273484776-elementary-algebra-problem-set.docx

Valenzuela Mathematics Science High school Elementary Algebra Entrance exam I.

Write TRUE if the statement is correct. Otherwise, write FALSE.

For any two sets 𝐴 and 𝐵, (𝐴 ∪ 𝐵)\(𝐴 ∩ 𝐵) = (𝐴\𝐵)(𝐵/𝐴). If 𝐴 ⊆ 𝐵 ⊆ 𝑋, then (𝑋\𝐵) ∪ (𝑋\𝐴) = 𝑋\𝐴. (ℤ×ℤ) ∩ ℤ = ∅. If A and B are disjoint sets, then so are A × B and B × A. There are sets A and B for which A ⊆ B and A ∩ B = ∅. If A ⊆ (B ∪ C), then A ⊆ B or A ⊆ C. If A, B, and C are sets then A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C). If A is a finite set, then the number of elements of A is always less than the number of subsets of A. 9. If E is the set of even counting numbers and O is the set of odd counting numbers, then (2, 1) ∈ E × O. 10. If S = {∅, 𝑎, 𝑏, 𝑐 c, then {∅} is an element of S. 11. All sets are disjoint with the empty set. 12. If A × B is an infinite set, then both A and B are infinite. 1. 2. 3. 4. 5. 6. 7. 8.

II.

Solve the following problems. Show all necessary solutions.

1. Let A = {a, b, c, d, e}, B = {a, e, i, o, u}, and C = {e, f, g, h, i}. Find (A ∩ B) ∪ (B ∩ C). 2. Let the universal set U be {1, 2, 3, …, 20}. If S = {2, 4, 6, 8, 10, 12}, E = {3, 6, 9, 12, 15, 18}, and T = {15, 16, 17, 18, 20}. Find (S ∪ E ∪ T)’. 3. Let the universal set U be the set of all prime numbers less than 20, A = {x | x is a prime number greater than 10}, and C = {3, 19}. Find A’ ∩ (B ∩ C)’. 4. Aldo cut a long piece of bamboo into four pieces. The second piece was one-third of the first piece, the third piece was one-third of the second, and finally the fourth piece was one-third of the third. If the smallest piece was 2 dm, how long, in inches, was the bamboo before it was cut?

5. Marvin is 10 % taller than Homer, and Homer is 10 % taller than August. How much in percent is Marvin taller than August? 6. Which is largest, a = 248 , b = 336, c = 524? 7. When 3n is divided by 7, the remainder is 4. What is the remainder when 2n is divided by 7? 8. If the sides of a cube are tripled, what percent of the original volume is the new volume? 12 1 9 9. If − ≤ 𝑥 ≤ − and 3 ≤ 𝑦 ≤ , what is the largest possible value of 𝑥−𝑦 𝑥+𝑦

5

2

2

?

10. How many positive factors does 62 × (633 + 632 + 63 + 1) + 1 have ? 11. Which is smaller, A = (2015)(2014)(2013)(2012)(2011) or B = 20135? 12. Given T = {1, 2} (a) If P is the set of subsets of T, give all the elements of P. (b) Write the set R = {(A, B) | (A, B) ∈ P × P such that A and B are disjoint} using the roster or enumeration method. 13. Jolly can do a job alone in 8 days. After Jolly has worked for 3 days, Happy joins Jolly and together they can complete the job in 3 more days. In how many ways could Happy have done the job alone? 14. Give the set of subsets of A = {a, {b, c}}. 15. Two numbers are in the ratio 3 : 5. If 60 is added to both of them, their ratio becomes 15 : 17. What are the two numbers? 16. The sum of the digits in a certain two-digit number is 11. If the number is reversed, the number is increased by 27. Find the original number. 17. Joseph’s father is now twice as old as he is. Sixteen years ago, he was four times as old. How old is Joseph now? 18. In a small locality, every family subscribes to newspaper A or to newspaper B. If 30 families subscribe to newspaper A, 40 to newspaper B, and 10 to both newspapers. How many families are there in this locality? How many localities subscribe only to newspaper A? 19. How many positive integers are divisible by 3 or 5?

20. A group of 102 students took examinations in Math, Chemistry, and English. Among them, 92 passed Chemistry, 75 English, and 63 Mathematics; at most 65 passed Chemistry and English, at most 54 Chemistry and Math, and at most 48 English and Math. Find the largest possible number of students who could have passed all three subjects. 21. Let U = {x | x is a natural number less than 15} A = {x ∈ U | x is an odd number} B = {x ∈ U | x is an even number} C = {1, 2, 3, 4, 5} D = {4, 5, 6, 7, 8} E = {7, 8, 9, 10, 11} F = {11, 12, 13, 14} Find (a) A ∩ B (b) B\(A ∩ C’) (c) (C ∪ D) ∪ (E ∩ F)’ (d) [(B ∩ E) ∪ (A ∪ F)]’ (e) (C ∪ D)\(A\E) (f) (D ∩ E)’ ∪ (C ∩ F)’ 22. List down all the subsets of {1, 2, a, b, 3 containing 3 elements} 23. List down all subsets of the set {∅, {∅}, {{∅}}}. 24. Find the number of positive integers not exceeding 300 that are divisible by 4 or 6. 25. Find the number of three-digit positive integers that are divisible by 2, 3, or 5. 26. Convert (a) 26 m/s2 to yd/min2 (b) 0.34532 acre to in2 (c) 67.67 gallons to cm3 (d) 0.98 dL/ft2 to in3/m2 (e) 98.99 cm3/s to ft3/day 27. Evaluate (a) –6 + 5{4–3[1-2(–9–7)]} (b) (x-a)(x-b)(x-c)…(x-z) (c)

45 210

1

1

1

1

− 2014 {(1 − ) (1 − ) (1 − ) … (1 − )} 2 3 4 2014

(d)

5

1+

8

(

1 1+

1

1+ 1 1+1) 1

−

−6{5−[−4−3(1−2)]} 66

III. Multiple Choice

1. Which of the following sets is not equal to the others? (a) {n | n ∈ N, n is neither prime nor composite} (b) {real numbers whose product with itself is itself} (c) {√1} 𝑥 (d) { | x ∈ R\{0} } 𝑥

For number 2, the symmetric difference of two sets A and B is defined as 𝐴 ∆ 𝐵 = (A ∪ B)\(A ∩ B) 2. Which of the following is an element of Z ∆ Q? (a) 3.141592653 (b) √2 (c) 0 (d) 3 + 7i