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 converg
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 determ
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
(
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 t
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
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
(i
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
(
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.
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 i
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 compu
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 i
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 compu
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 i
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
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
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 pro
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 orthogona
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
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:
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
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 =
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
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
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
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
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
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
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