Department of Mathematics Department of Mathematics
College of Science and Mathematics College of Science and Mathematics
MSU-Iligan Institute of Technology MSU-Iligan Institute of Technology
Math 17 Midterm Exam Math 17 Midterm Exam
February 8, 2013 February 8, 2013
I. Find the solution set of the following. I. Find the solution set of the following.
(2 + i)z z (2 + i)z z
1. = i35 + where z ∈ C 1. = i35 + where z ∈ C
3i 2−i 3i 2−i
√ √
2. 5 − x − x − 1 = 0 2. 5 − x − x − 1 = 0
3. x3 + 3x2 + 54 x − 3
2
=0 3. x3 + 3x2 + 54 x − 3
2
=0
x x
4. 1 − ≤4 4. 1 − ≤4
x+2 x+2
y ≥ x2 − 6x + 5
y ≥ x2 − 6x + 5
5. x+y ≤5 (sketch) 5. x+y ≤5 (sketch)
y+2>0 y+2>0
II. Perform as indicated: II. Perform as indicated:
1. Point M (x, −6) is the midpoint of the line 1. Point M (x, −6) is the midpoint of the line
segment joining points A(a, −10) and B(1, b). segment joining points A(a, −10) and B(1, b).
If A is on the third quadrant and |AB| = 10, If A is on the third quadrant and |AB| = 10,
find x, a and b. find x, a and b.
2. Let f r : Df → [0, +∞) be defined by 2. Let f r : Df → [0, +∞) be defined by
−x −x
f (x) = . f (x) = .
x+1 x+1
a. Determine if f is one-to-one or not. a. Determine if f is one-to-one or not.
b. Determine if f is onto or not. b. Determine if f is onto or not.
c. Define f −1 (x) if it exists. c. Define f −1 (x) if it exists.
√ √
3. If f (x) = 3 x + 1 and g(x) = 8x3 − 1, find 3. If f (x) = 3 x + 1 and g(x) = 8x3 − 1, find
(f ◦ g)(x). (f ◦ g)(x).
III. Prove the following statement using PMI: III. Prove the following statement using PMI:
an+1 − a an+1 − a
a1 + a2 + a3 + · · · + an = , ∀n ≥ 1. a1 + a2 + a3 + · · · + an = , ∀n ≥ 1.
a−1 a−1
IV. Prove or disprove using truth table: IV. Prove or disprove using truth table:
(p → q) → r ≡ (¬p → r) ∧ (q → r) (p → q) → r ≡ (¬p → r) ∧ (q → r)
V. Let f : Df ⊆ R → R be defined by V. Let f : Df ⊆ R → R be defined by
x−5 x−5
y = . Find the domain, range, intercepts, y = . Find the domain, range, intercepts,
x+5 x+5
and asymptotes; test for symmetry; and sketch its and asymptotes; test for symmetry; and sketch its
graph. graph.
College of Science and Mathematics College of Science and Mathematics
MSU-Iligan Institute of Technology MSU-Iligan Institute of Technology
Math 17 Midterm Exam Math 17 Midterm Exam
February 8, 2013 February 8, 2013
I. Find the solution set of the following. I. Find the solution set of the following.
(2 + i)z z (2 + i)z z
1. = i35 + where z ∈ C 1. = i35 + where z ∈ C
3i 2−i 3i 2−i
√ √
2. 5 − x − x − 1 = 0 2. 5 − x − x − 1 = 0
3. x3 + 3x2 + 54 x − 3
2
=0 3. x3 + 3x2 + 54 x − 3
2
=0
x x
4. 1 − ≤4 4. 1 − ≤4
x+2 x+2
y ≥ x2 − 6x + 5
y ≥ x2 − 6x + 5
5. x+y ≤5 (sketch) 5. x+y ≤5 (sketch)
y+2>0 y+2>0
II. Perform as indicated: II. Perform as indicated:
1. Point M (x, −6) is the midpoint of the line 1. Point M (x, −6) is the midpoint of the line
segment joining points A(a, −10) and B(1, b). segment joining points A(a, −10) and B(1, b).
If A is on the third quadrant and |AB| = 10, If A is on the third quadrant and |AB| = 10,
find x, a and b. find x, a and b.
2. Let f r : Df → [0, +∞) be defined by 2. Let f r : Df → [0, +∞) be defined by
−x −x
f (x) = . f (x) = .
x+1 x+1
a. Determine if f is one-to-one or not. a. Determine if f is one-to-one or not.
b. Determine if f is onto or not. b. Determine if f is onto or not.
c. Define f −1 (x) if it exists. c. Define f −1 (x) if it exists.
√ √
3. If f (x) = 3 x + 1 and g(x) = 8x3 − 1, find 3. If f (x) = 3 x + 1 and g(x) = 8x3 − 1, find
(f ◦ g)(x). (f ◦ g)(x).
III. Prove the following statement using PMI: III. Prove the following statement using PMI:
an+1 − a an+1 − a
a1 + a2 + a3 + · · · + an = , ∀n ≥ 1. a1 + a2 + a3 + · · · + an = , ∀n ≥ 1.
a−1 a−1
IV. Prove or disprove using truth table: IV. Prove or disprove using truth table:
(p → q) → r ≡ (¬p → r) ∧ (q → r) (p → q) → r ≡ (¬p → r) ∧ (q → r)
V. Let f : Df ⊆ R → R be defined by V. Let f : Df ⊆ R → R be defined by
x−5 x−5
y = . Find the domain, range, intercepts, y = . Find the domain, range, intercepts,
x+5 x+5
and asymptotes; test for symmetry; and sketch its and asymptotes; test for symmetry; and sketch its
graph. graph.
