PLC Papers Created For:
Algebra and proof 2 Grade 8 Objective: Use algebra to construct proofs Question 1 a) If n is a positive integer explain why the expression 2n + 1 is always an odd number. b) Use algebra to prove that the product of two odd numbers is also odd. (1) Question 2 (4) a) If x > 3 and prove that F > 1 b) Explain what happens if x = 3 (4) (1) Total /10
Approximate solutions to equations using iteration 2 Grade 9 Objective: Find approximate solutions to equations using iteration. Question 1. Find the first four iterations of each iterative formulae. Start each one with 1 = 6. a) +1 = 5 1 4... b) +1 = 2 + 5 (1)... c) +1 = 14 +1 (1)... (1) (Total 3 marks) Question 2. Starting with 1 = 5.3 verify that 5.37 is a solution, correct to 2 decimal places, of the quadratic equation 2 5 2 = 0 using iteration. (Total 3 marks)
Question 3. a) Show that = 4 9 can be rearranged into the equation 2 + 9 4 = 0. b) Use the iterative formula +1 = 4 9 and a starting value of 1 = 0.5 to obtain the solution to the equation correct to 2 decimal places. (1) (3) (Total 4 marks) TOTAL /10
Equation of a circle 2 Grade 9 Objective: Recognise and use + =. Question 1 (a) Write down the equation of a circle with centre (0, 0) and radius 1.5. (3) (b) Write down the centre and radius of the circle 2 + 2 =81. Centre = Radius = (2) (Total 5 marks) Question 2 On the grid, draw the graph of 2 + 2 =72. (3) (Total 3 marks)
Question 3 A graph has been drawn for you on the grid below. Write down the equation of this graph. (2) (Total 2 marks) TOTAL /10
Equation of a tangent to a circle 2 Grade 9 Objective: Find the equation of a tangent to a circle at a given point. Question 1 The grid below shows a circle with equation 2 + 2 =8. There are two tangents to this circle with gradient 1. (a) Draw these tangents on the graph above. (2) (b) Write down the equation of these tangents. y = y = (2) (Total 4 marks)
Question 2 Here is a circle, 2 + 2 =13, and a tangent to the circle. The tangent goes through the point B(2, -3) on the circle. Find the equation of the tangent at point B. (4) (Total 4 marks)
Question 3 The equation of a circle is 2 + 2 =. The line =7 is a tangent to the circle. Work out the value of k. (2) (Total 2 marks) TOTAL /10
Gradients and area under a graph 2 Grade 8 Objective: Calculate or estimate the gradient of a graph and the area under a graph Question 1 A straight line has been drawn on a grid. Calculate the gradient of the line. (2) (Total 2 marks) Question 2 Work out the gradient of the line 5 3 =20 (2) (Total 2 marks)
Question 3 The graph of = 3 +3 2 2 1 is drawn on the grid below. Calculate an estimate to the gradient of the curve at the point Q(-1, 3). (3) (Total 3 marks)
Question 4 The scatter graph shows the cost of cars in a used car showroom. (a) Draw a line of best fit and calculate the gradient of this line. (2) (b) Give an interpretation of this gradient. (1) (Total 3 marks) TOTAL /10
Quadratic equations (completing the square) 2 Grade 8 Objective: Solve quadratic equations by completing the square. Question 1. Rewrite 2 + 6 + 7 in the form ( + ) 2 (Total 1 mark) Question 2. Solve 2 10 + 9 = 0 by completing the square. (Total 2 marks)
Question 3. Solve 2 8 12 = 0 by completing the square. Leave your answers in surd form. (Total 3 marks) Question 4. Solve 4 2 + 28 24 = 0 by completing the square. Give your answers to 3 significant figures. (Total 4 marks) TOTAL /10
Trigonometric Graphs 2 Grade 8 Objective: Recognise, sketch, and interpret graphs of trigonometric functions Question 1 Sketch the graph of y = tan x for 0 360 (3) (Total 3 marks) Question 2 Here is the graph of y = cos x for 0 360 On the axes above, sketch the graph =cos(2 ) 2 for 0 360 (3) (Total 3 marks)
Question 3 The graph of y = sin x for 0 360 is shown below. What are the coordinates of the 4 points labelled on the graph? (, ) (, ) (, ) (, ) (4) (Total 4 marks) TOTAL /10
PLC Papers Created For:
Algebra and proof 2 Grade 8 Solutions Objective: Use algebra to construct proofs Question 1 a) If n is a positive integer explain why the expression 2n + 1 is always an odd number. 2n is a multiple of 2 so it must be even so 2n + 1 is the number after an even number so it must be odd. b) Use algebra to prove that the product of two odd numbers is also odd. Question 2 (2n + 1) (2m + 1) = 4mn + 2n + 2m + 1 = 2 ( 2mn + n + m) + 1 2 ( 2mn + n + m) must be even so 2 ( 2mn + n + m) + 1 must be odd a) If x > 3 and prove that F > 1 Expand and simplify brackets Factorise Explain why factorised part is even State result must be odd Factorise numerator Factorise denominator Simplify fraction Explain why F > 1 (1) (4) x + 2 > x so numerator is bigger than denominator hence F > 1 b) Explain what happens if x = 3 If x = 3 then x 3 = 0 If you divide by x 3 you are dividing by 0 so F is undefined (May write you can t divide by 0) (4) (1) Total /10
Approximate solutions to equations using iteration 2 Grade 9 SOLUTIONS Objective: Find approximate solutions to equations using iteration. Question 1. Find the first four iterations of each iterative formulae. Start each one with 1 = 6. a) +1 = 5 1 4 2 = 26, 3 = 126, 4 = 626, 5 = 3126, (A1) (1) a) +1 = 2 2 = 8, 3 = 9, 4 = 9.5, 5 = 9.75, (A1) (1) b) +1 = 14 +1 2 = 2, 3 = 14, 3 4 = 42, 17 5 = 238 59 (A1) (1) (Total 3 marks) Question 2. Starting with 1 = 5.3 verify that 5.37 is a solution, correct to 2 decimal places, of the quadratic equation 2 5 2 = 0 using iteration. +1 = 5 + 2 2 = 5.33853 3 = 5.35655 4 = 5.36495 5 = 5.36887 6 = 5.37069 5 = 6 to 2dp (C1) (Total 3 marks)
Question 3. a) Show that = 4 9 can be rearranged into the equation 2 + 9 4 = 0. = 4 9 2 = 4 9 2 + 9 4 = 0 b) Use the iterative formula +1 = 4 9 and a starting value of 1 = 0.5 to obtain a solution to the equation correct to 2 decimal places. 2 = 1 3 = 13 4 = 9.30769 5 = 9.42975 6 = 9.42418 7 = 9.42443 (1) 6 = 7 to 2dp = 9.42 (C1) (A1) (3) (Total 4 marks) TOTAL /10
Equation of a circle 2 Grade 9 Solutions Objective: Recognise and use + =. Question 1 (a) Write down the equation of a circle with centre (0, 0) and radius 1.5. 2 + 2 = where c>0 1.5 2 = 2.25 2 + 2 = 2.25 (A1) (3) (b) Write down the centre and radius of the circle 2 + 2 = 81. Centre = (0, 0) Radius = 9 (A1) (A1) (2) (Total 5 marks) Question 2 On the grid, draw the graph of 2 + 2 = 72. Centre at (0, 0) and attempt of circle in more than 3 quadrants Radius approx. 6 2 8.5 Fully correct graph (G1) (3) (Total 3 marks)
Question 3 A graph has been drawn for you on the grid below. Write down the equation of this graph. Sight of 4 2 = 16 2 + 2 = 16 (A1) (2) (Total 2 marks) TOTAL /10
Equation of a tangent to a circle 2 Grade 9 Solutions Objective: Find the equation of a tangent to a circle at a given point. Question 1 The grid below shows a circle with equation 2 + 2 =8. There are two tangents to this circle with gradient 1. (a) Draw these tangents on the graph above. Any line with gradient 1 (B1) y = x - 4 and y = x + 4 drawn (B1) (2) (b) Write down the equation of these tangents. y = x 4 y = x + 4 (B1) (B1) (2) (Total 4 marks)
Question 2 Here is a circle, 2 + 2 =13, and a tangent to the circle. The tangent goes through the point B(2, -3) on the circle. Find the equation of the tangent at point B. Grad of OB = -3/2 Grad of tangent = 2/3 +3= 2 ( 2) Use of this or y = mx + c 3 3 +9=2 4 3 =2 13 o.e. (A1) = 2 3 13 3 (4) (Total 4 marks)
Question 3 The equation of a circle is 2 + 2 =. The line =7 is a tangent to the circle. Work out the value of k. Radius of circle = 7 from diagram or explanation 2 + 2 =49 or k = 49 (A1) (2) (Total 2 marks) TOTAL /10
Gradients and area under a graph 2 Grade 8 Solutions Objective: Calculate or estimate the gradient of a graph and the area under a graph Question 1 A straight line has been drawn on a grid. Calculate the gradient of the line. = 4 2 m = -2 (A1) (2) (Total 2 marks) Question 2 Work out the gradient of the line 5 3 = 20 Correct attempt to make y the subject: = 3 5 + 20 = 3 5 (A1) (2) (Total 2 marks)
Question 3 The graph of = 3 + 3 2 2 1 is drawn on the grid below. Calculate an estimate to the gradient of the curve at the point Q(-1, 3). Consider points just above and just below, i.e. x = -1.1 and x = -0.9 (-1.1, 3.499) and (-0.9, 2.501) = 2.501 3.499 0.9+1.1 = 4.99 m = -4.99 (or -5) (A1) (3) (Total 3 marks)
Question 4 The scatter graph shows the cost of cars in a used car showroom. (a) Draw a line of best fit and calculate the gradient of this line. Using their line, = 8000 8 or use of any other points m = -1000 (A1) (2) (b) Give an interpretation of this gradient. The value of a car goes down by 1000 every year it gets older (or similar explanation) (C1) (1) (Total 3 marks) TOTAL /10
Quadratic equations (completing the square) 2 Grade 8 SOLUTIONS Objective: Solve quadratic equations by completing the square. Question 1. Rewrite 2 + 6 + 7 in the form ( + ) 2 2 + 6 + 9 9 + 7 ( + 3) 2 2 (A1) (Total 1 mark) Question 2. Solve 2 10 + 9 = 0 by completing the square. 2 10 + 25 25 + 9 = 0 ( 5) 2 16 = 0 ( 5) 2 = 16 5 = ±4 = 5 ± 4 = 9 = 1 (A1) (Total 2 marks)
Question 3. Solve 2 8 12 = 0 by completing the square. Leave your answers in surd form. 2 8 + 16 16 12 = 0 ( 4) 2 28 = 0 ( 4) 2 = 28 4 = ± 28 = 4 ± 28 = 4 ± 2 7 (A1) (Total 3 marks) Question 4. Solve 4 2 + 28 24 = 0 by completing the square. Give your answers to 3 significant figures. 2 + 7 6 = 0 2 + 7 + 12.25 12.25 6 = 0 ( + 3.5) 2 18.25 = 0 ( + 3.5) 2 = 18.25 + 3.5 = ± 18.25 = 3.5 ± 18.25 = 0.772 = 7.77 (A1) (Total 4 marks) TOTAL /10
Trigonometric Graphs 2 Grade 8 Solutions Objective: Recognise, sketch, and interpret graphs of trigonometric functions Question 1 Sketch the graph of y = tan x for 0 360 (3) (Total 3 marks) Question 2 Here is the graph of y = cos x for 0 360 1.5 1 0.5 0-0.5-1 -1.5-2 -2.5-3 -3.5 0 1 2 3 4 5 6 7 y = cos x y = cos(2x) -2 On the axes above, sketch the graph =cos(2 ) 2 for 0 360 (3) (Total 3 marks)
Question 3 The graph of y = sin x for 0 360 is shown below. What are the coordinates of the 4 points labelled on the graph? ( 0, 0 ) ( 90, 1 ) ( 270, -1 ) ( 360, 0 ) (4) (Total 4 marks) TOTAL /10