# CO 1102 University of London Binary Number and Rational Number Exercises

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Need an expert to solve easy four problems from 10 problems .I am attaching the sample.Will send the book when accepted.
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CO1102 ZB
BSc and CertHE Examination
COMPUTING AND INFORMATION SYSTEMS, CREATIVE COMPUTING
and COMBINED DEGREE SCHEME
Mathematics for Computing
Release date: Wednesday 15 July 2020
Time allowed: 3 hours and 30 minutes
There are TEN questions on this paper. Candidates should answer all TEN
questions. All questions carry equal marks and full marks can be obtained for
complete answers to TEN questions. The marks for each part of a question are
indicated at the end of the part in [.] brackets.
There are 100 marks available on this paper.
You may use any calculator for any appropriate calculations, but you may not
use computer software to obtain solutions. Credit will only be given if all
workings are shown.
UL20/0572
Page 1 of 6
Question 1
(a) Showing your working, convert the binary number:
(11000110000101000)2
ii. to octal.

(b) Working in binary and showing all carries, compute (110100)2 + (11101)2 .
(c) Consider the set:


S = { 2, 12 , 2, 4}.
Give each of the following sets by the listing method.
i. S ∩ Z;
ii. S ∩ Q;
iii. S − R.

(d) Showing your working, express the repeating decimal:
3.11111111 . . .
as a rational number in its simplest form.

Question 2
Let p and q be the following propositions concerning a positive integer n:
p : n is a factor of 60;
q : n is a factor of 20.
(a) List the truth sets for the two propositions p and q.

(b) For each of the following compound statements, express it using
propositions p and q and logical symbols, and give its truth set.
i. n is a factor of 60 but not a factor of 20;
ii. If n is a factor of 20 then n is a factor of 60;
iii. The contrapositive of the statement in part ii.

(c) Use truth tables to prove that:
(¬p ∨ q) ≡ p → q.

UL20/0572
Page 2 of 6
Question 3
(a) Let A, B and C be subsets of a universal set U and consider the two sets
X = (A ∩ B)0 − C and Y = (A ∩ B ∩ C)0 .
i. Draw a labelled Venn diagram depicting the sets A, B and C in such
a way that they divide U into 8 disjoint regions and shade the region
corresponding to the set X on this diagram.
ii. Draw a second labelled Venn diagram and shade the region
corresponding to the set Y on this diagram.

1 2 3 4
18
(b) Express the set { 15
, 20 , 25 , 30 , . . . , 100
} by using rules of inclusion.

(c) Let E = {t, e, a}. Give the power set P(E) by the listing method.

Question 4
Consider the function f : S → S where S = {1, 2, 3, 4, 5, 6} deﬁned by the table
x
1 2 3 4 5 6
.
f (x) 5 3 6 1 4 2
Let g : S → S be the function deﬁned by g(x) = f (f (x)).
(a) Compute f (3) and g(3).

(b) Complete the following table so it deﬁnes the function g(x).

x 1 2 3 4 5 6
g(x)
(c) Explain what it means for a function to be onto and hence prove that f is
onto.

(d) Explain what it means for a function to be one-to-one and hence prove
that f is one-to-one.

(e) Say what the function g must satisfy in order to be the inverse of the
function f and hence show that g is indeed the inverse of f .

UL20/0572
Page 3 of 6
Question 5
(a) Justifying your answer, say whether it is possible to construct a graph on
9 vertices in which every vertex has degree 3.

(b) Let K5 be the complete graph on the vertex set V = {u, v, x, y, z}. Further,
let G be the simple graph on the vertex set V with edge set deﬁned by the
u : x, y;
v : x, z;
x : u, v;
y : u, z;
z : v, y.
i. Draw the graph G.

ii. Let AK be the adjacency matrix of K5 and let AG denote the adjacency
matrix of the graph G. Compute the matrix AK − AG and hence draw
the graph G0 which has AK − AG as its adjacency matrix.
0
iii. Prove that the graph G deﬁned in part ii. is isomorphic to G.


Question 6
Let H be the simple graph on the vertex set V = {1, 2, 3, 4, 5, 6}

0
1
1
0
1
0
1
0
1
0
0
0
1
1
0
0
1
0
0
0
0
0
1
1
1
0
1
1
0
1
0
0
0
1
1
0

⎟.

(a) Explain how you can ﬁnd the degree of each vertex of H and the number
of edges of H from the adjacency matrix without drawing the graph.

(b) Draw the graph H.

(c) Draw two non-isomorphic spanning trees T1 and T2 for H each with degree
sequence 3, 2, 2, 1, 1, 1. Explain why your two trees are not isomorphic.

(d) Suppose that a tree has 18 vertices, how many edges has it got?

(e) Find a tree with the property that all vertices of the tree have the same
degree. Further, for each positive integer v ≥ 1, say how many such trees

UL20/0572
Page 4 of 6
Question 7
(a) Suppose that we have a group of children consisting of 4 boys and 8
girls. A basketball team with 5 players is chosen from this group. Find
the number of different teams possible and hence compute the probability
that the chosen team has:
i. all 4 boys in it;
ii. at least one boy in it.

(b) Let M be a set with 250 elements. Suppose that we choose three subsets
A, B and C of M such that each of them contains 100 elements, the
intersection of any two of them contains 32 elements, and the intersection
between all three of them contains 8 elements.
i. Given an element of M , what is the probability that it is in A?
ii. How many of the elements of M are in none of the three chosen
subsets?

Question 8
(a) Consider the augmented matrix:

1 2 : 7
.
5 1 : 3
i. Write down the system of equations corresponding to this augmented
matrix.

ii. Showing your working, use just two elementary row operations to
reduce the augmented matrix above to row echelon form.

iii. Showing your working, solve the system of equations corresponding
to the augmented matrix above.

(b) Consider the matrices A, B, C and D below, where a denotes a constant
real number. Compute the matrices A − B, B 2 and CD.

A=

1 5
,B=
2 1

a 5
3 2
1 1 3
,C=
, D = ⎝3 0 ⎠ .
a 2
0 a 2
4 2


UL20/0572
Page 5 of 6
Question 9
Let r1 , r2 , r3 , . . . , r12 be an ordered list of 12 records which are stored at the
internal nodes of a binary search tree T .
(a) Explain why record r6 is the one that will be stored at the root (level 0) of
the tree T .

(b) Construct the tree T showing where each record is stored.

(c) Let S = {r1 , r2 , r3 , . . . , r12 } denote the set of records stored at the internal
nodes of T , and deﬁne a relation R on S by:
ra R rb if ra and rb are stored at the same level of the tree T .
i. Show that R is an equivalence relation.
ii. List the equivalence class containing r7 .


Question 10
(a) Given a real number c 6= 1 and a positive integer n, express the formula:
cn+1 − 1
c0 + c 1 + c 2 + c 3 + · · · + c n =
c−1
by using Σ-notation.
(b) Use the formula from part (a) to compute the sum
12
X
2i+2 .


i=0
(c) Let the sequence {un } be deﬁned by the recurrence relation:
un+1 = un + 3n for n = 0, 1, 2, . . . ,
and the initial term u0 = 1.
Calculate u1 , u2 , u3 , u4 and u5 , showing all your working.

(d) Let the sequence {un } be as deﬁned in part (c) and let the sequence {wn }
be deﬁned by the recurrence relation:
wn+1 = wn + 3n for n = 0, 1, 2, . . . ,
and the initial term w0 = p, where p is a positive integer.
Prove by induction that wn − un = p − 1 for all n ≥ 0.
END OF PAPER
UL20/0572
Page 6 of 6


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Tags:
rational number

positive integer

binary number

decimal fraction

logical symbols

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