BASIC ALGEBRAIC TECHNIQUES
FINAL EXAM – PART 1
Name: _________________________
Directions:
Calculators are NOT permitted for this portion of the exam. For each question, show your work in the space
provided after that question. You must show your work to receive credit for your answer.
Multiple Choice Questions
2
1. If
𝑥𝑎
2
𝑥𝑏
a.
b.
c.
d.
= 𝑥16 , 𝑥 > 1, and 𝑎 + 𝑏 = 2, what is the value of 𝑎 − 𝑏?
8
14
16
18
3 − 5𝑖
2. Which of the following complex numbers is equivalent to 8 + 2𝑖? (Note: 𝑖 = √−1)
a.
b.
c.
d.
3
8
3
8
−
+
5𝑖
2
5𝑖
2
7
23𝑖
7
23𝑖
−
34
34
+
34
34
𝐹
𝑁+𝐹
3. A website uses the formula above to calculate a seller’s rating, 𝑅, based on the number of favorable
reviews, 𝐹, and unfavorable reviews, 𝑁. Which of the following expresses the number of favorable
reviews in terms of the other variables?
𝑅=
𝑅𝑁
a. 𝐹 = 𝑅 − 1
𝑅𝑁
b. 𝐹 = 1 − 𝑅
𝑁
c. 𝐹 = 1 − 𝑅
𝑁
d. 𝐹 = 𝑅 − 1
𝑦 = 𝑎(𝑥 − 2)(𝑥 + 4)
4. In the quadratic equation above, 𝑎 is a nonzero constant. The graph of the equation in the 𝑥𝑦-plane is
a parabola with vertex (𝑐, 𝑑). Which of the following is equal to 𝑑?
a.
b.
c.
d.
−9𝑎
−8𝑎
−5𝑎
−2𝑎
(𝑎𝑥 + 3)(5𝑥 2 − 𝑏𝑥 + 4) = 20𝑥 3 − 9𝑥 2 − 2𝑥 + 12
5. The equation above is true for all 𝑥, where 𝑎 and 𝑏 are constants. What is the value of 𝑎𝑏?
a.
b.
c.
d.
18
20
24
40
6. If 𝑥 > 3, which of the following is equivalent to
a.
b.
2𝑥 + 5
𝑥2
+ 5𝑥 + 6
𝑥 2 + 5𝑥 + 6
2𝑥 + 5
c. 2𝑥 + 5
d. 𝑥 2 + 5𝑥 + 6
1
1
𝑥+2
+
?
1
𝑥+3
8𝑥
7. If 3𝑥 − 𝑦 = 12, what is the value of 2𝑦 ?
a.
b.
c.
d.
212
44
82
The value cannot be determined from the information given.
8. If (𝑎𝑥 + 2)(𝑏𝑥 + 7) = 15𝑥 2 + 𝑐𝑥 + 14 for all values of 𝑥 and 𝑎 + 𝑏 = 8, what are the two possible
values for 𝑐?
a.
b.
c.
d.
3 and 5
6 and 35
10 and 21
31 and 41
9. Which of the following is equivalent to
a. 𝑥
b. 𝑥 + 4
2
c. 𝑥 − 4𝑥 + 2
2
d. 𝑥 + 1 − 4𝑥 + 2
4𝑥 2 + 6𝑥
4𝑥+2
?
2𝑥 2 − 4𝑥 = 𝑡
10. In the equation above, 𝑡 is a constant. If the equation has no real solutions, which of the following
could be the value of 𝑡?
a.
b.
c.
d.
–3
–1
1
3
11. The volume of right circular cylinder 𝐴 is 22 cubic centimeters. What is the volume, in cubic
centimeters, of a right circular cylinder with twice the radius and half the height of cylinder 𝐴?
a.
b.
c.
d.
11
22
44
66
3
12. Which of the following is equivalent to 94 ?
a.
b.
c.
d.
3
√9
√9
√3
3√3
4
2 + 3𝑥
13. Which expression is equivalent to 16 − 81𝑥 4?
a.
1
8 − 27𝑥3
b. 8 − 27𝑥 3
c.
1
(4 + 9𝑥 2 )(2 − 3𝑥)
d. (4 + 9𝑥 2 )(2 − 3𝑥)
14. If tan 𝑥 = 𝑚, what is sin 𝑥 in terms of 𝑚?
a.
b.
c.
d.
1
√𝑚2 + 1
1
√1 − 𝑚2
𝑚
√𝑚2 +1
𝑚
√1 − 𝑚2
15. An antique lamp was sold at an auction. The price 𝑝 of the lamp, in dollars, during the auction can be
modeled by the equation 𝑝 = 900 − 10𝑡, where 𝑡 is the number of seconds left in the auction.
According to the model, what is the meaning of the 900 in the equation?
a.
b.
c.
d.
The starting auction price of the lamp
The final auction price of the lamp
The increase in the price of the lamp per second
The time it took to auction off the lamp, in seconds
16. For two acute angles, ∠𝐴 and ∠𝐵, cos 𝐴 = sin 𝐵. The measures, in degrees, of ∠𝐴 and ∠𝐵 are 𝑥 + 8
and 4𝑥 + 2, respectively. What is the value of 𝑥?
a.
b.
c.
d.
37
34
16
2
17. Lucia and John will work together to make 60 paper flowers for a school party. The line shown
represents the possible combinations of time, in hours, spent by Lucia and John to fulfill this task.
According to the graph, on average, how many paper flowers will Lucia make per hour?
a.
b.
c.
d.
4
5
12
15
18. The function 𝑓 is defined by 𝑓(𝑥) = 𝑎 𝑥 + 𝑏, where 𝑎 and 𝑏 are constants. The graph of 𝑦 = 𝑓(𝑥) in
the 𝑥𝑦-plane passes through the points (0, −5) and (1, 2). What is the value of 𝑎?
a.
b.
c.
d.
8
7
6
5
Free Response Questions
19. How many liters of 10% salt solution must be added to 5 liters of 25% salt solution to obtain a 15% salt
solution?
−9𝑥 + 24𝑞𝑥 = 36
20. In the given equation, 𝑞 is a constant. The equation has no solution. What is the value of 𝑞?
21. If 𝑢 + 𝑡 = 5 and 𝑢 − 𝑡 = 2, what is the value of (𝑢 − 𝑡)(𝑢2 − 𝑡 2 )?
22. If 𝑥 3 + 𝑦 3 = 12 and 𝑥 3 − 𝑦 3 = 6, what is the value of 𝑥 6 − 𝑦 6 ?