Answers to Maths B (EE1.MAB) exam papers
[Pages:2]Answers to Maths B (EE1.MAB) exam papers
Spring 2006
1.
(a)
1-
1 4
x2
+
5 32
x4
-
15 128
x6
+ ? ? ?,
(b)
e2x cos 2x
=
1 + 2x -
8 3
x3
+ ? ? ?.
2.
(a)
1 4
,
(b)
1 3
.
3.
(i)
(a)
cosh x
=
13 5
,
(b)
tanh x
=
12 13
,
(c)
sinh 2x
=
312 25
.
(ii) sinh-1
x-1 3
+c
4.
(i)
y
=
1 x2-c
,
(ii)
y
=
tan
x3 3
+
4
,
(iii)
y
=
x 3
-
1 9
+
c e-3x
5.
(i)
y
=
A e-5x
+ B e-2x,
(ii)
y
=
e-x(A cos 2x + B sin 2x) +
1 5
x2
-
4 25
x
-
27 125
.
6. 1.164
7.
(i)
2 2
+ 2,
(ii)
13 8
8. (ii) because the function is even
(iii)
an
=
2 n
sin
n 2
if
n
= 1, 2, 3, . . .,
and
a0
=
.
Spring 2007
1. (i) 1 + 3x2 + 6x4 + 10x6 + ? ? ?
(ii)
ln(1
+
x)
=
x
-
x2/2
+
x3/3
-
?
?
?,
ln(1
+
2x)
=
2x
-
2x2
+
8 3
x3
+
?
?
?,
ln((1
+
2x)(1
+
x))
=
ln(1
+
2x)
+
ln(1
+
x)
=
3x
-
5 2
x2
+
3x3
+
?
?
?.
2.
(i)
1 12
,
(ii)
-1,
(iii)
9 2
.
3.
(i)
(a)
1 2
(x2
- 1/x2),
(b)0
(ii) 2 sinh-1(1) or 2 ln(1 + 2)
4.
(i)
y
= Aex4/4,
(ii)
y
=
, 2
1+e-2x
(iii)
y
=
1 2
+
c (x+1)2
.
Depending
on
how
you
do
the
calculations
you
might
instead
get
y
=
x2/2+x (x+1)2
+
d (x+1)2
which
is
equivalent.
5.
(i)
y
=
e-x(-2 cos 2x -
1 2
sin 2x),
(ii)
y
=
Ae-2x
+
Bex
-
3 20
cos
2x
+
1 20
sin
2x
6.
xn+1
=
xn
-
(x6n - 6x5n
2x2n - 1) - 4xn
root is 1.272
7.
(i)
4,
(ii)
4
(1
-
e-1
)
8. (ii) because the function is odd
(iii)
bn
=
2 n
(-1)n+1
Spring 2008
1. (i) 1 + 8x + 40x2 + 160x3 + 560x4 + ? ? ?,
(ii)
x2
-
1 6
x6
+
1 120
x10
+
?
?
?
1
sin(x2) dx 0.3103.
0
2.
(i)
8,
(ii)
3 5
,
(iii)
6
3. (i) tanh(ln x) = (x2 - 1)/(x2 + 1).
1
(ii)
sin2
2x
=
1 2
-
1 2
cos 4x
so,
by
Osborn's
rule,
sinh2
2x
=
1 2
cosh 4x
-
1 2
.
sinh2 2x dx = sinh 4x - x + c
8
2
dx
= cosh-1 x - 3 + c
x2 - 6x - 7
4
4. (i) y = -x2 + d, (ii) y =
3 2
x2
+
3x3
+
216
1/3
,
(iii)
y
=
-
cos 3x 3x2
+
c/x2.
5.
(i)
y
=
2 3
e-5x
+
1 3
e4x,
(ii)
y
=
e2x(A cos 3x + B sin 3x) +
3 25
e-2x.
6. 0.5049
7. (i) 1, (ii) /4.
8. the function is even so bn = 0 for all n.
4(-1)n
an =
, n2
22
a0 =
. 3
Spring 2009
1.
(i)
(1 + 3x)1/3
=
1 + x - x2 +
5 3
x3
+???
(ii)
e-x2
=
1
-
x2
+
1 2
x4
+
?
?
?,
cos
2x
=
1
-
2x2
+
2 3
x4
+
??
?,
e-x2
cos
2x
=
1
-
3x2
+
19 6
x4
+
?
?
?.
2.
(i)
6,
(ii)
1 2
,
(iii)
-6/.
3. (i) differentiate twice and use cosh2 - sinh2 = 1.
(ii)
1 2
x
+
1 12
sinh 6x + c,
(ii)
cosh-1
3 2
or
0.9624.
4. (i) y =
1 2(x2 -
, c)
(ii)
y
=
4ex e3(x +
, 1)
(iii)
y
=
-x
-
1 5
+
c e5x.
5. (i) y = Ae-7x + Be2x, (ii) y = e-x(2 cos x - 2 sin x) - sin 2x - 2 cos 2x.
6. 0.4638
7. (i) 1 - ln 2
2 2
(ii) after converting to polars, integral becomes
r sin r dr d.
01
8.
cn
=
1
3
f (t)e-2jnt/3 dt
30
=
1
3
e-2jnt/3 dt since f (t) = 0 for 0 < t < 1 & f (t) = 1 for 1 < t < 3
31
=
1
e-2jn/3 - 1
2jn
Putting n = 1 and using ej = cos + j sin gives result.
2
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