Engineering Mathematics: (4 Hrs): Prepared By: Engr. Isaiah James Maling

Engineering Mathematics: (4 hrs) 1.

From a group of 5 women and 7 men, how many different committees consisting of 3 women and 4 men can be formed if two of the men refuse to serve on any of the committee together? A. 250 C. 350 B. 300 D. 275 2. Kyla, Jericho and Trixie take turns flipping a coin in their respective order. The first one to flip head wins. What is the probability that Trixie will win? A. 1/8 C. 2/7 B. 1/7 D. 1/4 3. A tank contains 150 L of mango puree and 50 L of water. Then 40 L of the mixture is removed and is replaced by 40 L of water. What is the percentage of mango puree in the final mixture? A. 50% C. 60% B. 55% D. 65% 4. In how many ways can you arrange the word ALGEBRA such that the consonants are in alphabetical order? A. 2520 C. 210 B. 105 D. 5040 5. Mary is 42 years old. Mary was twice as old as Ann was when Mary was as old as Ann is now. How old is Ann? A. 24 C. 26 B. 21 D. 28 6. If electricity power failures according to a Poisson distribution with an average of 3 failures every twenty weeks, calculate the probability that there will not be more than 1 failure during a particular week. A. 0.86 C. 0.99 B. 0.92 D. 0.95 7. An ellipse, major axis 8 and minor axis 6 is revolved about its minor axis. Find the volume of the solid of revolution in terms of pi? A. 48𝜋 C. 32𝜋 B. 64𝜋 D. 24𝜋 8. James jogs every morning, while Dina cycles on the same route. If Dina’s speed is 3.5 times that of James, and Dina starts 2 hours after James, how many minutes does Dina cycle before she overtakes James? A. 42 mins C. 45 mins B. 48 mins D. 51 mins 9. A closed cylindrical tank container has a capacity of 128 pi cubic meters. Determine the minimum surface area. A. 96𝜋 C. 64𝜋 B. 32𝜋 D. 128𝜋 10. A curve has a parametric equation of y = 4(t+2) and x = 2𝑡 2. Find d𝑦 2/d𝑥 2 . A. −1/𝑡 2 C. −1/4𝑡 3 2 B. 1/𝑡 D. 1/4𝑡 3 11. Find the volume of the solid formed by revolving the curve defined by 9𝑥 2 + 4𝑦 2 = 36 about the line x + y = 6. A. 302.5 C. 502.5 B. 402.5 D. 602.5 12. A car headlight reflector is cut by a plane along its axis. The section is a parabola having the light center at the focus. If the distance of focus from vertex is 3/4 cm and if the diameter of the reflector is 10 cm, find its depth. A. 23/3 cm C. 29/3 cm B. 22/3 cm D. 25/3 cm

PREPARED BY: ENGR. ISAIAH JAMES MALING

13. For how many integers x does a triangle with side lengths 12, 25 and x has all its angles acute? A. 4 C. 6 B. 5 D. 7 1 2|. 14. Find the eigenvalues of the matrix | 4 3 A. 5 and -1 C. 3 and 1 B. -5 and 1 D. -3 and -1 15. A dice is loaded so that the probability of a number coming out is directly proportional to the number. When the die is rolled, what is the probability that a prime number comes out? A. 2/7 C. 3/7 B. 10/21 D. 8/21 16. Points A and B are 100 meters apart and are of the same elevation as the foot of the building from points A and B are 21 and 32 degrees respectively. How far is A from the building? A. 259.28m C. 158.62m B. 286.23m D. 345.41m 17. If the GCF of 2 numbers is 48 and their LCM is 336, what is the product of the numbers? A. 16128 C. 4032 B. 8064 D. 2016 18. A wooden cone of altitude 10 cm is to be cut into two parts of equal weight. How far from the vertex should the cut parallel to the base be made? A. 9.74cm C. 7.94cm B. 4.79cm D. 7.49cm 19. A triangle has 2 sides measuring 2018 cm and 2019 cm. How many triangles are possible with the 2 sides given such that the 3rd side is also having an integral length? A. 4034 C. 4035 B. 4036 D. 4037 20. Find the angle whose supplement exceeds six times its complement by 20 degrees A. 76° C. 64° B. 38° D. 72° 21. A 100 kg salt solution originally 4% by weight NaCl in water is evaporated until the concentration is 5% by weight NaCl. What percentage of water in the original solution is evaporated? A. 20% C. 80% B. 20.83% D. 79.17% 22. How many minutes after two o’clock will the hands of a clock form a 60 degrees angle? A. 22.85 C. 23.45 B. 21.82 D. 24.56 23. A circle is inscribed in rhombus whose diagonals are 30 cm and 40 cm. What is the value of the radius of the circle? A. 12.5cm C. 15cm B. 12cm D. 16cm 24. Find the minimum distance of the parabola 𝑦 2 = 8x from the point (4, 2). A. 4√2 C. 2√2 B. 3√2 D. √2 25. Find the smallest positive integer such that P(x) = 𝑥 4 - 2𝑥 3 - 10𝑥 2 + 40 x – 90, this integer is an upper bound. A. 5 C. 6 B. 4 D. 7

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Engineering Mathematics: (4 hrs)

26. 12 men can finish a job in 16 days. 5 men were working at the start and after 8 days, 3 men were added. How many days will it take to finish the whole job? A. 19days C. 27days B. 24days D. 21days 27. In how many equal parts should you divide 120 so that the continue product of those equal parts will be maximum? A. 43 C. 33 B. 44 D. 34 28. Albert and Katie are painting a room. Katie paints half the room red. Albert paints half of the unpainted area blue. Katie paints half of the unpainted area red, and so on. If this process continues infinitely, what fraction of the room will be painted red? A. 2/3 C. 3/4 B. 5/8 D. 4/7 29. The probability that A, B and C will hit the target are 0.2, 0.4 and 0.9 respectively. What is the probability that at least one of them will hit the bull’s eye? A. 0.812 C. 0.882 B. 0.922 D. 0.952 30. Find the radius of curvature at any point on the curve y = ln cos x. A. Sec x C. Tan x B. Cos x D. Cot x 31. What is the differential equation of all the straight lines with slope and y-intercept equal? A. ydx – (x + 1)dy = 0 B. ydx + (x + 1)dy = 0 C. (y + 1)dx + xdy = 0 D. (y + 1)dx – xdy = 0 32. Evaluate ln(3 + j4) A. 1.46 + j0.10 C. 1.77 + j0.84 B. 1.61 + j0.92 D. 1.95 + j0.112 33. Find the volume of the largest open box that can be made by cutting off equal squares from the corners of a 20 cm by 24 cm box and then turning up the sides. A. 774.16cc C. 456.12cc B. 325.55cc D. 875.12cc 34. A piece of wire 14 cm long is cut in two, one part being bent in the shape of an equilateral triangle and the other in the form of a circle. Find the length of the piece of wire used for the triangle if the sum of the area of these two figures is to be minimum. A. 8.725cm C. 10.561cm B. 12.771cm D. 11.453cm 35. What is the sum of all the distinct solutions of the equation 1 2

1

𝑥

𝑥

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3 (𝑥 + ) − 16 (𝑥 + ) + 20 = 0? A. 4 C. 13/3 B. 11/3 D. 10/3 36. Charles has a probability of 1/3 in winning a game against his younger sister. If they play four times, what is the probability that he will win at least twice? A. 16/27 C. 11/27 B. 13/27 D. 14/27 37. Determine the coordinates of the point which is three-fifths of the way from the point (2, -5) to the point (-3, 5). A. (-1, 1) C. (-1, -2) B. (-2, -1) D. (1, -1) 38. Find the eccentricity of the curve 9𝑥 2 − 4𝑦 2 − 36𝑥 + 8𝑦 − 4 = 0.

PREPARED BY: ENGR. ISAIAH JAMES MALING

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A. 1.6 C. 2.0 B. 1.8 D. 2.2 In how many ways can 3 distinct numbers be chosen from the set {1, 2, 3,…,10} such that the sum of the numbers is odd? A. 50 C. 60 B. 70 D. 80 What is the sum of all the positive factors of 144? A. 400 C. 402 B. 401 D. 403 How many regular polygons, with number of sides less than 100, have interior angle measures whose values are integer degrees? A. 21 C. 19 B. 20 D. 22 If a, b and c are the roots of 𝑥 3 − 2𝑥 2 + 3𝑥 − 4 = 0, find the numerical value of 𝑎3 + 𝑏 3 + 𝑐 3 . A. -2 C. 2 B. 3 D. 4 What is the area of a rhombus with side 12 cm whose diagonals differ by 6 cm? A. 135 sq.cm C. 128 sq.cm B. 144 sq.cm D. 169 sq.cm Triangle ABC has a right angle at B and contains a Point P for which PA = 10, PB = 8 and <APB =
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Engineering Mathematics: (4 hrs)

52. A point on the curve where the second derivative is greater than 0 is called: A. Maxima C. Inflection Point B. Minima D. Critical Point 8 53. Find the remainder when 𝑥 is divided by 𝑥 2 − 𝑥 − 2. A. 31x + 32 C. 63x + 64 B. 85x + 86 D. 15x + 16. 54. Two tangents from a circle form an angle of 100 degrees. What is the measurement of the larger intercepted arc? A. 240° C. 260° B. 250° D. 280° 55. Find the perimeter of a right triangle circumscribing a circle of radius 1.5 cm and hypotenuse 10 cm. A. 22cm C. 11cm B. 23cm D. 11.5cm 56. Three regular polygons fit exactly together around a point on a plane surface. One is a square and the other is a hexagon. How many sides does the third polygon have? A. 8 C. 15 B. 12 D. 10 57. Find the volume formed by revolving the triangle whose vertices are (1, 1), (2, 4) and (3, 1) about the line 2x – 5y = 10. A. 52 C. 56 B. 63 D. 60 58. A body moves such that its acceleration as a function of time is a = 2 + 12t, where a is in meters per second square. If its velocity after 1 second is 11m/s, find the distance traveled after 5 seconds. A. 290m C. 340m B. 256m D. 420m 59. In how many ways can you write 7 as the sum of 1’s and 2’s?. A. 20 C. 19 B. 21 D. 18 60. By selling a balut at 12 pesos each, a vendor gains 19.7%. The cost price of the the balut rises by 12.5%. If he sells the balut at the same price as before, what is his new gain in percent? A. 6% C. 6.4% B. 6.2% D. 6.6% 61. In triangle ABC, the length BC is the average of the lengths of the other sides. If cos A = AC/AB, find the numerical value of cos A. A. 1/3 C. 3/5 B. ½ D. 4/5 62. In square ABCD, a point P is chosen such that it is equidistant from three points: A, B and midpoint of CD. If this distance is 5 cm, find the side length of square ABCD A. 8cm C. 10cm B. 9cm D. 11cm 63. In how many ways can 7 people be lined up in a row for a picture if two of them should have at least 2 people between them? A. 1200 C. 3600 B. 2400 D. 4800 64. An MRT train 0.2 km long travels at a steady speed of 24 kph. It enters a tunnel 1 km long at exactly 4:00 PM. At what time will the tail-end of the train come out of the tunnel?

PREPARED BY: ENGR. ISAIAH JAMES MALING

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A. 4:04 PM C. 4: 05 PM B. 4:03 PM D. 4: 06 PM One candle will burn completely at a uniform rate of 4 hours while another in 3 hours. At what time should the 2 candles be simultaneously lighted so that one will be half the length of the other at 6:00 PM? A. 2: 24 PM C: 4: 00 PM B. 3: 36 PM D. 4: 48 PM Find the sum of all positive integers less than 45 which are not divisible by 3. A. 625 C. 675 B. 650 D. 700 Let x, y, and z be positive real numbers such that x(y + z) = 95, y(x + z) = 128 and z(x + y) = 143. Find xyz. A. 330 C. 400 B. 440 D. 500 The hypotenuse of a right triangle is 20 cm. What is the maximum possible area in square cm? A. 200 C. 150 B. 100 D. 200 The sum of two numbers is 4. Find the maximum possible value of the product of the square of one and the cube of the other. A. 26.3 C. 19.8 B. 35.4 D. 55.7 If 15 men can build a wall 108 meters long in 6 days, what length of similar wall can be built by 25 men in 3 days? A. 45m C. 60m B. 90m D. 120m A rubber ball is made to fall from a height of 80 feet and is observed to rebound 2/3 of the height from which it falls. How far will the ball travel before coming to rest if the ball continues to fall in this manner? A. 240 ft C. 400ft B. 320 ft D. 360ft If a dice is rolled once, what is the expected value of the sum of the faces? A. 3 C. 4 B. 3.5 D. 3.75 Two vertices of a triangle are (2, 4) and (-2, 3) and the area is 2 square units. What is the locus of the third vertex? A. 4x – y = 14 C. 4x + 4y = -14 B. x + 4y = 12 D. x – 4y = -18 How long is the shortest chord that can be drawn through a point 20 cm from the center of a circle whose radius is 29 cm long? A. 21cm C. 42cm B. 24cm D. 48cm 𝑥 What is the period of the graph of 𝑦 = tan 3 ? A. B.

𝜋 2𝜋

C. 3𝜋 D. 6𝜋

76. Alan, Ben and Carlo can do a piece of work in 20, 30 and 60 days respectively. In how many days can Alan do the work if he is assisted by Ben and Carlo on every third day? A. 12 C. 10 B. 15 D. 18 77. During an engineering board examination, there were 350 examinees from Luzon, 250 from the Visayas and 400 from Mindanao. The results of the exam revealed that the flunkers from Luzon, Visayas and Mindanao were 3%, 5%

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Engineering Mathematics: (4 hrs)

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and 7% respectively. If the name of a flunker is picked at random, what is the probability that he/she is from Luzon? A. 0.33 C. 0.55 B. 0.44 D. 0.64 Some birds are sitting in an oak tree. Ten more birds land. More birds arrive until there are a total of four times as many birds as the oak tree had after the ten landed. A nearby maple tree has 16 fewer than 12 times as many birds as the oak tree had before the 10 landed. If both trees now have the same number of birds, how many birds were originally in the oak tree before the first 10 landed? A. 4 C. 16 B. 7 D. 24 The perimeter of a right triangle is 60 cm and the altitude perpendicular to the hypotenuse is 12 cm. Find the longest side of the triangle. A. 15cm C. 18cm B. 12cm D. 20cm How many ways can you arrange the letters of the word DIVISIBLE such that no two I’s are adjacent to each other? A. 25200 C. 18900 B. 12600 D. 6300 In a regular polygon, we call the intersection of two diagonals a node. How many nodes are there in an octagon? A. 56 C. 28 B. 70 D. 35 How many ways can we group 9 people into three groups of 3 if three of them want to be in the same group? A. 30 C. 10 B. 20 D. 14 Point P is inside regular octagon ABCDEF GH so that triangle ABP is equilateral. How many degrees are in angle APC? A. 105° C. 125° B. 120° D. 112.5° A rhombus has a 60 degree angle. What is the ratio of its area to that of a circle inscribed inside it? A. 1.47 C. 1.24 B. 1.55 D. 1.75 In a group of five friends, the sums of the ages of each group of four of them are 124, 128, 130, 136, and 142. What is the age of the youngest? A. 21 C. 23 B. 22 D. 24 An old receipt has faded. It reads 88 candies with a total price of $A4.2B, where A and B are unreadable digits. How much (in cents) does a candy cost? A. 83 C. 93 B. 63 D. 73 What is the surface area generated by revolving the parabola 𝑦 = 𝑥 2 from x = 0 to x = √2 about the y – axis? A. 12.74 C. 11.65 B. 14.98 D. 13.61 Find the centroid of the region bounded by y = 𝑥 2 , y = 0 and x = 1. A. (1/4, 2/3) C. (3/4, 3/10) B. (2/3, 5/4) D. (3/5, 1/2) What is the largest possible remainder when a 2 digit number is divided by the sum of its digits? A. 14 C. 16

PREPARED BY: ENGR. ISAIAH JAMES MALING

B. 15 D. 13 90. If there are three positive integers x, y, and z such that 28x + 30y + 31z = 365. Find x + y + z. A. 12 C. 18 B. 14 D. 24 91. A 4-digit number is created by using each of the digits 3, 4, 5 and 8 exactly once. What is the probability that the number is not divisible by 4? A. 1/6 C. 5/6 B. 1/3 D. 2/3 92. Two fair 6-sided dice, colored red and white, are tossed. What is the probability that the number on the red die is at least as large as that on the white die? A. 1/2 C. 5/12 B. 2/3 D. 7/12 93. If Bill and Mary leave their houses at the same time, walking directly toward each other, each at their own constant rate, they will meet after 5 minutes. If Bill leaves 3 minutes later than Mary, they meet after he has walked for 3 minutes. How many minutes would it take him to walk all the way from his house to Mary’s? A. 12 C. 10 B. 15 D. 18 94. In triangle ABC, angle A is 120 degrees, BC + AB = 21, and BC + AC = 20. What is the length of BC? A. 11 C. 13 B. 12 D. 14 95. A flight of stairs has 9 steps. James can go 1 step or 2 steps at a time. The 4th step cannot be stepped since it is destroyed. How many ways are there for James to go up the stairs? A. 20 C. 21 B. 15 D. 13 96. Given that x and y are acute angles such that sin y = 3 cos(x + y) sin x. Find the maximum value of tan y. A. 4/3 C. 5/12 B. 3/4 D. 125 97. If x + 2y = 10, what is the minimum value of 𝑥 2 + 4𝑦 2? A. 50 C. 40 B. 100 D. 52 98. If tan A + tan B = 20 and cot A + cot B = 16, what is the numerical value of tan (A + B)? A. 40 C. 80 B. -40 D. -80 99. A, B, C and D are counting money in their savings. A has 18 pesos more than 9/7 of C. B has 108 pesos more than 2/3 of A. D has 20 pesos more than 9/8 of B. The total of their savings is 2018 pesos. How much does C have? A. 558 C. 390 B. 420 D. 650 100. What is the sum of all the roots of the equation 𝑥 3 − 3𝑥 + 1 = 0? A. 3 C. -3 B. 1 D. 0

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