Download Sample MAths exam with solutions

S228/318
DUBLIN INSTITUTE OF TECHNOLOGY
KEVIN STREET, DUBLIN 8.
_____________
BSc Computer Science
Year 3
______________
SUPPLEMENTAL EXAMINATIONS 2009
______________
Computational Mathematics
John Gilligan
Dr D. Lillis
Marking Scheme
Autumn 2009
Time allowed 2 Hours
Answer question 1 and 2 other questions.
1:
a) Find the distance between the points
i) (0, 2) and (3, 6) and ii) (-2, 3) and (4, -5)
(4 marks)
Solution/Question Rationale
Simple distance formula
Marking Scheme
2 marks for each distance.
b)
Find the values of a and b if
i) (4, 1) is the midpoint (a, b) and (-1, 5)
ii) (3, b) is the midpoint of (a, 2) and (0, 0).
(6 marks)
Solution/Question Rationale
Algebraic manipulation of midpoint formula
Marking Scheme
3 marks for each calculation
c) Sketch the image of △abc under an axial symmetry in K.
a
K
b
c
(6 marks)
Solution/Question Rationale
I introduced various transformations such as axial symmetry. Can they realise this
graphically.
Marking Scheme
2 marks for knowing what axial symmetry is
4 marks for correctly drawing image
d)
For each of the following letters, state whether it has a centre of symmetry and indicate
where it is.
H T A
Z
(4 marks)
Solution/Question Rationale
Do they know what the centre of symmetry is and where it is on these symbols
Marking Scheme
1 mark by 4
e)
What is [2]5 X [6]5
[12]5 = [2]5
(6 marks)
Solution/Question Rationale
Are they aware of the multiplication rule for modulo arithmetic and can it be applied?
Marking Scheme
4 marks for multiplication rule. 2 marks for reduction to modulo 5
f)
Using modulo 5 and the number 13 for illustration, define what is meant by the
equivalence class modulo n containing m
(6 marks)
{….,-7,-2,3,8,13,18,23,28,…..}
Marking Scheme
6 marks for correctly listing equivalence class.
g) Prove that A(2, 5), B(-4, 5) and C(-1, 2) are the vertices of a right-angled isosceles
triangle.
(6 marks)
Solution/Question Rationale
Can the students apply Pythagoras and Line distance formula.
Marking Scheme
3 marks for correctly using line distance formula and 3 marks for Pythagoras
2:
a) Given Vectors
A = (1,1,1) and B = (1,1,0) and C = (1,0,0),
Find the scalars α, β, and γ
if αA + βB + γC = (2,-3,4)
(5 marks)
Solution/Question Rationale
Do the students understand scalar multiplication and can they solve linear equations.
Marking Scheme
3 marks for correctly doing scalar multiplication and 2 marks for solving linear equations
Let
A = (1,3,6)
B = (4, -3, 3)
C = (2,1 ,5) be three vectors in V3
Determine the components of the following vectors
i) -3A +2B + 6C
(3 marks)
Solution/Question Rationale
This is basic vector addition
Marking Scheme
3 marks for correct answer.
ii) 2A + B – 3C
(3 marks)
Solution/Question Rationale
This is basic vector addition
Marking Scheme
3 marks for correct answer.
iii)
Draw these Geometrically.
(4 marks)
Solution/Question Rationale
This is just to remind them of graph plotting
Marking Scheme
2 marks for each graph.
iv)
|| A + B + C||
(A + B + C) = (7,1,14)
|| A + B + C|| =
2
7
 1  14
2
(4 marks)
Solution/Question Rationale
Can they calculate vector length
Marking Scheme
4 marks for correct answer.
v)
Calculate A.B and B X C
1*4 + 3*-3 + 6*3 = 13
-3*5 – 1*3 = -18
3*2 – 5*4 = -14
4*1 – 2*3 = -2 (-18,-14,-2)
(6 marks)
Solution/Question Rationale
These are basic vector operations of dot and cross product.
Marking Scheme
3 marks for correct answer. X2
vi)
Prove for any vectors u, v and w that:
(u+v)+w = u + (v+w)
(5 marks)
Solution/Question Rationale
Associativity is important within vector algebra.
Marking Scheme
4 marks for good attempt
1 marks for correct proof
3 a)
Suppose
3 7

A  
 2 5
1  1 3 

B  
 2 1 4
 1  1

C  
2 3 
Either calculate each of the following or explain why the calculation cannot be done.
i) 3A-C
ii) AB
ii) BA
iv) A-1
v) Bt
(15 marks)
Solution/Question Rationale
These are basic Matrix Operations.
Marking Scheme
3 marks for each X 5.
b) Using Gaussian elimination, solve the system
X+2Y-Z=6
2X+3Y+Z=11
X+3Y-2Z=2
(15 marks)
Question Rationale
This a major algorithmic technique for solving simultaneous equations.
Marking Scheme
3 marks for augmented matrix
8 marks for correctly applying row operations
4 marks for correct solution.
4:
a) Compute the greatest common divisor of 7,276,500, and 3,185,325
Gcd(7,276,500, 3,185,325) = gcd(3,185,325, 905,850)
= gcd(905,850 , 467,775)
= gcd(467,775, 438,075)
= gcd (438,075, 29,700)
= gcd(29,700, 22,275)
= gcd(22,275,7,425)
= gcd(7,425, 0)
Ans = 7,425
(10 marks)
Solution/Question Rationale
Can they compute GCD using Euclid
Marking Scheme
7 marks for Euclid
3 for correct answer
b) Encrypt “We will not win the All Ireland” using a columnar transposition Cipher with
key “Meath”
(10 marks)
Solution/Question Rationale
Do the students know how apply the basic columnar transposition.
Marking Scheme
6 marks for algorithm
2 for correct answer
c) Encrypt Plaintext = 1107300 using the RSA algorithm with two primes;
p= 563 and q = 2357.
(10 marks)
Solution/Question Rationale
Can the students apply RSA algorithm. Will they compute n, z,, d and e
Marking Scheme
7 marks for algorithm
3 for correct answer