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PROBLEM SET SUBMITTED TO: ENGR. REZEL STO. TOMAS SUMITTED BY: REA W. CUYOPAN
MARCH 2019
1. If the surface area of a cube is increased by 20%, by how many percent is the volume increased?
2. A company has x machines of equal capacity that can produce a total of 180 pieces each work day. If 2 machines break down, the work load of the remaining machines is increased by 3 pieces each per day to maintain the current production. Determine x.
3. The altitude of the hypotenuse of a right triangle divides the hypotenuse into two segments 8cm and 2 cm in length. Find the area of the triangle.
4. If versθ = x and 1 – sinθ = ½, find x if θ<90o.
5. A set has 5 items and it has a range of 7. The set is composed of the following: {1, 2, m, 5, m 2} with m>0. Find the average number in the set.
6. A man borrows ₱10,000 from a loan firm. The rate of simple interest is 15%, but the interest is to be deducted from the loan at the time the money is borrowed. At the end of one year he has to pay back ₱10,000. What is the actual rate of interest?
7. Justine was told to carry 20 cans in a store. Justine can carry only 3 cans at a time. How many trips would Justine have to make?
8. Determine the equation of the parabola where the vertex is at (4, 3) and the focus is at (4, -1).
9. An individual makes five annual deposits of ₱2000 in a savings account that pays interest at a rate of 4% per year. One year after making the last deposit, the interest rate changes to 6% per year. Five years after the last deposit, the accumulated money is withdrawn from the account. How much is withdrawn?
10. The area of a triangle inscribed in a circle having a radius of 9cm is equal to 43.23 cm2. If one of the sides of the triangle is 18cm, determine the length of one of the other sides.
11. A few books are laid out on a desk in the library. Two are Hydraulics, three are Mathematics, one is Design, and four are Surveying. Student A selects a Hydraulics book and student B then selects a Surveying book. Both students took their selections to the classroom to study. If student C then selects a book at random, what is the probability that he selects a Surveying book?
12. Determining the point of division of the line segment from A(5, 6) to B(-3, -2) that divides this line segment, starting from A, into two parts in the ratio 1:4.
13. An unknown amount invested at an unknown interest rate compounded semi-annually triples after 9 years. How many years will it take for the same unknown amount to double?
14. Two tangents OA and OB to a circle intersect at point O outside the circle. If the area of the smaller segment intercepted by the arc AB is 30% the area of the circle, find the measure of ∠BOA.
15. Boyle’s Law states that when a gas is compressed at a constant temperature, the product of its pressure and volume remains constant. If the pressure of a gas is 80psi, when the volume is 40 in3, find the rate of change of pressure with respect to volume when the volume is 20 in3.
16. The equation r= 2secθ is a/an:
17. A regular tetrahedron has a total surface area of 140 ft2. Determine the altitude of the tetrahedron.
18. A vat contains a mixture of acid and water. If 25 gallons of acid are added, the mixture will be 80% acid. If 25 gallons of water are added, the mixture will be 60% acid. Find the percentage of acid in the original mixture.
19. A small machine costing ₱80,000 has a salvage value of “X” at the end of its useful life of 5years. The book value at the end of the 4th year is ₱22,400. What is the value of “X” in pesos, using straight line method of depreciation?
20. What is the length of the latus rectum of the curve 16x2 + 25y2 – 64x – 50y – 311 = 0?
21. A bag contains 8 black, 5 red, 4 green and 9 yellow rubber bands. Three rubber bands are picked at random without replacement. Determine the probability that two are black rubber bands and the one is either green or yellow rubber band.
22. The cross section of a pipe is formed by two concentric circles such that the bigger one circumscribes a regular pentagon of sides measuring 23.20cm while the other one is inscribed in it. Find the crosssectional area of the pipe.
23. The legs of a right triangle change from 5cm and 12cm to 5.20c, and 12.30cm, respectively. Find the approximate change in area.
24. Two angles measuring P and Q are complementary. If 3P – 2Q = 40o, what is the measure of the larger angle?
25. Solve for one value of x in x3 – 8 = 0.
26. The probability that a student will pass an examination is 0.6, find the probability that of the 6 students who will undergo similar examination, half of them will pass.
27. A boy is entitled to 10 yearly endowments of ₱30,000 each year starting at the end of the eleventh year from now, the present value of these endowments is ₱93,241.78. What is the rate of interest if it is compounded annually?
28. How far is the centre of the circle x2 + y2 – 10x - 24y + 25 = 0 from the line y + 2 = 0?
29. X varies inversely as y. When x = 32, y = 68. Find x when y = 122.
30. Three circles with radii 3.0 cm, 5.0cm and 9.0cm are externally tangent to each other. Find the area of the triangle formed by connecting their centres.
31. A metal sphere with a specific gravity 8000kg/m3 originally weighs 24kg. It underwent a process of recasting and was formed into a solid cone with a base radius of 8cm. Assuming 15% of the metal was lost in the process, what is the height of the cone formed?
32. A square ABCD has a side equal to x. Point E is inside the square forming an equilateral triangle BEC having one side equal to the sides of the square. Determine the value of angle DEC.
33. If a stick is broken in two at random, what is the average length of the smaller piece?
34. A 20m high mast is placed on the top of a cliff whose height above sea level is not known. An observer at sea sees the top of the mast at an angle of elevation of 46o42’ and the foot of the mast at 38o23’. Which of the following most nearly gives the height of the cliff?
35. The points (1, 2, 6), (1, 6, 2) and (5, 2, z) are the vertices of a triangle. Determine the value of z if it is an equilateral triangle.
36. A central angle of 125o is subtended by an arc of a circle of radius 8.4cm. Which of the following most nearly gives the length of the major arc?
37. The upper portion of a sherbet glass is in the form of a right circular cone with a base of radius 2 inches and a slant height of 4 inches (inner dimensions). Find the volume of liquid it contains when filled to a depth of 3 inches.
38. The bases of a trapezoid are 150cm and 360c, respectively. The angles of inclination of the side are 60o and 48o, respectively. Determine the area of the trapezoid.
39. A point moves so that the difference between its distance from (0, 5) and (0, -5) is 8, what is the equation of its locus?
40. Two sides of a parallelogram are 68cm and 83cm and one of the diagonals is 42cm. Solve for the largest interior angle of the parallelogram.
41. The total surface area of a cue is 150ft2 . Determine a diagonal of the cube.
42. csc(520o) is equivalent to:
43. A fair coin is tossed 10 times. Compute the probability of getting at least 7 heads.
44. A hole 10cm in diameter is to be punched out from a sphere having a diameter of 16cm. Find the volume punched out.
45. A central circle has a radius of 10cm. Six circles of equal radius are to be arranged so that they are externally tangent to the central circle and each tangent to the adjacent circles. Determine the radius of each external circle.
46. Which of the following is not a prime number? A. 7 B. 107 C. 77 D. 109
47. If a right circular cone has a lateral surface area of 6π and a slant height of 6, what is the radius of its circular base?
48. Find the values of x that satisfy the inequality |x3 – 8| ≤5.
49. A quadrilateral having an area of 62m2 is inscribed in a circle. Three of its sides measure 15m, 7m and 13m, consecutively. Find the product of the diagonals of the quadrilateral.
50. Given f(x) = (x – 3)(x + 5) + 3. When f(x) is divided by (x – r), the remainder is r. Find the value of r.
51. A goat is tied outside and at one corner of a rectangular fenced property with 4m x 5m sides. If the rope with which the goat is tied is 6m, find the area over which the goat can gaze outside the fence.
52. The probabilities that a service station will pump gas into 0, 1, 2, 3, 4, or 5 or more cars during a certain 30-minute period are 0.03, 0.18, 0.24, 0.28, 0.10 and 0.17, respectively. Find the probability, that in this 30-minute period, at most 4 cars receive gas.
53. A man observes that the angle of elevation of the top of a tower from a certain point on a level ground id 30o. He then moves towards the tower by 25m and observes that the angle of elevation becomes 40o. How high is the tower?
54. A group of children playing with marbles placed 50 pieces of the marbles inside a cylindrical container with water filled to a height of 20cm. if the diameter of each marble is 1.5cm and the diameter of the cylindrical container is 6cm, what would be the new height of water inside the cylindrical container after the marbles were placed?
55. The complete factored form of (5 + x)2 – 14(5 + x) + 49 is
56. A bowl in the form of a spherical segment with two bases has a height of 0.10m. The upper base is a great circle with a diameter of 0.60m. Determine the capacity of the bowl.
57. A road is tangent to a circular lake. Along the road and 12miles from the point of tangency, another road opens towards the lake. From the intersection of the two roads to the periphery of the lake, the length of the new road is 11miles. If the new road will be prolonged across the lake, find the length of the bridge to be constructed.
58. Points A and C 1000, apart are plotted on a straight highway running East and West. From A, the bearing of a tower B is 32o E of N. From C, the bearing of the tower B is 64o W of N. Find the shortest distance of tower B from the highway.
59. A regular triangular pyramid has an altitude of 9m and a volume of 187.06m3. What is the length of the base edge?
60. It is amount which a willing buyer will pay to a willing seller for a property which each has equal advantage and is under no compulsion to buy or sell.
61. A man borrowed ₱300,000 from a lending institution which will be paid after 10 years at an interest rate of 12% compounded annually. If money is worth 7.72% compounded monthly, how much should he deposit to a bank at the end of each month in order to discharge his debt 10 years hence?
62. What is the number of permutations of the letter in the word BANANA?
63. If the area of a regular polygon is 50sq. m and its perimeter is 25m, determine the length of its apothem.
64. The logarithm of the product of M and N is 1.602059991 and the logarithm of their quotient is 0.397940008. Determine the value of N.
65. A lot is in the form of an equilateral triangle each of whose sides is 300m. Compute the length of the line parallel to one side that will divide the area into two equal parts.
66. A solid material is in the form of a rectangular parallelepiped 4ft x 6ft x 8ft. The solid is cut completely to form cubes 1ft x 1ft x 1ft. How many cubes will there be?
67. A man inherited a regular endowment of ₱100,000 every end of 3 months for x years. However, he may choose to get a single lump sum of ₱3,702,939.0 at the end of 4 years. If the rate of interest was 14% compounded quarterly, what is the value of x?
68. Telephone numbers in a certain locality start with 734. If the end and beginning of the next four digits cannot be zero, how many possible numbers can be formed?
69. A flagstaff standing on top of a tower 80ft high subtends an angle of arctan(1/9) at a point 100ft from the foot of the tower. Find the g=height of the flagstaff.
70. An ice cream cone is filled with ice cream and more ice cream in the form of hemisphere on top. The diameter of the hemisphere is also equal to the lateral surface area of the cone. If the diameter of the cone is 50mm, determine the total volume of ice cream.
71. How many circular arrangements can be made out of objects if 5 objects are taken at a time?
72. A car has a mass of 1000kg. A model of the car is made to a scale of 1:50. Determine the mass of the model if the car and its model are made of the same material.
73. An individual makes five annual deposits of ₱2000 in a savings account that pays interest at a rate of 4% per year. One year after making the last deposit, the interest rate changes to 6% per year. Five years after the last deposit, the accumulated money is withdrawn from the account. How much is withdrawn?
1
74. Evaluate: csc 𝑥+1 +
1 csc 𝑥−1
75. In a pre-licensure examination, an examinee may select 7 problems from a set of 10 questions. In how many ways can he make his choice?
MARCH 2019
1. If the surface area of a cube is increased by 20%, by how many percent is the volume increased?
2. A company has x machines of equal capacity that can produce a total of 180 pieces each work day. If 2 machines break down, the work load of the remaining machines is increased by 3 pieces each per day to maintain the current production. Determine x.
3. The altitude of the hypotenuse of a right triangle divides the hypotenuse into two segments 8cm and 2 cm in length. Find the area of the triangle.
4. If versθ = x and 1 – sinθ = ½, find x if θ<90o.
5. A set has 5 items and it has a range of 7. The set is composed of the following: {1, 2, m, 5, m 2} with m>0. Find the average number in the set.
6. A man borrows ₱10,000 from a loan firm. The rate of simple interest is 15%, but the interest is to be deducted from the loan at the time the money is borrowed. At the end of one year he has to pay back ₱10,000. What is the actual rate of interest?
7. Justine was told to carry 20 cans in a store. Justine can carry only 3 cans at a time. How many trips would Justine have to make?
8. Determine the equation of the parabola where the vertex is at (4, 3) and the focus is at (4, -1).
9. An individual makes five annual deposits of ₱2000 in a savings account that pays interest at a rate of 4% per year. One year after making the last deposit, the interest rate changes to 6% per year. Five years after the last deposit, the accumulated money is withdrawn from the account. How much is withdrawn?
10. The area of a triangle inscribed in a circle having a radius of 9cm is equal to 43.23 cm2. If one of the sides of the triangle is 18cm, determine the length of one of the other sides.
11. A few books are laid out on a desk in the library. Two are Hydraulics, three are Mathematics, one is Design, and four are Surveying. Student A selects a Hydraulics book and student B then selects a Surveying book. Both students took their selections to the classroom to study. If student C then selects a book at random, what is the probability that he selects a Surveying book?
12. Determining the point of division of the line segment from A(5, 6) to B(-3, -2) that divides this line segment, starting from A, into two parts in the ratio 1:4.
13. An unknown amount invested at an unknown interest rate compounded semi-annually triples after 9 years. How many years will it take for the same unknown amount to double?
14. Two tangents OA and OB to a circle intersect at point O outside the circle. If the area of the smaller segment intercepted by the arc AB is 30% the area of the circle, find the measure of ∠BOA.
15. Boyle’s Law states that when a gas is compressed at a constant temperature, the product of its pressure and volume remains constant. If the pressure of a gas is 80psi, when the volume is 40 in3, find the rate of change of pressure with respect to volume when the volume is 20 in3.
16. The equation r= 2secθ is a/an:
17. A regular tetrahedron has a total surface area of 140 ft2. Determine the altitude of the tetrahedron.
18. A vat contains a mixture of acid and water. If 25 gallons of acid are added, the mixture will be 80% acid. If 25 gallons of water are added, the mixture will be 60% acid. Find the percentage of acid in the original mixture.
19. A small machine costing ₱80,000 has a salvage value of “X” at the end of its useful life of 5years. The book value at the end of the 4th year is ₱22,400. What is the value of “X” in pesos, using straight line method of depreciation?
20. What is the length of the latus rectum of the curve 16x2 + 25y2 – 64x – 50y – 311 = 0?
21. A bag contains 8 black, 5 red, 4 green and 9 yellow rubber bands. Three rubber bands are picked at random without replacement. Determine the probability that two are black rubber bands and the one is either green or yellow rubber band.
22. The cross section of a pipe is formed by two concentric circles such that the bigger one circumscribes a regular pentagon of sides measuring 23.20cm while the other one is inscribed in it. Find the crosssectional area of the pipe.
23. The legs of a right triangle change from 5cm and 12cm to 5.20c, and 12.30cm, respectively. Find the approximate change in area.
24. Two angles measuring P and Q are complementary. If 3P – 2Q = 40o, what is the measure of the larger angle?
25. Solve for one value of x in x3 – 8 = 0.
26. The probability that a student will pass an examination is 0.6, find the probability that of the 6 students who will undergo similar examination, half of them will pass.
27. A boy is entitled to 10 yearly endowments of ₱30,000 each year starting at the end of the eleventh year from now, the present value of these endowments is ₱93,241.78. What is the rate of interest if it is compounded annually?
28. How far is the centre of the circle x2 + y2 – 10x - 24y + 25 = 0 from the line y + 2 = 0?
29. X varies inversely as y. When x = 32, y = 68. Find x when y = 122.
30. Three circles with radii 3.0 cm, 5.0cm and 9.0cm are externally tangent to each other. Find the area of the triangle formed by connecting their centres.
31. A metal sphere with a specific gravity 8000kg/m3 originally weighs 24kg. It underwent a process of recasting and was formed into a solid cone with a base radius of 8cm. Assuming 15% of the metal was lost in the process, what is the height of the cone formed?
32. A square ABCD has a side equal to x. Point E is inside the square forming an equilateral triangle BEC having one side equal to the sides of the square. Determine the value of angle DEC.
33. If a stick is broken in two at random, what is the average length of the smaller piece?
34. A 20m high mast is placed on the top of a cliff whose height above sea level is not known. An observer at sea sees the top of the mast at an angle of elevation of 46o42’ and the foot of the mast at 38o23’. Which of the following most nearly gives the height of the cliff?
35. The points (1, 2, 6), (1, 6, 2) and (5, 2, z) are the vertices of a triangle. Determine the value of z if it is an equilateral triangle.
36. A central angle of 125o is subtended by an arc of a circle of radius 8.4cm. Which of the following most nearly gives the length of the major arc?
37. The upper portion of a sherbet glass is in the form of a right circular cone with a base of radius 2 inches and a slant height of 4 inches (inner dimensions). Find the volume of liquid it contains when filled to a depth of 3 inches.
38. The bases of a trapezoid are 150cm and 360c, respectively. The angles of inclination of the side are 60o and 48o, respectively. Determine the area of the trapezoid.
39. A point moves so that the difference between its distance from (0, 5) and (0, -5) is 8, what is the equation of its locus?
40. Two sides of a parallelogram are 68cm and 83cm and one of the diagonals is 42cm. Solve for the largest interior angle of the parallelogram.
41. The total surface area of a cue is 150ft2 . Determine a diagonal of the cube.
42. csc(520o) is equivalent to:
43. A fair coin is tossed 10 times. Compute the probability of getting at least 7 heads.
44. A hole 10cm in diameter is to be punched out from a sphere having a diameter of 16cm. Find the volume punched out.
45. A central circle has a radius of 10cm. Six circles of equal radius are to be arranged so that they are externally tangent to the central circle and each tangent to the adjacent circles. Determine the radius of each external circle.
46. Which of the following is not a prime number? A. 7 B. 107 C. 77 D. 109
47. If a right circular cone has a lateral surface area of 6π and a slant height of 6, what is the radius of its circular base?
48. Find the values of x that satisfy the inequality |x3 – 8| ≤5.
49. A quadrilateral having an area of 62m2 is inscribed in a circle. Three of its sides measure 15m, 7m and 13m, consecutively. Find the product of the diagonals of the quadrilateral.
50. Given f(x) = (x – 3)(x + 5) + 3. When f(x) is divided by (x – r), the remainder is r. Find the value of r.
51. A goat is tied outside and at one corner of a rectangular fenced property with 4m x 5m sides. If the rope with which the goat is tied is 6m, find the area over which the goat can gaze outside the fence.
52. The probabilities that a service station will pump gas into 0, 1, 2, 3, 4, or 5 or more cars during a certain 30-minute period are 0.03, 0.18, 0.24, 0.28, 0.10 and 0.17, respectively. Find the probability, that in this 30-minute period, at most 4 cars receive gas.
53. A man observes that the angle of elevation of the top of a tower from a certain point on a level ground id 30o. He then moves towards the tower by 25m and observes that the angle of elevation becomes 40o. How high is the tower?
54. A group of children playing with marbles placed 50 pieces of the marbles inside a cylindrical container with water filled to a height of 20cm. if the diameter of each marble is 1.5cm and the diameter of the cylindrical container is 6cm, what would be the new height of water inside the cylindrical container after the marbles were placed?
55. The complete factored form of (5 + x)2 – 14(5 + x) + 49 is
56. A bowl in the form of a spherical segment with two bases has a height of 0.10m. The upper base is a great circle with a diameter of 0.60m. Determine the capacity of the bowl.
57. A road is tangent to a circular lake. Along the road and 12miles from the point of tangency, another road opens towards the lake. From the intersection of the two roads to the periphery of the lake, the length of the new road is 11miles. If the new road will be prolonged across the lake, find the length of the bridge to be constructed.
58. Points A and C 1000, apart are plotted on a straight highway running East and West. From A, the bearing of a tower B is 32o E of N. From C, the bearing of the tower B is 64o W of N. Find the shortest distance of tower B from the highway.
59. A regular triangular pyramid has an altitude of 9m and a volume of 187.06m3. What is the length of the base edge?
60. It is amount which a willing buyer will pay to a willing seller for a property which each has equal advantage and is under no compulsion to buy or sell.
61. A man borrowed ₱300,000 from a lending institution which will be paid after 10 years at an interest rate of 12% compounded annually. If money is worth 7.72% compounded monthly, how much should he deposit to a bank at the end of each month in order to discharge his debt 10 years hence?
62. What is the number of permutations of the letter in the word BANANA?
63. If the area of a regular polygon is 50sq. m and its perimeter is 25m, determine the length of its apothem.
64. The logarithm of the product of M and N is 1.602059991 and the logarithm of their quotient is 0.397940008. Determine the value of N.
65. A lot is in the form of an equilateral triangle each of whose sides is 300m. Compute the length of the line parallel to one side that will divide the area into two equal parts.
66. A solid material is in the form of a rectangular parallelepiped 4ft x 6ft x 8ft. The solid is cut completely to form cubes 1ft x 1ft x 1ft. How many cubes will there be?
67. A man inherited a regular endowment of ₱100,000 every end of 3 months for x years. However, he may choose to get a single lump sum of ₱3,702,939.0 at the end of 4 years. If the rate of interest was 14% compounded quarterly, what is the value of x?
68. Telephone numbers in a certain locality start with 734. If the end and beginning of the next four digits cannot be zero, how many possible numbers can be formed?
69. A flagstaff standing on top of a tower 80ft high subtends an angle of arctan(1/9) at a point 100ft from the foot of the tower. Find the g=height of the flagstaff.
70. An ice cream cone is filled with ice cream and more ice cream in the form of hemisphere on top. The diameter of the hemisphere is also equal to the lateral surface area of the cone. If the diameter of the cone is 50mm, determine the total volume of ice cream.
71. How many circular arrangements can be made out of objects if 5 objects are taken at a time?
72. A car has a mass of 1000kg. A model of the car is made to a scale of 1:50. Determine the mass of the model if the car and its model are made of the same material.
73. An individual makes five annual deposits of ₱2000 in a savings account that pays interest at a rate of 4% per year. One year after making the last deposit, the interest rate changes to 6% per year. Five years after the last deposit, the accumulated money is withdrawn from the account. How much is withdrawn?
1
74. Evaluate: csc 𝑥+1 +
1 csc 𝑥−1
75. In a pre-licensure examination, an examinee may select 7 problems from a set of 10 questions. In how many ways can he make his choice?
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