MAT102F - Introduction to Mathematical Proofs - Fall 2012 - UTM
Quiz 1 (Version A) - September 21, 2012 - Solutions
1. Complete the following sentences: (2 marks)
No explanation is need for this question.
(a) The geometric mean of 4 and 9 is 4▯9 = 36 = 6 and their arithmetic mean is
2▯(4 + 9) = 6:5 .
(b) A glass contains 5 ounces of wine and 3 ounces of water (mixed uniformly). If we remove
2 ounces of liquid from the glass and pour it into an empty cup, then the cup will contain
8▯2 = 4 = 1:25 ounces of wine.
2. Given a positive number b , under what conditions does there exist a rectangular carpet with
perimeter 2b and area 2 ? Show your work and simplify your ▯nal answer. (3 marks)
Denote the dimensions of the carpet by x and y. The given information implies that
xy = and 2x + 2y = 2b :
The second equation implies that y = b ▯ x and then the ▯rst equation becomes
x(b ▯ x) = b ) x ▯ b ▯ x +b = 0 :
This quadratic equation has a real solution if and only if (▯b) ▯ 2 ▯ 0, or equivalently: b ▯ 2,
and that is the required condition.
(Note: From the quadratic formula we can conclude that if b ▯ 2, then the solution(s) will be positive,
and thus no additional conditions are needed.)
2 ▯3 ▯
3. If 3