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Transformation of graphs
Question Paper 2
Level
Subject
Exam Board
Module
Topic
Sub Topic
Booklet
A Level
Mathematics (Pure)
AQA
Core 2
Algebra
Transformation of graphs
Question Paper 2
Time Allowed:
89 minutes
Score:
/75
Percentage:
/100
Grade Boundaries:
A*
>85%
A
777.5%
B
C
D
E
U
70%
62.5%
57.5%
45%
<45%
1
(a)
Sketch the curve with equation y = 2x, indicating the coordinates of any point where the
curve intersects the coordinate axes.
(2)
(b)
(i)
Use the trapezium rule with five ordinates (four strips) to find an approximate value
for
, giving your answer to three significant figures.
(4)
(ii)
State how you could obtain a better approximation to the value of the integral using
the trapezium rule.
(1)
(c)
Describe a geometrical transformation that maps the graph of y = 2x onto the graph of
y = 2x + 7 + 3.
(3)
(d)
The curve y = 2x + k + 3 intersects the y-axis at the point A(0, 8).
Show that k = logm n, where m and n are integers.
(2)
(Total 12 marks)
2
The diagram shows a sketch of the curve y = 24x.
The curve intersects the y-axis at the point A.
(a)
Find the value of the y-coordinate of A.
(1)
(b)
Use the trapezium rule with six ordinates (five strips) to find an approximate value
for
, giving your answer to two decimal places.
(4)
(c)
Describe the geometrical transformation that maps the graph of y = 24x onto the graph of
y = 24x–3.
(2)
Page 1 of 5
(d)
The curve y = 24x is translated by the vector
to give the curve y = g(x).
The curve y = g(x) crosses the x-axis at the point Q. Find the x-coordinate of Q.
(4)
(e)
(i)
Given that
loga k = 3 loga 2 + loga 5 – loga 4
show that k = 10.
(3)
(ii)
The line y =
crosses the curve y = 24x–3 at the point P. Show that the
x-coordinate of P is
.
(3)
(Total 17 marks)
3
The diagram shows a sketch of the curves with equations
.
The curves intersect at the origin and at the point A, where x = 4.
(a)
(i)
For the curve
, find the value of
when x = 4.
(2)
(ii)
Find an equation of the normal to the curve
at the point A.
(4)
Page 2 of 5
(b)
(i)
Find
.
(2)
(ii)
Find the area of the shaded region bounded by the two curves.
(4)
(c)
Describe a single geometrical transformation that maps the graph of
onto the graph of
.
(2)
(Total 14 marks)
4
(a)
Use the trapezium rule with four ordinates (three strips) to find an approximate value for
, giving your answer to four significant figures.
(4)
(b)
The curve with equation y =
factor
is stretched parallel to the x-axis with scale
to give the curve with equation y = f(x). Write down an expression for f(x).
(2)
(Total 6 marks)
5
(a)
Sketch the graph of y = 3x, stating the coordinates of the point where the graph crosses the
y-axis.
(2)
(b)
Describe a single geometrical transformation that maps the graph of y = 3x:
(i)
onto the graph of y = 32x;
(2)
(ii)
onto the graph of y = 3x+1.
(2)
Page 3 of 5
(c)
(i)
Using the substitution Y = 3x, show that the equation
9x – 3x+1 + 2 = 0
can be written as
(Y – 1)(Y – 2) = 0
(2)
(ii)
Hence show that the equation 9x – 3x+1 + 2 = 0 has a solution x = 0 and, by using
logarithms, find the other solution, giving your answer to four decimal places.
(4)
(Total 12 marks)
6
The diagram shows a sketch of the curve with equation y = 6x.
(a)
(i)
Use the trapezium rule with five ordinates (four strips) to find an approximate value
for
, giving your answer to three significant figures.
(4)
(ii)
Explain, with the aid of a diagram, whether your approximate value will be
an overestimate or an underestimate of the true value of
.
(2)
(b)
(i)
Describe a single geometrical transformation that maps the graph of y = 6x onto the
graph of y = 63x.
(2)
(ii)
The line y = 84 intersects the curve y = 63x at the point A. By using logarithms, find
the x-coordinate of A, giving your answer to three decimal places.
(4)
Page 4 of 5
(c)
The graph of y = 6x is translated by
to give the graph of the curve with
equation y = f(x). Write down an expression for f(x).
(2)
(Total 14 marks)
Page 5 of 5