MATHEMATICS, ENGINEERING SCIENCES AND LAWS (MASTERY PART 1)

1. Find the middle term of the expansion (𝑥 2 − 2𝑦)10 ? ft/sec^2, starting from rest at the first station and coming to a
A. −8064𝑥10 𝑦 5 C. −9064𝑥11 𝑦 5 stop at the end of the station, what is its maximum/minimum
B. −8864𝑥12 𝑦 5 D. −8164𝑥10 𝑦 5 speed in mph?
2. Find the coefficient of the term involving 𝑦 8 in the binomial A. 40.68 mph C. 49.22 mph
term, (3𝑥 2 + 2𝑦)10 B. 42.86 mph D. 70.55 mph
A. 153,680 C. 123,680 17. A can walk 7 miles in the time it takes B to walk 10 miles. A
B. 113,680 D. 103,680 requires 5 minutes longer than Patrick to walk for a mile. Find
3. In the binomial term, (𝑥 2 + 3𝑦)11 , find the sum of coefficients. the rate of B.
A. 4564304 C. 4294304 A. 2.56 mph C. 4.32 mph
B. 4194304 D. 4554304 B. 5.16 mph D. 4.16 mph
5 18. A commuter bought 23 pieces of school supplies for a total cost
4. Find the sum of the exponents for (𝑥 3 + 2𝑦 4 )
A. 103 C. 105 of P646.00. The notebooks costs P15.00 each piece, the crayons
B. 104 D. 100 costs P25.00 each pack and the pencil cases costs P36.00 each.
5. Eight times Larion’s age is 8 more than six times the age of his How many pencil cases did the commuter bought?
friend Borja. Ten years ago, the sum of their ages is 44. How A. 5 C. 11
old is Borja now? B. 7 D. 20
A. 42 C. 36 19. One number is seven times another number. The sum of the
B. 16 D. 28 numbers is 136. Find the smaller number.
6. Mary is 24 years old. Mary is twice as old as ann was when A. 15 C. 13
mary was as old as ann is now. how old is ann? B. 17 D. 19
A. 18 C. 6 20. Find the value of the 15th term of a series following arithmetic
B. 16 D. 8 progression, if the 9th term is 59 and the 3rd term is -37. Also
7. A and B are planning to mix alcohol solutions for their science find the common difference.
project. Two thousand kilograms of alcohol containing 8% A. 122, 13 C. 160, 16
solution is to be made by mixing an alcohol containing 14% B. 155, 16 D. 92, 13
solution with another containing 6% solution. How much of 21. From the previous problem, calculate the sum of the first 25
each solution is needed? terms.
A. 1000 & 1000 C. 200 & 1800 A. 4122 C. 3075
B. 1500 & 500 D. 1200 & 800 B. 3556 D. 2025
8. Joe has 24 ml of 68% sulfuric acid solution. How many liters 22. The 4th and 7th term of a geometric progression are 27/40 and
of pure sulfuric acid should he add to make 85% solution? 729/320, respectively. Find the value of the 2nd term and the
A. 4.6 ml C. 27.2 ml common ratio.
B. 51.2 ml D. 15.2 ml A. 9/20, 2/5 C. 5/4, 5/9
9. An Engineer can paint a condenser pipe in 30 hours. His wife B. 3/10, 3/2 D. 16/26, 1/2
can paint it in 35 hours. Together with their son, they can paint 23. A rubber ball was dropped from a height of 42m, and each time
in 12 hours. How long will their son finish the job working it strikes the ground it rebounds to a height of 5/6 of the distance
alone. from which it fell. Find the total distance travelled by the ball
A. 16.67 hrs C. 26.67 hrs before it comes to rest.
B. 36.67 hrs D. 46.67 hrs A. 551 C. 462
10. A paint job can be done by 70 men in 100 days. There were 80 B. 366 D. 325
men at the start of the project but after 50 days, 10 of them had 24. Find the 15th term of the harmonic progression 6/5, 4/3, 3/2
to be transferred to another project. How long will it take the ………….
remaining workforce to complete the job? A. -3 C. 7/5
A. 42.86 hrs C. 40.86 hrs B. 16 D. -15
B. 52.86 hrs D. 60.86 hrs 25. How many four digit numbers can be formed by the use of the
11. A Mechanical Engineer and his assistant can do a certain job in digits 1, 2, 3, 4, 5, 6, 7, 8 and 9 if the digits cannot be repeated.
3 hours. On a given day, they work together for 2 hours then A. 4042 C. 3422
his assistant left and the Engineer finishes the rest of the work B. 3559 D. 3024
in 8 more hours. How long will it take for the engineer to do 26. In how many ways can 5 engineers and 4 teachers be seated in
the job alone? a row if the teachers wants to sit next to each other?
A. 15 hours C. 36 hours A. 560 C. 20
B. 24 hours D. 6 hours B. 13520 D. 17280
12. What time between 10 and 11 o’clock will the hands of the 27. What is the permutation of the letters in the word SBEMEE
clock be directly opposite to each other? A. 120 C. 210
A. 10:20.54 C. 10:25.67 B. 150 D. 100
B. 10:21.82 D. 10:42.36 28. How many circular permutations are possible when seating
13. At what time after 12 o’clock will the hour hand and minute seven people around a table?
hand of the clock first form 120 degrees? A. 120 C. 24
A. 12:20.56 C. 12:21.82 B. 720 D. 5040
B. 12:23.54 D. 12:32.51 29. How many ways can you invite one or more of six friends to a
14. The time is past 2 o’clock. In 10 minutes, the minute hand will party?
be as much ahead of the hour hand as it is now behind it. What A. 63 C. 42
time is it? B. 31 D. 520
A. 2:05.909 C. 2:06.909 30. The time required for an elevator to lift a weight varies directly
B. 2:04.909 D. 2:07.909 with the weight and the distance through which it is to be lifted
15. It takes a plane one hour and forty-five minutes to travel 500 and inversely as the power of the motors. If it takes 30 seconds
miles upstream the current and covers the same distance in one for a 10 HP motor to lift 100 pounds through 50 ft, what size of
hour and fifteen minutes downstream. What is the speed of the motor is required to lift 800 lbs in 40 seconds through 12.2
current? meters?
A. 342.86 mph C. 411.43 mph A. 45 HP C. 48 HP
B. 650.50 mph D. 57.14 mph B. 30 HP D. 41 HP
16. A train travels between two stations ½ mile apart in a minimum 31. In board Examination of Mechanical Engineers, 119 examinees
time of 50 sec. If the train accelerates and decelerates at 8 took the MESL subject, 104 examinees took the PIPE subject
and 115 examinees took the MDSP Subject. Seventy-eight
 MATHEMATICS, ENGINEERING SCIENCES AND LAWS (MASTERY PART 1)
examinees took only MESL and PIPE subjects. Seventy-one 49. Determine the area of spherical triangle in sqm with given
examinees took only the PIPE and MDSP Subject. Eight-five angles A = 56deg, B = 65.9deg, C = 95deg with given radius of
examinees took only the MDSP and the MESL subject. Fifty- 20 m.
four examines took all three subjects. How many took the A. 251 C. 260
Mechanical Engineer Licensure Examination? B. 258 D. 269
A. 158 C. 200 50. Solve the other side c of the spherical triangle whose given
B. 140 D. 134 parts are:
32. Which of the following number is a PRIME NUMBER a = 72° b = 55° C = 90°
A. 477 C. 408 A. 80 deg C. 86 deg
B. 457 D. 435 B. 83 deg D. 78 deg
33. Which of the following statements is correct? 51. Determine the vertex angle B for spherical triangle whose sides
A. 5000 has four significant figures are given:
B. 0.00050 has two significant figures A. 55.57° C. 68.57°
C. 100050 has six significant figures B. 60.57° D. 62.57°
D. 500.00 has three significant figures 52. If two dice are tossed, what is the probability of rolling a sum
34. What is the greatest common factor of 156 and 180? of 6 or 9?
A. 10 C. 6 A. 1/5 C. 1/3
B. 11 D. 12 B. ¼ D. 1/8
53. A cat has litter of seven kittens. If the probability is 0.52 that
35. Solve for the value of x: 𝑥 = √5 − √5 − √5−. . . . . the kitten will be female, what is the probability that exactly
A. 1.25 C. 1.79 three of the seven will be male?
B. 2.26 D. 0.921 A. 0.50 C. 0.33
36. Which of the following number is a happy number? B. 0.48 D. 0.28
A. 97 C. 71 54. An urn contains four black balls and six white balls. What is
B. 48 D. 96 the probability of getting one black ball and one white ball in
37. Simplify the given expression: two consecutive draws from the urn?
5𝑥 𝑥+3 2𝑥 + 1 A. 0.04 C. 0.27
− + B. 0.24 D. 0.53
2𝑥 2 + 7𝑥 + 3 2𝑥 2 − 3𝑥 − 2 𝑥 2 + 𝑥 − 6
4 2 55. The probability that a married man watches a certain television
A. C.
x+3 x+5 show is 0.4 and the probability that a married woman watches
3 1
B. D. the show is 0.5. The probability that a woman watches the
x+6 x+3
38. What is the characteristic and mantissa of the common show, given that his man does is 0.66. Find the probability that
logarithm of 0.0071? a married couple watches the show.
A. -3, 0.85126 C. -3, 0.14874 A. 0.264 C. 0.55
B. -2, 0.85126 D. -2, 0.14874 B. 0.875 D. 0.25
39. Solve the value of x in the equation: 𝑥 3𝑙𝑜𝑔𝑥 =100𝑥 56. In the following set of numbers, 75, 85, 76, 55, 42, 91, 77.
A. 7 C. 10 Determine the standard deviation and variance.
B. 5 D. 9 A. 16.89, 252.53 C. 17.89, 252.53
40. WHAT IS THE ROOT MEAN SQUARE OF 11, 23, 35 and B. 15.89, 252.53 D. 12.89, 252.53
53? 57. In the following set of numbers, 75, 85, 76, 55, 42, 91, 77.
A. 33.22 C. 35.22 Determine the sample standard deviation and sample variance
B. 36.22 D. 34.22 A. 18.16, 294.6 C. 17.16, 294.6
41. Which of the following is equivalent to 7520 mils in radians? B. 17.16, 284.6 D. 15.16, 292.6
A. 47𝜋/20 C. 42𝜋/10 58. Find the probability that a couple having 5 children will have
B. 42𝜋/10 D. 47𝜋/10 at least one girl.
42. A certain angle has an explement 6 times the complement. Find A. 1/8 C. 15/8
the angle? B. 3/7 D. 31/32
A. 810 deg C. 139 deg 59. If seven fair coins are simultaneously tossed in the air, what is
B. 36 deg D. 24 deg the probability that at least one will land heads up?
43. The vertical angle to the top of a flagpole from point A on the A. 0.144 C. 0.971
ground is observed to be 37°11’. The observer walks 25 m B. 0.286 D. 0.992
directly away from point A and the flagpole to point B and find 60. In a deck of 52 playing cards, 2 cards are drawn at random.
the new angle to be 25°43’. What is the approximate height of What is the probability of getting an ace and a king?
the flagpole? A. 8/663 C. 5/353
A. 27 m C. 33 m B. 5/20 D. 2/663
B. 45 m D. 60 m
44. Simplify the equation 𝑠𝑖𝑛2 𝜃(1 + 𝑐𝑜𝑡 2 𝜃)
A. 1 C. sin2 θsec 2 θ
2
B. sin θ D. sec 2 θ
45. Coversine x is 0.134, find the value of x.
A. 60 deg C. 30 deg
B. 45 deg D. 20 deg
46. The sides of a triangle are 9 cm, 12 cm and 19 cm. Determine
the radius of the inscribed and circumscribing circle.
A. 2.10, 12.23 C. 4.10, 11.23
B. 3.10, 12.23 D. 2.33, 11.23
47. The corresponding sides of the two similar triangles are in the
ratio of 5:3. What is the ratio of their areas
A. 9:5 C. 5:3
B. 16:5 D. 25:9
48. What is the area of the circle inscribed in an equilateral triangle
with a side 10 cm long?
A. 32.18 cm^2 C. 126.18 cm^2
B. 5.18 cm^2 D. 26.18 cm^2
 MATHEMATICS, ENGINEERING SCIENCES AND LAWS (MASTERY PART 1)
TAKE HOME PROBLEMS 18. Two fair dice are thrown. What is the probability that the sum
1. Simplify: log 𝑥 + log 𝑥 3 + log 𝑥 5 shown on the dice is divisible by 5?
A. log(𝑥 + 𝑥 3 + 𝑥 5 ) C. 9 log 𝑥 A. 7/36 C. 1/9
B. log 9𝑥 D. 15 log 𝑥 B. 1/12 D. 1/4
2. What is the logarithm of negative number? 19. An urn contains 3 white balls and 1 black ball. Determine the
A. zero C. irrational probability of drawing two white balls in succession from the
B. rational D. complex urn without replacing the ball after each drawing.
3. Solve the value of x: 1.4 = (
0.0613 1.32
) A. 1/2 C. ¼
𝑥 B. 2/3 D. 3/2
A. 0.5471 C. 0.7541 20. A study has a sample size of 9, a standard deviation of 4.0, and
B. 0.04751 D. 0.4571 a sample standard deviation of 4.2. What is most nearly the
4. What is the discriminant of the equation: 4𝑥 2 − 8𝑥 + 5 = 0? sample variance?
A. 8 C. 16 A. 16 C. 34
B. -16 D. -8 B. 18 D. 36
5. If (𝑥 + 4) is a factor of 𝑥 3 + 2𝑥 2 − 7𝑥 + 𝑘, what is the value 21. A study has a sample size of 9, a standard deviation of 4.0, and
of k? a sample standard deviation of 4.2. What is most nearly the
A. 2 C. 3 sample variance?
B. 4 D. 5 A. 16 C. 34
6. What is the 4th term of the expansion (𝑥 + 𝑦)12 B. 18 D. 36
A. 220x9y3 C. 200x9y3 22. A coin is tossed three times. What is the probability of getting
9 3
B. 120x y D. 240x4y3 three heads?
7. A pump can pump out a tank in 11 hrs. Another pump can A. 1/8 C. ½
pump out the same tank in 20 hrs. How long will it take both B. 5/32 D. 3/2
pumps together to pump out the tank? 23. Find the area of spherical triangle ABC whose parts are A =
A. ½ hrs C. 5 hrs 93 deg 40 min, B = 64 deg 12 min, C = 116 deg 51 min. with
B. ¼ hrs D. 7 hrs radius of the sphere is 100 km.
8. At what time after 12 o’clock will the hour hand and minute A. 15314 km2 C. 13451 km2
hand of the clock first form 120 degrees? B. 16531 km 2
D. 14321 km2
A. 12:20.56 C. 12:29.32 24. Two cards are drawn at random from an ordinary deck of 52
B. 12:21.81 D. 12:30.45 cards. Find the probability P that one is a spade and one is a
9. The difference between two numbers is 12. If 2 is added to 7 heart.
times the smaller, the result is the same as when 2 is subtracted A. 3/51 C. 13/102
from 3 times the larger. Find the numbers. B. 1/26 D. 2/51
A. 20 and 6 C. 15 and 3 25. How many radians are there in the two successive numbers on
B. 20 and 8 D. 17 and 2 the face of a clock.
10. The product of ¼ and 1/5 of a number is 500. Find the number. A. π/2 C. π/6
A. 100 C. 300 B. π/3 D. π/4
B. 200 D. 150 26. Simplify 4cos(𝑦) sin(𝑦) (1 − 2 𝑠𝑖𝑛2 𝑦)
11. Three cats can kill 3 rats in 3 minutes. How many minutes will A. tan (4y) C. cos (4y)
it take for 100 cats to kill 100 rats? B. sec (4y) D. sin (4y)
A. 3 C. 100 27. The sides of a triangle are 8 cm, 10 cm and 14 cm. Determine
B. 33.33 D. 300 the radius of the inscribed and circumscribing circle.
12. How many committees can be formed by choosing 4 men A. 2.45, 7.14 C. 3.02, 6.82
from an organization of a membership of 15 men? B. 2.72, 8.21 D. 3.26, 7.82
A. 1390 C. 1435 28. An observer wishes to determine the height of a tower. He
B. 1240 D. 1365 takes sight at the top of the tower from A and B, which are 50
13. If the 4th term of GP is 8 and the 7th term is 1, find the 2nd term. ft apart at the same elevation on a direct line with the tower.
A. 45 C. 30 The vertical angle at point A is 30 deg and at point B is 40
B. 32 D. 25 deg. What is the height of the tower?
14. The vibration frequency of a string varies as the square root of A. 85.60 ft C. 110.29 ft
the tension and inversely as the product of the length and B. 143.97 ft D. 92.54 ft
diameter of the string. If the string is 3 feet long and 0.03-inch 29. Given a triangle with an angle C = 28.7°, side a = 132 units
in diameter vibrates at 720 times per second under 90 pounds and side b = 224 units. Solve for the side c.
tension, at what frequency will 2 feet long, 0.025-inch string A. 95 units C. 110 units
vibrate under 2500 pounds tension. B. 125.4 units D. 90 units
A. 6210 C. 7514 30. Simplify {cos 𝐴/[sin 𝐴 + 1]} + tan 𝐴
B. 6830 D. 5645 A. sec A C. sin A
15. A Mechanical Engineer bought 24 boxes of screws for P B. csc A D. cos A
2,200.00. There were three types of screws bought. Screw A 31. Joggers A and B, starting from the same point, jogging in
costs P 300 per box, screw B costs P 150 and screw C cost P opposite directions in a circular track field. Their circular path
50 per box. How many boxes of screw A did he buy? has a diameter of 800 meters. If the average speed of Jogger
A. 2 C. 7 A is 3 kph and that of Jogger B is to be 4 kph, compute the
B. 5 D. 5 time they will meet.
16. Three times the first of three consecutive odd integers is three A. 6.85 min C. 14.54 min
more than twice the third. Find the third integer. B. 21.54 min D. 43.23 min
A. 12 C. 10 32. Mary is three times as old as Ann now. Four years ago, she
B. 15 D. 18 was four times as old as Ann was at that time. How old is Ann?
17. A witness to a hit-and-run accident told the police that the license A. 12 C. 15
number contained the letter RLH followed by 3 digits, the first of
B. 10 D. 20
which is a 5. If the witness cannot recall the last two digits, but he
33. The electrical resistance of a wire made of a certain material
is certain that all 3 digits are different, find the maximum number
of automobile registrations that the police may have to check.
varies as its length and inversely as the square of the diameter.
A. 300 C. 72 If a wire 100 m long and 1.25 mm in diameter has a resistance
B. 45 D. 87 of 30 ohms, find the length of a wire of the same material
 MATHEMATICS, ENGINEERING SCIENCES AND LAWS (MASTERY PART 1)
whose resistance and diameter are 25 ohms and 0.75 mm
respectively.
A. 26 m C. 19 m
B. 30 m D. 56 m
34. Find the 4th term of the HARMONIC progression ½, 0.2,
0.125…
A. 1/10 C. 1/11
B. 0.102 D. 0.099
35. Points A and B are 100 m apart and are of the same elevation
as the foot of the tower. The angles of elevation of the top of
the tower from the points A and B are 21 degrees and 32
degrees respectively. How far is A from the tower in meters?
A. 259.28 C. 271.62
B. 265.42 D. 277.92
36. Solve for x in the equation: arctan(𝑥𝑥 + 1) + arctan(𝑥𝑥 − 1) =
arctan(12)
A. 1.50 C. 1.20
B. 1.34 D. 1.25
37. How many triangles are determined by the vertices of a
regular hexagon?
A. 10 C. 20
B. 15 D. 25
38. What is the standard deviation of 1, 4, and 7?
A. 3.8 C. 4.5
B. 5.2 D. 2.45
39. Douglas can paint a fence of 50% faster than Nonoy and 20%
faster than Jerome and together they can paint a given fence
in 4 hours. How long will it take Douglas to paint the same
fence if he had to work alone?
A. 9 C. 8
B. 10 D. 11
40. The product of two numbers is increased by 4 is P. If one
number is D, find the other number.
A. P/D-4 C. (P+4)/D
B. (P-4)/D D. D/P-4