AQA Level 2 Further Maths June 2018 Paper 1 (non-calculator) · old spec 8360

20 questions · 70 marks. Every question from the paper, recreated, with the official mark scheme, an annotated worked solution and a shadow question (the same skill with new numbers). Open the paper in the Ultimate Exam Question Finder to present it, print it or build a set.

1. Question 1 (2 marks)

   y = (x⁶)/2 + (x⁴)/4

   Work out dfracdydx

   Simplify your answer.

   (2 marks)

2. Question 2 (3 marks)

   P is the point (-12, b)

   Q is the point (a, 4)

   R is the point (6, -2)

   Q is the midpoint of PR.

   Work out the values of a and b.

   (3 marks)

3. Question 3 (2 marks)

   mathbfA = ( 2 & 4 , 3 & -1 ) and mathbfB = ( -2 & 6 , 2 & 1 )

   Work out mathbfAB.

   (2 marks)

4. Question 4 (4 marks)

   P = 4x and Q = 7x

   P increases by 25%

   Q decreases by 40%

   Now, P is 28 greater than Q.

   Work out the value of x.

   (4 marks)

5. Question 5 (3 marks)

   In the expansion and simplification of (x – 3)(x² + 5x + k) the coefficient of x² is equal to the coefficient of x.

   k is a constant.

   Work out the value of k.

   (3 marks)

6. Question 6 (1 mark)

   A circle has centre (-1, 2) and radius 5

   Which of these is the equation of the circle?

   Tick one box.

   (x + 1)² + (y – 2)² = 5
   (x – 1)² + (y + 2)² = 5
   (x + 1)² + (y – 2)² = 25
   (x – 1)² + (y + 2)² = 25

   (1 mark)

7. Question 7 (3 marks)

   Points A, B and C lie on a circle, centre O.

   Angle AOC = x + 75°

   Angle ABC = 2x

   Diagram: A circle with centre O. C is on the circle level with O, to the right; A is on the circle below and to the left of O, and B is on the circle at the bottom, between A and C. OA, OC, AB and BC are drawn. Angle AOC, on the side towards B, is x + 75° and angle ABC is 2x.

   Work out the value of x.

   (3 marks)

8. Question 8 (2 marks)

   Write (1 + 2√5)(4 – √5) in the form a + b√5 where a and b are integers.

   (2 marks)

9. Question 9 (4 marks)

   f(x) = 14 – x² for all real values of x.

   Solve f(2x) = 5

   You must show your working.

   (4 marks)

10. Question 10 (3 marks)

    Rearrange 1/xy = 4 – 3/y to make x the subject.

    (3 marks)

11. Question 11 (4 marks)

    A curve has equation y = 2x² + 3x – 9

    At a point P on the curve, the tangent is parallel to the line y = 4 – 5x

    Work out the coordinates of P.

    You must show your working.

    (4 marks)

12. Question 12 (4 marks)

    In the diagram,

    A is the point (15, 0) and B lies on the y-axis.

    Angle ABC = 90° and tan theta = 5/3

    Diagram: Axes with O at the origin. A is on the x-axis at (15, 0) and B is on the y-axis above O. The line AB goes up to the left from A to B, and the line BC goes up to the right from B to C, at right angles to AB. Angle OAB, at A, is marked θ.

    Work out the equation of the line BC.

    (4 marks)

13. Question 13 (6 marks)

    Solve the simultaneous equations

    xy = 2 and y = 3x + 5

    Do not use trial and improvement.

    You must show your working.

    (6 marks)

14. Question 14 (3 marks)

    Work out the value of (3^(1/2) + 3^(3/2))²

    You must show your working.

    (3 marks)

15. Question 15 (4 marks)

    Here is the graph of y = 3x – x² for values of x from –1 to 4

    Diagram: A grid with x from –1 to 4 and y from –4 to 3. The curve y = 3x – x squared is drawn from (–1, –4), up through O to its highest point (1.5, 2.25), and down through (3, 0) to (4, –4).

    By drawing a suitable linear graph on the grid, work out approximate solutions to

    x² – 4x + 2 = 0

    (4 marks)

16. Question 16 (4 marks)

    y = f(x) is a cubic curve with a maximum and a minimum stationary point.

    dfracdydx = x² + 2x – 3

    The y-coordinate of the minimum point is 21/3

    The y-coordinate of the maximum point is 13

    (0, 4) is a point on the curve.

    The tangent at (0, 4) has a negative gradient.

    Sketch the curve on the grid below.

    Show the coordinates of the stationary points.

    Diagram: Blank axes: the x-axis and the y-axis crossing at O, near the middle, with no scale marked.

    (4 marks)

17. Question 17 (4 marks)

    (a)

    Use the factor theorem to show that (x – 2) is a factor of x³ + 8x² + 5x – 50

    (1 mark)

    (b)

    Hence, factorise fully x³ + 8x² + 5x – 50

    (3 marks)

18. Question 18 (5 marks)

    D, E, F and S are points on a circle.

    RST is a tangent.

    The straight line EDT is parallel to FS.

    DS = DT

    Diagram: A circle through D at the top right, E on the left, F at the bottom and S on the right. RST is the tangent at S, with R below S and T above and to the right. The straight line EDT goes from E through D to T, and is parallel to FS. DS, DF and EF are drawn. DS and DT are marked equal, and angle DTS is x.

    Prove that FD is parallel to RST.

    Use angle DTS as x to help you.

    (5 marks)

19. Question 19 (4 marks)

    Write 2x² – 16x + 13 in the form a(x + b)² + c where a, b and c are integers.

    (4 marks)

20. Question 20 (5 marks)

    In triangle ABC,

    AB = 6√2 cm, angle ABC = 45° and angle ACB = 60°

    Diagram: Triangle ABC with B at the bottom left, C at the bottom right and A at the top, nearer C. AB is 6√2 cm, AC is x, angle ABC is 45° and angle ACB is 60°.

    Work out the value of x.

    Give your answer in the form a√b, where a and b are integers.

    You must show your working.

    (5 marks)

Questions © AQA, recreated by Mr Barton Maths for teaching. Each question is credited to its board, series, paper and number.