1. Session 1 Paper 2 Questions and Answers Calculator Harris Academy Supported Study

2. Question 1 (Unit 1 LO1 Straight Line) Triangle ABC has as its vertices A(-18,6), B(2,4) and C(10,-8). (a) Find the equation of the median from A to BC (b) Find the equation of the perpendicular bisector of side AC. (c) Find the coordinates of T, the point of intersection of these lines. A B C marks 3, 4, 3

3. Solution 1(a) mid-point of BC gradient of median equation of line (a) ans:

4. Solution 1(b) mid-point of AC gradient of AC perpendicular gradient equation of line (b) ans:

5. Solution 1(c) solving a system of equations x -coordinate y -coordinate ans: T(-3,1)

6. Question 2 (Unit 1 LO3 Differentiation) The diagram below shows the parabola with equation and a line which is a tangent to the curve at the point T(1,5). Find the size of the angle marked θ, to the nearest degree. marks (4)

7. Solution 2 Know to differentiate Find gradient of tangent at x = 1 Use m = tan θ Complete calculations ans:

8. Question 3 (Unit 2 LO4 Circle ) The circle in the diagram has equation. The line AB is a chord of the circle and has equation. (a) Show that the coordinates of A and B are (-1, -8) and (6, -1) respectively. (b) Establish the equation of the circle which has AB as its diameter. marks (4,3) x A B y O

9. Solution to question 3a ans: substituting into circle equation multiplying brackets and tidying up factorising and values of y corresponding values of x A(-1, -8) and B(6, -1)

10. Solution to question 3b ans: knowing to find midpoint of AB finding radius substituting into equation (2  5, -4  5)

11. The graph of the cubic function y = f (x) is shown in the diagram. There are turning points at (1,1) and (3,5). Sketch the graph of y = f '(x) Question 4 (Unit 1 LO3 Differentiation) x y (3,5) (1,1) marks (3)

12. Solution 4 ans: Interpret stationary points Parabola Maximum TP 31

13. Session 2 Paper 2 Questions and Answers Calculator Harris Academy Supported Study

14. Question 5 ( Unit 2 LO3 Trigonometry) Solve algebraically the equation where marks 5

15. Solution 5 double angle formula re-arrange to zero and factorise find roots answers from ans : 19  5 0, 160  5 0, 210 0, 330 0

16. An open box is designed in the shape of a cuboid with a square base. The total surface area of the base and four sides is 1200cm 2 x x Question 6 (Unit 1 LO3 Differentiation) (a) If the length of the base is x centimetres, show that the volume V (x) is given by (b) Find the value of x that maximises the volume of the box. h marks (3,5)

17. Solution 6 (a) ans: Equation for surface area Rearrange with h = ……. Find V

18. knowing to differentiate differentiate set derivative to zero solve for x nature table Solution 6 (b) ans: max TP at x = 20

19. Question 7 (Unit 1 LO3 Differentiation Unit 2 LO1 Polynomials) Also shown is the tangent to the curve at the point P where Part of the curve is shown in the diagram (a) Find the equation of the tangent. (b) The tangent meets the curve again at Q. Find the coordinates of Q. marks (4,4) x y P Q O x=1

20. Solution 7(a) differentiate gradient y -coordinate equation (a) ans:

21. Solution 7(b) form equation rearrange to zero factorise coordinates of Q (a) ans: Q (8,64 ) 11-1017-8 1 -98 1 80

22. Question 8 (Unit 2 LO2 Integration ) marks (4, 4) The diagram shows the parabolas and (a) Find the coordinates of the point A (b) Calculate the area enclosed between the two curves. 4 A x y O

23. Solution 8a Form equation Rearrange to = 0 Factorise and solve Coordinates of A (a) ans:

24. Solution 8b (b) ans: Integrate Substitute limits Answer 4 A x y O

25. Session 3 Paper 2 Questions and Answers Calculator Harris Academy Supported Study

26. Question 9 (Unit 2 LO4 Circle ) A circle, centre C, has equation. Show that the line with equation 2y = x + 8 is a tangent to the circle and find the coordinates of the point of contact P marks (5) C P

27. Solution 9 ans: substitute into circle equation multiply out brackets and simplify factorise and solve for y complete proof point of contact One point of intersection so line is a tangent P(4, 6)

28. Question 10 (Unit 2 LO2 Integration) The diagram shows a sketch of the graph of y = ( x + 2)( x – 1)( x – 2) and the points P and Q x y PQ (0,4) O (-2,0) (a) Write down the coordinates of P and Q (b) Find the total shaded area marks (2,6)

29. Solution 10a ans: Coordinates of P Coordinates of Q

30. Solution 10b two integrals multiply out brackets integrate integral from 0 to 1 integral from 1 to 2 total area ans:

31. Question 11 (Unit 2 LO3 Trigonometry) The diagram shows a sketch of part of the graph of a trigonometric function whose equation is of the form Find the values of a, b and c x 5 -3 0 π y marks (3)

32. Solution 11 ans: Interpret amplitude Interpret period Interpret vertical displacement

33. (b) The second diagram shows lines OA and OB. The angle between these two lines is 30 0 Question 12 (Unit 1 LO1 Straight Line) marks 3,1.(a) The diagram shows line OA with equations. The angle between OA and the x-axis is Find the value of a.. Calculate the gradient of line OB correct to 1 decimal place

34. Solution 12a ans: gradient of line gradient = tan (angle) and apply process

35. Solution 12b ans: angle = tan -1 (angle)