These collection of math questions were collected from Melbourne
University Mathematics & Statistics Society.
http://ms.unimelb.au.edu/~mums/
1. Mary has sat for 10 tests this term and has
an average score of 68. What mark must she gain in the next one to raise her
average to 70?
A. 85
B. 90
C. 95
D. 100
E. 105
2. Norm and Denise each wish to buy an ice cream
which costs a whole number of dollars.
However Norm needs seven more dollars to
buy an ice cream, while Denise needs one more dollar. They decide to buy only
one ice cream together, but discover that they do not have enough money. How
much does one ice cream cost (in dollars)?
A. 6
B. 11
C. 7
D. 9
E. 8
3. Norm and Geordie each roll a die. What is the
probability that the product of the two numbers rolled is less than 6? (Express
your answer as a fraction in simplest form.)
4. On the planet Dankuhn lives a being which can
be one of three sex types: male, female and emale. Any two different sexes may
breed and the offspring from such a union is of the third sex. How many of an
emale's great great great great grandparents were emales?
A. 18
B. 22
C. 26
D. 28
E. 32
5. One mathematician once said, "Today I
will prove more theorems than two days ago but fewer than one week ago."
What is the greatest number on consecutive days that she could have said this
and still be telling the truth?
A. 4
B. 5
C. 6
D. 7
E. 12
6.
Simplify
7. Simplify
A. 12
B. 14
C. 18
D. 24
E. 28
8. Solve the following equation for the integer n:
A. 208
B. 202
C. 220
D. 280
E. 820
9. Suppose where f is a
function defined for all non-zero real values of x. What is the value of
f(2)?
10. The digits 3, 1, 4 and 1 can be arranged to
form how many different 4-digit numbers?
A. 12
B. 16
C. 18
D. 24
E. 32
1.
B
Let x denote the mark that Mary
must receive in the next test. In the previous 10 tests, the total number of
marks that Mary received was 68 x 10 = 680. To raise her average to 70 in the
next test, the following equation must hold:
2.
C
Suppose an ice cream costs x, Norm
has n dollars and Denise has d dollars. Then x−n
= 7, x − d = 1, n + d < x. Adding
these equations together gives 2x < 8 + x, so x < 8.
Now if
x < 7, and x
− n = 7, then n < 0, which is impossible if we assume
Norm does not have a negative amount of money. Since the cost of an ice cream
is smaller than 8 but not smaller than 7, and a whole number, it must be 7
dollars.
3.
C
Let the number that Norm rolls be N
and the number that Geordie rolls G. Since there are six possibilities
for N and for G, all equally likely, there are 6 x 6 = 36
possibilities for the pair (N,G). The only possible pairs whose
product is less than 6 are
(1, 1), (1, 2), (2, 1), (1, 3), (3, 1),
(1, 4), (4, 1), (1, 5), (5, 1) and (2, 2).
Since there are 10 of these pairs, the
required probability is .
4.
B
The following table shows the number of
emales, females and males for each generation in the past six generations of an
emale. So we can see that every emale has 22 great great great great
grandparents.
1 2 3 4 5 6
emales 0
2 2 6 10 22
females 1
1 3 5 11 21
males 1
1 3 5 11 21
5.
C
6.
A
7.
B
8.
D
Let and . Then the equation given to us is a + b = 8.
We can also find the product of a and b.
Now we note that:
2n = (a3 + b3)
= (a + b)3 – 3ab(a + b)
= 83 – 3 x (-2) x 8
= 512 – (-48)
= 560
Therefore, n = 280.
9.
B
Let , then . Let , then . Taking
twice the second equation and subtracting
the first equation yields , giving
10.
A
If we treat the two 1's as two different
numbers, then there would be 4! = 24 ways of rearranging the four digits.
(There would be one of four di_erent numbers which we could choose for the
thousands digit, one of three numbers for the hundreds digit, one of two
numbers for the tens digit, and only one choice for the units digit.) However,
since the two 1's can be interchanged, we have counted each number twice.
Therefore, there are only different 4-digit
numbers that can be made from the digits 3, 1, 4 and 1.
