1000 Math Questions LET 1-78-221 PDF

1. How many line segments can be made from 30 5. Which of the following is ALWAYS true? non-collinear points? A. Vertical pairs of angles are supplementary. A. 900 B. 870 B. Vertical pairs of angles are complementary. C. 450...

1. How many line segments can be made from 30 5. Which of the following is ALWAYS true? non-collinear points? A. Vertical pairs of angles are supplementary. A. 900 B. 870 B. Vertical pairs of angles are complementary. C. 450 D. 435 C. Linear pairs of angles are congruent. D. Linear pairs of angles are supplementary. Solution: 30(29) 30C2 = 435; or = 435 Explanation: 2................................................ Linear pairs are supplementary, while vertical pairs are congruent. 2. Calculate the mean absolute deviation of the................................................ following numbers: 60, 80, 100, 75 and 95 A. 12.4 B. 14.2 6. The average of 5 different counting numbers is C. 16.1 D. 18.9 20. What is the highest possible value that one of the numbers can have? Solution: A. 20 B. 40 C. 30 D. 90 Mean = (60 + 80 + 100 + 75 + 95)/5 = 82 Solution: Mean absolute deviation daw, ibig sabihin, mean The 5 different counting numbers will assume or average ng absolute value ng x-𝑥̅. the values of 1, 2, 3, 4, and N. Since the average is MAD = (|60-82| + |80 – 82| + | 100 – 82| + 20, the sum is 5(20) or 100. |75 – 82| + |95 – 82|) / 5 = 62/5 = 12.4 1+2+3+4+N = 100................................................ 10 + N = 100 N = 90 3. Which of the following is the factorization of................................................ the binomial x2 - 42? A. (x + 4)(x + 2) 7. Three brothers inherited a cash amount of B. (x – 4)2 P62,000 and they divided it among themselves in C. x(x + 2x + 2) the ratio of 5:4:1. How much more is the largest D. (x – 4)(x + 4) share than the smallest share? A. P75,000 B. P30,000 Explanation: C. P24,800 D. P37,200 The factors of the difference of two squares are the sum and difference of their roots. Solution:................................................ Let the three numbers 5x, 4x, and x so that the ratio will still be 5:4:1. 4. What value of x will satisfy the equation: 0.4(5x – 1470) = x? 5x + 4x + x = 62000 A. 490 B. 2,130 10x = 62000; C. 1470 D. 588 x = 6200 Difference: 5x – x = 4x; 4x = 4(6200) = 24,800 Solution:................................................ 0.4(5x - 1470) = x 2x – 588 = x 8. What are the missing terms in the series 2x – x = 588 5, 20, 80, ,1280, , 20480? x = 588 A. 50; 210 B. 40; 160 C. 35; 135 D. 320; 5120 Solution: 12. The vertex angle of an isosceles triangle is 20°. What is the measure of one of the base Since the common ratio is 4, then next terms angles? should be 80(4) and 1280(4), or 320 and 5120. A. 150° B. 60° C. 75° D. 80°................................................ Solution: 9. At what rate per annum should P2400 be invested so that it will earn an interest of P800 in (180-20)/2 = 160/2 = 80 8 years?................................................ A. 6 ½ % B. 5 ½ % C. 4.17 % D. 6 % 13. Ana and Beth do a job together in three hours. Working alone, Ana does the job in 5 Solution: hours. How long will it take Beth to do the job alone? i=PRT A. 3 and 1/3 hours B. 2 and 1/3 hours 800 = 2400 x R x 8 C. 3 hours D. 7 and 1/2 hours 800 = 19200 R 0.0416666 = R Solution:................................................ 𝐴𝐵 Time to finish a job by working together = 𝐴+𝐵 10. The area of a rectangle is (x2 + 2x - 8). If its 𝐴𝐵 =3 where A = 5 length is x + 4, what is its width? 𝐴+𝐵 5𝐵 A. x + 2 B. x - 2 =3 5+𝐵 C. x + 1 D. x + 6 5B = 15 + 3B 2B = 15 Explanation: B = 7.5 Just factorize................................................................................................. 14. How much greater is the sum of the first 100 11. What is the value of 12¼ - 3 ⅜ - 5 ⅔ + 20 ¾? counting numbers than the sum of the first 50 1 7 A. 21 B. 22 C. 23 D. 21 counting numbers? 8 A. 110 B. 3,775 C. 3,155 D. 1200 Solution: LCD = 24 Solution: 4 9 16 18 𝑁 +𝑁 12 –3 –5 + 20 Sum of the first N counting numbers = 24 24 24 24 4 18 9 16 Sum of the first 100 counting numbers: = 12 + 20 –3 –5 24 24 24 24 (1002 + 100)/2 = 5050 22 25 = 32 –8 Sum of the first 50 counting numbers: 24 24 −3 −1 (502 + 50)/2 = 1275 = 24 or 24 24 8 5050 – 1275= 3775 =23 7................................................ 8................................................ 15. Which of the following has the largest value? A. 85 B. 39 C. 65 D. 94 Explanation: (just use your calculator) 19. Ruben’s grades in 6 subjects are 88, 90, 97, 85 = 32,768 39 =19,683 90, 91 and 86. What is the grade that he should 6 = 7,776 5 94 = 6,561 aim for in the 7th subject if he has to have an................................................ average of 91? A. 97 B. 95 C. 92 D. 89 16. A water tank contains 18 liters when it is 20% full. How many liters does it contain when Solution: 50% full? 91(7) – (88+90+97+90+91+86) = N A. 60 B. 30 C. 58 D. 45 637 – 542 = 95................................................ Solution: 18:20 = n : 50 20. On a certain day, three computer technicians 18(50) = 20n took turns in manning a 24-hour internet shop. 900 = 20n The number of hours Cesar, Bert, and Danny n = 45 were on duty was in the ratio 3:4:5, respectively................................................. The shop owner pays them P50 per hour. How much would Danny receive for that day? 17. The edges of a rectangular solid have these A. P 230 B. P500 measures: 1.5 feet by 1½ feet by 3 inches. What C. P160 D. P480 is its volume in cubic inches? A. 324 B. 225 C. 972 D. 27 Solution: Let their respective times be 3x, 4x, and 5x for a Solution: total of 24 hours. Convert the side measures from feet to inches 3x + 4x + 5x = 24 before proceeding with multiplication 12x = 24 1.5 ft = 1.5(12) or 18 in x=2 Vol = 18 (18) (3) = 972.: Danny works for 10 hours at P50/hr, or P500 for that day................................................................................................. 18. In a certain school, the ratio of boys to girls is 21. A retailer buys candies for P90.25. The pack 5 is to 7. If there are 180 boys and girls in the has 35 pieces of candies. If she sells each candy school, how many boys are there? for P3.25, how much profit does she make? A. 105 B. 90 A. P11.50 B. P23.50 C. 45 D. 75 C. P37.50 D. P18.75 Solution: Solution: Let 5x = boys Profit = 35(3.25) – 90.25 = 113.75 – 90.25 7x = girls Profit = 23.50 5x + 7x = 180................................................ 12x = 180 x = 15 22. Determine the midpoint of the line segment joining the points (7, -3) and (-1, 6). 5x = 5(15) = 75 A. (2, 3/2) B. (2, -3/2)................................................ C. (3, 3/2) D. (1, 5/2) Solution: 26. Which of these has the longest perimeter? x = (7+ -1)/2 = 3 A. A square 21 cm on a side y = (6 + -3)/2 = 3/2 B. A rectangle 19 cm long and 24 cm wide................................................ C. An equilateral triangle whose side is 28 cm D. A right triangle whose two legs are 24 and 32 23. One side of a 45° - 45° - 90° triangle cm measures x cm. What is the length of its hypotenuse? Solution: A. x √3 cm B. x cm A. P = 4S; 4(21) = 84 C. (x √3)/2 cm D. x √2 cm B. P = 2(L+W) 2(24+19) = 86 C. P = 3S 3(28) = 84 Explanation: D. P = L1 + L2 + H 24 + 32 + 40 = 96 In a 45-45-90 triangle, the hypotenuse is √2................................................ times of the leg................................................. 27. How many square inches are in 2 square yards? 24. The legs of one right triangle are 9 and 12, A. 900 B. 144 while those of another right triangle are 12 and C. 1296 D. 2,592 16. How much longer is the perimeter of the larger triangle than the perimeter of the smaller Solution: triangle? 1 yard = 3 feet = 3(12) or 36 inches A. 84 B. 7 C. 12 D. 14 1 square yard = 362 or 1296 square inches.: 2 square yards = 2(1296) = 2592 sq in Solution:................................................ Solve for the hypotenuse of the two triangles. 28. In a playground for Kindergarten kids, 18 The first one will have 15, while the other will children are riding tricycles or bicycles. If there have 20. are 43 wheels in all, how many tricycles are Get their respective perimeters. The first triangle there? has a perimeter of 9+12+15 or 36. The other A. 8 B. 9 C. 7 D. 11 triangle’s perimeter is 12+16+20 or 48. 48 – 36 = 12 Solution:................................................ T + B = 18  2T + 2B = 36 25. An online shop sells a certain calculator for 3T + 2B = 43  3T + 2B = 43 P950 and charges P150 for shipping within T=7 Manila, regardless of the number of calculators................................................ ordered. Which of the following equations shows the total cost (y) of an order as a function of the 29. Aira takes ¾ hour to dress and get ready for number of calculators ordered (x)? school. It takes 4/5 hour to reach the school. If A. y = (950 + 150)x B. y = 150x +950 her class starts promptly at 8:00 am; what is the C. x = 950y + 150 D. y = 950x + 150 latest time she can jump out of bed in order not to be late for school? Explanation: A. 6:42 am B. 6:27 am C. 6:57 am D. 7:02 am The cost of each calculator is P950, so x calculators cost P950x. Add the constant shipping cost which is P150 and that’s D. Solution: Then, solve algebraically. ¾ hr = 45 mins, while 4/5 hr = 48 mins (3x – 4) + (x – 4) = 36 45+48 = 93 mins, 93 mins = 1 hr 33 mins 4x – 8 = 36 8:00 7:60 4x = 44; x = 11; - 1:33  1:33 3x = 3(11) = 33 6:27.................................................: 1hr 33 mins before 8:00 AM is 6:27 AM................................................ 33. What is the least common multiple of 12, 24 and 72? 30. Which common fraction is equivalent to A. 12 B. 72 C. 144 D. 36 0.215? A. 43/200 B. 27/125 Explanation: C. 21/50 D. 108/375 Use continuous division. 12 24 72 Explanation: 6 2 4 12 0.215 is read as 215 thousandths. In fraction 2 1 2 6 215 43 form, that’s 1000. In simplest form, 200. 2 1 1 3 6 x 2 x 2 x 1 x 1 x 3 = 72 Alternative Method:................................................ Just use your calculator. 34. The hypotenuse of a right triangle is 25 feet................................................. If one leg is 24 feet, what is the length of the other leg? 31. What are the next three terms in the A. 6 ft. B. 5 ft. C. 20 ft. D. 7 ft. progression 1, 4, 16 …? A. 64, 256, 1024 B. 67, 259, 1027 Explanation: C. 48, 198, 1026 D. 65, 257, 1025 Use the Pythagorean Theorem. Explanation:................................................ Each term is 4 times its precedent. 35. If two variables X and Y are directly................................................ related, which of these is NOT true? A. When X is low, Y is also low. 32. A man is 3 times as old as his son now. Four B. As X increases, Y also increases. years ago, the sum of their ages was 36. Find the C. When X increases, Y decreases. man’s age now. D. A high Y is associated with a high X. A. 33 B. 11 C. 29 D. 36 Explanation: Solution: C refers to an inverse or indirect relation................................................. First, create a table. Age Now Age 4 Yrs Ago 𝑥2−4 Man 3x 3x-4 36. Find the domain of f(x) =. 𝑥+1 Son x x-4 A. x ∈ ℝ B. x = 1 C. x = -1 D. x ∈ ℝ, x ≠ -1 Explanation: 40. Factorize (x4 – 81) completely. A. (x-3)4 The given function is a rational algebraic B. (x – 3)2 (x + 3)2 expression (RAE). When facing RAE, just look at C. (x+3) (x-3) (x2+9) the denominator and see if it can be equated to 0 D. (x+3)3 (x-3) to make the RAE undefined. The RAE will have an undefined value at Solution: x = -1. Otherwise, it will always be equal to a real number. (x4 – 81) = (x2 – 9) (x2 + 9)................................................ (x4 – 81) = (x+3) (x-3) (x2 + 9)................................................ 37. A car travels D km in H hours. Which of the following expressions shows 41. √8 + √18 − √2 = the distance travelled by the car after M A. 4√2 B. 5√2 C. √24 D. 2√6 minutes? A. MD/H B. 60MD/H Solution: C. MD/60H D. 60HD/M √8 + √18 − √2 = 2√2 + 3√2 − √2 = 4√2 Solution:................................................ Distance = Speed x Time (the unit of time should 42. By which property can we be consistent) state the following: The car is traveling at a speed of D/H km per hr. “If ax + b = c, then ax + b - b = c – b.” The time is M minutes or M/60 hrs (for A. transposition B. transitive consistency). C. additive inverse D. addition property Distance = (D/H) (M/60) = MD/60H................................................ Explanation: 38. Find the surface area of a We added –b to both sides of the equation, thus rectangular box whose dimensions are 30 we used APE (addition property of equality). cm x 40 cm x 50 cm................................................. A. 4700 cm2 B. 7050 cm2 C. 9400 cm 2 D. 11750 cm2 43. The midpoint of P and (-7, 4) is (-3, 1). What are the coordinates of P? Solution: A. (-5, 5/2) B. (-11, 7) C. (1, -2) D. (-2, 3/2) SA = 2 (LW + WH + LH) SA = 2 (50x40 + 40x30 + 30x50) Solution: SA = 2 (2000 + 1200 + 1500) = 9400................................................ Let P be at (x,y). By Midpoint formula: (-7 + x)/2 = -3 (4 +y)/2 = 1 39. If x – y = 3, then (y-x)-3 =. -7 + x = -6 4+y=2 A. 9 B. -9 x = -6 + 7 y=2–4 C. 1/27 D. -1/27 x=1 y = -2................................................ Solution: 44. What is the slope of the line 3x – y = 11? Since x – y = 3, then y – x = -3. A. -1/3 B. 1/3 C. -3 D. 3 (-3)-3 = 1/(-3)3 = 1/-27 or -1/27................................................ Solution: 48. ∠A and ∠B form a vertical pair. If m∠A = 3x and m∠B = 5x – 44, what is the Isolate y on one side of the equation to rewrite value of x? the equation in the form y = mx + b. A. 50.5 B. 28 C. 22 D. 16.75 3x – y = 11 -y = -3x + 11 Solution: y = 3x – 11................................................ Since the two angles form a vertical pair, then they are congruent. 45. What is the minimum 3x = 5x – 44 value of f(x) = 3x2 + 6x + 7? 44 = 5x – 3x A.1 B. -1 C. 4 D. -4 44 = 2x; 22 = x................................................ Solution: 49. The angle of elevation from an Min Value = c – b2/4a observer to the top of a building is 30o. If That’s 7 – 36/12 or 7-3=4 the building is 50 meters high, how far is................................................ the observer from the building? 46. If xy = 23 and x2 + y2 = 75, find x + y. A. 25 B. 25√3 C. 50√3 D. 100 A. 10.7845 B. 11 C. 11.2155 D. 11.7845 Solution: Use a 30-60-90 triangle. The side opposite of the Solution: 30o angle will represent the building. x2 + 2xy +y2 = x2 + y2 + 2xy................................................ x2 + 2xy +y2 = 75 + 2(23) x2 + 2xy +y2 = 121 50. ∠1 and ∠3 are opposite angles x + y = 11 in a parallelogram. If m∠1 = 40o, what................................................ is m∠3? A. 40o B. 50o C. 70o D. 140o 47. How much water must be evaporated from 90 ml of a 50% salt Explanation: solution to increase its concentration to Opposite angles of a parallelogram are 75%? congruent. A. 40 ml B. 38 ml................................................ C. 35 ml D. 30 ml 51. Two parallel lines are cut by a Solution: transversal, forming ∠H and ∠K. If the two V1 C1 + V2 C2 = VR CR angles are exterior angles on the same Since we are evaporating water, we will be side of the adding a NEGATIVE volume of water (or simply transversal, what is the measure of ∠H if the put, we are subtracting water, diba?) measure of ∠K is 50o? 90(50) + (-X)(0) = (90-X)(75) A. 25o B. 50o 4500 + 0 = 6750 – 75X C. 100 o D. 130o 75X = 6750 – 4500 75X = 2250; X = 30 Explanation:................................................ Exterior angles on the same side of the transversal are supplementary. *Mnemonic: SST (same side of transversal) means supplementary. ALTERNATE (either interior or exterior) means congruent. Also, CORRESPONDING angles are congruent.  52. There are 33 red bags, 25 green The nearest multiple of 7 to 125 is 126. That bags, and 17 blue bags in a store. What means 126 days after today is Saturday again, percent of the bags is red? and 125 days after today should be Friday. A. 33% B. 44%................................................ C. 66% D. 67% 56. Car A is traveling towards the east at Solution: a speed of 35 kph, while car B is traveling towards the west at 45 kph. If they left the 33/(33+25+17) = 33/75 or 11/25 same point at 1:00 PM, how far apart are they 11/25 in percent is 44% at 3:45 PM?................................................ A. 240 km B. 220 km C. 200 km D. 180 km 53. Given sin θ = 0.28, which of the following could possibly be cos θ? Solution: A. 0.72 B. -0.86 C. 0.96 D. 1.14 Time spent driving: 1:00 to 3:45 = 2.75 hrs (45 mins in decimals is 45/60 since there are 60 Solution: mins in 1 hr) Car A distance from mid: 2.75 (35) = 96.25 sin2 θ + cos2 θ = 1 Car B distance from mid: 2.75 (45) = 123.75 (0.28)2 + cos2 θ = 1 Total distance: 123.75 + 96.25 = 220 km cos2 θ = 1 – 0.0784 cos2 θ = 0.9216 Alternative Solution: cos θ = √0.9216 = ±0.96................................................ Since the two cars are traveling in two opposite directions, add their speeds and multiply by 54. If the sum of the supplement and elapsed time. the complement of an angle is 130 degrees, 2.75 (45+35) = 2.75 (80) = 220 km what is the angle?................................................ A. 65o B. 70o C. 50o D. 25o 57. Mr. Santos left the house at 1:00 PM and traveled east at an average speed of 40 Solution: kph. His wife Mrs. Santos left the at 2:00 PM and traveled west at an average speed of 30 (90-x) + (180-x) = 130 kph. How far apart are they at 4:00 PM? 270 – 2x = 130 A. 180 km B. 140 km 270 – 130 = 2x C. 100 km D. 60 km 140 = 2x 70 = x Solution:................................................ Mr. Santos’s data: 55. If today is a Saturday, what day is Speed: 40 kph 125 days from now? Elapsed time: 1PM to 4PM = 3 hrs A. Thursday B. Friday Distance: 40kph (3hrs) = 120 km C. Sunday D. Monday Mrs. Santos’s data: Solution: Speed: 30 kph Elapsed time: 2PM to 4PM = 2 hrs Every 7 days, it would be a Saturday again. Distance: 30 kph (2hrs) = 60 km Total Distance: 60 + 120 = 180 km  2+2𝑥 58. Five consecutive even numbers have a sum of 61. Given f(x) = ln 𝑒𝑥 , what is f ‘(x)? 120. What is the sum of the 2nd and 5th even 2𝑥+2 𝑥2+2𝑥 A. 𝑥2+2𝑥 B. 2𝑥+2 numbers? A. 46 B. 48 C. 50 D. 52 C. (2x+2) ln (x2+2x) D. 2x + 2 Solution: Solution: 2 You can rewrite ln 𝑒𝑥 +2𝑥 as x2 + 2x since ln is Let x = lowest even number the natural logarithm (the logarithm whose base x + (x+2) + (x+4) + (x+6) + (x+8) = 120 is the natural number, e). 5x + 20 = 120 5x = 100 Remember: ln eu = u, wherein u is the exponent x = 20 to which e is being raised..: numbers are 20, 22, 24, 26, 28 The derivative of x2 + 2x is, of course, 2x + 2................................................. 22 + 28 = 50 62. Which of the following could be the value of x Alternative Solution: if x ≅ 3(mod 11)? The middle (3rd) even number is 120/5 or 24. A. 33 B. 47 C. 52 D. 2 That means the 2nd even number is 24 – 2 or 22, and the 5th is 24 + 2(2) or 28. Solution:................................................ Just divide the numbers by 11 and see which one gives a remainder of 3. 59. If x = 3, which of the following is equal to 13?................................................ A. 5x + 2 B. x2 + 2x + 1 C. x – 4x – 2 3 D. x2 + x + 2 𝑑𝑢 63. If 𝑑𝑥 = 6x2 + 8x – 7, which could be u? Explanation: A. 12x + 8 B. 3x3 + 4x2 – 7x + 11 C. 2x3 + 4x2 -7x +1 D. 12x2 + 8x - 10 Just substitute x with 3................................................. Explanation: 60. If f(x) = x2 + 4x + 3, which of the Anti-derivatives. If you already forgot how to do following is equal to 99? that, simply check which choice has a derivative A. f(11) B. f(-12) of 6x2 + 8x – 7. C. f(12) D. f(-8)................................................ 64. What is the center of x2 + y2 – 8x + 6y = 0? Solution: A. (-8.6) B. (8, -6) x2 + 4x + 3 = 99 C. (-4, 3) D. (4, -3) x2 + 4x + 3 + 1 = 99 + 1 x2 + 4x + 4 = 100 Solution: √(𝑥 + 2)2 = 100 The center, C(h,k) is given as h = -D/2 and x + 2 = ± 10 k = -E/2 wherein D and E are from the equation x = -2 ± 10 x2 + y2 + Dx + Ey + F = 0. That’s -2+10 or 8, and -2-10 or -12 65. Which of the following is a parabola that Explanation: opens to the right? Since A and B are the roots, then AB pertains to A. 6y = (x+9)2 - 8 B. -4y = (x-6)2 + 3 the product of the roots which is given as c/a. C. -5x + 3 = (y-2)2 D. 2x + 6 = (y+3)2................................................ Explanation: 69. 1 + 2 + 4 + 8 + … + 2048 = When x is the squared variable, the parabola A. 4095 B. 4096 opens upward when the coefficient of y is C. 4097 D. 4098 positive (example: A). When x is the squared variable, the parabola Solution: opens downward when the coefficient of y is You may use the Geometric Series formula which negative (example: B). is ∑𝑛 1−𝑟𝑛 When y is the squared variable, the parabola 𝑖=1 𝑎𝑖 = 𝑎 1 ( ), where r is the common 1−𝑟 opens to the left when the coefficient of x is ratio, n is the number of terms, and a1 is the first negative (example: C). term. When y is the squared variable, the parabola opens to the right when the coefficient of x is Alternative Solution: positive (example: D). In this problem, however, you cannot easily use................................................ the GS formula since you don’t know n, the number of terms. 66. Factorize: 12x2 – 7x – 10. I will personally use the shortcut for the sum of a A. (6x + 5) (2x – 2) B. (6x – 2) (2x + 5) geometric sequence wherein the ratio is 2 or ½. C. (3x + 2) (4x – 5) D. (3x – 2) (4x + 5) The shortcut is SUM = 2(largest) – smallest. In................................................ this problem, that’s 2(2048)-1 = 4095. You may also apply this in the next item, #70. 67. For which value of k does 4x2 + kx + 49 have only one root? 70. 24 + 12 + 6 + 3 + 1.5 + … = A. -28 B. -14 C. 7/2 D. -7/4 A. 48 B. 50 C. 54 D. 60 Explanation: Solution: You may use Completing Square Trinomials. The You may use the Infinite Geometric Series middle term is twice the product of the square 𝑎 𝑖=1 𝑎𝑖 = ( 1−𝑟 ), where r is the 1 roots of the first and third terms. In the problem, formula which is ∑∞ the middle term is twice the product of √4𝑥2 and common ratio and a1 is the first term. √49. That’s 2(2x)(7) or 28x. Don’t forget that the middle term could be positive or negative. Alternative Solution: In this problem, I would still be using the You may also use the discriminant to answer shortcut since the ratio is ½. Since this is an this: infinite geometric sequence, then the last term b2 – 4ac = 0 when there’s only one root, won’t have any significant value. Thus, the sum is b2 – 4ac > 0 when there are two real roots simply twice the first term. That’s 2(24) = 48. b2 – 4ac < 0 when there are no real roots 68. If A and B are the roots of x2 + 7x + 15, what is AB? A. 7√3 + 2 B. 2√3 + 7 C. 3√2 + 2√3 D. 15 71. How many terms are there in the sequence 74. C is the midpoint of ̅A̅B̅where A is at (-3,4) 5, 13, 21, 29, …, 357? and B is at (7,-10). Find the coordinates of C. A. 40 B. 44 A. (5,-7) B. (-5,7) C. (2,-3) D. (-2,3) C. 45 D. 70 Solution: Solution: 𝑥 +𝑥 𝑦1+𝑦2 Midpoint Formula: ( 1 2 , ) 2 2 An = A1 + (n – 1)d −3+7 4−10 Midpoint: ( , ) 357 = 5 + (n – 1)(8) 2 2 357-5 = 8(n – 1)................................................ 352 =8(n – 1) 44 = n – 1 75. It is a line segment formed by connecting two 45 = n non-consecutive vertices of a polygon. A. side B. apothem Alternative Solution: C. altitude D. diagonal (this is the “y=mx+b” solution I taught my grade 3 student for Singapore. Yes, Grade 3.) Explanation: Before anything else, since this might be “new” A side is formed by connecting two consecutive to you, your d is our m, your An is our y, your n is vertices of a polygon. our x, and b is your A1 – d. The apothem is only for regular polygons. It is 357 = 8x +(5 – 8) the perpendicular bisector of one of its sides, 357 = 8x – 3 passing through the center. 360 = 8x A diagonal is a line segment formed by 45 = x connecting two non-consecutive vertices of a................................................ polygon................................................. 72. How many ways can a group of 5 be selected from 5 boys and 5 girls if the group must contain 76. Find the equation of the line perpendicular to 3 boys and 2 girls? 2x – 3y = 7, passing through (1,2). A. 151,200 B. 1200 A. 2x + 3y = 8 B. 3x + 2y = 7 C. 252 D. 100 C. 2x – 3y = -4 D. 3x – 2y = -1 Solution: Solution: A group, committee, or team (any set with no Simply interchange the numerical coefficients of hierarchy of members) calls for Combinations. x and y in the original equation, then change the To pick 3 boys from a total of 5 boys, use 5C3 operation between them. 2x – 3y becomes and that’s 10. To pick 2 girls from a total of 5 3x + 2y. girls, use 5C2 and that’s 10. Lastly, 10x10 = 100. For the constant, simply substitute the................................................ x and y values of the point ((1 ,2) in this problem) and solve for the constant. 73. What is the probability of getting a sum of 9 3(1) + 2(2)=7. when rolling 2 dice? Thus, 3x+2y=7. A. 1/9 B. 5/36 C. 1/6 D. 7/36 The only pairs with a sum of 9 are (3,6), (4,5), (5,4), and (6,3). There are only 4 pairs out of 36. 77. Two parallel lines are cut by a transversal to Solution: form ∠X, ∠Y, and ∠Z. Given that ∠X and ∠Y are Usually, people would alternate interior angles while ∠Y and ∠Z are straight go for the Test interior angles on the same side of the Point Table method which transversal, find m∠Z if m∠X = 40o. we use in Calculus. A. 40o B. 50o However, since this is the C. 130 o D. 140o licensure exam, I’d prefer that you use a simpler and Explanation: quicker approach to this Alternate, corresponding, and vertical pairs problem. automatically suggest that the two angles are First, identify the zeros of congruent. Linear pairs and angles on the same the inequality by equating side of transversal (SST) are supplementary. each factor to 0. Our zeros m∠X = 40,.: m∠Y = 40 since alternate interior are -9 and 3. angles Next, identify the opening m∠Z = 180-40 = 140 since ∠Y and ∠Z are of the parabola. Since the interior angles on the same side of the leading coefficient would transversal. be positive, then the................................................ parabola opens upwards. Since the parabola opens upwards, then the 78. The measure of each interior angle of a parts less than 0 should be between the zeros of regular polygon is 144o. How many vertices does the inequality. That means x should be between it have? -9 and 3. A. 36 B. 24 C. 12 D. 10................................................ Solution: 80. The product of two consecutive even counting numbers is 3248. Find the smaller MIA = number. A. 42 B. 46 C. 52 D. 56 Alternative Solution: Solution: Personally, I always go for the exterior angle first to get the number of sides or vertices. Since the x(x+2) = 3248 exterior and interior are supplementary, then x2 + 2x = 3248 each exterior measures 180-144 or 36. The x2 + 2x + 1 = 3249 formula for number of sides or vertices given the √𝑥2 + 2𝑥 + 1 = √3249 measure of each exterior is 360÷MEA, so that’s x + 1 = 57 360÷36 or 10 vertices. x = 56 By the way, you may derive this solution by 180(𝑛−2) WAIS Solution: manipulating the formula for MIA: 𝑛. That 360 360 becomes 180 –. That means = 180 – MIA, Get your scientific calculator, extract √3248 and 𝑛 𝑛 360 then scrape the decimals or round down. or = n. 180−MIA #2EZ4U................................................................................................ 79. Solve: (x + 9) (x – 3) < 0 81. Solve for x: 2log2 3 – log2 18 = x A. -9 < x < 3 B. x < -3 ∪ x > 9 A. ½ B. -1 C. -2 D. 1 C. x < -9 ∪ x > 3 D. x ∈ ℝ; x ≠ -9, 3 Solution: surface area. Just remember that the slant height is always longer than the height. The slant height Rewrite the logarithm as a single logarithm by is the hypotenuse, while the height is one of the applying the rules of logarithms. 32 legs with the radius as the other. Just use the 2log2 3 becomes log2 , or log 2 ½ Pythagorean formula to solve for whichever is 18 log2 ½ = -1 missing................................................. The height is 12 cm (after using Pythagorean formula). 1 1 82. Twinkle Bucks has four serving sizes for their Vol = π r2 h = π (92) (12) = 324 π cm2 3 milk tea: Small, Medium, Large, and Extra Large................................................. What level of data are they using for their serving sizes? 86. If f(x) = x2 + 4x + 4 and g(x) = x-2, find A. nominal B. ordinal f(g(x)). C. interval D. ratio A. x2 B. x3 – 6x2 + 6x – 9................................................ C. x2 + 8x + 16 D. x2 – 8x + 16 83. After receiving a 20% markup, a bag was sold Solution: for P960. How much was it originally? A. P1152 B. P4800 f(g(x)) = f(x-2) C. P800 D. P1200 = (x-2)2 + 4(x-2) + 4 = (x2 – 4x + 4)+ (4x – 8) + 4 = x2 Solution:................................................ Selling Price = Original Price (1 + Markup Rate) 87. A 10 ft ladder leans against a wall, forming a 960 = OP (1 + 0.20) 30o angle with it. How high on the wall does it 960/1.2 = OP reach? 800 = OP................................................ A. 5 ft B. 5 √3 ft C. 10 √3 ft D. 10 √6 ft 84. Given ̅B̅T̅bisects ∠ABC and m∠ABT = 40o, Solution: find m∠ABC. A. 20o B. 40o C. 60o D. 80o Draw the problem first. The ladder and the wall form a 30o angle with each other and the wall is Explanation: of course perpendicular to the ground. ∠ABT is formed after the bisection of ∠ABC. That That means the ladder forms a 60o angle with the means ∠ABT is half of ∠ABC, or ∠ABC is twice of ground. The ladder is the hypotenuse, while its ∠ABT. reach on the wall is adjacent to the 30o angle or................................................ simply put, the longer side. The smaller side measures half of 10 or 5 ft, therefore the longer 85. A cone has a radius of 9 cm and a slant height side must be 5√3 ft. of 15 cm. Find its volume................................................. A. 243 π cm3 B. 324 π cm3 C. 405 π cm 3 D. 486 π cm3 88. How many ways can a committee of 5 be selected from 9 people? Explanation: A. 126 B. 120 C. 3024 D. 15120 Be careful with cones. Tendency kasi sa LET that they will give the slant height while looking for volume and the height while looking for the Explanation: 93. How many 3-digit numbers can be formed using the digits 0, 1, 2, 3, 4 and 5 if repetition is Just use your scientific calculator: 9C5. not allowed?................................................ A. 60 B. 80 C. 100 D. 120 89. What is 60% of 80% of 500? Solution: A. 480 B. 240 C. 120 D. 60 Use FCP (Fundamental Counting Principle): x x Solution: For the first digit, we cannot use 0. That means we only have 5 choices for the first digit. (0.6)(0.8)(500) = 240 For the second digit, we can now use 0. Since we................................................ have already used one digit for the first, that means we have 5 choices for the second digit. 90. If 3x = 7 and 2y = 5, what is 6(x-y)? For the last digit, since we have already used two A. -1 B. 1-√35 digits, we only have 4 choices. 47 C. √7 - √5 D. 5 x 5 x 4 = 100 5................................................ Solution: 94. How many ml of 20% acid must be added to 6(x-y) = 6x – 6y = 2(3x) – 3(2y) 400 ml of 50% acid to make a 30% acid solution? 2(3x) – 3(2y) = 2(7) – 3(5) = 14-15 = -1 A. 1000 ml B. 900 ml................................................ C. 800 ml D. 750 ml 91. If two numbers have a product of 71 and the Solution: sum of their squares is 147, what is their sum? A. -17 B. 5 C1 V1 + C2 V2 = CR VR C. 12√3 + √5 D. 12 + √3 20 (V) + 50 (400) = 30 (V + 400) 20V + 20,000 = 30V + 12,000 Solution: 20,000 – 12,000 = 30V – 20 V 8,000 = 10V; 800 = V Let A and B be our two numbers................................................. AB = 71; A2 + B2 = 147.: A2 + B2 + 2AB = 147 + 2(71) = 289 95. How many ml each of 10% and 50% solution (A + B)2 = 289; A + B = ±17 should be mixed to make 500 ml of 18%................................................ solution? A. 400 ml of 10% and 100 ml of 50% 92. Find the median: 7, 9, 11, 10, 9, 13, 17, 14 B. 350 ml of 10% and 150 ml of 50% A. 10 and 11 B. 9 and 10 C. 300 ml of 10% and 200 ml of 50% C. 10.5 D. 9.5 D. 200 ml of 10% and 300 ml of 50% Solution: Solution: Rearrange the numbers from least to greatest: Since our resultant volume is 500, then our two 7, 9, 9, 10, 11, 13, 14, 17  there are 8 nos. volumes will be x and (500-x). 8+1 The median is the 2 th or 4.5th number. That C1 V1 + C2 V2 = CR VR means we have to get half the sum of our 4th and 10(x) + 50(500-x) = 18(500) 5th numbers. (10+11)/2 = 10.5 10x + 25,000 – 50x = 9,000 25,000 – 9,000 = 50x – 10x 16,000 = 40x; 400 = x 96. It takes 28 men a total of 24 days to build a A rectangle’s diagonals are congruent and they house. How long would it take 32 men to build a bisect each other. However, they are not similar house? perpendicular. A. 28 days B. 27 3 days An isosceles trapezoid has congruent diagonals, 7 2 however, they do not bisect each other, nor are C. 21 days D. 19 days 7 they perpendicular. A rhombus has diagonals that are perpendicular Solution: and that bisect each other. However, they are not This is an indirect or inverse proportion. congruent. Let x = number of days it would take the 32 men PS: A square has diagonals that are congruent, to build the house perpendicular, and that bisect each other. 28(24) = 32 x................................................ 672 = 32 x 21 = x 100. A pipe can fill a pool in 6 hours while................................................ another pipe can drain empty the pool in 15 hours. How long will it take to fill the pool if both 𝑥2−16 pipes are open? 97. Evaluate: lim𝑥→4 𝑥−4 A. 9 hours B. 9.125 hours A. undefined B. limit does not exist C. 9.45 hours D. 10 hours C.8 D. +∞ Solution: Explanation: This is similar to our “Working Together” You may simplify the function first before problem, except instead of adding their times, we substituting x with 4. will subtract (since the draining pipe is doing the................................................ opposite of helping). AB/(A-B) = 15(6)/(15-6) = 90/9 = 10 hrs 98. A box contains 7 red, 8 blue, and 9 white................................................ balls. When taking two balls in succession, what is the probability that both balls are white? 101. If log n – 1 = 2, find n. A. 9/64 B. 9/69 A.3 B. 1000 C. e3 D. 3e C. 7/64 D. 7/69 Solution: Solution: log n – 1 = 2 First white ball: 9/24 log n = 2 + 1 = 3 Second white ball: 8/23 (note that the base of the log is 10) 9/24 x 8/23 = 9/69 log n = 3 translates to 103 = n................................................ Therefore n = 1000................................................ 99. Which of the following has two diagonals that are perpendicular bisectors of each other? 102. log2 3 + 2 log2 7 – log2 5 =. 42 9 A. kite B. rectangle A. log2 B. log2 5 5 C. rhombus D. isosceles trapezoid 147 142 C. log2 D. log2 5 Explanation: Explanation: A kite’s diagonals are perpendicular but only one diagonal will bisect the other. Just apply the laws of logarithms. 103. The surface areas of two spheres are 12 π 106. Find the volume of a steel cylinder of radius cm2 and 108 π cm2. What is the ratio of their 5 cm and height 12 cm. volumes? A. 300 π cm3 B. 250 π cm3 A. 1:3√3 B. 1:9 C. 200 π cm3 D. 100 π cm3 C. 1:27 D. 2:3√3 Solution: Solution: Vol = π r2 h = 52 (12) π = 300 π cm3 Ratio of surface areas: 12:108 or 1:9................................................ Ratio of radii: √1: √9 or 1:3 Ratio of volumes: 13:33 or 1:27 107. A cube sits perfectly inside a sphere of................................................ volume 108 √3 π cm3. Find the volume of the cube. 104. The volume of a regular hexahedron is 64 A. 27 cm3 B. 54 cm3 in3. How long is each side? C. 108 cm3 D. 216 cm3 A. 2 in B. 4 in C. 6 in D. 8 in Solution: Explanation: Volume of sphere = 108√3 π cm3 A regular hexahedron is simply a cube. 4 π r3 = 108√3 π 3................................................ 3 r3 = (108)√3 4 105. Which of the following statements is r3 = 81√3; r = 3√3; d = 6√3 ALWAYS true? Diagonal of cube = s√3 = 6√3 A. The square of a prime number is odd..: s = 6; volume = s3 = 63 = 216 B. The sum of two consecutive even numbers is divisible by 4. Alternative Solution: C. Any even number is composite. Ratio of volume of cube to sphere (cube is inside D. The product of two consecutive even numbers is divisible by 8. sphere) = 2 : √3 π N : 108 √3 π = 2 : √3 π Explanation: (108√3)(2) N= = 216 √3 A. Counterexample: The prime number, 2. The................................................ square of 2 is 4 which is even. B. Always false. One example is 2 and 4. Their 108. Find the distance in cm of an 80 cm chord sum, 6, is not divisible by 4. from the center of a circle whose radius is 41 cm. C. Counterexample: The prime even number, 2. A. 41 - 2√10 B. 41 - 4√10 D. Proof by Algebra: Let the first even number be C. 9√2 D. 9 2x. The second even number will be 2x + 2. Their product will be 4x2 + 4x. Solution: If x is an odd number, x = 2y + 1 where y is a counting number. 4x2 + 4x = 4(4y2 + 4y + 1) + The chord is perpendicularly bisected by a 4(2y + 1) = 16y2 + 16y + 4 + 8y + 4 = 16y2 + segment connected to the center of the circle, 24y + 8, which is divisible by 8. whose length is the distance we are looking for. If x is even, x = 2y where y is any counting If the radius is drawn connected to one endpoint number. 4(4y2) + 4(2y) = 16y2 + 8y, which is of the chord, we can form a right triangle whose also divisible by 8. hypotenuse is the radius and one leg is half of the Either ways, the statement holds true. chord. Using the Pythagorean theorem, the distance is √412 − 402 or simply, 9. 109. Which quadrilateral has two congruent 113. Find the intersection of y = 2x + 3 and diagonals that bisect each other? y = 4x – 11. A. kite B. isosceles trapezoid A. (-4/3, 0) B. (4/3, 0) C. rectangle D. rhombus C. (7, 17) D. (-7,-17)................................................ Solution: 110. What is the longest side of ∆MTC if m∠M = y = 2x + 3 40o and m∠C = 60o? - y = 4x – 11 A. ̅M ̅C B. ̅T̅C C. ̅M ̅T̅ D. ̅C̅T̅ 0 = -2x + 14 2x = 14; x=7 Explanation:................................................ m∠T = 180-(40+60) = 80 The longest side is opposite the largest angle, ∠T. 114. Find the area of the triangle whose vertices................................................ are (1,4), (2,3), and (3,0). A. 0 B. 1 C. 5/3 D. 3/4 111. Find the altitude to the hypotenuse of a right triangle whose legs measure 10 cm and 24 Solution: cm. 1 1 2 3 1 1 A. 120 cm B. 120 cm | | | = (−2) = −1 2 4 3 0 4 2 13 C. 120√2 cm D. 24√5 cm Note: If the result is negative, that means your points are simply arranged clockwise. Just get Solution: the absolute value of the answer. Find the hypotenuse first. That would be 26 cm................................................. Altitude to the Hyp = (L1 L2)/Hyp = 24(10)/26 Altitude to the Hyp = 120/13 115. Find the tenth term: 3, 10, 17, 24, …................................................ A. 66 B. 67 C. 68 D. 69 112. Find the inverse of y = x2 + 10x. Solution: A. y-1 = √𝑥 − 25 + 5 A10 = 3 + (10-1) (7) = 3 + 63 = 66 B. y-1 = √𝑥 − 25 – 5................................................ C. y-1 = √𝑥 + 25 + 5 116. Find the remainder when D. y-1 = √𝑥 + 25 – 5 x4 – 5x3 + 6x2 + 2x + 1 is divided by (x – 2). Solution: A. 17 B. 13 C. 9 D. 5 y = x2 + 10x Solution: y + 25 = x2 + 10x + 25 y + 25 = (x+5)2 24 – 5(23) + 6(22) + 2(2) + 1 = 16 – 40 + 24 + 4 + 1 = 5 √𝑦 + 25 = x + 5................................................ √𝑦 + 25 – 5 = x √𝑥 + 25 – 5 = y-1 117. The sum of Fe’s age and Sita’s age is 60................................................. Twelve years ago, Fe was twice as old as Sita. How old is Sita now? A. 18 B. 24 C. 30 D. 36 Solution: By transitive property of equality, 8 8x = 15z, or x = z Age Now Age 12 yrs ago 15 Fe x x – 12................................................ Sita 60-x (60 – x) – 12 or 48 – x 121. Victor had an average of 94 on his first four x – 12 = (2)(48 – x) Math tests. After taking the next test, his average x – 12 = 96 – 2x dropped to 93. Find his most recent grade. x + 2x = 96 + 12 A. 88 B. 89 C. 90 D. 91 x + 2x = 108 x = 36 Solution: 60 – x = 60 – 36 = 24 New Score =(New Number)(New Average) –................................................ (Old Number)(Old Average) New Score = 5(93) – 4(94) = 465 – 376 = 89 118. If the length of a rectangle is increased by................................................ 20% while the width is decreased by 10%, what will happen to its area? 122. X is 3 of Y and Y is 56 of Z. What part of Z is X? A. B. decreased increased byby10% 10% 4 3 5 A. X = Z B. X = Z 5 C. increased by 8% 5 C. X = Z8 D. X = Z D. decreased by 2% 5 3 Solution: Solution: 3 (L x 1.2) (W x 0.9) = (1.08 x LW) X= Y 4................................................ 3 5 X = (5 Z) = 15 Z or Z 4 6 24 8 119. The 19th term of an arithmetic sequence is................................................ 85 and the 12th term is 43. Find the common difference. 123. Two buses leave the same station at 8:00 A. 5 B. 6 C. 7 D. 8 pm. One bus travels north at the rate of 30 kph and the other travels east at 40 kph. How many Solution: kilometers apart are the buses at 10 pm? 𝐴𝑚−𝐴𝑛 85−43 42 A. 140 km B. 100 km d= = = =6 C. 70 km D. 50 km 𝑚−𝑛 19−12 7 Solution: 120. If 2x = 3y and 4y = 5z, what is z in terms of From 8 to 10 PM is 2 hours. After two hours, one x? bus will have travelled 60 km while the other, 80 6 A. z = x B. z = 15 x km. Since the two buses are traveling on 5 8 5 D. z = 8 x perpendicular directions, we can use the C. z = 6 x Pythagorean Theorem to find their distance. Solution: D = √602 + 802 = 100 km................................................ Make two equations wherein y will have the same numerical coefficients. 124. A bus drove for 6 hours at 75 kph and 4 2x = 3y  8x = 12y hours at 80 kph. What was its average speed? 4y = 5z  12y = 15z A. 76 kph B. 77 kph C. 77.5 kph D. 78 kph Solution: 128. Mr. Tondo has P100,000 to invest, from which he wants to earn P5600 per year. Bank A Get the total distance and the total time first. offers 5% per annum while Bank B offers 6%. 6 hrs x 75 kph = 450 km How much should he invest at Bank B? 4 hrs x 80 kph = 320 km Total distance = 770 km, total time = 10 hrs A. P45,000 B. P50,000 𝑡𝑜𝑡𝑎𝑙 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 770 𝑘𝑚 C. P55,000 D. P60,000 Average spd = 𝑡𝑜𝑡𝑎𝑙 𝑡𝑖𝑚𝑒 = 10 ℎ𝑟𝑠 = 77 kph................................................ Solution: Let x = investment in Bank B 125. 18 students failed a quiz. They represent.: 100,000 – x = investment in Bank A 30% of the class. How many students passed the 0.05(100,000 – x) + 0.06x = 5,600 quiz? (5,000 – 0.05x) + 0.06x = 5,600 A. 60 B. 42 C. 36 D. 24 0.01x = 600; x = 60,000................................................ Solution: 𝑥+𝑦 5 18:30% = N:70% 129. Evaluate when x = ¾ and y =. 18(70)/30 = N 𝑥−𝑦 6 42 = N A. -38 B. -19 C. 19 D. 38................................................ Solution: 2 126. Rationalize: 3 5 18+20 38 3 5 18−20 −2 + = = ; − = = √5+2 4 6 24 24 4 6 24 24 2√5 38 −2 38 24 A. +1 B. 2√5 – 4 ÷ = x = −19 5 24 24 24 −2 C. 2√5+4 D. 2√5................................................ 9 3 130. Today, Vic is 11 years old while his father is Solution: 37. How many years from now will his father be To rationalize this, multiply both numerator and twice as old as he? denominator by the conjugate of the A. 15 B. 13 C. 11 D. 10 denominator. By doing this, we are sure to have a rational denominator. Solution: 2 √5+2 x √5−2 √5−2 = 2√5−4 2 2 = 2√5−4 1 Let x = number of years from now √5 −2 2(11+x) = 37+x................................................ 22 + 2x = 37 + x 2x – x = 37 – 22; 15 = x 127. RNHS has 130 quizzers. 67 of them are................................................ Math, 60 are Science, and 20 are quizzers for both Math and Science. How many quizzers are 131. Carla and Diana are on a seesaw. Carla neither Math nor Science? weighs 50 kg and sits 168 cm to the left of the A. 0 B. 13 C. 17 D. 23 fulcrum. If Diana weighs 60 kg, how far to the Solution: right of the fulcrum must she sit to balance the seesaw? (A∪B)’ = U - (A∪B) A. 140 cm B. 170.8 cm (A∪B)’ = 130 – [A + B – (A∩B)] C. 201.6 cm D. 210 cm (A∪B)’ = 130 – (67 + 60 – 20) = 130 – 107 = 23 Solution: 136. Find the largest area of a rectangle whose perimeter is 100 cm. Seesaw problems call for inverse or indirect A. 2500 cm2 B. 2499 cm2 proportion. C. 625 cm2 D. 624 cm2 50(168) = 60N 8400 = 60N Solution: 140 = N................................................ Instead of jumping to differential calculus (minima and maxima) to solve this, simply make 132. Twenty guests shake hands with each other. it a square. That’s the shortcut for this kind of If each guest is to shake hands with all the other question. guests, how many handshakes will be made?................................................ A. 400 B. 380 C. 200 D. 190 137. What time is 200 minutes past 10:30 PM? Solution: A. 12:30 AM B. 12:30 PM C. 1:50 AM D. 1:50 PM 20C2 = 190................................................ Solution: 133. How many line segments can be made from 200 minutes = 3 hrs 20 mins 30 non-collinear points? 10:30 PM A. 900 B. 870 C. 450 D. 435 + 3:20 13:50 PM  1:50 AM Solution:................................................ 30C2 = 435 138. Find the product of two numbers whose................................................ GCF is 24 and LCM is 120. A. 2880 B. 1440 C. 720 D. 360 134. The longest chord of a circle is 80 cm. How long is its radius? Explanation: A. 20 cm B. 30 cm C. 20√2 cm D. 40 cm The product of two numbers is equal to the product of their LCM and GCF. Explanation: 24 x 120 = 2880................................................ The longest chord is the diameter, and the radius is half the diameter. 139. The salary of 4 men for 5 days is P9,000................................................. How much is the salary of 5 men for 6 days? A. P12,000 B. P12,600 135. Find k such that 34k67 is divisible by 9. C. P13,500 D. P14,400 A. 5 B. 6 C. 7 D. 8 Solution: Solution: First, find the cost of each “man-day”. Remember that for a number to be divisible by 9, 4 men x 5 days = 20 man-days the sum of its digits must be equal to 9. P9,000 ÷ 20 man-days = P450 per man-day 3+4+k+6+7 = 20+k 2+0+k = 2 + k = 9; k=7 You may now solve the problem................................................. 5 men x 6 days = 30 man-days 30 man-days x P450 per man-day = P13,500 140. The average grade of eleven students is 83. 144. The average of x+5, 2x-4, and x+7 is 20. If the average of six of these students is 88, what Find x. is the average of the other 5 students? A. 18 B. 13 C. 9 D. 8 A. 77 B. 78 C. 79 D. 80 Solution: Solution: (x+5) + (2x−4) + (x+7) 3 = 20 Sum of grades of 11 students: 11 x 83 = 913 (x + 5) + (2x – 4) + (x + 7) = 60 Sum of grades of 6 students: 6 x 88 = 528 4x + 8 = 60 Sum of grades of other five: 913 – 528 = 385 4x = 52 Average of grades of other five: 385 ÷ 5 = 77 x = 13................................................................................................ 141. If x is 80% of y, what percent of y is x? 145. Mia is 16 years younger than Kia. 13 years A. 120% B. 125% ago, Kia was thrice as old as Mia. What is Kia’s C. 130% D. 135% present age? A. 43 B. 40 C. 37 D. 34 Solution: x = 0.8 y Solution: x 0.8 =y  1÷0.8 = 1.25 Age today Age 13 years ago 1.25x = y Kia x x – 13................................................ Mia x – 16 x – 16 – 13 or x – 29 (x – 13) = 3(x – 29) 142. Bus X left the terminal at 1 PM and traveled x – 13 = 3x – 87 at a speed of 60 kph. Bus Y left the same terminal -13 + 87= 3x - x 2 hours later and traveled 80 kph on the same 74 = 2x route. What time will Bus B catch up with Bus A? 37 = x A. 6 PM B. 9 PM................................................ C. 11 PM D. 1 AM 146. Insert one term between 18 and 32 to make Solution: a geometric sequence. Let x = running time for Bus X A. 20 B. 24 C. 25 D. 27 60x = 80(x-2)  Bus Y left 2 hrs later Solution: 60x = 80x – 160 160 = 20x; Shortcut for inserting one term is √AB. This is 8=x also the formula for the geometric mean. 8 hours after Bus X left the terminal is 9AM. √18(32) = √576 = 24................................................................................................ 143. What is the degree of the polynomial 147. There are 100 pigs and chickens in a farm, -3 x2y3 + 21 x3y4 – 7 x5y6 – 15? all of which are healthy. If there are 340 legs in A.4 B. 5 C. 11 D. 21 total, how many pigs are there? A. 70 B. 65 Explanation: C. 60 D. 55 The degree of a polynomial is the highest sum of exponents in a term. Solution: 150. Solve for x: 49x = 343 A. 1.142857 B. 7 Let P = number of pigs; C = number of chickens C. 1.5 D. √7 P + C = 100 Solution: 4P + 2C = 340  since pigs have four legs and chickens have two First, express both numbers as powers of the 2(P + C = 100)  2P + 2C = 200 same base. 4P + 2C = 340 - 4P + 2C = 340 49x = 343  (72)x = 73 -2P = - 140 Next, apply the laws of exponents. P = 70 (72)x = 73................................................ 72x = 73 2x = 3; 148. Adam can do a job alone in 8 hours, while x = 3/2 or 1.5 Bam can do the same job in 12 hours. One day,................................................ they worked together for 1 hour before Bam left Adam to finish the job. How long will it take 151. What is the highest possible product of two Adam to finish the remaining job? numbers if their sum is 45? A. 6 hrs 50 mins B. 6 hrs 40 mins A. 506 B. 506.25 C. 6 hrs 30 mins D. 6 hrs 20 mins C. 506.5 D. 506.725 Solution: Solution: 𝐴𝐵−𝑇(𝐴+𝐵) Instead of jumping straight to minima and 8(12)−1(8+12) 96−20 76 4 hrs maxima under differential calculus, simply make = = or 6 your numbers equal to maximize their product. 12 12 12 12 That’s 6 hrs. and 20 mins. 45/2 = 22.5 Both numbers will be 22.5, so their product is Mnemonic: 22.5 x 22.5 = 506.25................................................ For questions like this (about working together and then someone leaves), use PuTS U. When 152. Which statistical test is used for comparing someone leaves you, “PuTS U” !! observed frequencies to expected frequencies? PuTS U stands for P(u)roduct, Time, Sum, Umalis A. ANOVA B. t-test................................................ C. Pearson R D. Chi Square 149. Find x if 2748 = 9x. Explanation: A. 144 B. 81 C. 72 D. 60 Observed vs Expected: Chi Square Solution: Relationship: Pearson R (R for relationship) Group differences: ANOVA (variance = Express both sides as a power of 3. differences) (33)48 = (32)x Comparing sets of normal distributions: T-test 3144 = 32x................................................ 144 = 2x 72 = x 153. The product of two consecutive odd................................................ counting numbers is 1443. What is their sum? A. 76 B. 78 C. 80 D. 82  1 1 Solution: 157. Find + given x + y = 20 and xy = 81. 𝑥 𝑦 81 40 81 20 Let x = first number; x+2 = next number A. 40 B. 81 C. 20 D. x(x+2) = 1443 x2 + 2x = 1443 Solution: x2 + 2x + 1 = 1443 + 1 1 1 𝑦 𝑥 𝑥+𝑦 20 √𝑥2 + 2x + 1 = √1444 +𝑦 = 𝑥 𝑥𝑦 + 𝑥𝑦 = 𝑥𝑦 = 81 x+1 = 38; x = 37 x+2 = 39................................................................................................ 158. What is the remainder when 2𝑥 + 1; 𝑥 < 4 534,214,557,989,215 is divided by 4? 154. Given 𝑓(𝑥) = { 4; 𝑥 = 4, A. 0 B. 1 C. 2 D. 3 𝑥 − 7; 𝑥 > 4 2 find lim𝑥→4 𝑓(𝑥). Explanation: A.4 B. 9 C. 0 D. limit does not exist The divisibility rule for 4 tells us that our concern would only be the last 2 digits. Solution: 15 ÷ 4 = 3 r. 3 Limit from the left: 2(4) + 1 = 9................................................ Limit from the right: 42 – 7 = 9 Since both limits are equal, then the limit is 9. 159. Dividing by 0.125 is the same as multiplying................................................ by which number? A.5 B. 8 C. 10 D. 16 155. If today is a Saturday, what day is 125 days from now? Explanation: A. Friday B. Sunday Just use 1 as your test number. C. Monday D. Tuesday 1÷ 0.125 = 8 Solution:................................................ This is an application of modulo. 160. Find the surface area of a sphere whose 125 ≅ 6 (mod 7) or 125÷7 = 17 r. 6 radius is 6 cm. 6 days after Saturday is Friday A. 72 π cm2 B. 108 π cm2................................................ C. 144 π cm2 D. 192 π cm2 156. If the sum of the supplement and the Solution: complement of an angle is 124, what is the angle? Surface Area = 4 π r2 = 4 (62) π = 144 π A. 71 B. 72 C. 73 D. 74 161. Which of the following is the reference angle of 216o? Solution: A. 84o B. 66o C. 54o D. 36o (180 – x) + (90 – x) = 124 Explanation: 270 – 2x = 124 270 – 124 = 2x The reference angle for angles from the different 146 = 2x; 73 = x quadrants are as follows:................................................ QI: the angle θ itself QII. 180 – θ QIII. θ – 180 QIV. 360 – θ 162. Which of the following angles in standard 165. Find the equation of the line passing position is coterminal with 40o? through the point of origin and (3,4). 4 3 A. 2200o B. 1760o A. y = x B. y= x C. 1520o D. 1360o 4 3 7 C. y = x + D. y = x + 1 4 4 Explanation: Solution: Coterminal angles are congruent, modulo 360. 𝑦 −𝑦 y – y1 = 2 1 (x − x ); That means they will leave the same remainder (0,0) and (3,4) 𝑥2−𝑥1 1 when divided by 360. 4−0 In textbooks, θ is coterminal with any angle y–0= (x − 0) 3−0 4 expressed as 360N + θ wherein N is an integer. y= x 3 To easily tackle this question, simply subtract 40................................................ from each of the choices, then see if any of those is divisible by 360 (or leaves a remainder of 0 166. Find the range of f(x) = -2x2 + 4x. when divided by 360) using your calculator. A. y ≤ 2 B. y ≥ 2 2200 – 40 = 2160 C. y ≤ -2 D. y ≥ -2 2160 ÷ 360 = 6.: 2200o is coterminal with 40o Explanation:................................................ Since this is a quadratic function, you need to know two things to determine its range: its 163. Find the equation of the line passing opening and k of its vertex (h,k). through (2,7) and (-3,-3). A. y = 4x -1 B. y = 3x + 1 The parabola opens downward since a = -2. C. y = 3x + 6 D. y = 2x + 3 k = c – (b2/4a) = -16/(-8) = 2 Since the parabola opens downward, the graph Solution: starts from -∞, going to k which is 2. Two-point form of linear equations: Thus, y ≤ 2. y – y1 = (x − x )................................................ −3−7 y–7= (x − 2) −3−2 167. If a3/2 – 1 = 7, what is a? y – 7 = 2(x – 2) A.4 B. 8 C. 9 D. 18 y – 7 = 2x – 4 y = 2x + 3 Solution:................................................ a3/2 – 1 = 7 a3/2 = 8 164. In which quadrant can we find θ if tan θ < 0 (a3/2 = 23)2/3 and sin θ > 0? a = 22 = 4 A. First Quadrant................................................ B. Second Quadrant C. Third Quadrant 168. Which of the following is true? D. Fourth Quadrant A. A rectangle is a square. B. A rhombus is a rectangle. Explanation: C. A trapezoid is a rhombus. Use the CAST D. A square is a rhombus. mnemonic. 169. What is the measure of each exterior angle Solution: of a pentagon? P250 = discount A. 108o B. 72o 10% = discount rate C. 60o D. 36o Original Price (OP) = ??? Selling Price = ??? Solution: DC = OP x DC Rate MEA = 360/N = 360/5 = 72o................................................ 250 = OP x (0.1) 250/0.1 = OP 170. How many diagonals does a nonagon have? 2500 = OP A. 27 B. 36 C. 45 D. 54 Selling Price = OP – DC Selling Price = 2500 – 250 = P2250 Solution:................................................ Diagonals = N(N-3)/2 = 9(6)/2 = 27................................................ 173. A book was sold for P270 after a 10% discount was given. How much was the book 171. What is the fractional equivalent of originally? 0.123123123123…? A. P330 B. P300 123 123 41 41 A. 1001 B. 1000 C. D. C. P297 D. P280 321 Solution: Algebraic Solution: SP = OP (1-DC Rate) Let x = 0.123123123123… 270 = OP (0.9) 1000x = 123.123123123123… 270/0.9 = OP 1000x – x = 123 300 = OP 999x = 123................................................ x = 123/999 x = 41/333 174. Find the area of an equilateral triangle whose sides measure 12 cm each. Alternative Solution: A. 36√3 cm2 B. 48√3 cm2 Write a fraction whose numerator is the C. 60√3 cm2 D. 72√3 cm2 repeated digits (123) and whose denominator has the same number of digits but is made of 9s Solution: (123 is 3-digit, so use 999). 𝑠2√3 144√3 Area EqTri = = = 36√3 Thus, 123/999 or 41/333. 4 4................................................................................................ 172. Mrs. Pasay saved P250 after buying a phone 175. This is located at the intersection of the with a 10% discount. How much did she pay for angle bisectors of a triangle. the phone? A. Incenter A. P2500 B. P2250 B. Circumcenter C. P2000