Oxford Cambridge and RSA
A Level Further Mathematics B
(MEI)
Y420/01 Core Pure
Practice Paper – Set 1
Time allowed: 2 hours 40
... [Show More] minutes
INSTRUCTIONS
• Use black ink. HB pencil may be used for graphs and diagrams
only.
• Complete the boxes provided on the Printed Answer
Booklet with your name, centrenumber and candidate
number.
• Answer all the questions.
• Write your answer to each question in the space
provided in the Printed AnswerBooklet. Additional
paper may be used if necessary but you must clearly
show your candidate number, centre number and
question number(s).
• Do not write in the barcodes.
• You are permitted to use a scientific or graphical calculator in
this paper.
• Final answers should be given to a degree of accuracy
appropriate to the context.
INFORMATION
• The total number of marks for this paper is 144.
• The marks for each question are shown in brackets [ ].
• You are advised that an answer may receive no marks
unless you show sufficient detailof the working to indicate
that a correct method is used. You should communicate
your method with correct reasoning.
• The Printed Answer Booklet consists of 24 pages. The
Question Paper consists of
8 pages.
Turn over
2
3 + x
2
y =
1
3 +x
2
O
n
0 4 e
0
1+2x
1
Section A (33 marks)
Answer all the questions.
1 Using standard summation formulae, find / r^r + 2h, giving your answer in a fully factorised form. [5]
r=1
2 (i) Describe the transformation of the plane represented by the matrix M = c
1 k
m, where k is a non-zero
constant.
0 1
[2]
(ii) The image of the point (2, 3) under the transformation in part (i) lies on the y-axis. Find k. [3]
3 In this question you must show detailed reasoning.
Given that z = 2 is a root of the equation z
3
+ kz2
+ 7z- 6 = 0, find, in exact form, the other two roots. [6]
4 In this question you must show detailed reasoning.
Fig. 4 shows the region bounded by the curve y =
1
, the x-axis, the y-axis and the line x = 1.
y
x
Fig. 4
This region is rotated through 2r radians about the x-axis. Find, in an exact form, the volume of the solid of
revolution generated. [4]
5 In this question you must show detailed reasoning.
Show that y
1
sinh 2xdx =
1
ae -
1
k
2
. [3]
6 Verify that the lines x - 2
=
y + 3
=
z
and x + 6
=
y + 4
=
z- 6
intersect, stating clearly the coordinates of
2 3 2 -3 1 4
the point of intersection. [6]
7 In this question you must show detailed reasoning.
3
Evaluate y ^ h
2
dx. [4]
3
r +r
c m
Section B (111 marks)
Answer all the questions.
8 In this question you must show detailed reasoning.
100
1
Find / 2
, expressing your answer as an exact fraction. [5]
r=10
9 (i) On an Argand diagram, draw the locus of points defined by z- 3 - 4i = 3. [2]
(ii) Find the complex number z which lies on the locus in part (i) and has the smallest possible argument.
Give your answer in the form a + bi. [7]
10 Given that the points A (1, 0, 2), B ( C (0, 6, 3) and D ( m, 1) are coplanar, find m. [8]
11 Matrix M is given by M =
1 c
.
-c 1
2
2 cosh 2x -sinh x = 5,
giving the answers in exact logarithmic form. [6]
14 (i) Show that the planes with equations
mx+ y + z = a
x + my + 2z = b
2x - y+ z = c
where m, a, b and c are real constants, always meet at a point. [5]
(ii) You are now given that a = m
2 +2, b = 0 and c = 0. Show that, for different values of m, the point of
intersection of the planes always lies on a fixed line. [7]
Turn over
(i) Given that detM = 0,find the possible values of c. [2]
(ii) Prove by induction that Mn
= 2
n-1M, for all positive integers n. [7]
12 (i) Determine, from first principles, the Maclaurin series up to the x
3
term for the function arcsin 1
x. [6]
(ii) Use the series in part (i) to find a rational approximation for r. [3]
13 (i)
(ii)
Prove, using exponential functions, that cosh 2x = 1 + 2 sinh2
x.
In this question you must show detailed reasoning.
[3]
Use the result in part (i) to solve the equation
4
z 8 2 8
15 You are given that x and y satisfy the simultaneous differential equations
dx
= x - y+sint
dt
and
dy
dt
= 6x - 4y + cost.
(i) Show that d
2
x
3
dx
2x 4 sin t. [4]
dt
2
+ dt
+ =
(ii) Find the general solution of the differential equation in part (i). [7]
(iii) Find the particular solution of this differential equation, given that, when t = 0, x = 0 and dx
= 0. [4]
dt
(iv) For the particular solution found in part (iii), find the limit of the amplitude of x as t 3. [2]
16 (i) Given that z = cosi +i sin i, express z
n
+
1
z
n
and z
n
-
1
z
n
in terms of trigonometric functions of ni
[3]
(ii) By considering az1
k
4
, show that sin4
i =
1
cos 4i1
cos 2i +
3
. [5]
(iii) (A) Sketch the curve with polar equation r = a sin2
i, for 0 G i G r, where a is a positive constant.
[2]
(B) In this question you must show detailed reasoning.
Find, in terms of a and r, the area of the region enclosed by the curve. [Show Less]