MA1S12: SOLUTIONS TO TUTORIAL 9
1. Find the radius of convergence for the power series
k=0
(2x)k
k!
k=1
(1)k xk
k
Solution: Recall that the Maclaurin series for ex is
k=0
xk
k!
and this series converges for all values of x. The rst series is the
Maclaurin
MA1S12 (Timoney) Tutorial sheet 5c
[February 1721, 2014]
Name: Solutions
1. Show matrix
1
0
0
R = 0 cos(/3) sin(/3)
0 sin(/3) cos(/3)
is a rotation matrix. [Hint: Is it orthogonal? What is its determinant?]
Solution: Rotation matrices are exactly orthogo
MA1S12 (Timoney) Tutorial sheet 9a
[March 2631, 2014]
Name: Solutions
1. A loaded die has the following probabilities of showing the numbers 16 after a throw:
3 2 3 2 1 4
,
,
,
,
,
15 15 15 15 15 15
(in that order). Find the probability that a number 4 wi
MA1S12 (Timoney) Tutorial sheet 8a
[March 1924, 2014]
Name: Solution
1. Find the equation of the line that is the best least squares t to the data points (2, 3), (3, 2),
(5, 1), (6, 1).
Solution: We take
x1 1
2
x2 1
3
X= .
=
.
5
.
6
xn 1
1
1
,
1
1
MA1S12 (Timoney) Tutorial sheet 8b
[March 1924, 2014]
Name: Solutions
1. Find the equation of the line that is the best least squares t to the data points (2, 1),
(3, 1), (5, 2), (6, 3).
Solution: We take
x1 1
2
x2 1
3
X= .
=
.
5
.
6
xn 1
1
1
,
1
1
MA1S12 (Timoney) Tutorial sheet 9c
[March 2631, 2014]
Name: Solution
1. A loaded die has the following probabilities of showing the numbers 16 after a throw:
3 2 3 2 1 4
,
,
,
,
,
15 15 15 15 15 15
(in that order). Find the probability that one of the num
MA1S12 (Timoney) Tutorial sheet 9b
[March 2631, 2014]
Name: Solutions
1. A loaded die has the following probabilities of showing the numbers 16 after a throw:
3 2 3 2 1 4
,
,
,
,
,
15 15 15 15 15 15
(in that order). Find the probability that an odd number
MA1S12 (Timoney) Tutorial/exercise sheet 10
[March 31, 2014]
Name: Solutions
1. The number of alpha particles hitting a certain detector per second is found to obey a
Poisson distribution with mean 4. What then is the probabilty of 3, 4 or 5 alpha particl
MA1S12 (Timoney) Tutorial sheet 7c
[March 1014, 2014]
Name: Solutions
1. Find the eigenvalues for the matrix
A=
1 4
2 4
Solution: We need to solve the chracteristic equation, det(A I2 ) = 0 and that is
det(A I2 ) = det
1
4
= ( + 1)( 4) 8 = 0
2
4
or
2 3 1
MA1S12 (Timoney) Tutorial sheet 6b
[March 37, 2014]
Name: Solutions
1. Find the eigenvalues for the matrix
7 0 0
A = 0 5 1
0 1 5
Solution: The eigenvalues are the solutions of det(A I3 ) = 0. We compute
7
0
0
5
1
A I3 = 0
0
1
5
and if we expand the deter
MA1S12 (Timoney) Tutorial sheet 7b
[March 1014, 2014]
Name: Solutions
1. Find the eigenvalues for the matrix
0 1
3 2
A=
Solution: We need to solve the chracteristic equation, det(A I2 ) = 0 and that is
det(A I2 ) = det
1
= ( 2) 3 = 0
3 2
or
2 2 3 = 0
( 3)
MA1S12 (Timoney) Tutorial sheet 6c
[March 37, 2014]
Name: Solutions
1. Find the eigenvalues for the matrix
6 0 2
A = 0 2 0
2 0 6
Solution: The eigenvalues are the solutions of det(A I3 ) = 0. We compute
6
0
2
2
0
A I3 = 0
2
0
6
and if we expand the deter
MA1S12 (Timoney) Tutorial sheet 7a
[March 1014, 2014]
Name: Solutions
1. Find the eigenvalues for the matrix
A=
0 4
3 1
Solution: We need to solve the chracteristic equation, det(A I2 ) = 0 and that is
det(A I2 ) = det
4
= ( 1) 12 = 0
3 1
or
2 12 = 0
( 4)
Chapter 3. Eigenvalues, diagonalisation and some applications
This material is a reduced treatment of what is in Anton & Rorres chapter 6, chapter 5 (mostly
for the case of 3 dimensions) and sections 5.4, 6.5 and 10.5.
3.1
Orthogonal diagonalisation
We mo
Chapter 2. Linear transformations
This material is a reduced treatment of what is in Anton & Rorres chapter 4, chapter 6 (mostly
for the case of 3 dimensions) and chapter 7. Rotations are mentioned in section 7.1 along with
orthogonal matrices. See also s
MA1S12 (Timoney) Tutorial sheet 4b
[February 1014, 2014]
Name: Solutions
1. If A and B are orthogonal n n matrices, show that their product AB is also orthogonal.
[Hint: What is the transpose of a product?]
Solution: The transpose of a product is the prod
MA1S12 (Timoney) Tutorial sheet 4c
[February 1014, 2014]
Name: Solutions
1. Let u = (2/3)i (2/3)j + (1/3)k and v = (1/ 5)j + (2/ 5)k
Show that u and v are both unit vectors and that they are orthogonal to one another.
Solution:
4 4 1
+ + = 1 (which implie
MA1S12 (Timoney) Tutorial sheet 5a
[February 1721, 2014]
Name: Solutions
For the rst two questions, let
(1 + 4 3)/9
(5 3)/9
(1 +
3)/9
A = (1 3)/9 (8 + 5
3)/18 (11 + 3)/18
4
(5 + 3)/9 (5 + 4 3)/18 (8 + 5 3)/18
Then A is in fact an orthogonal matrix and d
MA1S12: SOLUTIONS TO TUTORIAL 10
1. Find an equation for the tangent line to the ellipse
x = 3 cos t,
3
at the point P ( 2 ,
y = 2 sin t,
t [0, 2]
2).
Solution: To nd an equation for the tangent line at P we rst need
to nd its slope.
dy
=
dx
3
The point P
MA1S12: SOLUTIONS TO TUTORIAL 7
1. Write the nth term un of the following sequences and state if they
are convergent, divergent, bounded or monotone.
1, 1, 1, 1, 1 . . .
1, 3, 5, 7, 9 . . .
Solution: The rst sequence has nth term un = (1)n . This is a
bou
MA1S12: SOLUTIONS TO TUTORIAL 4
1. Let R be the region bounded by the graph of
f (x) = 1 +
x3
,
2
x [0, 2]
Find the volume of the solid generated by revolving R about the
y-axis.
Solution: We use the method of cylindrical shells.
2
2 x f (x) dx
V olume =
MA1S12: SOLUTIONS TO TUTORIAL 8
1. Determine if the following series are convergent.
5
3k
k=0
k=1
1
k
Solution: The rst series is a geometric series since it has the form
ark
k=0
1
with a = 5 and r = 3 . Since r < 1 it is convergent.
The second series is
MA1S12: SOLUTIONS TO TUTORIAL 6
1. Find the rst three approximations given by Eulers method with
increment
x = 0.1 for the initial value problem
y = x2 y
y(0) = 1
Solution: From the initial condition we have
x0 = 0,
y0 = y(x0 ) = 1
The sample points are
x
MA1S12: SOLUTIONS TO TUTORIAL 2
1. Use a trigonometric substitution to compute the indenite integral
3
dx
4 + x2
Solution: We use the trigonometric substitution
x = 2 tan
= dx =
dx
d = 2 sec2 d
d
We also have
4 + x2 =
4 + 4 tan2 = 2 1 + tan2 = 2 sec
and
UNIVERSITY OF DUBLIN
XMA1S121
TRINITY COLLEGE
Faculty of Engineering, Mathematics
and Science
school of mathematics
JF Natural Science
JF Human Genetics
JF Medicinal Chemistry
JF Physics and Chemistry of Advanced
Materials
JF Chemistry with Molecular Mode
MA1S12: SOLUTIONS TO TUTORIAL 3
1. Evaluate the improper integral
1
0
1
dx
x
Solution: There is a vertical asymptote at x = 0 and so
1
0
1
1
dx =
x
lim
+
a0
a
1
dx
x
1
=
lim 2x 2
+
a0
1
a
lim 2 2 a
+
=
a0
= 2
2. Evaluate the improper integral
(x2
2x
dx
MA1S12: SOLUTIONS TO TUTORIAL 5
1. Solve the initial value problem
y = x2 y,
y(0) = 1
Solution: This is a separable rst-order ordinary dierential equation. We rst rewrite the equation as
1
y 2
dy
= x2
dx
Now we integrate both sides
1
y 2 dy =
1
= 2y 2 =
x
MA1S12: SOLUTIONS TO TUTORIAL 1
1. Use integration by parts to compute the indenite integral
ex sin x dx
Solution: Let
dv = ex dx
u = sin x,
Then
du =
v=
du
dx = cos x dx
dx
ex dx = ex
dv =
Now
ex sin x dx =
u dv = uv
= ex sin x
ex cos x dx
We have to u
MA1S12 (Timoney) Tutorial sheet 4
[February 1014, 2014]
Name: Solutions
In this sheet, we consider 3 orthonormal vectors u = u1 i + u2 j + u3 k, v = v1 i + v2 j + v3 k
and w = w1 i + w2 j + w3 k in R3 . Let
u1 v1 w1
P = u2 v2 w2
u3 v3 w3
1. If A and B ar