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AQA Level 2 Further Maths June 2016 Paper 2 (calculator) · old spec 8360
26 questions · 105 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.
Question 1 (3 marks)
A triangle has vertices
A(2, 5) B(2, 0) and C(-4, 3)
Work out the area of triangle ABC.
You may use the grid to help you.Diagram: A grid with x from –6 to 4 and y from –2 to 7, one square to a unit.
(3 marks)
Question 2 (5 marks)
The function f(x) = x² – 4x + 1 has domain -1 ≤ x ≤ 5
Here is the graph of y = f(x)
Diagram: The graph of y = x squared – 4x + 1 on graph paper, x from –1 to 5. The curve starts at (–1, 6), comes down through about (0.3, 0) to its lowest point (2, –3), and rises through about (3.7, 0) to (5, 6).
(a)
Write down the equation of the line of symmetry of the graph.
(1 mark)
(b)
Use the graph to work out the solutions of x² – 4x + 1 = 5
Give your answers to 1 decimal place.
(2 marks)
(c)
Write down the range of f(x) for domain -1 ≤ x ≤ 5
(2 marks)
Question 3 (2 marks)
L is a straight line with equation ax + by = c
where a, b and c are non-zero integers.(a)
At which point does L intersect the x-axis?
Circle your answer.(a/c, 0) (c/a, 0) (b/c, 0) (c/b, 0) (1 mark)
(b)
What is the gradient of a line parallel to L?
Circle your answer.-b/a b/a -a/b a/b (1 mark)
Question 4 (4 marks)
Work out the point of intersection of the lines
2x + 3y = 11 and 2y = 13 – 3x
(4 marks)
Question 5 (4 marks)
a, b and c are numbers such that
a < 0
b > 1
-1 < c < 1Tick the correct box for each statement.
Always true Sometimes true Never true a³ < 0 b < 10a² ab > 0 b – c > 1 (4 marks)
Question 6 (3 marks)
For the curve y = f(x),
dfracdydx = 3/2x – kx⁴ + k where k is a constant.
When x = -2 the gradient of the curve is 12
Work out the value of k.
(3 marks)
Question 7 (2 marks)
Simplify fully (2/3x³y)³
(2 marks)
Question 8 (3 marks)
D(-6, 4) and E(-2, 9) are joined by a straight line.
P is a point on DE.
DP : PE = 3 : 5Work out the coordinates of P.
(3 marks)
Question 9 (4 marks)
PQ is parallel to BC.
Diagram: PQ and BC are parallel horizontal lines, with arrows, PQ above BC. P is joined to B, below and slightly to its left. A line goes from B down to the left to A. Angle x is at P, between PQ and PB. At B, the angle between BP and BA is y + 100°, and the angle between BA and BC, below B, is 2y + 80°.
Prove that x = 3y
(4 marks)
Question 10 (4 marks)
(a)
Simplify (x² – 7x + 10)/(x² – 2x – 15)
(2 marks)
(b)
Factorise fully w⁵x³y² + w²x⁶y³
(2 marks)
Question 11 (3 marks)
The x² term in the expansion of (3x + 4)(x² + px + 5) is -23x²
Work out the value of p.
(3 marks)
Question 12 (5 marks)
Here are the first four terms of linear sequences X and Y and quadratic sequence Z.
Sequence X 7 9 11 13 …. Sequence Y 2 5 8 11 …. Sequence Z 14 45 88 143 …. (a)
Work out the nth term of sequence X.
(2 marks)
(b)
The nth term of sequence Y is 3n – 1
Using your answer to part (a), or otherwise, work out the nth term of sequence Z.
Give your answer in the form an² + bn + c where a, b and c are integers.(3 marks)
Question 13 (4 marks)
Here is a sketch of y = a + bx – 2x² where a and b are constants.
The graph intersects the x-axis at (-1, 0) and (7/2, 0) and the y-axis at point P.
Diagram: A sketch of an upside-down U-shaped curve. It crosses the x-axis at –1, left of the origin, and at 7/2, right of the origin, and crosses the y-axis at P, above the origin. Its highest point is just to the right of the y-axis.
Work out the coordinates of point P.
You must show your working.(4 marks)
Question 14 (4 marks)
P, Q, R and S are points on the circumference of a circle.
w : y = 7 : 5
PQ = PSDiagram: A circle through P at the top left, S at the top right, R on the right and Q at the bottom left. PQ, PS, QS, QR and RS are drawn, and PQ and PS are marked equal. Angle w is at P, between PQ and PS. Angle x is at Q, between QP and QS. Angle y is at R, between RS and RQ.
Work out the size of angle x.
(4 marks)
Question 15 (4 marks)
(a)
Solve 2/5√x = 1
(2 marks)
(b)
Solve x³ = 5x²
(2 marks)
Question 16 (4 marks)
Rearrange y = (8(w – x))/x to make x the subject.
(4 marks)
Question 17 (3 marks)
A cylinder has base radius x cm and height y cm
A hemisphere has radius 6y cmThe cylinder and hemisphere have equal volumes.
Work out the value of x/y
You must show your working.(3 marks)
Question 18 (5 marks)
Angle y is acute.
tan y = (p + 1)/(p – 1) where p is a constant greater than 1
(a)
Which of the statements below is correct?
Circle your answer.y = 45° y < 45° y > 45° y could be any acute angle (1 mark)
(b)
Work out the expression for sin y
Give your answer in the form dfracap + b√(cp² + d) where a, b, c and d are integers.
You may use a diagram to help you.(4 marks)
Question 19 (3 marks)
The continuous curve y = g(x) has exactly two stationary points.
The stationary points are
• a point of inflection at P(1, 2)
• a minimum point at Q(a, b) where a > 1 and b < 0
On the axes below, sketch the curve.
Label points P and Q on your sketch.Diagram: Blank axes: an x-axis and a y-axis crossing at O, with no scale.
(3 marks)
Question 20 (4 marks)
Under the transformation represented by ( -1 & -3 , 2 & 4 ),
the image of point P(a, 2) is point Q.
Can point Q be the same as point P?
You must show your working.(4 marks)
Question 21 (6 marks)
Solve 3/(x – 2) + 2/(x – 1) = 5
Do not use trial and improvement.
Write your solutions to 3 significant figures.(6 marks)
Question 22 (4 marks)
Pyramid VABCD has a horizontal rectangular base.
X is the centre of the base.
V is vertically above X.VB = VC = 17 cm AB = 22 cm BC = 16 cm
Diagram: A pyramid VABCD with a rectangular base. AB is the front edge of the base and BC goes back on the right; D is at the back on the left. V is at the top, directly above X, the centre of the base, which is marked with a dot. The edges AD, DC and VD are hidden (dashed).
Work out the angle between the planes VBC and ABCD.
(4 marks)
Question 23 (5 marks)
Shape A maps to shape B by an enlargement, scale factor 3, centre the origin.
Shape B maps to shape C by a rotation through 180°, centre the origin.Shape A can be mapped to shape C by a single transformation.
Use matrices to show that the single transformation is an enlargement, centre the origin.
State the scale factor of the enlargement.(5 marks)
Question 24 (5 marks)
f(x) = x/(2x + 1) for positive values of x.
Work out f(x + 1) – f(x)
Give your answer as a fraction in its simplest form.
You must show your working.(5 marks)
Question 25 (6 marks)
The curve y = 2x³ – 5 intersects the y-axis at C.
The tangent to the curve at P(2, 11) intersects the y-axis at D.
Diagram: Axes with the curve y = 2x³ – 5, which rises from the bottom left, flattens out as it crosses the y-axis at C, below the origin, then rises steeply to the top right through P (2, 11). The tangent at P is a straight line that crosses the y-axis at D, below C.
Work out the length CD.
(6 marks)
Question 26 (6 marks)
(a)
Prove that sin² x – 3cos² x equiv 4sin² x – 3
(2 marks)
(b)
Hence, or otherwise, work out the values of x between 0° and 360° for which
sin² x – 3cos² x = 0
(4 marks)
Questions © AQA, recreated by Mr Barton Maths for teaching. Each question is credited to its board, series, paper and number.