高中数学必修2考试答案解析-Ex期望高中数学
Problem 1
Each third-grade
classroom at Pearl Creek Elementary has 18
students and 2 pet
rabbits. How many more
students than rabbits are there in all 4 of the
third-grade
classrooms?
Solution
Problem 2
A circle of radius 5 is
inscribed in a rectangle as shown. The ratio of
the length of the
rectangle to its width is
2:1. What is the area of the rectangle?
Solution
Problem 3
The point in the
xy-plane with coordinates (1000, 2012) is
reflected across the line
y=2000. What are the
coordinates of the reflected point?
Solution
Problem 4
When Ringo places
his marbles into bags with 6 marbles per bag, he
has 4 marbles
left over. When Paul does the
same with his marbles, he has 3 marbles left over.
Ringo and Paul pool their marbles and place
them into as many bags as possible,
with 6
marbles per bag. How many marbles will be left
over?
Solution
Problem 5
Anna enjoys dinner at a restaurant
in Washington, D.C., where the sales tax on
meals is 10%. She leaves a 15% tip on the
price of her meal before the sales tax is
added, and the tax is calculated on the pre-
tip amount. She spends a total of 27.50
dollars for dinner. What is the cost of her
dinner without tax or tip in dollars?
Solution
Problem 6
In order to
estimate the value of x-y where x and y are real
numbers with x > y >
0, Xiaoli rounded x up by
a small amount, rounded y down by the same amount,
and
then subtracted her rounded values. Which
of the following statements is
necessarily
correct?
A) Her estimate is larger than x-y
B) Her estimate is smaller than x-y C) Her
estimate equals x-y D) Her estimate equals y -
x E) Her estimate is 0
Solution
Problem
7
For a science project, Sammy observed a
chipmunk and a squirrel stashing acorns
in
holes. The chipmunk hid 3 acorns in each of the
holes it dug. The squirrel hid 4
acorns in
each of the holes it dug. They each hid the same
number of acorns,
although the squirrel needed
4 fewer holes. How many acorns did the chipmunk
hide?
Solution
Problem 8
What is the sum of all integer solutions to
Solution
?
Problem 9
Two integers have a sum of 26.
When two more integers are added to the first two
integers the sum is 41. Finally when two more
integers are added to the sum of the
previous
four integers the sum is 57. What is the minimum
number of even integers
among the 6 integers?
Solution
Problem 10
How
many ordered pairs of positive integers (M,N)
satisfy the equation
Solution
=
Problem 11
A dessert chef prepares the
dessert for every day of a week starting with
Sunday.
The dessert each day is either cake,
pie, ice cream, or pudding. The same dessert
may not be served two days in a row. There
must be cake on Friday because of a
birthday.
How many different dessert menus for the week are
possible?
Solution
Problem 12
Point B is due east of point A. Point C is due
north of point B. The distance between
points
A and C is , and = 45 degrees. Point D is 20
meters due North
of point C. The distance AD
is between which two integers?
Solution
Problem 13
It
takes Clea 60 seconds to walk down an escalator
when it is not operating, and
only 24 seconds
to walk down the escalator when it is operating.
How many seconds
does it take Clea to ride
down the operating escalator when she just stands
on it?
Solution
Problem 14
Two equilateral triangles are contained in
square whose side length is
a rhombus. What is
the area of the rhombus?
. The
bases of
these triangles are the opposite side of the
square, and their intersection is
Solution
Problem 15
In a round-
robin tournament with 6 teams, each team plays one
game against each
other team, and each game
results in one team winning and one team losing.
At the
end of the tournament, the teams are
ranked by the number of games won. What is
the
maximum number of teams that could be tied for the
most wins at the end on
the tournament?
Solution
Problem 16
Three circles with radius 2 are mutually
tangent. What is the total area of the circles
and the region bounded by them, as shown in
the figure?
Solution
Problem 17
Jesse cuts a circular paper disk of radius 12
along two radii to form two sectors, the
smaller having a central angle of 120 degrees.
He makes two circular cones, using
each sector
to form the lateral surface of a cone. What is the
ratio of the volume of
the smaller cone to
that of the larger?
Solution
Problem 18
Suppose that one of every 500
people in a certain population has a particular
disease, which displays no symptoms. A blood
test is available for screening for this
disease. For a person who has this disease,
the test always turns out positive. For a
person who does not have the disease, however,
there is a
other words, for such people,
false positive rate--in
of the time the
test will turn out negative, but
of the time
the test will turn out positive and will
incorrectly indicate that the person
has the
disease. Let be the probability that a person who
is chosen at random from
this population and
gets a positive test result actually has the
disease. Which of the
following is closest to
?
Solution
Problem 19
In
rectangle , ,
to point
?
, and is the
midpoint of . Segment
and is extended 2 units
beyond
. What is the area of
, and is the
intersection of
Solution
Problem 20
Bernardo and Silvia play the
following game. An integer between 0 and 999,
inclusive, is selected and given to Bernardo.
Whenever Bernardo receives a number,
he
doubles it and passes the result to Silvia.
Whenever Silvia receives a number, she
adds 50
to it and passes the result to Bernardo. The
winner is the last person who
produces a
number less than 1000. Let be the smallest initial
number that results
? in a win for Bernardo.
What is the sum of the digits of
Solution
Problem 21
Four distinct points are
arranged on a plane so that the segments
connecting them
have lengths , , , , , and .
What is the ratio of to ?
Solution
Problem 22
Let
either
be a list of
the first 10 positive integers such that for each
or or both appear somewhere before in the
list.
How many such lists are there?
Solution
Problem 23
A solid
tetrahedron is sliced off a wooden unit cube by a
plane passing through two
nonadjacent vertices
on one face and one vertex on the opposite face
not adjacent
to either of the first two
vertices. The tetrahedron is discarded and the
remaining
portion of the cube is placed on a
table with the cut surface face down. What is the
height of this object?
Solution
Problem 23
A solid
tetrahedron is sliced off a wooden unit cube by a
plane passing through two
nonadjacent vertices
on one face and one vertex on the opposite face
not adjacent
to either of the first two
vertices. The tetrahedron is discarded and the
remaining
portion of the cube is placed on a
table with the cut surface face down. What is the
height of this object?
Solution
Problem 24
Amy, Beth, and Jo listen to
four different songs and discuss which ones they
like. No
song is liked by all three.
Furthermore, for each of the three pairs of the
girls, there
is at least one song liked by
those girls but disliked by the third. In how many
different ways is this possible?
Solution
Problem 25
A bug travels
from to along the segments in the hexagonal
lattice pictured
below. The segments marked
with an arrow can be traveled only in the
direction of
the arrow, and the bug never
travels the same segment more than once. How many
different paths are there?
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