Mathcad - CAPE - 2008 - Math Unit 2 - Paper 02
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7/29/2019 Mathcad - CAPE - 2008 - Math Unit 2 - Paper 02
1/7
CAPE 2008 - Pure Mathematics Unit2
Paper 2
1 a( ) i( )
d
dx e
4 x
cos .
x e
4 x
sin .
x 4 e
4 x
cos x.d
dx e
4 x
cos .
x
x
x
x
x
e4 x
4 cos . x sin . x( )
ii( )d
dxln
x2
1
x
. d
dxln x
21.
d
dx
1
2ln. x( ).
d
dxln
x2
1
x
. d
dxx
d
dxx
2 x
x2
1
1
2 x
3 x2
1
2 x x2
1.
2 x
x2
1
1
2 x
x
x x
Alternatively:d
dxln x
21. ln x.
2 x
x2
1
1
2 x
d
dxln x
21. ln x.
x
x x
3 x2
1
2 x x2
1.
b( ) ln y. x ln. 3.ln y. x1
y
dy
dxln 3.
1
y
dy
dx
dy
dxy ln. 3.
dy
dxy
dy
dx3
xln 3.
dy
dx
x
c( ) i( )2 x
23 x 4
x 1( ) x
2
1
A
x 1
Bx C
x
2
1
Bx C
x
Bx
expands in partial fractions to3
2 x 1( ).( )
1
2
5 x( )
x2
1
.
ii( ) x3
2 x 1( )
x
2 x2
1
5
2 x2
1
d
3
2
ln x 1( ).1
4
ln x2
1.5
2
arctan x( ). C
2 x2
3 x 4
x 1( ) x2
1
by integration, yields3
2ln x 1( ).
1
4ln x
21.
5
2arctan x( ). C
1
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7/29/2019 Mathcad - CAPE - 2008 - Math Unit 2 - Paper 02
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2 a( ) I e
x1d x
I exx
ex dy
dx
. ex
y e3 x
ex dy
dx
. ex
yx
xd
dxe
xy. d xe
3 xdx
d
dxe
xy. d x
ex
y1
3e
3 xCe
3 xC
xy
1
3e
2 x C
ex
e2 x
ex
x
x
b( )
1
y
y1 d
0
x
xe4 x
d
1
y
y1 d
x
y1
4e
4 xx
0
. 1x
x
1
4e
4 x3.
c( )
1
e
xx2
ln x. dx
3
3ln x. e
1
.
1
e
xx
3
3
1
x
. d
1
e
xx2
ln x. dx
x
x3
3ln x.
x3
9e
1
. 1
92 e
31
x3
3ln x.
x3
9e
1
.
d( ) i( ) dv dudv du vv
1
2d 2 v. Cv Cv 2 1 u( ). C
ii( ) du cos x. dx.x dx du 1 sin2
x dxx dx du 1 u2
dxu dx
limits in terms of u are 0, 1
0
1
u
1 u( )
1 u2
d0
1
u1 u( )
1
2
d
0
1
u
1 u( )
1 u2
d
2 1 u( )
1
21
0
. 22 1 u( )
1
21
0
.
2
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7/29/2019 Mathcad - CAPE - 2008 - Math Unit 2 - Paper 02
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Section B (Module 2)
3 a( ) i( ) u1
3 u2
4 u3
6 u4
9
ii( ) let the statement Pn
n2
n 6
2
n nn
be assumed true
P1
1 1 6
2P
13 P
2
4 2 6
2P
24
Assume Pn
is true for n = k Then Pk
k2
k 6
2
k kk
Pk 1
Pk
kk
kk
k2
k 6
2 k
k2
k 6
2
k 1( )2
k 1( ) 6
2
k2
k 6
2
k k
Since Pk 1
is of the form of Pk
for n = k + 1 and
Pn
is true for n = 1 Pn
is true for n = 2 Pn
is true for all n N
Alternatively: for n > 1 un
un 1
n 1( )n
nn
from given defn of un
un
un 1
n 1un
un 1
n un 1
un 2
n 2un 1
un 2
n un 2
un 3
n 3un 2
un 3
n
u2
u1
1u2
u1
un
u1
1 2 ... n 1( )... un
u1
n
2n 1( )u
nu
1
nn arith prog
un
n
2n 1( ) 3
nn
nu
n
n2
n 6
2
n nn
b( )a
1 r81
a
1 r
a 1 r4.
1 r65
a 1 r4.
1 r1 r
4 65
811 r
4r
4 16
81r
4r
2
3
a 27
c( ) i( ) ln 1 x( ). xx
2
2
x3
3
x4
4
x5
5......x
x x x x1 x< 1
ii( ) a( ) ln 1 x( ). xx
2
2
x3
3
x4
4
x5
5ln 1 x( ). x
x x x x1 x 1
