SEPTEMBER 2024 LICENSURE (LET) MATH MAJORS DIAGNOSTIC TEST PDF

SEPTEMBER 2024 LICENSURE EXAMINATION FOR TEACHERS (LET) REVIEW FOR MATH MAJORS Diagnostic Test (200 Items) KIM JAY C. ENCIO, Ph.D. Discussant DIRECTIONS: Shade the letter of the correct answer before the number. If the correct answer is not found...

SEPTEMBER 2024 LICENSURE EXAMINATION FOR TEACHERS (LET) REVIEW FOR MATH MAJORS Diagnostic Test (200 Items) KIM JAY C. ENCIO, Ph.D. Discussant DIRECTIONS: Shade the letter of the correct answer before the number. If the correct answer is not found among the choices, shade E. Item #1 He is considered the founder of trigonometry, but is most famous for his incidental discovery of the precession of the equinoxes. Who is this person? a. Hipparchus c. Euclid b. Pythagoras d. Thales Item #1 b. Pythagoras - famous for the Pythagorean Theorem c. Euclid - associated with “The Elements” d. Thales - one of the 7 Sages of Greece Item #1 Hipparchus was a Greek astronomer, geographer, and mathematician. He is considered the founder of trigonometry, but is most famous for his incidental discovery of the precession of the equinoxes. Hipparchus was born in Nicaea, Bithynia, and probably died on the island of Rhodes, Greece. Answer: A Item #2 What do you call symbols that represent numbers? a. ordinals c. numerals b. scribbles d. cardinals Item #2 Numerals are symbols that represent numbers. Answer: C Item #3 The earliest form of numbers were tally marks found on prehistoric a. bones c. mud b. temples d. paper Item #3 The earliest form of numbers were tally marks found on prehistoric bones and artifacts. Answer: A Item #4 Which of the following is the Filipino term for “Mathematics”? a. Logpaulit c. Sukgisan b. Sipnayan d. Tumbasan Item #4 Which of the following is the Filipino term for “Mathematics”? a. Logpaulit (Logarithm) b. Sipnayan c. Sukgisan (Geometry) d. Tumbasan (Equation) Answer: B Item #5 Which of the following is “setenta y nueve”? a. 69 c. 89 b. 79 d. 97 Item #5 Which of the following is “setenta y nueve”? a. 69 (sesenta y nueve) b. 79 c. 89 (ochenta y nueve) d. 97 (noventa y siete) Answer: B Item #6 This mathematician is famous for the equation a2 + b2 = c2 where a, b, and c are the sides of a right triangle and c is the hypotenuse. a. Anaxagoras b. Euclid c. Plato d. Pythagoras Item #6 a. Anaxagoras - Greek philosopher of nature remembered for his cosmology and for his discovery of the true cause of eclipses b. Euclid - famous for his “The Elements” c. Plato - What is Plato known for? Plato's most famous work is the Republic, which details a wise society run by a philosopher. Item #6 In antiquity, Pythagoras was credited with many mathematical and scientific discoveries, including the Pythagorean theorem. Answer: D Item #7 Which of the following is one of the contributions of Archimedes? a. pioneered the study of graph theory b. the equation V – E + F = 2 c. the most beautiful equation 𝑒 + 1 = 0 𝑖𝜋 d. the principle of buoyancy related to fluids Item #7 Which of the following is one of the contributions of Archimedes? a. pioneered the study of graph theory b. the equation V – E + F = 2 c. the most beautiful equation 𝑒 +1=0 𝑖𝜋 d. the principle of buoyancy related to fluids Answer: D Item #8 Who was the first computer programmer in the world? a. Ada Lovelace b. Hypatia c. Valentina Tereshkova d. Sophie Germain Item #8 Who was the first computer programmer in the world? a. Ada Lovelace b. Hypatia - She is one of the earliest known female mathematicians who invented the astrolabe for ship navigation and devices for measuring the density of fluids. c. Valentina Tereshkova - She is known for being the first and youngest woman in space, having flown a solo mission on the Vostok 6 on 16 June 1963. d. Sophie Germain - She discovered this identity in her explorations related to Fermat's last theorem and primality testing. Answer: A Item #9 If x is an integer, which of the following is always even? a. x 2 c. 2x + 1 b. x 3 d. x (x + 1) Item #9 x2 and3 x are odd when x is odd 2x + 1 is always odd For x (x + 1) If x is even, say 2, then 2(3) = 6 (even) If x is odd, say 3, then 3(4) = 12 (even) x (x + 1) is EVEN when x is an integer. Answer: D Item #10 There are 10 people in the market. How many handshakes will there be in all if each one will shake hands with each other exactly once? a. 45 c. 60 b. 55 d. 70 Item #10 You have an arithmetic series! Number of Number of Pattern For 10 people, People Handshakes there are 1 0 0 0+1+2+3+…+9 𝑛 2 1 0+1 = (a1 + an) 2 10 3 3 0+1+2 = (0 + 9) 2 = 5(9) 4 6 0+1+2+3 = 45 handshakes Answer: A Item #11 If 12 = 1, 112 = 121, and 1112 = 2 12,321, what is 11,111 ? a. 1,234,321 b. 123,454,321 c. 12,345,654,321 d. 1,234,567,654,321 Item #11 1 2 =1 2 11 = 121 2 111 = 12,321 2 What is 11,111 ? 2 11,111 = 123,454,321 Answer: B Item #12 What is the8 th number in the sequence 1, 3, 2, 6, 5, 15, … ? a. 42 c. 44 b. 43 d. 45 Item #12 × 3 −1 × 3 −1 × 3 −1 ×3 1, 3, 2, 6, 5, 15,14, 42 Answer: A Item #13 A learner is still trying to figure out a formula that works on the problem. What stage of the investigation process is he in? a. deriving c. summarizing b. proving d. verifying Item #13 A learner is still trying to figure out a formula that works on the problem. What stage of the investigation process is he in? a. deriving b. proving - formula is true within its specified domain c. summarizing - stating the main points of the investigation d. verifying - checking if the conjecture is true for some cases Answer: A Item #14 A learner restates the salient points of his mathematical investigation process. What is he doing? a. deriving c. summarizing b. proving d. verifying Item #14 A learner restates the salient points of his mathematical investigation process. What is he doing? a. deriving - finding formula/conjecture b. proving - formula is true within its specified domain c. summarizing - stating the main points of the investigation d. verifying - checking if the conjecture is true for some cases Answer: C Item #15 Which of the following refers to the exploration of an open-ended situation? a. Nonroutine Problem Solving b. Mathematical Problem Solving c. Arithmetic Juxtaposition d. Mathematics Investigation Item #15 A mathematics investigation allows students to examine situations using various techniques and in the process of their exploration, develop skills that can be applied to other problems. Answer: D Item #16 To prove p→q, a mathematician assumed ¬q (not q) and arrived to ¬p. What do you call this type of proof? a. direct b. proof by contrapositive c. proof by contradiction d. mathematical induction Item #16 In mathematics, proof by contrapositive, or proof by contraposition, is a rule of inference used in proofs, where one infers a conditional statement from its contrapositive. In other words, the conclusion "if A, then B" is inferred by constructing a proof of the claim "if not B, then not A" instead. Answer: B Item #17 1 𝑏+3 If 𝑎 = 𝑏 + = , what is 2 2 the value of 𝑎? 1 3 a. c. 2 4 1 5 b. d. 3 2 Item #17 2𝑏 = 4 1 𝑏+3 𝑎=𝑏+ = 2 2 𝑏=2 2𝑏 + 1 𝑏 + 3 2+3 = 𝑎= 2 2 2 𝟓 4𝑏 + 2 = 2𝑏 + 6 𝑎= 𝟐 Answer: D Item #18 Which of the following 0.25 is equivalent to 𝑥 ? 4 a. 𝑥 c. 𝑥 5 2 5 b. 𝑥 d. 𝑥 Item #18 0.25 𝑥 1 =𝑥 4 𝟒 = 𝒙 Answer: C Item #19 50 If 𝑥 = 100, what is 25 the positive value of 𝑥 ? a. 5 c. 25 b. 10 d. 50 Item #19 50 𝑥 = 100 1 1 50 𝑥 2 = 100 2 25 𝑥 = 100 25 𝑥 = 𝟏𝟎 Answer: B Item #20 𝑦 If x = and 5x = 7, 5 What is y? a. 7 c. 28 b. 2 d. 14 Item #20 𝑦 If x = and 5x = 7, What is y? 5 𝑦 =x 5 y = 5x If y = 5x and 5x = 7, then by transitive property of equality, we could say that y = 7. Answer: A Item #21 Twenty eight students passed while twenty percent failed in a departmental exam. How many students were tested? a. 38 c. 44 b. 35 d. 32 Item #21 100% – 20% = 80% passed Let 𝑥 be no. of students 80% of x = 28 4 𝑥=28 5 4𝑥 = 140 𝑥 = 35 Answer: B Item #22 If you are currently earning Php22, 000 a month, how much will your new salary (in PhP) be you are to receive a 4% increase next month? a. 22, 880 c. 26, 560 b. 25, 440 d. 24, 330 Item #22 Since you will have 4% increase, then your new salary will be 100% + 4% = 104% of your current salary. 104% of Php 22,000 = 1.04 x Php 22,000 = Php 22,880 Answer: A Item #23 If a + b = 3 and ab = 4, 2 what is the value of a + b ?2 a. 1 c. 3 b. 2 d. 4 Item #23 a + b = 3 and ab = 4 2 (a + b) = (3) 2 2 2 a + 2ab + b = 9 2 2 a + 2(4) + b = 9 2 a +8+b =9 2 2 2 a +b =1 Answer: A Item #24 The property of real numbers which describes (5 + 6) + (– 3) = (-3) + (5 + 6). a. reflexive c. distributive b. Associative d. commutative Item #24 Reflexive Property a=a Associative Property of Addition a + (b + c) = (a + b) + c Distributive Property of Multiplication over Addition a (b + c) = ab + ac Commutative of Addition a+b=b+a Item #24 (5 + 6) + (– 3) = (-3) + (5 + 6) Let a = (5 + 6) and b = (-3) a+b=b+a Commutative Answer: D Item #25 Factor: ab – 1 – b + a. a. (a – 1) (b + 1) b. (a + 1) (b – 1) c. (a – 1) (b – 1) d. (1 – a) (b + 1) Item #25 ab – 1 – b + a = ab – b + a – 1 = b (a – 1) + 1 (a – 1) = (a – 1) (b + 1) Answer: A Item #26 Which of the following has a prime factorization 3 2 of 2 x 3 x 5 x 7? a. 2,520 c. 2,800 b. 2,780 d. 2,850 Item #26 2 3 2 x x5x7 3 =8x9x5x7 = 2,520 Answer: A Item #27 Which of the following numbers can be expressed as the sum of 3 consecutive integers? a. 173 c. 203 b. 183 d. 301 Item #27 Let x, x + 1, and x + 2 be the 3 integers (x) + (x + 1) + (x + 2) = 3x + 3 (x) + (x + 1) + (x + 2) = 3 (x + 1) The sum of 3 consecutive integers is divisible by 3. Among the choices, only 183 is divisible by 3. In fact, 183 = 60 + 61 + 62 Answer: B Item #28 The number of hours spent for review in Algebra, Chemistry and English is in the rate 4:3:1. How many hours does each student spend for Chemistry in a 24 hours review package? a. 12 c. 6 b. 9 d. 3 Item #28 We could see that the 24-hour period is divided into 4 + 3 + 1 = 8 equal parts. 24 ÷ 8 = 3 hrs/part Three parts are dedicated for Chemistry. 3x3=9 Answer: B Item #29 If 𝑚 < 0 and 𝑛 > 0, which of the following is true? 𝑚 a. = 0 c. 𝑚𝑛 > 0 𝑛 1 1 1 1 b. < d. > 𝑚 𝑛 𝑚 𝑛 Item #29 m is negative while n is positive say, m = –1 and n = 1. 𝑚 a. =0 𝑛 −1 = 0 → -1 = 0 (false) 1 1 1 b. 𝑚 < 𝑛 1 1 −1 < → -1 < 1 1 (true) c. 𝑚𝑛 > 0 (-1) (1) > 0 → -1 > 0 (false) 1 1 d. >𝑛 𝑚 1 1 −1 > 1 → -1 > 1 (false) Answer: B Item #30 If 2x + 4y – 5z = 19 and z = 3, what is the value of x + 2y + z? a. 5 c. 15 b. 10 d. 20 Item #30 2x + 4y – 5z = 19 With z = 3 2x + 4y – 5(3) = 19 x + 2y + z = 17 + z 2x + 4y – 15 = 19 x + 2y + z = 17 + 3 2x + 4y = 34 x + 2y + z = 20 x + 2y = 17 Answer: D Item #31 The product of 2 two digit numbers is 351. What is their sum? a. 30 c. 40 b. 35 d. 45 Item #31 This could be solved by factoring… 351 = 3 x 3 x 3 x 13 351 = (3 x 3 x 3) x 13 351 = 27 x 13 Thus, the two numbers are 27 and 13 and their sum is 27 + 13 = 40. Answer: C Item #32 There are 9 people in a party and each one will shake hands with each other exactly once. How many handshakes are there in all? a. 27 c. 45 b. 36 d. 54 Item #32 The formula to determine the number of 9 8 Handshakes if n distinct HS = 2 people will shake hands with HS = 9 (4) each other exactly once is HS = 36 Answer: B Item #33 If 25% of a number is 50, what is 75% of the number? a. 75 c. 125 b. 100 d. 150 Item #33 25% of a number = 50 x3 x3 75% of a number = 150 Answer: D Item #34 A certain amount of money was invested at 4% simple interest and it gained Php500 in a span of 2 years. How much (in Php) was invested then? a. 6,250 c. 6,750 b. 6,500 d. 7,850 Item #34 I = Prt 500 = P (0.04) (2) 500 = 0.08P 500 =P 0.08 6,250 = P Answer: A Item #35 After investing Php20,000 at 3% simple interest for x years, it became Php24,200. What is x? a. 5 c. 7 b. 6 d. 8 Item #35 I = 24,200 – 20,000 = 4,200 I = Prt 4,200 = 20,000 (0.03) (x) 4,200 =x 20,000(0.03) 7=x Answer: C Item #36 Mrs. Parisan would like to buy a divider at Php50,000. She is required of a downpayment of Php15,000 and she will pay the rest at 5% interest for 18 months. How much will her monthly payment (in Php) be? a. 2,090.28 c. 2,250.33 b. 2,150.10 d. 2,650.59 Item #36 Php50,000 – Php15,000 = Php35,000 FA = Php35,000 + Php35,000 (0.05) (1.5) FA = Php37,625 Mrs. Insular’s monthly payment should be Php37,625 ÷ 18 ≈ Php2,090.28 Answer: A Item #37 After investing a certain amount of money at 8% simple interest rate for 5 years, my money became Php35,000 in all. How much was invested? a. Php15,000 c. Php25,000 b. Php20,000 d. Php30,000 Item #37 FA = P + Prt FA = P (1 + rt) 𝐹𝐴 =P 1+𝑟𝑡 35,000 =P 1+(0.08)(5) 25,000 = P Answer: C Item #38 How much will an amount (in Php) be after 5 years when Php100,000 is invested at 3% interest compounded annually? a. 113,927.29 c. 115,927.41 b. 114,837.15 d. 117,905.18 Item #38 𝑟 𝑠𝑡 𝐹 = 𝑃0 1+ 𝑠 1(5) 0.03 𝐹 = 𝑃ℎ𝑝100,000 1 + 1 5 𝐹 = 𝑃ℎ𝑝100,000 1.03 𝐹 ≈ 𝑃ℎ𝑝115,927.41 Answer: C Item #39 How much will an amount (in Php) be after 15 years when Php18,000 is invested at 2% compounded quarterly? a. 24,279.30 c. 27,559.10 b. 25,209.40 d. 28,554.93 Item #39 𝑟 𝑠𝑡 𝐹 = 𝑃0 1+ 𝑠 4(15) 0.02 𝐹 = 𝑃ℎ𝑝18,000 1 + 4 60 𝐹 = 𝑃ℎ𝑝18,000 1.005 𝐹 ≈ 𝑃ℎ𝑝24,279.30 Answer: A Item #40 A certain product was given successive discounts of 30% and 40%, in this order. Overall, the original price of the product was decreased by how many percent? a. 30 c. 58 b. 55 d. 70 Item #40 A certain product was given successive discounts of 30% and 40%, in this order. 1 – (0.70) (0.60) = 1 – 0.42 = 0.58 Overall, the original price of the product was decreased by 58%. Answer: C THANK YOU VERY MUCH!