ADVANCED PROBLEMS
 
 
Question: 12.  From the list of numbers below, two different numbers will be chosen. Of the following, which could be the average (arithmetic mean) of the two numbers? 1, 3, 4, 5, 11
                       
  I. 3
 II. 4
III. 6
 
 
 
 
 
ARITHMETIC
 
 
Question: 13.  The average (arithmetic mean) of 5 positive (no two are equal) integers is 350. Two of the integers are 99 and 100 and the other integers are greater than 102. If all 5 integers are different, what is the greatest possible value for any of the 5 integers?
 
 
 
 
 
 
 
 
 
 
Word Problems
 
 
Question: 14. Which of the following could be the graph of !x! >=y?
 
 
 
 
 
 
 
 
Functions and Graphs
 
 
 
 
Question: 15.  Which of the following system of inequalities is illustrated by the picture below?
 
 
 
 
Functions and Graphs
 
 
 
 
Question: 16. How many real roots can have a parabola on the picture below?
 
 
 
Functions and Graphs
 
 
 
 
 
Question: 17. ((sin x)^4 - (cos x)^4)/ ((sin x)^2 - (cos x)^2) =
 
 
Geometry and Trigonometry
 
 
 
 
 
 
Question: 18.  Rectangle ABCD on the picture below is inscribed in the circle. If the length of side AB is 5 and the length of side BC is 12, what is the area of the shaded region?
 
 
 
 
 
Geometry and Trigonometry
 
 
 
 
Question: 19.  What of the following equations have two distinct real solution?
I !x! = 5
IIx = 16^(1/2)
IIIx^2 - 1 = 0
 
 
Algebra
 
 
 
Question: 20.  If (x + 8)^1/3 = -0.5, then x =
 
 
 
Algebra
 
 
 
Question: 21.  A bag contains a number of pieces of candy of which 78 are red, 24 are brown, and the rest are yellow. If the probability of selecting a yellow piece of candy from this bag at random is 1/3, how many yellow pieces of candy are in this bag?
 
 
 
Sets, Statistics & Probability
 
 
 
 
 
Question: 22.  A certain set of disks contains only blue disks, and green disks. If the probability of randomly choosing a red disk is 1/5 and the probability of randomly choosing a blue disk is 1/3, what is the probability of randomly choosing a green disk?
 
 
 
 
Sets, Statistics & Probability
 
 
 
 
Question: 23.  Let us assume that new born baby can and should have one of three: red hair, blond hair or black hair. It is known that probability for a newborn baby to have the same color as a natural color of its mother’s hair is twice as in opposite case. There are three women in the hospital, each one of whom just gave birth to a baby. What is the probability that at least one baby has red hair?
 
 
 
Sets, Statistics & Probability
 
 
 
All Answers, Explanation and all necessary knowledge to handle such kinds of problems
can be found in the first volume
 
 
 
Problems for those who have become familiar with the second volume
 
 
Question: 24.  What kind of feasible region creates the following systems of inequalities? Explain your choice.
 
1.  "Y>=X + 2"
 "Y>=2"
 "Y>=-X -3"
 
A.   Incompatible
B. Inbound to - infinity
C.  Inbound to infinity
2. "Y>=-3"
 "Y<=5"
 "Y>=7"
 "Y<=3X + 2"
 
A.      Incompatible
B. Inbound to - infinity
C.  Inbound to infinity
3. Y>=3 + x
 Y>=0
 
A.     Incompatible
B.  Inbound to - infinity
C.  Inbound to infinity
 
 
Arithmetic I
 
 
 
Question: 25.  To maximize a sum (a)^(1/2) + (b)^(1/2) for two non negative numbers a and b when a sum (a + b) is given.
 
A.  Linear Programming
B.  Convex Programming
C. Concave Programming
D. Integer Programming
E.    Neither
 
 
Arithmetic I
 
 
 
Question: 26.  In the following exercises calculate the value of a $1000 investment at a yearly interest rate of 10% compounded in the following way?
 
I.Annually
A $1105
B $1200
 
II.Continuously
A $1200
B $1105
 
 
Arithmetic I
 
 
 
Question: 27.  Find complex solutions of an equation x^4 -x^3 - x^2 - x - 2, if you know that it has a real roots -1 and 2.
 
A. i, -i
B. 1+i, 1 -i
 
 
 
 
Algebra I
 
 
 
Question: 28. Find whether the following sequences have or have not limits:
 
I.xn = (-1)/5^n
A. No
B. Yes, the limit is 1.
 
 
II.xn = (-1)/5^n
A. Yes, 0
B. No
 
 
III.xn = 1/2
A. No, infinity
B. Yes, 1/2
 
 
Algebra I
 
 
 
Question: 29. Everyone who used to work with LOGO can recognize the figure on the picture below. Which one of three colors illustrates this function R = sin 5F on the picture below? Explain your answer.
 
A. Black
B. Orange
C. Pink
 
 
Functions and Graphs I
 
 
 
Question: 30. Is it possible to build an inverse function for non monotone function?
 
A. Yes
B. No
 
Functions and Graphs I
 
 
 
Question: 31. Can a projection of the four- dimensional sphere on any on its boundaries be anything but a circle?
 
A. Yes
B. No
 
 
Functions and Graphs I
 
 
 
Question: 32. Can someone investigate the equation sin(2x) + sin(3x) = a sin (x), where a is a real constant graphically?
If “ yes”, choose the picture below, which illustrates this investigation. Explain your choice.
 
A. Yes
B. No
 
First Picture
 
Second Picture
 
Geometry and Trigonometry I
 
 
 
Question: 33. Choose the correct answer and provide the solution of the equation sin(x) = cos (2x).
 
A. sin(x) = -1, sin(x) = 1/2; x = -pi/4 + 2kpi, x = pi/6 + 2kpi
B. sin(x) = 1, sin(x) = -1/2; x = pi/4 + 2kpi, x = -pi/6 + 2kpi
 
Geometry and Trigonometry I
 
 
 
Question: 34. The locus of points equidistant from two given points.
 
A. Line
B. Segment
C. Circle (or a part)
D. None
 
Geometry and Trigonometry I
 
 
 
Question: 35. The rat can choose at random one of five ways with exits. The probabilities to get out of these five ways within three minutes are correspondingly 0.6, 0.3, 0.2, 0.1, 0.1. What is the probability that the rat chose the second way?
 
A. 3/13
B. 6/13
 
(Sets, Statistics & Probability) (Bayes theorem)
 
 
 
Question: 36. Here is a Marcov chain with the matrix of transitional probabilities:
 
P =(1/2 1/2)
(1/3 2/3)
and two possible stages a1 and a2 correspondingly.
 
A. 4/9
B. 5/12
 
(Sets, Statistics & Probability) ( Marcov chains)
 
 
 
Question: 37. What will be a probability for a sequence of four experiments with four different results at random to have at least one occurrence of each results.
 
A. 4! (1/4)^4
B. 6/134!/3! (1/4)^3
 
(Sets, Statistics & Probability) (Stochastic processes, continued distribution)
 
 
 
Question: 38. If there are possible to have not less than 1 and not more then three balls then can we say that statements: “ There is one ball”, “ There are two balls”, and “ There are three balls” is a full set of alternative?
 
A. No
B. Yes
 
(Word Problems I) (Boole's Algebra)
 
 
 
Question: 39. Describe in symbolic form, choose correct answer and prove your choice.
 
“If Jones is the murderer, then he knows the exact time of death and the murder weapon. Therefore, if he doesn’t know the exact time or does not know the weapon, then he is not the murderer.”
 
A. p -> ( q V r)
B. p -> (q ^ r)
 
(Word Problems I)
 
 
 
Question: 40. Let p, q, and r be a complete set of alternatives ( see previous lectures of this course.)What can we say about the truth set of r V p V q ?
 
A. It is always true.
B. It is not always true.
 
(Word Problems I) (Boole Algebra & Theory of Predicates)
 
Question: 41. A word is chosen at random from the set of words U = {men, bird, ball, field, book}. Let p, q, and r be the statements:
p : The word has two vowels.
q : The first letter of the word is b.
r : The word rhymes with cook.
 
Find the probability of the following statements.
 
(a) p.
(b) q.
(c) r.
(d) p ^ q.
(e) (p v q) ^ - r.
(f) p -> q.
 
(Appendix. Finite Math)
 
 
 
Question: 42. Suppose that there are n applicants for a certain job. Three interviewers are asked independently to rank the applicants according to their suitability for the job.
It is decided that an applicant will be hired if he ranked first by at least two of the three interviewers the question is :
What is the probability for each candidate to be accepted for this job?
 
(Appendix. Finite Math)
 
All Answers, Explanation and all necessary knowledge to handle such kinds of problems can be found in the second volume
 
 
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