7/29/2019 Mathcad - CAPE - 2007 - Math Unit 2 - Paper 01
1/9
CAPE 2007Pure Mathematics Unit 2 - Paper 01
Section A - Module 1
1 a( ) In the diagram below the curves y 2 e2 xx
and y exx
intersect
at P (p, q)
2 e2 x
ex
x
1 0 1 2
1
1
2
Determine the values of p and q [5 marks]
b( ) Solve 22 x 1
622 x 1
for x R [3 marks]
a( ) 2 e 2 x e x2 e 2 x x 2e
2 x1e
x02
e2 x
1e
x2 ex 02 ex x = ln 2
y1
eln 2.
y1
eln 2.
y1
2P ln 2.
1
2,.
b( ) 2 x 1( ) ln 2. ln 6.2 x 1( ) ln 2. x1
2
ln 6.
ln 2.1.x
1
2
ln 6.
ln 2.1. x = 0.792 (3 d.p)
Alternatively: 2 22 x. 62 22 x.
x1
2
lg 3.
lg 2.
lg
lgx = 0.792 (3 dec places)
1
7/29/2019 Mathcad - CAPE - 2007 - Math Unit 2 - Paper 01
2/9
2 a( ) The parametric equations of a curve are x4
ttand y 2 t
22t
Find the gradient of the curve at the point (4, 4) [5 marks]
b( ) Differentiate with respect to x y tan2
3 x ln x3.x x [4 marks]
a( )dy
dx
4 t
4
t2
dy
dx
t
t
simplifies to t3
x = 4 t = 1dy
dx1
1dy
dx
b( )dy
dx2 tan. 3 x( ). 3 sec
23 x( )..
1
x3
3 x2
3 sec2
3 x( ). 3 x2
x x
x
x
dy
dx6 tan. 3 x( ). sec
2. 3 x( )3
x3 x( )
xx x
x
3 a( ) Express5
3 x( ) 2 x( ).in the form
P
3 x
Q
2 xwhere P and Q are
constants
[3 marks]
b( ) Hence find x5
3 x( ) 2 x( ).d [3 marks]
a( ) 5 P 2 x( ). Q 3 x( ).5 P x Q x P 1 Q 1
5
3 x( ) 2 x( ).
1
3 x
1
2 x
5
3 x( ) 2 x( ). x x
b( ) x1
3 xd
1
2 xln 3 x( ). ln 2 x( ). ln K.ln 2 x( ). Kx x
I ln k 3 x
2 x.. k
x
x
2
7/29/2019 Mathcad - CAPE - 2007 - Math Unit 2 - Paper 01
3/9
4 Obtain the following
a( ) xx 2 x 5( )4. d by substituting u = 2x - 5 [5 marks]
b( ) xx sec2
x. d using integration by parts [4 marks]
a( )1
2du dx
1
2du dx x
1
2u 5( )u
I u1
2
u 5( ) u4 1
2
. du I1
4
uu5
5 u4
d. u
I1
4
u6
6u
5. Ku
6
6u
5K
uu I
u5
24u 6( ) Ku 6( ) K
uu I
2 x 5( )5
2 x 1( )
24K
24K
x x
b( ) I x tan. x. xtan x. dx x x I x tan. x. ln cos x.. Kln cos x.. Kx x x
5 The cost $c of manufacturing x items may be modelled by the differential equation
dc
dx2 c 10 x
dc
dx2 c x
By using a suitable integrating factor solve the differential equation given that there is a costof $100 when no items are produced
[8 marks]
I e
x2 d x
I e2 xx
e2 x dc
dx. 2 e
2 xc e
2 x10 x.e
2 x dc
dx. 2 e
2 xc
xx
xd
dxc e
2 x. d xe2 x
10 xdxd
dxc e
2 x. d x
c e2 x.
1
2e
2 x10 x x
1
2e
2 x10( ) dc e
2 x. x x x
3
7/29/2019 Mathcad - CAPE - 2007 - Math Unit 2 - Paper 01
4/9
c e2 x.
1
2e
2 x10 x
5
2e
2 xK
5
2e
2 xK
xx
xx = 0 c = 100
100
1
2 0( )
5
2 1( ) K
5
2 1( ) K K
205
2 c 5 x
5
2
205
2 e2 xx x
Section B - Module 2
6 A sequence {un} is defined by un 1 un 2n
un 1
un
nn 1
a( ) Prove that un 2
2 un
unn
[4 marks]
b( ) If u1 2 find u3 and u5 [4 marks]
a( ) un 1
2n
un
n
nn
un 2
2n 1
un 1
n
nn
un 2
22
n
un 1
.n
nn
2n
un 1
un
2n
un 1
nu
n 22 u
n.
nn
b( ) u1
2 u3
2 2( ). yields 4 u3
2 u1
.
u5
2 4( ). yields 8 u5
2 u3
.
7 The sum of the first and third terms of a GP is 50 and the sum of the second and fourth termsis 150. For this GP find
a( ) the common ratio [4 marks]
b( ) the first term [2 marks]
c( ) the sum of the first five terms [2 marks]
a( ) a 1 r2. 50a 1 r2. ar 1 r
2. 150ar 1 r2. r = 3
b( ) a = 5 c( ) S5
5 35
1.
3 1yields 605
4
7/29/2019 Mathcad - CAPE - 2007 - Math Unit 2 - Paper 01
5/9
8 Find the term independent of x in the expansion of 2 x2 5
x3
10
[7 marks]
10( ) 9( ) 8( ) 7( )
242( )
65( )
4yields 8400000
9 a( ) Use the fact that nC
k
n !
k! n k( ) !.Ck
n
k n kkto express in terms of factorials
i( ) the coefficient u of xn
in the expansion of 1 x( )2 n
[2 marks]
ii( ) the coefficient v of xn
in the expansion of 1 x( )2 n 1
[3 marks]
b( ) Hence show that u = 2v [4 marks]
a( ) i( )2 n( ) !
n ! 2 n n( ) !.x
n 2 n( ) !
n !( ) n !( )x
n.2 n( ) !
n ! 2 n n( ) !.x
n n
n nx
nu
2 n( ) !
n !( ) n !( ).
n
n n
ii( )2 n 1( ) !
n ! 2 n 1 n( ) !.x
n 2 n 1( ) !
n ! n 1( ) !.x
n2 n 1( ) !
n ! 2 n 1 n( ) !.x
n n
n nx
nv
2 n 1( ) !
n ! n 1( ) !.
n
n n
iii( )2 n( ) 2 n 1( ) !
n !( ) n. n 1( ) !.u
2 n( ) 2 n 1( ) !
n !( ) n. n 1( ) !.u
2 2 n 1( ) !.
n ! n 1( ) !.
n
n nu 2 vv
10 a( ) Given that f r( ).2 r 1
r 1( ) r 2( ).f r( ).
r
r rprove that
f r( ). f r 1( ).2 r 3( ).
r r 1( ). r 2( ).f r( ). f r 1( ).
r
r r r[4 marks]
b( ) Hence find
3
n
r
r 3
r r 1( ). r 2( ).=
[4 marks]
a( )2 r 1
r 1( ) r 2( ).
2 r 3
r r 1( ).
r 2 r 1( ). r 2( ) 2 r 3( )
r r 1( ). r 2( ).
2 r 1
r 1( ) r 2( ).
2 r 3
r r 1( ).
r r r r
r r r
f r( ). f r 1( ).2 r 3( ).
r r 1( ). r 2( ).f r( ). f r 1( ).
r
r r r
5
7/29/2019 Mathcad - CAPE - 2007 - Math Unit 2 - Paper 01
6/9
b( )
2 Sn
7
2
9
6
9
6
11
12...
2 n 1
n 1( ) n 2( ).
2 n 3
n n 1( )....
Sn
1
2
7
2
2 n 3
n n 1( ).
n
n nnS
n
7 n2
11 n 6
4 n n 1( ).
n n
n nn
Section C - Module 3
11 a( ) Determine the number of ways in which the letters of the word S T A T I S T I C S
may be arranged so that the vowels are placed together
[3 marks]
b( ) A team of five is chosen at random from 4 boys and 6 girls. Calculate the numberof ways that this team can be chosen to include at least 3 girls
[5 marks]
a( ) vowels are A I I considered as one item
1 7( ) ! 3 !
3 ! 3 !. 2 !.3360 ways
1 7( ) ! 3 !
3 ! 3 !. 2 !.ways
b( )
(3 girls from 6 and 2 boys from 4) or (4 girls from 6 and 1 boy from 4)or (5 girls from 6 and no boys)
6C
34
C2
. 6C
44
C1
. 6C
54
C0
.
120( ) 60( ) 6( ) 186120( ) 60( ) 6( )
6
7/29/2019 Mathcad - CAPE - 2007 - Math Unit 2 - Paper 01
7/9
12 Two unbiased dice each with six faces are tossed randomly one after another
a( ) Determine the set of possible outcomes [2 marks]
b( ) Find the probability that
i( ) the product of the numbers on the two dice is a multiple of 5 [2 marks]
ii( ) the second die shows the number 2 [1 mark]
iii( ) the product of the numbers on the two dice is a multiple of 5 or the seconddie shows the number 2
[2 marks]
a( )
{(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6),(3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6),(5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)}
b( ) i( )11
36ii( )
1
6iii( )
11
36
6
36
1
36
4
9
11
36
6
36
1
36
13 The matrices X and Y are given by X =
4
5
7
2
6
9
1
8
3
and Y =
1
5
4
6
2
6
Calculate
a( ) the determinant of X [4 marks]
b( ) YT
X [3 mark]
a( ) X 4 18( ) 72( )( ). 2 15( ) 56( )( ). 45( ) 42( )X yields 365
4
5
7
2
6
9
1
8
3
yields 365
b( ) 1
6
5
2
4
6
4
5
7
2
6
9
1
8
3
. yields7
56
68
78
29
4
7
7/29/2019 Mathcad - CAPE - 2007 - Math Unit 2 - Paper 01
8/9
14 Y and X are 3 x 1 matrices and are related by the equation Y = AX where
A
1
7
3
0
5
2
3
0
1
is a non-singular matrix
a( ) Find A1
[6 marks]
b( ) X, when Y =
10
12
8
[3 marks]
a( )
A1
1
7
3
0
5
2
3
0
1
1
A1
yields
5
2
7
2
1
2
3
4
1
15
2
21
2
5
2
b( ) 10
12
8
1
7
3
0
5
2
3
0
1
X.
10
12
8
X
X
5
2
7
2
1
2
3
4
1
15
2
21
2
5
2
10
12
8
. yields
1
1
3
8
7/29/2019 Mathcad - CAPE - 2007 - Math Unit 2 - Paper 01
9/9
15 Air is pumped into a spherical balloon of radius r cm at the rate of 275 metres cube persecond. When r = 10 calculate the rate of increase of
a( ) the radius r [5 marks]
b( ) the surface area S [4 marks]
[The volume V and surface area S of a sphere of radius r are given by
V4 r
3
3
rand S 4 r
2r respectively]
a( )dr
dt
dr
dV
dV
dt.
dr
dt
dr
dV
dV dr
dt
1
4 r2
275( )dr
dt r
dr
dt10
275
4 10( )2
dr
dtyields
11
16 .( )cm
b( ) dSdt
dSdr
drdt
.dSdt
dSdr
dr dSdt
8 r 1116
.dSdt
r
dS
dt10
8 . 10( ).11
16
dS
dtyields 55 cm
2
9
Top Related