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ENTRANCE EXAM FEBRUARY 22, 2014

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Department: International School of Economics and Social Sciences
Discipline: Calculus AB
Instructors: S.N. Kharin, S.V. Khruschev, A. Islyami
Date:
Friday, December 23, 2011
Time:
10.00 – 13.15
Time: 2 hours
Max Marks=10 points
ENTRANCE EXAM
FEBRUARY 22, 2014
Final Examination
Name and ID:
Section I Multiple-Choice Questions
Section I A, time : 55 min, 28 Questions. No calculator allowed
x3 − 6 x 2 + 11x − 6
=
x →1
x 2 − 3x + 2
ln(1 − 3 x)
2. 1. Solve
=
lim
thex equation
x →0
tan
1.
3.
a) 0
lim
b) -3
a) 1
b) -3
c) -2
d) -1
c) 1/2
d) 0
e) 1
e) 4
− the
2| +
|x −of
4|A−and
|2x B−to6|provide
= 1. continuity of a function
Using above results |x
find
values

ln(1 − 3 x) family drank an 8-ounce mixture
2. One morning each member
of Angela’s
for
x<0

tan x and milk varied from cup to cup,
of coffee with milk. The amount
of coffee

f ( x)  drankAx
B
for amount
0 ≤ x ≤of
1 milk and a
but were never zero.=
Angela
a +quarter
of total

3
2
sixth of the total amount of coffee. xHow
are in Angela’s family?
− 6 xmany
+ 11x people
−6
 3
2
x >1
for
x 2 − 3x + 2
3. A polynomial P (x) = x + ax + bx + c has the property that the mean
of its zeros,
product
its−1,zeros,
its2 coefficients
a) A the
c)the
d) A = 1, are
=
2,=
B 1 b) of
A=
B=
1and
=
A sum
2,=
Bof 1/
B = −all
3
equal. If the y-intercept of the graph of y = P (x) is 2, what is b?
e) do not exist
4. The speedometer of an automobile registers 50 mi/hr as it passes a mileage marker
4. Mr. Earl Bird leaves his house for work at exactly 8 : 00 A.M. every
along a highway. Four minutes later, as the automobile passes a marker 5 mi from the
morning.
When
he averages
40 55mi/hr.
miles per
hour,
arrives
his workplace
first, the
speedometer
registers
Using
the he
mean
value at
theorem
indicate the
threegreatest
minutes
late.which
When
he surely
averages
60 miles
perthehour,
he arrives three
velocity
could
achieved
between
two markers.
a) 60 mi/hr
b) 55 mi/hr
1c) 75 mi/hr
d) 80 mi/hr
e) 100 mi/hr
5. The function f ( x)= ( x 2 + x + 2)e x has points of inflection at:
a) -1 and -2
b) -2 and -3
6. If e y + xy =
e , then y′′(0) =
c) -3 and -4
a)
1
e
b) e
d) 0 and 1 e) There are no points of inflection
c) 0
d)
1
e2
e) does not exist
minutes early. At what average speed, in miles per hour, should Mr. Bird
drive to arrive at his workplace precisely in time?
5. The number of geese in a flock increases so that the difference between
the populations in year n + 2 and year n is directly proportional to the
population in year n + 1. Find the population in 2012, if the populations in
the years 2010, 2011, and 2013 were 39, 60, and 123, respectively.
6. Each week a company produces 1000 LCD TV panels and sells 20%
of its stock. Initially the warehouse of the company was empty. What is
the minimal size of the warehouse so that the company would never have a
shortage of space?
7. In the figure, polygons A, E, and F are isosceles right triangles; B, C,
and D are squares with sides of length 1; and G is an equilateral triangle.
The figure can be folded along its sides to form a polyhedron having the
polygons as faces. Find the volume of this polyhedron.
8. Solve the equation
log1.75 (2 − sin2 x − sin x − cos 2x) = 1.
9. Solve the inequality
log|3x+2| (x2 − 5x + 4) ≥ 2.
10. A circle centered at A with radius 1 and a circle centered at B with
a radius of 4 are externally tangent. A third circle is tangent to the first two
and to one of their common external tangents as shown. What is the radius
of the third circle?
2
B
4
A
1
Vocabulary
Problem:
2
2
2
3
English
ounce
quarter
sixth
mean
3
y-intercept
4
5
5
5
6
6
7
7
7
7
7
average
geese
flock
population
stock
warehouse
isosceles
equilateral
polyhedron
polygon
face
Russian
1 унция = 28.3495 грамма
четверть
шестая часть
среднее значение
y − координата точки пересечения
графика с осью OY
делать в среднем
гуси
стая
численность
иметь в наличии
товарный склад
равнобедренный
равносторонний
многогранник
многоугольник
сторона
3
1/--страниц
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