Gabay Sinta 2025 - PUPCET Math Mock Exam PDF

PUP SENTRAL NA KONSEHO NG MAG-AARAL GABAY SINTA 2025 PUP STA. MESA CAMPUS PUPCET 2025 MOCK EXAM MATHEMATICS (ANSWER KEY AND SOLUTION) PREPARED BY: PUBLIC INFORMATION AND AFFAIRS COMMITTEE ACADEMIC AND RESEARCH COMMITTEE...

PUP SENTRAL NA KONSEHO NG MAG-AARAL GABAY SINTA 2025 PUP STA. MESA CAMPUS PUPCET 2025 MOCK EXAM MATHEMATICS (ANSWER KEY AND SOLUTION) PREPARED BY: PUBLIC INFORMATION AND AFFAIRS COMMITTEE ACADEMIC AND RESEARCH COMMITTEE VOLUNTEERS Question 1. Seph decided to invest Php 500,000 in a computer shop and a coffee shop. The computer shop and the coffee shop pay monthly interest of 5% and 3%, respectively. If he wants to earn a total of Php 18,000 monthly from the two businesses, how much must he allot to each business? A) Computer shop: Php 250,000, Laundry shop: Php 250,000 B) Computer shop: Php 150,000, Laundry shop: Php 350,000 ✔ C) Computer shop: Php 200,000, Laundry shop: Php 300,000 D) Computer shop: Php 350,000, Laundry shop: Php 150,000 Solution: Topic: Systems of Linear Equations (Grade 8) Let: x be the amount invested in the computer shop (at 5% monthly interest) y be the amount invested in the coffee shop (at 3% monthly interest) We can illustrate the word problem as: 1. x + y = 500,000 (total amount of money Seph invested) 2. 0.05x + 0.03y = 18,000 (total monthly interest earned from both investments) Step 1: Solve for y in the first equation: 𝑥 + 𝑦 = 500,000 𝑦 + 𝑥 − 𝑥 = 500,000 − 𝑥 (Subtraction Property of Equality, subtract both sides) 𝒚 = 𝟓𝟎𝟎, 𝟎𝟎𝟎 − 𝒙 Now we got our y, we can now substitute it to the second equation, which is the total monthly interest earned by Seph from both investments 0.05𝑥 + 0.03𝑦 = 18,000 0.05𝑥 + 0.03 (500,000 − 𝑥) = 18,000 0.05𝑥 + 15,000 − 0.03𝑥 = 18,000 Combine like terms. 0.02𝑥 + 15,000 = 18,000 Transpose 15,000 to the right side. (15,000 will become negative) 0.02𝑥 = 18,000 − 15,000 0.02𝑥 = 3,000 Divide both sides by 0.02, in order to get the value of x. 0.02𝑥 = 3,000 0.02𝑥 3,000 = 0.02 0.02 Gabay Sinta 2025 2 In cooperation with Public Information and Affairs Committee & Academic and Research Committee  𝒙 = 𝟏𝟓𝟎, 𝟎𝟎𝟎 total amount Seph invested in computer shop In order to get the value of y, use the first equation: 𝑥 + 𝑦 = 500,000 𝑥 + 𝑦 = 500,000 𝑦 = 500,000 − 𝑥 𝑦 = 500,000 − 150,000 (substitute the obtained x value) 𝒚 = 𝟑𝟓𝟎, 𝟎𝟎𝟎 total amount Seph invested in coffee shop Question 2. The fourth term of an arithmetic sequence is 2 and the seventeenth term is 7. What is the general term of the sequence? A. 𝑎𝑛 = 6n+2 / 13 B. 𝑎𝑛 = 5n + 3 /13 C. 𝑎𝑛 = 5n + 6 / 13 ✔ D. 𝑎𝑛 = 3n + 5 / 14 Solution Remember that the general term of an arithmetic sequence is given by: 𝑎𝑛 = 𝑎1 + (𝑛 − 1) 𝑑 Step I. Let us define the given facts into an equation: I. Fourth term of an arithmetic sequence is 2. 𝑎4 = 2, 𝑎1 = ? , 𝑛 = 4, 𝑑 = ? 𝑎4 = 𝑎1 + (4 − 1)𝑑 𝑎4 = 𝑎1 + 3𝑑 𝒂𝟏 + 𝟑𝒅 = 𝟐 (Eq. 1) II. Seventeenth term is 7. 𝑎17 = 7, 𝑎1 = ? , 𝑛 = 17, 𝑑 = ? 7 = 𝑎1 + (17 − 1) 𝑑 𝑎1 + 16𝑑 = 7 (Eq. 2) Step II. Solve for d, by subtracting the equation 1 from 2. (𝑎1 + 16𝑑) − (𝑎1 + 3𝑑) = 7 − 2 𝑎1 − 𝑎1 + 16𝑑 − 3𝑑 = 5 13𝑑 = 5 Gabay Sinta 2025 3 In cooperation with Public Information and Affairs Committee & Academic and Research Committee  13𝑑 5 = 13 13 (Divide both sides by 13) 𝟓 𝒅 = 𝟏𝟑 3. To solve for a1 (first term), use the first equation. 𝑎1 + 3𝑑 = 2 5 𝑎1 + 3( ) = 2 13 15 𝑎1 + = 2 13 15 𝑎1 = 2 − 13 26 15 𝑎1 = − 13 13 𝟏𝟏 𝒂𝟏 = 𝟏𝟑 Now we got all the missing values, we can now determine the general term of the sequence. 𝑎𝑛 = 𝑎1 + (𝑛 − 1) 𝑑 11 5 𝑎𝑛 = + (𝑛 − 1) ( ) 13 13 11 5 5 𝑎𝑛 = + ( 𝑛 − ) 13 13 13 11 + 5𝑛 − 5 𝑎𝑛 = 13 𝟓𝒏 + 𝟔 𝒂𝒏 = 𝟏𝟑 The answer is Letter C. Question 3. In the expansion of (2𝑥 + 𝑦)^8, what is the 5th term? A. 896𝑥 4 𝑦 4 B. 2560𝑥 4 𝑦 4 C. 𝟏𝟏𝟐𝟎𝒙𝟒 𝒚𝟒 ✔ D. 1280𝑥 4 𝑦 4 Solution: To find the fifth term of the binomial expansion (2x + y)^8, use the formula: 𝑛 𝑛 Tk+1 = ∑ ( ) (𝑎)𝑛−𝑘 𝑦 𝑘 𝑘 𝑘 Gabay Sinta 2025 4 In cooperation with Public Information and Affairs Committee & Academic and Research Committee In this time, we use k = 4 to get the fifth term since the binomial coefficient starts at 0. 4 8 T5 = ∑ ( ) (2𝑥)8−𝑘 𝑦 𝑘 𝑘 𝑘=4 4 8 T5 = ∑ ( ) (2𝑥 )8−4 𝑦 4 4 𝑘=4 4 8 T5 = ∑ ( ) (2𝑥)4𝑦 4 4 𝑘=4 Simplify. T5 = 70(24 𝑥 4 )(𝑦 4) T5 = 70(16𝑥 4 )(𝑦 4) 𝐓𝟓 = 𝟏𝟏𝟐𝟎𝒙𝟒 𝒚𝟒 Question 4. Which of the following statements is/are TRUE? I. Given the sequence 2, 6, 18, 54, 160,... is an geometric sequence. II. Given the sequence x, 2x + y, 3x+2y, …. Is an arithmetic sequence. III. Given the sequence 1, 5, 9, 13, 17, … the sum of the first 10 terms is 175. A. II only ✔ C. III only B. I and III only D. All of the above Answer: The answer is Letter A. II only since statements I and III shows false statements. Here’s why: In statement 1, the common ratio is 3. If we multiply 3 by its fourth term (54), the answer would be 162, not 160. Hence, the first statement is false. In statement 2, the common difference is x + y 𝑥 + (𝑥 + 𝑦) = 2𝑥 + 𝑦 2𝑥 + 𝑦 + (𝑥 + 𝑦) = 3𝑥 + 2𝑦 Since we calculated the same answer as above, the statement is TRUE.. In statement 3, we use the series formula. But first we get the common difference of the sequence, since it is an arithmetic sequence. D = 5 - 1 = 4 (common difference) Since we are finding the sum of the first 10 terms, the n will be 10. Given: d = 4, a1 = 1, n = 10 Gabay Sinta 2025 5 In cooperation with Public Information and Affairs Committee & Academic and Research Committee  𝑛 𝑆𝑛 = [2𝑎1 + (𝑛 − 1) 𝑑] 2 10 𝑆10 = [2(1) + (10 − 1) (4)] 2 𝑆10 = 5 [2 + 9(4)] 𝑆10 = 5 (2 + 36) 𝑆10 = 5 (38) 𝑆10 = 190 The answer is 190, and not 175. The answer is FALSE. In conclusion, the answer is Letter A, only statement II is correct. Question 5. In how many ways can the letters of the word PUPIAN be arranged? A. 360 ways ✔ C. 120 ways B. 720 ways D. 240 ways Solution: Topic: Permutation The word “PUPIAN” has six (6) letters: P, U, P, I, A, N Now, determine the letters of the word that occur multiple times. In this moment, the letter P occur twice; the rest of the letters occur once. P = 2, U = 1, I = 1, A = 1, N = 1 Determine the number of arrangements by means of this formula: n! / n1!n2!n3!... Solve. 6! 6! 720 = 2! = = 360 𝑤𝑎𝑦𝑠 2! 1! 1! 1! 1! 2 The answer is letter A, 360 ways. Question 6.) Mx. Cruz has PHP 70,000 to invest. They want to earn PHP 15,000 in interest. They are considering a savings and loans bank that is offering them 3.8% interest annually. For how long will Mx. Cruz has to leave their money in the bank to reach their goal of PHP 15,000? A. 5.5 years B. 6.2 years C. 5.64 years ✔ D. 7 years Solution: Gen-Math: Simple Interest Gabay Sinta 2025 6 In cooperation with Public Information and Affairs Committee & Academic and Research Committee Use the formula: I = PRT Determine the given facts first. P = PHP 70,000; R = 3.8% or 0.038 (in decimal); I = PHP 15,000; T = ? (unknown) To get the value of term (t), manipulate the formula of simple interest. I = PRT I / PR = PRT / PR (Divide both sides by PR) T = I / PR Substitute all the given facts. T = I / PR T = PHP 15,000 / (70,000) (0.038) T = PHP 15,000 / 2660 T = 5.64 years Therefore, Letter C is the answer. Question 7.) Determine the slope-intercept form of the equation of the line passing through the points (5, 4) and (6, 9). A. y = 5x - 19 B. y = 5x - 21 ✔ C. y = 4x + 1 D. y = 5x - 20 Solution: Linear Equation, Grade 8 Point A (x1, y1): (5, 4), Point B (x2, y2): (6, 9) Since m is equals to rise over run (rise/run) and y2 - y1 / x2 - x1 We can use two-point form in this: 𝑦2 − 𝑦1 𝑦 − 𝑦1 = ( ) (𝑥 − 𝑥1 ) 𝑥2 − 𝑥1 9 − 4 𝑦 − 4 = ( ) (𝑥 − 5) 6−5 5 𝑦 − 4 = 1 (𝑥 − 5) Simplify. 𝑦 − 4 = 5 (𝑥 − 5) Distribute. 𝑦 − 4 = 5𝑥 − 25 (Transpose, Addition Property of Equality) 𝑦 = 5𝑥 − 25 + 4 𝒚 = 𝟓𝒙 − 𝟐𝟏 Therefore, the answer is Letter C. Gabay Sinta 2025 7 In cooperation with Public Information and Affairs Committee & Academic and Research Committee 8.) Joshua decided to invest in a savings account that offers an annual interest rate of 6%. He deposited Php 14,300 into the account, and the interest is compounded monthly. How much money will Joshua have in the account after 4 years? A. Php 17,864.12 C. Php 18,243.45 B. Php 18,167.99 ✔ D. Php 18,023.90 Solution: Gen-Math: Compound Interest Determine first the given facts of the problem: R = 6% or 0.06, P = PHP 14,300; n = 12 (interest is compounded monthly), t = 4 years Find: A = ? Solution: A = 14,300 (1 + 0.06/12)^(12)(4) A = 14,300 (1 + 0.005)^48 A= 14,300 (1.005)^48 A = 18,167.99 The answer is Letter B. 9.) Find three consecutive odd integers such that the sum of the smallest and largest is 62, and the sum of all three integers is 93. A. 25, 27, 29 B. 27, 31, 35 C. 31, 29, 33 D. 29, 31, 33 ✔ Solution: Let x be the smallest odd Integer, x+2 Is the middle odd Integer, and x +4 as the largest odd Integer. 1. Determine and solve the first statement In the word problem: sum of the smallest and largest Integers is 62. 𝑥 + (𝑥 + 4) = 62 2𝑥 + 4 = 62 2𝑥 = 58 2𝑥 58 = 2 2 𝒙 = 𝟐𝟗 2. Get the remaining integers. Since the smallest integer is 29, then we can now substitute the value of x into the equation we determine earlier. Gabay Sinta 2025 8 In cooperation with Public Information and Affairs Committee & Academic and Research Committee Smallest odd integer: x = 29 Middle odd integer: x + 2 = 29 + 2 = 31 Largest odd integer: x + 4 = 29 + 4 = 33 Sum of all integers: 29 + 31 + 33 = 93 The answer is letter D. 10.) In a normal distribution, which of the following is true about the mean, median, and mode? A) The mean, median, and mode are all equal and located at the center of the distribution. ✔ B) The mean is greater than the median, which is greater than the mode. C) The mean is less than the median, which is less than the mode. D) The mean and median are equal, but the mode is located to the left of the mean. Explanation: In a normal distribution, the distribution is perfectly symmetric, meaning the mean, median, and mode all lie at the center. The mean is the average, the median is the middle value, and the mode is the value that occurs most frequently. Due to the symmetry, all three measures are equal and located at the center of the distribution, making statement A the only true one. 11. According to the pattern shown below, what is the number in the blank? 1, 2, 7, 34, 203, 1420, ___. a. 10354 b. 10359 c. 11354 d. 11359 ✔ SOLUTION: 1(2) − 1 = 1 1(3) − 1 = 2 2(4) − 1 = 7 7(5) − 1 = 34 34(6) − 1 = 203 203(7) − 1 = 1420 1420(8) − 1 = 11359 12. If 2𝑥 4 − 9𝑥 3 + 𝑚𝑛2 + 𝑛𝑥 + 54 = 0 is divisible by 𝑥 2 + 4𝑥 + 3, find m + n. a. 9 b. -23 ✔ c. -45 d. 67 SOLUTION: Gabay Sinta 2025 9 In cooperation with Public Information and Affairs Committee & Academic and Research Committee  𝑥 2 + 4𝑥 + 3 = (𝑥 + 1)(𝑥 + 3) 2(−1)4 − 9(−1)3 + 𝑚(−1)2 + 𝑛(−1) + 54 = 0 (when x = -1) 2 + 9 + 𝑚 − 𝑛 + 54 = 0 (n = m + 65) 4 3 2 2(−3) − 9(−3) + 𝑚(−3) + 𝑛(−3) + 54 = 0 (when x = -3) 459 + 9𝑚 − 3𝑛 = 0 459 + 9𝑚 = 3𝑛 but n = m + 65 So, 459 + 9𝑚 = 3(𝑚 + 65) 459 + 9𝑚 = 3𝑚 + 195; 𝑚 = −44; 𝑛 = −44 + 65; 𝑛 = 21 𝑚 + 𝑛 = −44 + 21 = −𝟐𝟑 13. Factorize: 4𝑥 2 − 4𝑦 2 + 40𝑥 − 28𝑦 + 51. a. (2x - 2y + 17) (2x + 2y - 3) b. (2x + 2y + 17) (2x - 2y + 3) ✔ c. (2x + 2y - 17) (2x + 2y - 3) d. (2x - 2y - 17) (2x - 2y + 3) SOLUTION: 4𝑥 2 − 4𝑦 2 + 40𝑥 − 28𝑦 + 51. = 4(𝑥 2 + 10𝑥 + 25) − 4𝑦 2 − 28𝑦 − (51) − 100 = 𝑥 2 + 40𝑥 + 100 − 4𝑦 2 − 28𝑦 − 49 = (2𝑥 + 10)2 − (2𝑦 + 7)2 = (2𝑥 + 10 + 2𝑦 + 7)[2𝑥 + 10 − (2𝑦 + 7)] = (2𝑥 + 10 + 2𝑦 + 7) (2𝑥 + 10 − 2𝑦 − 7) = (𝟐𝒙 + 𝟐𝒚 + 𝟏𝟕) ( 𝟐𝒙 − 𝟐𝒚 + 𝟑) 14. Find the sum of all positive factors of 756. a. 2240 ✔ b. 2475 c. 2970 d. 3225 SOLUTION: 756 = 22 (33 )(7) (1 + 2 + 22 ) (1 + 3 + 32 + 33 ) (1 + 7) = 7(40)(8) = 𝟐𝟐𝟒𝟎 15. If a straight line P passes through U(4, 65) and the slope of P is -5, find the x-intercept of P. a. 14 b. 15 c. 16 d. 17 ✔ Gabay Sinta 2025 10 In cooperation with Public Information and Affairs Committee & Academic and Research Committee SOLUTION: Let (x, 0) be the x-intercept. 0 − 65 −5 = 𝑥−4 0 − 65 − 5(𝑥 − 4) = (𝑥 − 4) 𝑥−4 −5𝑥 + 20 = −65 𝒙 = 𝟏𝟕 Hence, we have (17, 0) or 17. 16. If the lengths of the sides of a triangle are 9, 40, and 41, find the area of the triangle. a. 90 b. 180 ✔ c. 270 d. 360 SOLUTION: Note that the triangle is a right triangle since 92 + 402 = 412. Hence, the area is 1 𝐴 = (9)(40) = 𝟏𝟖𝟎 2 17. Jandrik goes 15 km west and then goes 36 km south. How far in km is he now from the original position? a. 32 b. 35 c. 37 d. 39 ✔ SOLUTION: √152 + 362 = 𝟑𝟗 18. Given a and b are real numbers and satisfy equations a + b = 15 and ab = 45, find the value of 𝑎2 + 𝑏2. a. 135 ✔ b. 140 c. 145 d. 150 Gabay Sinta 2025 11 In cooperation with Public Information and Affairs Committee & Academic and Research Committee SOLUTION: (𝑎 + 𝑏)2 = 𝑎2 + 2𝑎𝑏 + 𝑏2 (15)2 = 𝑎2 + 2(45) + 𝑏2 225 = 𝑎2 + 90 + 𝑏2 𝟏𝟑𝟓 = 𝑎2 + 𝑏2 19. Given ΔABC is an isosceles right-angled triangle (two sides are equal) and ∠ACB = 90°, if the distance from B to C is equal to or larger than 13 cm, find the minimum value of area of ΔABC. a. 71 sq. cm. b. 84.5 sq. cm. ✔ c. 96.5 sq. cm. d. 103 sq. cm. A SOLUTION: (13)(13) / 2 = 169 / 2 = 84 ½ or 84.5 ≥ 13 C B ≥ 13 20. There are 4 distinct Models A’s cars and 4 distinct Models B’s cars lining up in a row in a car show. Assume cars of the same model cannot be placed next to each other, how many different arrangement(s) is/are there? a. 1152 ✔ b. 1243 c. 1375 d. 1420 SOLUTION: 4! 4! (2) = 24(24)(2) = 𝟏𝟏𝟓𝟐 21. It is a line segment drawn from the center of the polygon perpendicular to a side. a. Radius b. Central Angle c. Altitude d. Apothem ✔ EXPLANATION: Radius - a straight line from the center to the circumference of a circle or sphere. Central Angle - is the angle formed at the center of a circle by any two radii. Altitude - (triangle) is the segment perpendicular from the base to the opposite vertex. Apothem - a line segment drawn from the center of the polygon perpendicular to a side. Gabay Sinta 2025 12 In cooperation with Public Information and Affairs Committee & Academic and Research Committee 22. Which of the following is not a Pythagorean Triple? a. 3-4-5 b. 7-21-22 ✔ c. 9-40-41 d. 11-60-61 EXPLANATION: Any odd number can be part of a Pythagorean triple. The other two sides just have to be consecutive numbers adding up to the square of the odd number. 3^2 is 9, then the next two numbers shall be consecutive numbers adding up to 9, which are 4 and 5. Therefore, 3-4-5 is a Pythagorean Triple. As for 7-21-22, it isn’t a Pythagorean Triple as 21 and 22 does not sum the square of 7, which is 49. The correct triple for this is 7-24-25. 23. A right triangle is formed on the second quadrant in which sin θ = 7/25. What is sec θ? a. -25/24 ✔ b. 1 c. 25/24 d. 0 SOLUTION: Since sin θ = 7/25, it means that opp/hyp = 7/25. The hypotenuse can be solved using Pythagorean Theorem. Hypotenuse = √(𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 )2 + (𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡)2 Substituting, 25 = √(7)2 + (𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡)2 Solving for the adjacent, we would get 24. We know that sec θ = 1/cos θ = hyp/adj. We computed that adjacent = 24, and the hypotenuse measures 25. Thus, sec θ = 25/24. However, it was stated in the problem that it lies on the second quadrant. Following the sign notations, sec θ is negative on Q2. Thus, sec θ = -25/24. 24. A piggy bank contains 35 coins consisting of 10-centavo and 25-centavo coins. If the value of the coins in the piggy bank is P5.90, how many of each kind are there? a. Eighteen 10-centavo coins and seventeen 25-centavo coins. b. Nineteen 10-centavo coins and sixteen 25-centavo coins. ✔ c. Twenty 10-centavo coins and fifteen 25-centavo coins. d. Twenty-one 10-centavo coins and fourteen 25-centavo coins. Gabay Sinta 2025 13 In cooperation with Public Information and Affairs Committee & Academic and Research Committee SOLUTION: Let x be the number of 10-centavo coins and 35 - x = the number of 25-centavo coins. The value of the total number of 10-centavo coins is (0.10)x while the value of the total number of 25-centavo coins is (0.25)(35-x). The total value of money in the piggy bank is therefore, (0.10)x + (0.25)(35-x) = 5.90 -> x = 19. Therefore, there are nineteen 10-centavo coins and sixteen 25-centavo coins. 25. The difference of the reciprocal of two positive numbers is ⅙, and one of the numbers is 3 times the other. Find the numbers. a. 4 and 12 ✔ b. 6 and 18 c. 12 and 36 d. 15 and 45 SOLUTION: Let x = the lesser number and 3x = the greater number. 1 1 Since 𝑥 > 0, 3𝑥 > 𝑥. Thus, 𝑥 𝑎𝑛𝑑 3𝑥 1 1 1 6𝑥( − ) = 6𝑥( ) 𝑥 3𝑥 6 [𝐺𝑟𝑒𝑎𝑡𝑒𝑟 𝑅𝑒𝑐𝑖𝑝𝑟𝑜𝑐𝑎𝑙] − [𝐿𝑒𝑠𝑠𝑒𝑟 𝑅𝑒𝑐𝑖𝑝𝑟𝑜𝑐𝑎𝑙] = 𝐷𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 6𝑥 2𝑥 − = 𝑥 𝑥 𝑥 6 − 2 = 𝑥 4 = 𝑥 Thus x = 4, and 3x = 12 Therefore, the numbers are 4 and 12. 26. One-half of Heather’s age two years from now plus one-third of her age three years ago is twenty years. How old is she now? a. 6 b. 12 c. 18 d. 24 ✔ SOLUTION: Age now: H Age two years from now: H + 2 Age three years ago: H - 3 Gabay Sinta 2025 14 In cooperation with Public Information and Affairs Committee & Academic and Research Committee  1 One-half of age two years from now: ( )(𝐻 + 2) = 𝐻/2 + 1 2 1 One-third of age three years ago: ( )(𝐻 − 3) = 𝐻/3 − 1 3 The sum of these two numbers is twenty, so we’ll add them and set this equal to 20. 𝐻 𝐻 + 1 + − 1 = 20 2 3 𝐻 𝐻 + = 20 2 3 𝐻 𝐻 (2)(3)( + ) = 20(2)(3) 2 3 3𝐻 + 2𝐻 = 120 5𝐻 = 120 𝐻 = 𝟐𝟒 27. How many kilograms of rice which sells at P10 per kilo must be mixed with rice which sells at P7.50 to make a 100-kilo mixture which sells at P8.50? a. 30 b. 40 ✔ c. 50 d. 60 SOLUTION: Let x = the number of kilos of P10 per kilo rice and 100 - x = the number of kilos of P7.50 per kilo rice. 10𝑥 + 7.5(100 − 𝑥) = 8.5(100) Therefore, x = 40. 28. Ji leaves home for the PUP Entrance Exam, riding his bicycle at a rate of 12 mph. Twenty minutes after he leaves, his mother sees Ji’s PUPCET Test Permit on his bed and leaves to bring it to him. If his mother drives at 36 mph, how far must she drive before she reaches Ji? a. 3 miles b. 4 miles c. 5 miles d. 6 miles ✔ Gabay Sinta 2025 15 In cooperation with Public Information and Affairs Committee & Academic and Research Committee SOLUTION: RATE TIME DISTANCE Ji 12 x 12x Mother 36 X-⅓ 36(x - ⅓) Notice that the 20 minutes have been changed to ⅓ of an hour. In this problem, the times are not equal, but the distances are. 12𝑥 = 36𝑥 − 12 1 12𝑥 = 36(𝑥 − ) 3 12 = 24𝑥 𝟏 𝒙 = 𝟐 Therefore, if Ji rode for ½ hour at 12 mph, the distance covered was 6 miles. 29. If dangerous fires are rare (1%) but smoke is fairly common (10%), and 90% of dangerous fires make smoke, determine the probability that a fire occurs when there is smoke. a. 0.09 ✔ b. 0.9 c. 9 d. 90 SOLUTION: [ 𝑃(𝑓𝑖𝑟𝑒)𝑥 (𝑃 (𝑠𝑚𝑜𝑘𝑒 | 𝑓𝑖𝑟𝑒)] 𝑃(𝑓𝑖𝑟𝑒 | 𝑠𝑚𝑜𝑘𝑒) = 𝑃 (𝑠𝑚𝑜𝑘𝑒) 0.01𝑥0.90 = 0.10 = 𝟎. 𝟎𝟗 30. Green’s Diner offers a meal special: 100-Php for a side item, juice, and pasta. For a side item, you can choose one of the following: regular fries, chicken nuggets, cheese fries, and hash brown. For juice, you can choose one of the following: orange juice, lemon juice, mango juice, and strawberry juice. For pasta, you can choose between carbonara or spaghetti. Find the number of possible meal combos. a. 24 b. 32 ✔ c. 40 d. 48 Gabay Sinta 2025 16 In cooperation with Public Information and Affairs Committee & Academic and Research Committee SOLUTION: Since there are four choices for a side item, four choices for a juice, and two choices for a pasta, then the number of possible meal combos can be determined as: (no. of side item choices)(no. of juice choices)(no. of pasta choices) = (4)(4)(2) = 32. Therefore, there are 32 possible meal combos. 31. Solve for x: 𝑥 3 − 6𝑥 2 + 11𝑥 − 6 = 0 a. 1, 2, 3 ✔ b. 2, 3, 4 c. 1, -2, 3 d. -1, -2, -3 SOLUTION: The polynomial can be factored using the Rational Root Theorem. The possible roots are ±1, ±2 ,±3, ±6. Testing these: 𝑥 = 1; (1)3 − 6(1)2 + 11(1) − 6 = 0. Where it confirms that x = 1 is a root. Factoring (x-1) from the polynomial, it becomes: 𝑥 − 6𝑥 + 11𝑥 − 6 = (𝑥 − 1)(𝑥 2 − 5𝑥 + 6) 3 2 Factoring (𝑥 2 − 5𝑥 + 6) and you will get: (𝒙 − 𝟐)(𝒙 − 𝟑) Therefore, the roots are x = 1, 2, 3 Gabay Sinta 2025 17 In cooperation with Public Information and Affairs Committee & Academic and Research Committee 32. Solve for x in 𝑙𝑜𝑔3 (𝑥 − 1) + 𝑙𝑜𝑔3 (𝑥 + 1) = 2. a. 1 b. √5 c. 3 d. √𝟏𝟎 ✔ SOLUTION: Using the logarithmic property 𝒍𝒐𝒈𝒃 (𝑨) + 𝒍𝒐𝒈𝒃 (𝑩) = 𝒍𝒐𝒈𝒃 (𝑨𝑩): log 3 ( (𝑥 − 1)(𝑥 + 1)) = 2 log 3 (𝑥 2 − 1) = 2 By rewriting, 𝑥 2 − 1 = 32 = 9 𝑥 2 = 10 ⇒ 𝑥 = ±√10 Since logarithms require positive arguments, x - 1 > 0, so x > 1. Thus, x = √10 33. Find the radius of the circle inscribed in a triangle with sides 13, 14, and 15. a. 5 b. 4 ✔ c. 7 d. 6 SOLUTION: The semi-perimeter of a triangle is the sum of all of its sides divided by 2. Following the formula of, (𝑎 + 𝑏 + 𝑐 ) 𝑠 = 2 Using this, 13 + 14 + 15 𝑠 = = 21. 2 Using the Heron’s Formula, we can find the area A of the triangle. 𝐴 = √𝑠(𝑠 − 𝑎)(𝑠 − 𝑏)(𝑠 − 𝑐) 𝐴 = √21(21 − 13)(21 − 14)(21 − 15) 𝐴 = √21(8)(7)(6) 𝐴 = √7056 𝑨 = 𝟖𝟒 The radius of the circle inscribed is the quotient of the Area of the triangle divided by its semi- perimeter. 𝐴 84 𝑟 = = = 𝟒 𝑠 21 Gabay Sinta 2025 18 In cooperation with Public Information and Affairs Committee & Academic and Research Committee  Thus, r = 4. 34. If 𝑧 = 2 + 3𝑖, 𝑓𝑖𝑛𝑑 |𝑧 2 |. a. 11 b. 12 c. 13 ✔ d. 14 SOLUTION: First, calculate 𝑧 2 : 𝑧 2 = (2 + 3𝑖 )2 = 4 + 12𝑖 + 9𝑖 2 = 4 + 12𝑖 − 9 = −𝟓 + 𝟏𝟐𝒊 The magnitude |z^2| is: |𝑧 2 | = √(−5)2 + 122 |𝑧 2 | = √25 + 144 |𝑧 2 | = √169 |𝑧 2 | = 13 Thus, |z^2| = 13 35. Find the number of diagonals in a regular icosagon. a. 20 b. 35 c. 135 d. 170 ✔ SOLUTION: The formula for the number of diagonals in a n-sided polygon is: 𝒏 (𝒏 − 𝟑) 𝑫 = 𝟐 For icosagon, 20(20 − 17) 𝐷= 2 20(17) 𝐷= 2 340 𝐷= 2 𝐷 = 170 Thus, D = 170. Gabay Sinta 2025 19 In cooperation with Public Information and Affairs Committee & Academic and Research Committee 36. If X follows a uniform distribution on [0,10], find 𝑃(3 < 𝑋 < 7). a. 0.2 b. 0.4 ✔ c. 0.6 d. 0.8 SOLUTION: The probability of a uniform random variable is proportional to the length of the interval: P(3