Save My Exams! – The Home of Revision
For more awesome GCSE and A level resources, visit us at www.savemyexams.co.uk/
Transformation of graphs
Question Paper 3
Level
Subject
Exam Board
Module
Topic
Sub Topic
Booklet
A Level
Mathematics (Pure)
AQA
Core 2
Algebra
Transformation of graphs
Question Paper 3
Time Allowed:
83 minutes
Score:
/69
Percentage:
/100
Grade Boundaries:
A*
>85%
A
777.5%
B
C
D
E
U
70%
62.5%
57.5%
45%
<45%
1
(a)
Describe the geometrical transformation that maps the curve with equation
y = sinx onto the curve with equation:
(i)
y = 2 sinx;
(2)
(ii)
y = –sinx;
(2)
(iii)
y = sin(x – 30°).
(2)
(b)
Solve the equation sin(ș – 30°) = 0.7, giving your answers to the nearest 0.1° in the interval
᷆ᵿͻᶐͻșͻᶐͻ᷿᷉᷆ᵿ᷄
(3)
(c)
Prove that (cosx + sinx)2 + (cosx – sinx)2 = 2.
(4)
(Total 13 marks)
2
The diagram shows a sketch of the curve with equation y = 27 – 3x.
The curve y = 27 – 3x intersects the y-axis at the point A and the x-axis at the point B.
(a)
(i)
Find the y-coordinate of point A .
(2)
Page 1 of 4
(ii)
Verify that the x-coordinate of point B is 3.
(1)
(b)
The region, R, bounded by the curve y = 27 – 3x and the coordinate axes is shaded. Use
the trapezium rule with four ordinates (three strips) to find an approximate value for the
area of R.
(4)
(c)
(i)
Use logarithms to solve the equation 3x = 13, giving your answer to four
decimal places.
(3)
(ii)
The line y = k intersects the curve y = 27 – 3x at the point where 3x = 13.
Find the value of k.
(1)
(d)
(i)
Describe the single geometrical transformation by which the curve with
equation y = – 3x can be obtained from the curve y = 27 – 3x.
(2)
(ii)
Sketch the curve y = – 3x.
(2)
(Total 15 marks)
3
(a)
Describe the single geometrical transformation by which the curve with equation
y = tan x can be obtained from the curve y = tan x.
(2)
(b)
Solve the equation tan x = 3 in the interval 0 < x < 4π, giving your answers in
radians to three significant figures.
(4)
(c)
Solve the equation
cos ș(sin ș – 3 cos ș) = 0
in the interval 0 < ș < 2π , giving your answers in radians to three significant figures.
(5)
(Total 11 marks)
Page 2 of 4
4
The diagram shows the graph of y = cos 2x for 0° ᶐ x ᶐ 360°.
(a)
Write down the coordinates of the points A, B and C marked on the diagram.
(4)
(b)
Describe the single geometrical transformation by which the curve with equation
y = cos 2x can be obtained from the curve with equation y = cos x.
(2)
(c)
Solve the equation
cos 2x = 0.37
giving all solutions to the nearest 0.1° in the interval 0° ᶐ x ᶐ 360°.
(No credit will be given for simply reading values from a graph.)
(5)
(Total 11 marks)
5
The diagram shows a curve C with equation y =
. The point O is the origin (0, 0).
The region bounded by the curve C, the x-axis and the vertical lines x = 1 and x = 4 is shown
shaded in the diagram.
(a)
(i)
Write
in the form xp, where p is a constant.
(1)
Page 3 of 4
(ii)
Find
dx.
(2)
(iii)
Hence find the area of the shaded region.
(3)
(b)
The point on C for which x = 4 is P. The tangent to C at the point P intersects the x-axis
and the y-axis at the points A and B respectively.
(i)
Find an equation for the tangent to the curve C at the point P.
(4)
(ii)
Find the area of the triangle AOB.
(3)
(c)
Describe the single geometrical transformation by which the curve with equation
y=
can be obtained from the curve C.
(2)
(d)
Use the trapezium rule with four ordinates (three strips) to find an approximation
for
, giving your answer to three significant figures.
(4)
(Total 19 marks)
Page 4 of 4