Math 3.2
問題一覧
1
What is the area of the largest rectangle that can be inscribed in an ellipse whose
equation is 4x^2 + y2 = 4.
4
2
A box contains 9 red balls and 6 blue balls. If two balls are drawn in succession, what is the probability that one of them is red and the other is blue?
18/35
3
A fair coin is tossed 10 times. Find the probability that exactly 3 heads will come out.
15/128
4
Five cards are drawn from a pack of 52 well-shuffled cards. Find the probability that
3 are 10's and 2 are queens:
1/108,290
5
If any 10 of the 26 letters of the alphabet are chosen to represent the ten digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. How many possible coding patterns are there? Powerline Review Center
1.93 х 10^13
6
In a 30-60-90 right triangle the bisector of the 60 degrees angle divides its opposite side in the ratio
1:2
7
The parabolic reflector of a solar power plant is 12m wide and 4m deep. Where should the stream pipe be located (distance from the vertex) so that all incident rays will be concentrated on it?
2.25
8
When discriminant of a conic section, D = B2 - 4AC = 1, the curve is a
hyperbola
9
The eccentricity of a given curve is equal to zero, the given curve is:
circle
10
The logarithm of a negative number is:
complex number
11
In the chemical processing of a certain mineral, the rate of change of the amount of mineral present varies as the amount of the mineral remaining. If after 8 hours, 100 pounds of mineral have been reduced to 70 pounds, what quantity of the mineral will remain after 24 hours?
34.3
12
How many positive factors (including 1 and 1989) does the number 1989 has?
12
13
What is the value of A between 270° and 360° if 2sin 2A - sinA = 1
330°
14
Solve for x in the equation: arctan (+ 1) + arctan (x - 1) = arctan (12).
1.33
15
The number of newspaper distributed is given by C = 50t 2 - 200t + 1,000 where t.
is in years. Find the minimum number of copies distributed from 1995 to 2002.
800
16
Find the area enclosed by y= sin (n x /2) and y = x.
0.137
17
The kilowatt that can be transmitted safely by a shaft varies directly as the number of revolution it makes per minute and the cube of its diameter. If a shaft 3 centimeters in diameter making 200 revolutions per minute can safely transmit 60 kilowatt, what kilowatt can be safely transmitted by a 2-centimeter shaft making 300 revolutions per minute?
26.67
18
Find the height of a right circular cylinder of maximum volume than can be inscribed in a sphere whose radius is 10 cm.
11.55 cm
19
A father and his son can dig a well if the father works 6 hours and his son works for
12 hours or they can do the job if the father works 9 hours and the son works 8 hours.
How long will it take for the son to dig the well alone?
20
20
A small line truck hauls poles from substation stockyard to pole sites along a proposed distribution line. The truck can handle only one pole at a time. The first pole site is 150 meters from the substation and the poles are to be 50 meters apart.
Determine the total distance travelled by the line truck, back and forth, after returning from delivering the 30th pole.
52.5 km
21
At the minimum point, the slope of the tangent line is
zero
22
In conic section, if discriminant. B2 - 4AC = 1, the locus is a
hyperbola
23
Which of the following is a prime number?
101
24
If seven coins are tossed simultaneously, find the probability that there will be at least five heads.
29/128
25
Find the derivative of cosh^3 (4x).
12cosh^2 4x sinh 4x
26
A farmer estimates that if he digs his potatoes now he will have 120 bushels, which he can sell at $1.75 per bushels. If he expects his crop to increase 8 bushels per week, but the price to drop 5 cents per bushels per week, in how many weeks should he sell to realize the maximum amount for his crop?
10 weeks
27
The Re of one of the two square roots of 5 - j12 is
3
28
The sum of the cube of numbers from 32 to 100 is
25,256,484
29
There are 13 teams in a tournament. Each team is to play with each other only once.
What is the minimum number of days can they all play without any team playing more than one game in any day?
13
30
If seven coins are tossed simyltaneously, find the probability that they will just have three heads.
35/128
31
Determine the area of a regular 6-star polygon if the inner regular hexagon has 10 cm sides.
519.60 cm2
32
A goat is tied to a corner of a 30 ft by 35 ft building. If the rope is 40 ft long and the goat can reach 1 ft farther than the rope length. What is maximum area the goat can cover?
4084.07
33
The sum of the ages of two men equals 99. If the inverted age of the elder is added to the age of the younger, the sum is 108. However, if the age of the younger is inverted and subtracted from the age of the older, the difference is 44. Find the age of the older man.
67
34
A man bought 20 chickens for P2,000. The cocks cost P300 each, the hens P150 each, and the chicks at P50 each. How many cocks did he buy.
2
35
What is the maximum area of a rectangle that can be inscribed in a right triangle with a base of 8 cm and height of 6 cm?
12 cm2
36
Solve for x if x is raised to x raised to x and so on raise to 3 is equal to 3.
cube root of 3
37
Given 8 French books, 12 Spanish books, 9 German books, 15 Russian books and $18 Italian books, how many books be chosen to guarantee that there are 10 books of the same language.
above 30
38
What decimal value is nearest to x if x is equal to the square root of the quantity one minus the square root of the quantity one minus the square root of the quantity one minus the square root of quantity one ... and so on.
0.62
39
A contractor wishes to build 5 houses, each different in design. In how many ways can he place these houses on a street if 6 lots are on one side on the street and 3 lots are on the opposite side.
362,880
40
Point P which is 10m distance from the vertices of a triangle with sides 4, 5 and 6.
Find the distance from point P to the plane of the triangle.
9.53
41
Determine the inverse Laplace transform of; I (s) = 100 / (s + 20)^2
i (t) = 100te^-20t
42
Evaluate j^174.
-1
43
What is -i^i?
- 0.21
44
Determine the inverse Laplace Transform of:
(s+ a) / ((s+ a)^2+ ω^2)
exp (- at) cos ωt
45
A and B are independent events. The probability that event A occurs is 0.2 and the probability that A and B occur is 0.04. What is the probability that event B will occur?
0.20
46
Simplify (COSB - 1) (COSB + 1).
- 1/CSC^2 B
47
Find k so that A = <3, - 2> and B = <1, k> are perpendicular.
3/2
48
For a complex number z = 3 + j4 the modulus is:
5
49
Simplify the trigonometric expression:
(cscθ + cotθ ) / cscθ
1 - secθ
50
A horizontal line has a slope of
Zero
51
What section is formed by a cone cut by a plane perpendicular to its axis?
Circle
52
In a conic section if the discriminant D = 1, then the locus is a
hyperbola
53
Evaluate lim sin 3x / x as x→0
3
54
Find the 95th term of an AP is 1.03, 1.02, 1.01,
0.09
55
Evaluate limit of 9x2-×2+x / 2x3 + x as x approaches infinity.
9/2
56
Find the rectangular coordinates of the spherical coordinate (2, π/4, π/2).
(V2, V2, 0)
57
What theorem is used to solve for centroid?
Pappus' theorem
58
If a bus weighs 2.5 tons, how much pounds does it weigh? (1 ton = 2000 Ibs)
5000
59
A basketball coach has a total of 10 players. In how many ways can he field a team of 5 players if the team captain is always included?
126
60
Find the work done by the force F = 3 + 10j newtons in moving an object 10 meters
north.
100 J
61
A cone with 4-in base radius and 8-in depth, if water flows inside the cone 1 cu. in/sec, find the rate on which the depth changes when the water is 2-in from the base.
1/9π
62
Sand is pouring at 25m ft 3/min and forming a conical pile where the radius is always twice its height. Find the rate when the height is 5ft and height is increasing at the rate of 2 in/min?
0.58 ft/min
63
Find the area of the largest region that a 10m string could enclose.
7.96
64
In conic section, if eccentricity e = 0, the locus is a
Circle
65
Write the vector of length 2 and direction 150 degrees in the form ai + bj.
-1.731 + j
66
A rubber ball is dropped from a height of 15 meters. On each rebound, it rises 2/3 of the height from which it last fell. Find the distance traversed by the ball before it comes to rest. The geometric progression occurs after the first rebound.
75 m
67
A die is rolled and a coin is tossed. What is the probability that a three and a head will appear?
1/12
68
What is the sum of the coefficients of the expansion of (2x - 1)^21
2
69
In how many ways can 2 integers be selected from the integers 1, 2, 3, ..., 100 so that their difference is exactly 7?
93
70
At the inflection point, the value of the second derivative of the function is
zero
71
Evaluate the integral of two log e to base 10, over x times dx from 1 to 10.
2
72
178. In how many ways can a deck of 52 cards be distributed to 5 players if each player will receive 4 cards?
4.6196 x10^27
73
The largest volume of a rectangular box that can be made of a cardboard 20 sq. in by cutting identical squares on its edges.
592.60 cu. cm
74
The segment from (-1, 4) to (2, - 2) is extended 3 times its own length. The terminal point is
(11, - 20)
75
When William was as old as Mae is now, the sum of their ages was 51. When Mae will be as old as William is now, the sum of their ages will be 103. How many years older is William than Mae?
13
76
If any 10 of the 26 letters of the alphabet are chosen to represent the ten digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. How many possible coding patterns are there?
1.93 x10^13
77
What is the value of A between 270° and 360° if 2sin 2A - sinA = 1
330°
78
Find the largest rectangle that can be inscribed in x2= - (y - 26) and x2= y -3.
4 cm x 16 cm
79
Two lines pass through (5, 5) and separate tangents to the circle x2 + y2 = 9.
Determine the distance between the x-intercepts of the two lines.
12.01
80
The cost of renting a ballroom for a school dance is different at each of the three establishments. The Ambassador charges $100 plus $3 per person. The Belmont charges $150 plus $2 per person, while the Coliseum charges $250 plus $1 per person. Under what condition would the Belmont be the best place to hold the dance?
bet. 50-100
81
The roots of equation x2 + x + 1 = 0 are:
No real roots
82
Determine the sum of the internal angles of a hexagon
720 degrees
83
A can plow the field in 4 hours less than B: If both will work together, the job is done in 10 hours. Find the number of hours for B to do the job alone.
22.2
84
The lotto uses number 1 - 42. A winning number consists of six different numbers in any order. What are your chances of winning it?
1/5,245,786
85
The laplace transform of t^2sin4t
8(3s2 - 16)/(s2 + 16)3
86
Suppose the dimension (in inches) of a rectangular box change from 9, 6, 4.to 902,
5.97, and 4.01 respectively. Find the exact change in volume.
- 0.063906
87
Find the equation of the line passing thru (6, 8) and pass thru a point 5 units distance from the origin.
0.393x - 0.919y + 5 = 0
88
Find the radius of curvature at any point of the curve y + In (cosx) = 0.
sec x
89
The roots of equation x2 + x + 1 = 0 are:
No real roots
90
The equation x^3 + y^3 - 3axy = 0 represents a curve called
Folium of Descartes
91
In how many ways can 6 people be seated in a row of 9 seats?
60,480
92
Determine the inverse Laplace transform of; 1 (s) = 100 / (s + 20) 2
i (t) = 100te- 20t
93
A can plow the field in 4 hours less than B: If both will work together, the job is done in 10 hours. Find the number of hours for B to do the job alone.
22.2
94
The lotto uses number 1 - 42. A winning number consists of six different numbers in any order. What are your chances of winning it?
1/5,245,786
95
Suppose the dimension (in inches) of a rectangular box change from 9, 6, 4.to 902,
5.97, and 4.01 respectively. Find the exact change in volume.
- 0.063906
96
Find the equation of the line passing thru (6, 8) and pass thru a point 5 units distance from the origin.
0.393x - 0.919y + 5 = 0
97
Find the radius of curvature at any point of the curve y + In (cosx) = 0.
sec x
98
Find the area inside the cardioid r = a(1 + cosθ) and outside the arc r = a
(π/4 + 2) a^2
99
Evaluate (1 - i)^-6
0.125 cis -90°
100
Evaluate [sq. root of 3 - i]^- 10
0.0009766 cis - 60°
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19問 • 2年前問題一覧
1
What is the area of the largest rectangle that can be inscribed in an ellipse whose
equation is 4x^2 + y2 = 4.
4
2
A box contains 9 red balls and 6 blue balls. If two balls are drawn in succession, what is the probability that one of them is red and the other is blue?
18/35
3
A fair coin is tossed 10 times. Find the probability that exactly 3 heads will come out.
15/128
4
Five cards are drawn from a pack of 52 well-shuffled cards. Find the probability that
3 are 10's and 2 are queens:
1/108,290
5
If any 10 of the 26 letters of the alphabet are chosen to represent the ten digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. How many possible coding patterns are there? Powerline Review Center
1.93 х 10^13
6
In a 30-60-90 right triangle the bisector of the 60 degrees angle divides its opposite side in the ratio
1:2
7
The parabolic reflector of a solar power plant is 12m wide and 4m deep. Where should the stream pipe be located (distance from the vertex) so that all incident rays will be concentrated on it?
2.25
8
When discriminant of a conic section, D = B2 - 4AC = 1, the curve is a
hyperbola
9
The eccentricity of a given curve is equal to zero, the given curve is:
circle
10
The logarithm of a negative number is:
complex number
11
In the chemical processing of a certain mineral, the rate of change of the amount of mineral present varies as the amount of the mineral remaining. If after 8 hours, 100 pounds of mineral have been reduced to 70 pounds, what quantity of the mineral will remain after 24 hours?
34.3
12
How many positive factors (including 1 and 1989) does the number 1989 has?
12
13
What is the value of A between 270° and 360° if 2sin 2A - sinA = 1
330°
14
Solve for x in the equation: arctan (+ 1) + arctan (x - 1) = arctan (12).
1.33
15
The number of newspaper distributed is given by C = 50t 2 - 200t + 1,000 where t.
is in years. Find the minimum number of copies distributed from 1995 to 2002.
800
16
Find the area enclosed by y= sin (n x /2) and y = x.
0.137
17
The kilowatt that can be transmitted safely by a shaft varies directly as the number of revolution it makes per minute and the cube of its diameter. If a shaft 3 centimeters in diameter making 200 revolutions per minute can safely transmit 60 kilowatt, what kilowatt can be safely transmitted by a 2-centimeter shaft making 300 revolutions per minute?
26.67
18
Find the height of a right circular cylinder of maximum volume than can be inscribed in a sphere whose radius is 10 cm.
11.55 cm
19
A father and his son can dig a well if the father works 6 hours and his son works for
12 hours or they can do the job if the father works 9 hours and the son works 8 hours.
How long will it take for the son to dig the well alone?
20
20
A small line truck hauls poles from substation stockyard to pole sites along a proposed distribution line. The truck can handle only one pole at a time. The first pole site is 150 meters from the substation and the poles are to be 50 meters apart.
Determine the total distance travelled by the line truck, back and forth, after returning from delivering the 30th pole.
52.5 km
21
At the minimum point, the slope of the tangent line is
zero
22
In conic section, if discriminant. B2 - 4AC = 1, the locus is a
hyperbola
23
Which of the following is a prime number?
101
24
If seven coins are tossed simultaneously, find the probability that there will be at least five heads.
29/128
25
Find the derivative of cosh^3 (4x).
12cosh^2 4x sinh 4x
26
A farmer estimates that if he digs his potatoes now he will have 120 bushels, which he can sell at $1.75 per bushels. If he expects his crop to increase 8 bushels per week, but the price to drop 5 cents per bushels per week, in how many weeks should he sell to realize the maximum amount for his crop?
10 weeks
27
The Re of one of the two square roots of 5 - j12 is
3
28
The sum of the cube of numbers from 32 to 100 is
25,256,484
29
There are 13 teams in a tournament. Each team is to play with each other only once.
What is the minimum number of days can they all play without any team playing more than one game in any day?
13
30
If seven coins are tossed simyltaneously, find the probability that they will just have three heads.
35/128
31
Determine the area of a regular 6-star polygon if the inner regular hexagon has 10 cm sides.
519.60 cm2
32
A goat is tied to a corner of a 30 ft by 35 ft building. If the rope is 40 ft long and the goat can reach 1 ft farther than the rope length. What is maximum area the goat can cover?
4084.07
33
The sum of the ages of two men equals 99. If the inverted age of the elder is added to the age of the younger, the sum is 108. However, if the age of the younger is inverted and subtracted from the age of the older, the difference is 44. Find the age of the older man.
67
34
A man bought 20 chickens for P2,000. The cocks cost P300 each, the hens P150 each, and the chicks at P50 each. How many cocks did he buy.
2
35
What is the maximum area of a rectangle that can be inscribed in a right triangle with a base of 8 cm and height of 6 cm?
12 cm2
36
Solve for x if x is raised to x raised to x and so on raise to 3 is equal to 3.
cube root of 3
37
Given 8 French books, 12 Spanish books, 9 German books, 15 Russian books and $18 Italian books, how many books be chosen to guarantee that there are 10 books of the same language.
above 30
38
What decimal value is nearest to x if x is equal to the square root of the quantity one minus the square root of the quantity one minus the square root of the quantity one minus the square root of quantity one ... and so on.
0.62
39
A contractor wishes to build 5 houses, each different in design. In how many ways can he place these houses on a street if 6 lots are on one side on the street and 3 lots are on the opposite side.
362,880
40
Point P which is 10m distance from the vertices of a triangle with sides 4, 5 and 6.
Find the distance from point P to the plane of the triangle.
9.53
41
Determine the inverse Laplace transform of; I (s) = 100 / (s + 20)^2
i (t) = 100te^-20t
42
Evaluate j^174.
-1
43
What is -i^i?
- 0.21
44
Determine the inverse Laplace Transform of:
(s+ a) / ((s+ a)^2+ ω^2)
exp (- at) cos ωt
45
A and B are independent events. The probability that event A occurs is 0.2 and the probability that A and B occur is 0.04. What is the probability that event B will occur?
0.20
46
Simplify (COSB - 1) (COSB + 1).
- 1/CSC^2 B
47
Find k so that A = <3, - 2> and B = <1, k> are perpendicular.
3/2
48
For a complex number z = 3 + j4 the modulus is:
5
49
Simplify the trigonometric expression:
(cscθ + cotθ ) / cscθ
1 - secθ
50
A horizontal line has a slope of
Zero
51
What section is formed by a cone cut by a plane perpendicular to its axis?
Circle
52
In a conic section if the discriminant D = 1, then the locus is a
hyperbola
53
Evaluate lim sin 3x / x as x→0
3
54
Find the 95th term of an AP is 1.03, 1.02, 1.01,
0.09
55
Evaluate limit of 9x2-×2+x / 2x3 + x as x approaches infinity.
9/2
56
Find the rectangular coordinates of the spherical coordinate (2, π/4, π/2).
(V2, V2, 0)
57
What theorem is used to solve for centroid?
Pappus' theorem
58
If a bus weighs 2.5 tons, how much pounds does it weigh? (1 ton = 2000 Ibs)
5000
59
A basketball coach has a total of 10 players. In how many ways can he field a team of 5 players if the team captain is always included?
126
60
Find the work done by the force F = 3 + 10j newtons in moving an object 10 meters
north.
100 J
61
A cone with 4-in base radius and 8-in depth, if water flows inside the cone 1 cu. in/sec, find the rate on which the depth changes when the water is 2-in from the base.
1/9π
62
Sand is pouring at 25m ft 3/min and forming a conical pile where the radius is always twice its height. Find the rate when the height is 5ft and height is increasing at the rate of 2 in/min?
0.58 ft/min
63
Find the area of the largest region that a 10m string could enclose.
7.96
64
In conic section, if eccentricity e = 0, the locus is a
Circle
65
Write the vector of length 2 and direction 150 degrees in the form ai + bj.
-1.731 + j
66
A rubber ball is dropped from a height of 15 meters. On each rebound, it rises 2/3 of the height from which it last fell. Find the distance traversed by the ball before it comes to rest. The geometric progression occurs after the first rebound.
75 m
67
A die is rolled and a coin is tossed. What is the probability that a three and a head will appear?
1/12
68
What is the sum of the coefficients of the expansion of (2x - 1)^21
2
69
In how many ways can 2 integers be selected from the integers 1, 2, 3, ..., 100 so that their difference is exactly 7?
93
70
At the inflection point, the value of the second derivative of the function is
zero
71
Evaluate the integral of two log e to base 10, over x times dx from 1 to 10.
2
72
178. In how many ways can a deck of 52 cards be distributed to 5 players if each player will receive 4 cards?
4.6196 x10^27
73
The largest volume of a rectangular box that can be made of a cardboard 20 sq. in by cutting identical squares on its edges.
592.60 cu. cm
74
The segment from (-1, 4) to (2, - 2) is extended 3 times its own length. The terminal point is
(11, - 20)
75
When William was as old as Mae is now, the sum of their ages was 51. When Mae will be as old as William is now, the sum of their ages will be 103. How many years older is William than Mae?
13
76
If any 10 of the 26 letters of the alphabet are chosen to represent the ten digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. How many possible coding patterns are there?
1.93 x10^13
77
What is the value of A between 270° and 360° if 2sin 2A - sinA = 1
330°
78
Find the largest rectangle that can be inscribed in x2= - (y - 26) and x2= y -3.
4 cm x 16 cm
79
Two lines pass through (5, 5) and separate tangents to the circle x2 + y2 = 9.
Determine the distance between the x-intercepts of the two lines.
12.01
80
The cost of renting a ballroom for a school dance is different at each of the three establishments. The Ambassador charges $100 plus $3 per person. The Belmont charges $150 plus $2 per person, while the Coliseum charges $250 plus $1 per person. Under what condition would the Belmont be the best place to hold the dance?
bet. 50-100
81
The roots of equation x2 + x + 1 = 0 are:
No real roots
82
Determine the sum of the internal angles of a hexagon
720 degrees
83
A can plow the field in 4 hours less than B: If both will work together, the job is done in 10 hours. Find the number of hours for B to do the job alone.
22.2
84
The lotto uses number 1 - 42. A winning number consists of six different numbers in any order. What are your chances of winning it?
1/5,245,786
85
The laplace transform of t^2sin4t
8(3s2 - 16)/(s2 + 16)3
86
Suppose the dimension (in inches) of a rectangular box change from 9, 6, 4.to 902,
5.97, and 4.01 respectively. Find the exact change in volume.
- 0.063906
87
Find the equation of the line passing thru (6, 8) and pass thru a point 5 units distance from the origin.
0.393x - 0.919y + 5 = 0
88
Find the radius of curvature at any point of the curve y + In (cosx) = 0.
sec x
89
The roots of equation x2 + x + 1 = 0 are:
No real roots
90
The equation x^3 + y^3 - 3axy = 0 represents a curve called
Folium of Descartes
91
In how many ways can 6 people be seated in a row of 9 seats?
60,480
92
Determine the inverse Laplace transform of; 1 (s) = 100 / (s + 20) 2
i (t) = 100te- 20t
93
A can plow the field in 4 hours less than B: If both will work together, the job is done in 10 hours. Find the number of hours for B to do the job alone.
22.2
94
The lotto uses number 1 - 42. A winning number consists of six different numbers in any order. What are your chances of winning it?
1/5,245,786
95
Suppose the dimension (in inches) of a rectangular box change from 9, 6, 4.to 902,
5.97, and 4.01 respectively. Find the exact change in volume.
- 0.063906
96
Find the equation of the line passing thru (6, 8) and pass thru a point 5 units distance from the origin.
0.393x - 0.919y + 5 = 0
97
Find the radius of curvature at any point of the curve y + In (cosx) = 0.
sec x
98
Find the area inside the cardioid r = a(1 + cosθ) and outside the arc r = a
(π/4 + 2) a^2
99
Evaluate (1 - i)^-6
0.125 cis -90°
100
Evaluate [sq. root of 3 - i]^- 10
0.0009766 cis - 60°