AQA Level 2 Further Maths June 2019 Paper 1 (non-calculator) · old spec 8360

19 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 (3 marks)

   A straight line passes through the points (-2, 11) and (1, 2)

   Work out the equation of the line.

   Give your answer in the form    y = mx + c

   (3 marks)

2. Question 2 (2 marks)

   Write    5/6a + a/4    as a single fraction.

   Give your answer in its simplest form.

   (2 marks)

3. Question 3 (2 marks)

   Work out the smallest integer value of x that satisfies the inequality    8 – 5x < 26

   (2 marks)

4. Question 4 (4 marks)

   p(x – 1) + 2(3x + k) equiv 4(x + 2)    where p and k are integers.

   Work out the values of p and k.

   (4 marks)

5. Question 5 (3 marks)

   Solve    3√((2√x – 10)) = 2

   (3 marks)

6. Question 6 (5 marks)

   The transformation matrix    ( 2a & b , -b & -a )    maps the point (3, 4) onto the point (8, -7)

   Work out the values of a and b.

   (5 marks)

7. Question 7 (4 marks)

   A function is given by    beginaligned f(x) &= -2x && -1 ≤ x < 0 \\ &= x(4 – x) && 0 ≤ x < 3 \\ &= 2x – 3 && 3 ≤ x ≤ 4 endaligned

   Draw the graph of    y = f(x)    on the grid.

   Diagram: A blank grid with x from –2 to 5 and y from –2 to 6, with axes through O.

   (4 marks)

8. Question 8 (4 marks)

   ABC is a straight line.

   A is the point (-4, 5)

   C is the point (20, -7)

   AB : BC = 5 : 3

   Diagram: A straight line from A (-4, 5) at the top left down to C (20, -7) at the bottom right, with B marked on the line between them, nearer C.

   Work out the coordinates of B.

   (4 marks)

9. Question 9 (1 mark)

   y = 2x(x² – 5x)

   Circle the expression for    dy/dx

   2(2x – 5)        6x² – 20        3x² – 10x        6x² – 20x

   (1 mark)

10. Question 10 (3 marks)

    Factorise fully    6x² + 26xy – 20y²

    (3 marks)

11. Question 11 (5 marks)

    A cone has base radius r cm, perpendicular height h cm and slant height l cm

    The curved surface area is 60π cm²
    l = 3r

    Work out the value of h.

    Give your answer in the form    a√10    where a is an integer greater than 1

    You must show your working.

    (5 marks)

12. Question 12 (5 marks)

    A curve has the equation    y = x³ + ax² – 7    where a is a constant.

    The gradient of the curve when x = 4 is twice the gradient of the curve when x = -1

    Work out the value of a.

    You must show your working.

    (5 marks)

13. Question 13 (4 marks)

    Prove that    (3x + 5)² – 5x(x + 10) ≥ 0    for all values of x.

    (4 marks)

14. Question 14 (4 marks)

    Here are two transformations.

    A        Rotation 90° clockwise about the origin.

    B        Reflection in the line    y = x

    Use matrix multiplication to work out the single matrix which represents the combined transformation A followed by B.

    (4 marks)

15. Question 15 (2 marks)

    Here is a sketch graph of    y = cos x    for    0° ≤ x ≤ 360°

    Diagram: A sketch of y = cos x from 0° to 360°. It starts at 1, crosses the x-axis at 90°, reaches –1 at 180°, crosses the x-axis again at 270° and is back at 1 at 360°.

    You are given that    cos 36° = 0.8090

    Solve    cos x = -0.8090    for    0° ≤ x ≤ 360°

    (2 marks)

16. Question 16 (4 marks)

    Rationalise the denominator and simplify fully    dfrac21 – 11√53 – √5

    (4 marks)

17. Question 17 (4 marks)

    A, B and C are points on the circumference of a circle, centre O.
    ECD is a tangent to the circle at C.
    Angle AOB = 2x + 46°
    Angle OBC = 37°
    Angle ACD = 3x

    Diagram: A circle with centre O. A is near the top, B on the left and C on the lower right. Triangle ABC is drawn, with radii OA and OB. The line ECD touches the circle at C, with D above C and E below. Angle AOB is 2x + 46°, angle OBC is 37° and angle ACD (between the chord CA and the tangent CD) is 3x.

    Work out the value of x.

    (4 marks)

18. Question 18 (5 marks)

    ADEF is a trapezium.
    ABCD is a straight line.
    BCEF is a square of side √6 cm

    Diagram: A trapezium ADEF with AD along the bottom and FE along the top. B and C are on AD, with the square BCEF standing on BC. The side BF is labelled √6 cm. The angle at A is 60° and the angle at D is 30°.

    (a)

    Show that    AB = √2 cm

    (1 mark)

    (b)

    Show that    DE = 2√6 cm

    (1 mark)

    (c)

    Work out the perimeter of the trapezium ADEF.

    Give your answer in the form    t√2 + w√6    where t and w are integers.

    You must show your working.

    (3 marks)

19. Question 19 (6 marks)

    f(x) = (x – 3)/2x

    Solve    f(x + 1) – f(2x) = 0.5

    You must show your working.

    (6 marks)

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