MATH 107 College Algebra Final Examination Solutions - Summer 2019

This document presents the solutions to a College Algebra final examination, identified as MATH 107 from the Summer of 2019. The exam is an open-book assessment and includes a mix of multiple-choice and short-answer questions. The solutions cover a wide range of topics within college algebra, including complex number operations, solving inequalities, exponential and logarithmic functions, compound interest calculations, properties of parabolas (finding vertex, range, decreasing intervals), polynomial functions (end behavior, y-intercept, zeros), and rational functions (domain, asymptotes). The solutions are step-by-step, showing the mathematical processes and formulas used to arrive at the correct answers for each problem. The assignment provides a detailed guide and complete answers to the MATH 107 final exam.

Q 13.
Answer
(9 + i)(5 + 2i) = 45 + 18i + 5i + 2i2
= 45 + 23i + 2i2
[i2 = −1]
= 45 + 23i − 2
= 43 + 23i
Q 14.
Answer
x + 4
x − 2≥ 0
The numerator x + 4 is 0 at x = −4, the denominator x − 2 is 0 at x = 2.
To check the interval of solution the number line is divided into three regions:
x < −4, −4 < x < 2 and x > 2.Each region is tested to check if satisfies
the inequality.
region x f (x) check
x < −4 −5 1
7 ≥ 0
−4 < x < 2 1 −5 ≤ 0
x > 2 3 7 ≥ 0
Thus the region x < −4 and x > 2 satisfy the inequality.Additionally,at
x = −4, f (x) = 0, therefore the inequality is satisfied.The solution interval
is
(−∞, −4] ∪ (2, ∞)
Q 15.
Answer
T (t) = 75 + 110e−0.075t
when t = 12
T (12) = 75 + 110e−0.075×12≈ 120◦F
1

Q 16.
Answer
log2
1
16
Property of logarithms:
loga (an) = n
log2
1
16 = log2
1
24
= log2 (2−4)
= −4
Q 17.
Answer
95x−2 = 81
95x−2 = 92
=⇒ 5x − 2 = 2
∴ x = 0.8
Q 18.
Answer
Compound interest formula:
A = P 1 + r
n
nt
where, A is the amount accumulated at the end of the period, r is the rate
of interest,n is the frequency of compounding and t is the totaltime over
which the interest was compounded.P is the principal amount.
For double growth:A/P = 2. The given rate of interest is 3.2 %.Interest
2

is compounded annually which implies n = 1.Therefore,
2 = 1 + 3.2
100
t
[Taking log on both sides]
ln 2 = t ln (1.032)
=⇒ t = ln 2
ln 1.032
≈ 22.0
Therefore, it takes 22 years to double the investment at the given rate.
Q 19.
Answer
y = x2 + 6x + 13
Vertex (h, k) of a parabola y = ax2+ bx + c is obtained by using the formula:
h = − b
2a
k = ah2 + bh + c
Comparing
ax2 + bx + c' x 2 + 6x + 13
we get, a = 1, b = 6 and c = 13.Therefore,
h = − 6
2 × 1= −3
k = (−3)2 + 6(−3) + 13 = 4
A parabola with positive a opens up and it ranges from k to infinity.There-
fore,
Range:[4, ∞)
An upward opening parabola has minima at the vertex.It decreases up to
the vertex and thereafter increases.
Therefore, the given parabola decreases up to the point x = −3.The interval
in which the function is decreasing is
(−∞, −3)
3

Q 20.
Answer
P (x) = 1
8x4 − 3
4x3 + 3
8x2 + 13
4 x − 3 =1
8(x + 2)(x − 1)(x − 3)(x − 4)
a)
For P (x) the degree is even (4) and the leading coefficient is positive,this
implies
x → ∞ =⇒ f(x) → ∞ & x → −∞ =⇒ f(x) → ∞
This end behaviour is given by option C.
b)
y- intercept is obtained at x = 0.Therefore,
y = 1
8(0 + 2)(0 − 1)(0 − 3)(0 − 4) = −3
The y- intercept therefore is (0, −3).
c)
Zeros of the function are obtained by solving
P (x) = 0
Therefore, 1
8(x + 2)(x − 1)(x − 3)(x − 4) = 0
Implies,
x = −2, 1, 3, 4
are all zeros of the function.
d)
All the above information is correctly represented in graph A, which is the
correct choice.
Q 21
Answer
f (x) = 3x2 + 3
x2 − 4
4

a)
The function f (x) is not defined when the denominator becomes 0, that is
x2 − 4 = 0 =⇒ x = ±2
Therefore, the domain of the function is
{x ∈ R | x 6= ±2}
b)
Horizontal asymptote(s) of f (x) are obtained by solving
y = lim
x→∞ f (x)
Therefore,
y = lim
x→∞
3x2 + 3
x2 − 4 = 3
The horizontal asymptote is y = 3.
c)
Vertical asymptote(s) are the values for which the denominator becomes 0,
therefore
x = ±2
are the vertical asymptotes.
d)
All of the above information is correctly represented in graph C,which is
the correct choice.
5