Department of Mathematics and Statistics Department of Mathematics and Statistics
College of Science and Mathematics College of Science and Mathematics
MSU-Iligan Institute of Technology MSU-Iligan Institute of Technology
Math 17 Preliminary Exam Math 17 Preliminary Exam
July 17, 2013 July 17, 2013
I. Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, A = {1, 3, 6, 8, 10}, I. Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, A = {1, 3, 6, 8, 10},
B = {2, 4, 5, 6, 8}, C = {1, 4, 6, 10} and let B = {2, 4, 5, 6, 8}, C = {1, 4, 6, 10} and let
D = the set of all elements in U which are perfect D = the set of all elements in U which are perfect
squares. squares.
1. Write the set D using the Roster method and Rule 1. Write the set D using the Roster method and Rule
method. [2 pts] method. [2 pts]
2. Illustrate the relationship among sets U , A, C and 2. Illustrate the relationship among sets U , A, C and
D using the Venn diagram. [5 pts] D using the Venn diagram. [5 pts]
3. Find and tabulate the following: [2 pts each] 3. Find and tabulate the following: [2 pts each]
a. (A ∪ B)0 d. (B\C) × (C ∩ D) a. (A ∪ B)0 d. (B\C) × (C ∩ D)
b. C ∩ D e. subsets of (A ∪ B)0 b. C ∩ D e. subsets of (A ∪ B)0
c. B\C c. B\C
II. Perform as indicated. [4 pts each] II. Perform as indicated. [4 pts each]
1. Simplify 6x2 − 3x{2x − 4[x + 4(x − 1)] + 20x} + 48x. 1. Simplify 6x2 − 3x{2x − 4[x + 4(x − 1)] + 20x} + 48x.
2. Find the product of the following expressions. 2. Find the product of the following expressions.
2 2
1 x 1 4 1 1 x 1 4 1
a. x + n + +x −x a. x + n + +x −x
3 32n 34n 3n 3 32n 34n 3n
b. (2s − 3t2 )4 b. (2s − 3t2 )4
3. Factor the following completely. 3. Factor the following completely.
a. 4x2 + 4x − 9y 2 − 6y a. 4x2 + 4x − 9y 2 − 6y
b. 64p6 − 1 b. 64p6 − 1
32x5 + 8x2 + 4x 32x5 + 8x2 + 4x
4. Find the quotient of . 4. Find the quotient of .
2x + 1 2x + 1
2x3 3x2 5x2 2x3 3x2 5x2
5. + 2 − 2 5. + 2 − 2
x3 +y 3 x + 2xy + y 2 x − xy + y 2 x3 +y 3 x + 2xy + y 2 x − xy + y 2
m4 + 6m2 n2 + 9n4 3m2 + 12mn − 24n2 m2 − 4mn + 8n2 m4 + 6m2 n2 + 9n4 3m2 + 12mn − 24n2 m2 − 4mn + 8n2
6. ÷ ÷ 6. ÷ ÷
m4 n − 9n5 3mn2 − m3 m2 n + 3n3 m n − 9n
4 5 3mn − m
2 3 m2 n + 3n3
0
−1 0
−1
−1 y (2x + y) −1 y (2x + y)
y + x2 − x y + x2 − x
7. 7.
x−2 y 2 + xy −1 x−2 y 2 + xy −1
√ r r √ r r
a 100a5 3a2 9a3 a7 a 100a5 3a2 9a3 a7
8. − − 8. − −
b2 b b2 b4 b2 b b2 b4
p6
√ p6
√
32xy 5 · 2xy 32xy 5 · 2xy
9. p3
9. p3
2y 3 2y 3
10. Let x ∈ (1, +∞). Rationalize the denominator and 10. Let x ∈ (1, +∞). Rationalize the denominator and
x−1 x−1
simplify p √ . simplify p √ .
6
(1 − x)4 6
(1 − x)4
III. Let p, q, and r be statements. Then there are 8 III. Let p, q, and r be statements. Then there are 8
combinations of their truth values. How many of these
combinations of their truth values. How many of these
College of Science and Mathematics College of Science and Mathematics
MSU-Iligan Institute of Technology MSU-Iligan Institute of Technology
Math 17 Preliminary Exam Math 17 Preliminary Exam
July 17, 2013 July 17, 2013
I. Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, A = {1, 3, 6, 8, 10}, I. Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, A = {1, 3, 6, 8, 10},
B = {2, 4, 5, 6, 8}, C = {1, 4, 6, 10} and let B = {2, 4, 5, 6, 8}, C = {1, 4, 6, 10} and let
D = the set of all elements in U which are perfect D = the set of all elements in U which are perfect
squares. squares.
1. Write the set D using the Roster method and Rule 1. Write the set D using the Roster method and Rule
method. [2 pts] method. [2 pts]
2. Illustrate the relationship among sets U , A, C and 2. Illustrate the relationship among sets U , A, C and
D using the Venn diagram. [5 pts] D using the Venn diagram. [5 pts]
3. Find and tabulate the following: [2 pts each] 3. Find and tabulate the following: [2 pts each]
a. (A ∪ B)0 d. (B\C) × (C ∩ D) a. (A ∪ B)0 d. (B\C) × (C ∩ D)
b. C ∩ D e. subsets of (A ∪ B)0 b. C ∩ D e. subsets of (A ∪ B)0
c. B\C c. B\C
II. Perform as indicated. [4 pts each] II. Perform as indicated. [4 pts each]
1. Simplify 6x2 − 3x{2x − 4[x + 4(x − 1)] + 20x} + 48x. 1. Simplify 6x2 − 3x{2x − 4[x + 4(x − 1)] + 20x} + 48x.
2. Find the product of the following expressions. 2. Find the product of the following expressions.
2 2
1 x 1 4 1 1 x 1 4 1
a. x + n + +x −x a. x + n + +x −x
3 32n 34n 3n 3 32n 34n 3n
b. (2s − 3t2 )4 b. (2s − 3t2 )4
3. Factor the following completely. 3. Factor the following completely.
a. 4x2 + 4x − 9y 2 − 6y a. 4x2 + 4x − 9y 2 − 6y
b. 64p6 − 1 b. 64p6 − 1
32x5 + 8x2 + 4x 32x5 + 8x2 + 4x
4. Find the quotient of . 4. Find the quotient of .
2x + 1 2x + 1
2x3 3x2 5x2 2x3 3x2 5x2
5. + 2 − 2 5. + 2 − 2
x3 +y 3 x + 2xy + y 2 x − xy + y 2 x3 +y 3 x + 2xy + y 2 x − xy + y 2
m4 + 6m2 n2 + 9n4 3m2 + 12mn − 24n2 m2 − 4mn + 8n2 m4 + 6m2 n2 + 9n4 3m2 + 12mn − 24n2 m2 − 4mn + 8n2
6. ÷ ÷ 6. ÷ ÷
m4 n − 9n5 3mn2 − m3 m2 n + 3n3 m n − 9n
4 5 3mn − m
2 3 m2 n + 3n3
0
−1 0
−1
−1 y (2x + y) −1 y (2x + y)
y + x2 − x y + x2 − x
7. 7.
x−2 y 2 + xy −1 x−2 y 2 + xy −1
√ r r √ r r
a 100a5 3a2 9a3 a7 a 100a5 3a2 9a3 a7
8. − − 8. − −
b2 b b2 b4 b2 b b2 b4
p6
√ p6
√
32xy 5 · 2xy 32xy 5 · 2xy
9. p3
9. p3
2y 3 2y 3
10. Let x ∈ (1, +∞). Rationalize the denominator and 10. Let x ∈ (1, +∞). Rationalize the denominator and
x−1 x−1
simplify p √ . simplify p √ .
6
(1 − x)4 6
(1 − x)4
III. Let p, q, and r be statements. Then there are 8 III. Let p, q, and r be statements. Then there are 8
combinations of their truth values. How many of these
combinations of their truth values. How many of these
