Sample Questions
Dont use calculators.
1. A tutorial group has 7 members, of whom 4 are female and 3 are male. Each day the members
all volunteer to clean the blackboard/whiteboard for the tutor, and one of them is selected
at random to do the job.
(a) Wh
Sample Questions
You may use calculators.
1. A fair coin is tossed 4 times. Find the expectation, variance and standard deviation for the
number of heads.
2. A biased coin, with probability p = 2/3 of getting a head, is tossed 5 times. Find the
expectatio
Answers to Sample Questions
A calculator was used sparingly.
1. Since this is a binomial distribution with n = 4, p = 1/2 and (therefore) q = 1/2, we
can use the fact that a binomial distribution has expected value E (X ) = np and variance
Var (X ) = npq
Answers to Sample Questions
No calculator was used.
1. Here is the binomial probability formula for k successes in n trials when the probability of
success is p. Let X be the number of successes. Then
P r(X = k ) = n Ck pk q nk
where q = 1 p.
Also we have
Sample Questions
Dont use calculators.
1. It is known that 20% of students suer from Mathematical Exposure Shaky Syndrome
(MESS), which causes them to start shaking whenever they hear the the word mathematics.
(a) If 5 students are selected at random, wha
Answers to Sample Questions
No calculator was used.
1. (a) | 3.57| = 3.57
(b) Yes, when x = 0 or x is negative, |x| = x.
2. (a)
(b)
(c)
(d)
(e)
3.8 = 4
3.8 = 3
3.8 = 3
3.8 = 4
Yes, x = x whenever x is an integer, in which case both functions have value x.
Answers to Sample Questions
No calculator was used.
1. (a) Let the particular person be p1 . Let the other people be p2 , p3 , p4 , p5 , p6 and p7 . So the
sample space is S = cfw_p1 , p2 , p3 , p4 , p5 , p6 , p7 . Then the probability that p1 is selected
Sample Questions
Dont use calculators.
1. Let X be a 3-element set and Y be a 5-element set. If X and Y are disjoint, how many
elements are there in their union X Y ?
2. Suppose that each of four pieces of paper has one of the four compass points (N for n
Answers to Sample Questions
No calculator was used.
1. |X Y | = |X | + |Y | = 8 (since X and Y are disjoint)
2. (a) Since the possible outcomes from tossing the coin are H (for head) and T (for tail), the
two events (tossing a coin and selecting a compass
Sample Questions
Dont use calculators.
1. Suppose that the predicate P (x) means x is positive. Let the domain of interpretation D
be the set R of all real numbers. Express each of the following sentences in the notation of
predicate logic.
(a) Some real
Answers to Sample Questions
No calculator was used.
1. Suppose that the predicate P (x) means x is positive. Let the domain of interpretation D
be the set R of all real numbers. Express each of the following sentences in the notation of
predicate logic.
(
Sample Questions
Dont use calculators.
1. Let W mean the wind is blowing, P mean Lara passes her exam and H mean Dad is happy.
Express the following statements symbolically, using the notation of propositional logic.
(a) The wind is not blowing.
(b) Lara
Answers to Sample Questions
No calculator was used.
1. Let W mean the wind is blowing, P mean Lara passes her exam and H mean Dad is happy.
Express the following statements symbolically, using the notation of propositional logic.
(a) The wind is not blowi
Sample Questions
Dont use calculators.
1. (a) Write down formulae for the quotient q and remainder r produced by the division algorithm when an integer a is divided by a positive integer b (so that a = qb + r where
0 r < b).
(b) Apply the division algorit
Sample Questions
Dont use calculators.
1. Consider the matrix A =
3 5 7
0 1 4
.
(a) How many rows does A have?
(b) How many columns does A have?
(c) What is the order of A?
(d) Find the matrix A.
(e) Find the tranpose AT of A.
2. Consider the matrix B =
3
Answers to Sample Questions
No calculator was used.
1. Consider the matrix A =
3 5
7
0
1 4
.
(a) There are 2 rows in A.
(b) There are 3 columns in A.
(c) The order of A is (2 3).
3
5 7
0 1
4
3
0
1
(e) AT = 5
7 4
(d) A =
2. Consider the matrix B =
3 5
0
1
Sample Questions
Dont use calculators.
1. (a) What is | 3.57|?
(b) Can |x| ever equal x? (Yes or No)
2. (a)
(b)
(c)
(d)
(e)
(f)
What is 3.8 ?
What is 3.8 ?
What is 3.8 ?
What is 3.8 ?
Can x ever equal x ? (Yes or No)
Can you nd a general formula for x in