METU
DEPARTMENT OF MATHEMATICS
2013-2014 Spring Semester
Math 112 Discrete Mathematics
Exercise 1
1)
In how many ways can you arrange 7 different books, so that a specific book is on the
third place?
2)
In how many ways can you take 3 marbles out of a box
CHAPTER TEST 4C
14.
How long will it take for an amount invested at
12 percent compounded annually to double?
In Problems 15-18 solve for x.
15.
log
16.
log
32 = - 5
(log
3
17.
i o
q x
x)
= 1
3
e = \
18.
44
CHAPTER TEST 4C
In Problems 1-3, if log 2 = w, log 3 = V, and log 5 = W,
evaluate the given logarithm in terms of u, V, and W.
1.
log 12
2.
log 1.2
3.
l o g
/l5
I n Problems 4-7
3
4.
log
5.
log
6.
log
7.
log3r_4
7
9
5
simplify
/f
3
125
8. Bacteria, follow
CHAPTER TEST 3D
1.
Find the distance in terms of the letter h from the point
(3, 5) to the point (a, b) on the line x - 2y = 1.
2.
Draw the graph of y = \x2 -
2x\.
3. Determine the symmetry of x3 - with respect to the
#-axis, the z/-axis, and the origin.
CHAPTER TEST 2E
In Problems 1 and 2 find the solutions.
1.
5cfw_x
- 2) - 3(x
2.
3
-5
+
x - 2
x - 2
3.
Michael and Eric can paint a room in 3 hours.
- 1) = -5
twice as fast as Eric.
Michael paints
How long will it take Eric to paint
the room by himself?
4.
CHAPTERS 1-3, CUMULATIVE TEST E
In Problems 1-5 perform the indicated operations and simplify.
1
- 4
2
1.
2.
3
i
+
1
1
+
x_
y
6x + 8
3.
x
4.
- 5x + 6
I)
x
+ x - 20
3/2
5.
(1 + 2i)
6.
Rationalize the denominator:
7.
Find all solutions to \3x - l| = 8.
8.
CHAPTERS 1-3, CUMULATIVE TEST A
In Problems 1-5 perform the indicated operations and simplify.
1
9
2
2 3
4 1.
x +
2.
3.
2
JL
x - 2_
y
+ 4x + 1 . x2
3x2
x
z
- x - 30
2
x
+ 3x + 2
- 4x - 12
,-V3
4.
(64]
5.
(-5 +
6.
Rationalize the denominator :
7.
Find a l
CHAPTERS 1-3, CUMULATIVE TEST A
15.
The surface area of an object varies directly as the two-thirds
power of its volume.
The constant of proportionality is 2.
If the volume of the object is 8, what is the surface area?
In Problems 16-20 fcfw_sc)
= ^11
Sx
CHAPTER TEST 2E
15.
16.
Find the set of x such that
17.
The length of the hypotenuse of a right triangle is 10.
of the triangle is twice as long as the other leg.
length of the shortest side of the triangle.
20
One leg
Find the
CHAPTER TEST 4B
In Problems 14 and 15 write each expression as a single logarithm.
14.
2 In x - 3 In y + 1
15. i ln(a; + 1) + j InQ/ + 1) 2 - 2
In Problems 16-18 solve for x.
16.
In(In x) = 0
17.
log C2x - 1) = 2
18.
log (3x - 2) = 2
42
CHAPTER TEST 4E
In Problems 1 and 2, if log 3 = x and log 5 = y evaluate the
given logarithm in terms of x and y.
(I)
1.
log
2.
log [(45)(75) ]
In Problems 3-6 simplify
3. log8(l)
4.
log (42)
2
5.
-.
6. 1 0 , ^ 6 )
In Problems 7 and 8 write each expressio
CHAPTER TEST 3E
14.
Find (g of)
15. Find f1(x)
(x) .
.
16.
Find (/ o /)(x).
17.
Prove that f(x)
18.
Find the equation of the line parallel to x = -2y
is one-to-one.
through (4, 2 ) .
30
that passes
METU
DEPARTMENT OF MATHEMATICS
2013-2014 Spring Semester
Math 112 Discrete Mathematics
Exercise 2
1)
Using the letters of English alphabet, in how many different ways is it possible to write a
7 letter string so that
a) no two letters are the same,
b) no
METU
DEPARTMENT OF MATHEMATICS
2013-2014 Spring Semester
Math 112 Discrete Mathematics
Exercise 3
1) Find the number of positive integers not larger than 1000 which are divisible by 3 or 5.
2) Find the number of five-digit combinations from the set
a) Som
CHAPTER TEST 4D
In Problems 1-5 simplify.
1.
log
64
2
2.
3.
log
9
1/3
1
log T
9 3
4.
log
216
3S
5.
log
36
216
In Problems 6-8, if log 5 = a: and log 7 = y,
the given logarithm in terms of x and
6.
log(35)
evaluate
y.
2
7 10
^
8.
log 1.4
In Problems 9 and
CHAPTER TEST 3B
7. Find the formula for V(x) .
8. Find the domain of V(x).
9.
Draw the graph of V(x).
10.
Find the range of V(x).
11.
Let x be directly proportional to the sum of y and z and
inversely proportional to the product of y and z.
If
x = 4 when
CHAPTER TEST 2D
In Problems 14 and 15 find the solutions.
14.
15.
16.
Find the set of x such that
17.
The lengths of two legs of a right triangle are in the
proportion 3:2.
If the hypotenuse is ^/S2,
find the
length of the longest leg.
18
CHAPTER TEST 3E
1.
Determine the longest side of the triangle whose coordinates
are (-1, - 2 ) , (2, 2),
2.
and (-3, 4 ) .
Determine the symmetry of x2y3
- xhy
= 3y with respect to
the a:-axis, the zy-axis, and the origin.
Problems 3-6 refer to the follow
CHAPTER TEST 3D
16.
If
I
determine where fix)
x2
x < 0
x
0 < x < 1
1
X > 0
is increasing, decreasing, and
constant.
17.
The force of attraction between two bodies is inversely
proportional to the square of the distance between them.
Find the constant of p
CHAPTER TEST 3A
1. Find the equation of the line parallel to Sx + lOy - 13 = 0
that passes through (6, 1/3).
2.
The distance from (1, 2) to (x, y) is equal to the distance
from (3, 5) to cfw_x, y).
Find the relationship between
x and y.
Problems 3-8 refer
CHAPTER TEST 3B
Problems 1-3 refer to the following principle:
The distance from a point to a line is measured by the length
of the perpendicular to that line from the point in question.
1. Find the equation of the line perpendicular to x + 2y = 3
that pa
CHAPTER TEST 2D
In Problems 1 and 2 find the solutions.
1.
2.
3.
2 (.3a; + 2 ) = 1 (3x - 1) - 19
x + 1
9
+ 4 =
Michael, Cheryl, and Eric can paint a room in two hours.
Cheryl and Eric paint at the same rate, and Michael paints
twice as fast.
How long will