next Comment = £1.40 + £2.50 Sim Eqs Menu = £3.90 Back to Home

  38. Simultaneous Equations : Question 3B A company plan to introduce a new blend of tropical fruit drink made from bananas & kiwis. They produce three blends of the drink for market research purposes. Blend 1 uses 70% banana syrup, 30% kiwi syrup and the cost per litre is 74p. Blend 2 has 55% banana syrup, 45% kiwi syrup and costs 71p per litre to produce. A third blend is made using 75% banana syrup and 25% kiwi. How does its cost compare to the other two blends? Get hint Reveal answer Go to Comments EXIT Go to full solution Go to Sim Eq Menu

  39. Simultaneous Equations : Question 3B A company plan to introduce a new blend of tropical fruit drink made from bananas & kiwis. They produce three blends of the drink for market research purposes. Blend 1 uses 70% banana syrup, 30% kiwi syrup and the cost per litre is 74p. Blend 2 has 55% banana syrup, 45% kiwi syrup and costs 71p per litre to produce. A third blend is made using 75% banana syrup and 25% kiwi. How does its cost compare to the other two blends? Eliminate either c’s or d’s by making coefficient same. Form two equations, eliminating decimals wherever possible. Substitute found value into either of original equations. Remember to answer the question!!! What would you like to do now? Reveal answer Go to Comments EXIT Go to full solution Go to Sim Eq Menu

  40. Simultaneous Equations : Question 3B A company plan to introduce a new blend of tropical fruit drink made from bananas & kiwis. They produce three blends of the drink for market research purposes. Blend 1 uses 70% banana syrup, 30% kiwi syrup and the cost per litre is 74p. Blend 2 has 55% banana syrup, 45% kiwi syrup and costs 71p per litre to produce. A third blend is made using 75% banana syrup and 25% kiwi. How does its cost compare to the other two blends? So this blend is more expensive than the other two. Go to Comments EXIT Go to full solution Go to Sim Eq Menu

  41. Question 3B (x10) (x100) 0.70B + 0.30K = 74 0.55B + 0.45K = 71 1. Form two equations, keeping costs in pence to avoid decimals. Blend 1 uses 70% banana syrup, 30% kiwi syrup and the cost per litre is 74p. Blend 2 has 55% banana syrup, 45% kiwi syrup and costs 71p per litre to produce. A third blend is made using 75% banana syrup and 25% kiwi. How does its cost compare to the other two blends? Let one litre of banana syrup cost B pence & one litre of kiwi syrup cost K pence. 1 2 2. Get rid of decimals (x10) 1 = 3 (x100) 2 4 = Begin Solution Continue Solution Comments Sim Eq Menu Back to Home

  42. Question 3B 3. Eliminate either B’s or K’s by making coefficient same. Blend 1 uses 70% banana syrup, 30% kiwi syrup and the cost per litre is 74p. Blend 2 has 55% banana syrup, 45% kiwi syrup and costs 71p per litre to produce. A third blend is made using 75% banana syrup and 25% kiwi. How does its cost compare to the other two blends? (x15) 3 4 (x15) 5 3 = 4 5 - 4 Begin Solution Continue Solution Comments Sim Eq Menu Back to Home

  43. Question 3B 4. Substitute found value into an equation without decimals. Blend 1 uses 70% banana syrup, 30% kiwi syrup and the cost per litre is 74p. Blend 2 has 55% banana syrup, 45% kiwi syrup and costs 71p per litre to produce. A third blend is made using 75% banana syrup and 25% kiwi. How does its cost compare to the other two blends? (x15) 3 4 Substitute 80 for B in equation 3 560 + 3K = 740 3K = 180 K = 60 5. Use these values to answer question. What would you like to do now? 75%B+25%K = (0.75 x 80p)+(0.25 x 60p) Begin Solution = 60p + 15p Continue Solution = 75p Comments So this blend is more expensive than the other two. Sim Eq Menu Back to Home

  44. Markers Comments (x10) (x100) 0.70B + 0.30K = 74 0.55B + 0.45K = 71 Step 1 Form the two simultaneous equations first by introducing a letter to represent the cost per litre of banana syrup ( B) and a different letter to represent the cost per litre of kiwis fruit (K). 1. Form two equations, keeping costs in pence to avoid decimals. Let one litre of banana syrup cost B pence & one litre of kiwi syrup cost K pence. 1 2 i.e. 0.70 B + 0.30K = 74 0.55B + 0.45K = 71 2. Get rid of decimals (x10) 1 = 3 Multiply all terms by 100 to remove decimals. Note change to pence eases working. (x100) 2 4 = Next Comment Sim Eqs Menu Back to Home

  45. Markers Comments Step 2 Solve by elimination. Choose whichever variable it is easier to make have the same coefficient in both equations. 3. Eliminate either B’s or K’s by making coefficient same. (x15) 3 4 (x15) 5 3 = 4 - 5 4 Next Comment Sim Eqs Menu Back to Home

  46. Markers Comments Step 3 Once you have a value for one variable you can substitute this value into any of the equations to find the value of the other variable. It is usually best to choose an equation that you were given in question. 4. Substitute found value into an equation without decimals. (x15) 3 4 Substitute 80 for B in equation 3 560 + 3K = 740 3K = 180 Step 4 Remember to answer the question!!! K = 60 5. Use these values to answer question. 75%B+25%K = (0.75 x 80p)+(0.25 x 60p) = 60p + 15p Next Comment = 75p Sim Eqs Menu So this blend is more expensive than the other two. Back to Home

  47. Simultaneous Equations : Question 4 • A Fibonacci Sequence is formed as follows……. • Start with two terms add first two to obtain 3rd • add 2nd & 3rd to obtain 4th • add 3rd & 4th to obtain 5th etc • eg starting with 3 & 7, next four terms are 10, 17, 27 & 44. • If the first two terms of such a sequence are P and Q then find expressions for the next 4 terms in their simplest form. • If the 5th & 6th terms are 8 and 11 respectively then write down two equations in P and Q and hence find the values of P and Q. Get hint Reveal answer Go to Comments EXIT Go to full solution Go to Sim Eq Menu

  48. Write down expressions using previous two to form next. Simultaneous Equations : Question 4 Solve and substitute found value into either of original equations. • A Fibonacci Sequence is formed as follows……. • Start with two terms add first two to obtain 3rd • add 2nd & 3rd to obtain 4th • add 3rd & 4th to obtain 5th etc • eg starting with 3 & 7, next four terms are 10, 17, 27 & 44. • If the first two terms of such a sequence are P and Q then find expressions for the next 4 terms in their simplest form. • If the 5th & 6th terms are 8 and 11 respectively then write down two equations in P and Q and hence find the values of P and Q. Establish two equations.Eliminate either P’s or Q’s by making coefficient same. To find values of P & Q: Match term from (a) with values given in question. What would you like to do now? Go to Comments Reveal answer EXIT Go to full solution Go to Sim Eq Menu

  49. Simultaneous Equations : Question 4 • A Fibonacci Sequence is formed as follows……. • Start with two terms add first two to obtain 3rd • add 2nd & 3rd to obtain 4th • add 3rd & 4th to obtain 5th etc • eg starting with 3 & 7, next four terms are 10, 17, 27 & 44. • If the first two terms of such a sequence are P and Q then find expressions for the next 4 terms in their simplest form. • If the 5th & 6th terms are 8 and 11 respectively then write down two equations in P and Q and hence find the values of P and Q. P + Q, P + 2Q 2P + 3Q 3P + 5Q P = 7 Go to Comments Try another like this EXIT Go to full solution Go to Sim Eq Menu

  50. Question 4 1. Write down expressions using previous two to form next. • If the first two terms of such a • sequence are P and Q then find • expressions for the next 4 terms • in their simplest form. (a) First term = P & second term = Q 3rd term = P + Q 4th term = Q + (P + Q) = P + 2Q 5th term = (P + Q) + (P + 2Q) = 2P + 3Q 6th term = (P + 2Q) + (2P + 3Q) = 3P + 5Q Begin Solution Try another like this Comments Sim Eq Menu Back to Home

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