UNIVERSITY OF DUBLIN
MA1S21
TRINITY COLLEGE
Faculty of Science
school of mathematics
JF Natural Science
JF Human Genetics
JF Medicinal Chemistry
Annual Examination
Trinity Term 2005
Mathematics 1S2
Date
Examination Location
Time
Dr. Buttimore
SAMPLE PAPER
Attempt Six Questions
Log tables are available from the invigilators if required
Non-programmable calculators are permitted for this examination
Please indicate the make and model of your calculator on each answer book used
The inside covers of answer books are available for writing
You may not start this examination until you are instructed to do so by the Invigilator
Page 2 of 4
MA1S21
1. Two vectors in three dimensional space are given by
~v = (1, −2, 2),
w
~ = (4, 4, 7).
(a) Evaluate the vectors 4~v + w
~ and 3~v − w.
~
(b) Find the length of the vector ~v and the length of the vector w.
~
(c) Obtain unit vectors in the same direction as the vectors ~v and w.
~
(d) Determine the cross product of the vector ~v and the vector w.
~
2. In a sample of people taken from a particular region R, a = 81 people had blood group
A, b = 36 had blood group B, and c = 324 had blood group O.
(a) Find the components of the comparison vector, ~v =
√
a,
√
√ b, c .
(b) Evaluate the norm (that is, the length) of the vector ~v and show that the square
of the norm corresponds to the size of the sample taken.
(c) A sample of people from a second region R0 of the world indicated the levels,
a0 = 25 (group A), b0 = 16 (group B), and c0 = 400 (group O).
A further sample of people from a third region R00 revealed that the levels were,
a00 = 121 (group A), b00 = 64 (group B), and c00 = 256 (group O).
Find the components, and the norms, of the vectors for regions R0 & R00
~v 0 =
√
a0 ,
√
b0 ,
√ c0 ,
~v 00 =
√
a00 ,
√
b00 ,
√ c00 .
(d) Obtain the angle in radians between the following pairs of regional vectors
(i) ~v 0 and ~v 00 ,
(ii) ~v and ~v 00 ,
(iii) ~v and ~v 0 .
(e) Suppose that the angles between the vectors form a measure of genetic disparity.
Among the three regions {R, R0 , R00 }, which two regions are
(i) most closely related ? (ii) most distantly related ?
Page 3 of 4
MA1S21
3. A system of three linear equations in three unknowns is given by
2x + 4y + 8z = 42
3x + 7y + 6z = 47
5x + 9y + 2z = 49
(a) Exhibit the augmented matrix for the system of equations.
(b) Reduce the augmented matrix by using elementary row operations.
(c) Obtain the solution of the linear system from the reduced matrix.
(d) Verify the result obtained for the solution set of the equations.
4. Explain the terms, transpose of a matrix, symmetric matrix, trace of a matrix. Consider
the following matrix

6 2 1


A = 
 1 7 4

3 1 5



.


(a) Obtain the determinant det A of the matrix A. Is det(A2 ) = (det A)2 ?
(b) Write down the transpose AT and the trace tr A of the matrix A.
(c) Calculate the product of the matrix A with its transpose matrix AT .
(d) Is A AT a symmetric matrix?
5. Consider the following matrix with three rows

1
1 

A=
 3
10 
4
and three columns

2 7


1 6 

5 1
(a) Write the matrix A in a form without the pre-factor of 1/10.
(b) Show that the sum of each row of the matrix A is unity.
(c) Obtain the square of the matrix A and find the sum of each row of A2 .
(d) Write down the transpose of the row vector u = (1, 1, 1) and explain why it is
an eigenvector of the matrix A. What is the corresponding eigenvalue? Is uT an
eigenvector of A2 ? of A3 ? Explain.
Page 4 of 4
6. Consider the following three-by-three matrix B

0
0



B=
0
 15


17
1
MA1S21
1
0
1




.



(a) Find the eigenvalues of the matrix B. Are the eigenvalues distinct?
(b) Obtain the eigenvectors of the matrix B for each eigenvalue of B.
(c) Verify that each eigenvector V of B and its corresponding eigenvalue λ obeys the
matrix equation
BV = λV .
7. Explain why the signs of the coefficients in the following system of differential equations
are appropriate for describing the development in time of the population levels, x and
y, of two symbiotic species
dx
= −x + 4y
dt
dy
= 2x − 8y .
dt
Determine the general solution of the system of equations. Hence obtain the particular
solution induced by the initial conditions x0 = 5 and y0 = 8 at t = 0 .
c UNIVERSITY OF DUBLIN 2005