📘 Precalculus Practice — Mixed Set 01
Recommended: 60–75 minutes. No calculator.
Problems
1.
Tags: Exponents · Easy · source: Original (AMC-style)
What is $2^3 \cdot 2^4$?
A) $2^7$
B) $2^{12}$
C) $4^7$
D) $8^7$
E) $16^7$
Answer & Solution
Answer: A
Using the exponent rule: $2^3 \cdot 2^4 = 2^{3+4} = 2^7$.
2.
Tags: Logarithms · Easy · source: Original (AMC-style)
What is $\log_2(8)$?
A) $2$
B) $3$
C) $4$
D) $8$
E) $16$
Answer & Solution
Answer: B
Since $2^3 = 8$, we have $\log_2(8) = 3$.
3.
Tags: Trigonometry · Easy · source: Original (AMC-style)
What is $\sin(30°)$?
A) $\frac{1}{4}$
B) $\frac{1}{2}$
C) $\frac{\sqrt{2}}{2}$
D) $\frac{\sqrt{3}}{2}$
E) $1$
Answer & Solution
Answer: B
$\sin(30°) = \frac{1}{2}$.
4.
Tags: Complex Numbers · Easy · source: Original (AMC-style)
What is $i^2$?
A) $-1$
B) $0$
C) $1$
D) $i$
E) $2i$
Answer & Solution
Answer: A
By definition, $i^2 = -1$.
5.
Tags: Sequences · Easy · source: Original (AMC-style)
What is the 5th term of the arithmetic sequence $2, 5, 8, 11, \ldots$?
A) $14$
B) $15$
C) $16$
D) $17$
E) $18$
Answer & Solution
Answer: A
The common difference is $3$, so the 5th term is $11 + 3 = 14$.
6.
Tags: Exponents · Easy · source: Original (AMC-style)
What is $\frac{3^5}{3^2}$?
A) $3^2$
B) $3^3$
C) $3^7$
D) $9^3$
E) $27$
Answer & Solution
Answer: B
$\frac{3^5}{3^2} = 3^{5-2} = 3^3$.
7.
Tags: Logarithms · Easy · source: Original (AMC-style)
What is $\log_3(27)$?
A) $2$
B) $3$
C) $4$
D) $9$
E) $27$
Answer & Solution
Answer: B
Since $3^3 = 27$, we have $\log_3(27) = 3$.
8.
Tags: Trigonometry · Easy · source: Original (AMC-style)
What is $\cos(60°)$?
A) $\frac{1}{4}$
B) $\frac{1}{2}$
C) $\frac{\sqrt{2}}{2}$
D) $\frac{\sqrt{3}}{2}$
E) $1$
Answer & Solution
Answer: B
$\cos(60°) = \frac{1}{2}$.
9.
Tags: Complex Numbers · Easy · source: Original (AMC-style)
What is $(1+i)(1-i)$?
A) $0$
B) $1$
C) $2$
D) $2i$
E) $1+i$
Answer & Solution
Answer: C
$(1+i)(1-i) = 1 - i^2 = 1 - (-1) = 2$.
10.
Tags: Sequences · Easy · source: Original (AMC-style)
What is the 4th term of the geometric sequence $3, 6, 12, \ldots$?
A) $18$
B) $24$
C) $36$
D) $48$
E) $72$
Answer & Solution
Answer: B
The common ratio is $2$, so the 4th term is $12 \times 2 = 24$.
11.
Tags: Exponents · Medium · source: Original (AMC-style)
What is $2^{10}$?
A) $512$
B) $1024$
C) $2048$
D) $4096$
E) $8192$
Answer & Solution
Answer: B
$2^{10} = 1024$.
12.
Tags: Logarithms · Medium · source: Original (AMC-style)
What is $\log_2(16) + \log_2(4)$?
A) $4$
B) $5$
C) $6$
D) $7$
E) $8$
Answer & Solution
Answer: C
$\log_2(16) + \log_2(4) = \log_2(16 \cdot 4) = \log_2(64) = 6$.
13.
Tags: Trigonometry · Medium · source: Original (AMC-style)
What is $\sin^2(30°) + \cos^2(30°)$?
A) $\frac{1}{4}$
B) $\frac{1}{2}$
C) $\frac{3}{4}$
D) $1$
E) $\frac{5}{4}$
Answer & Solution
Answer: D
By the Pythagorean identity: $\sin^2(30°) + \cos^2(30°) = 1$.
14.
Tags: Complex Numbers · Medium · source: Original (AMC-style)
What is $|3 + 4i|$?
A) $3$
B) $4$
C) $5$
D) $7$
E) $25$
Answer & Solution
Answer: C
$|3 + 4i| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$.
15.
Tags: Sequences · Medium · source: Original (AMC-style)
What is the sum of the first 5 terms of the arithmetic sequence $1, 4, 7, 10, \ldots$?
A) $25$
B) $35$
C) $45$
D) $55$
E) $65$
Answer & Solution
Answer: B
The first 5 terms are $1, 4, 7, 10, 13$. Sum = $1 + 4 + 7 + 10 + 13 = 35$.
16.
Tags: Exponents · Medium · source: Original (AMC-style)
What is $\log_3(81)$?
A) $2$
B) $3$
C) $4$
D) $9$
E) $27$
Answer & Solution
Answer: C
Since $3^4 = 81$, we have $\log_3(81) = 4$.
17.
Tags: Logarithms · Medium · source: Original (AMC-style)
What is $\log_2(8) - \log_2(2)$?
A) $1$
B) $2$
C) $3$
D) $4$
E) $6$
Answer & Solution
Answer: B
$\log_2(8) - \log_2(2) = \log_2\left(\frac{8}{2}\right) = \log_2(4) = 2$.
18.
Tags: Trigonometry · Medium · source: Original (AMC-style)
What is $\tan(45°)$?
A) $0$
B) $\frac{1}{2}$
C) $1$
D) $\sqrt{2}$
E) $\sqrt{3}$
Answer & Solution
Answer: C
$\tan(45°) = 1$.
19.
Tags: Complex Numbers · Medium · source: Original (AMC-style)
What is $(2+i)^2$?
A) $3 + 4i$
B) $4 + 4i$
C) $5 + 4i$
D) $3 + 2i$
E) $4 + 2i$
Answer & Solution
Answer: A
$(2+i)^2 = 4 + 4i + i^2 = 4 + 4i - 1 = 3 + 4i$.
20.
Tags: Sequences · Medium · source: Original (AMC-style)
What is the sum of the first 4 terms of the geometric sequence $2, 6, 18, \ldots$?
A) $26$
B) $40$
C) $54$
D) $80$
E) $162$
Answer & Solution
Answer: D
The first 4 terms are $2, 6, 18, 54$. Sum = $2 + 6 + 18 + 54 = 80$.
21.
Tags: Exponents · Hard · source: Original (AMC-style)
What is $2^{100} \pmod{7}$?
A) $1$
B) $2$
C) $3$
D) $4$
E) $5$
Answer & Solution
Answer: B
By Fermat's Little Theorem, $2^6 \equiv 1 \pmod{7}$. Since $100 = 16 \cdot 6 + 4$, we have $2^{100} = (2^6)^{16} \cdot 2^4 \equiv 1^{16} \cdot 2^4 \equiv 16 \equiv 2 \pmod{7}$.
22.
Tags: Logarithms · Hard · source: Original (AMC-style)
If $\log_2(x) = 3$ and $\log_2(y) = 4$, what is $\log_2(xy)$?
A) $7$
B) $12$
C) $24$
D) $64$
E) $128$
Answer & Solution
Answer: A
$\log_2(xy) = \log_2(x) + \log_2(y) = 3 + 4 = 7$.
23.
Tags: Trigonometry · Hard · source: Original (AMC-style)
What is $\sin(75°)$?
A) $\frac{\sqrt{2}}{4}$
B) $\frac{\sqrt{6} - \sqrt{2}}{4}$
C) $\frac{\sqrt{6} + \sqrt{2}}{4}$
D) $\frac{\sqrt{3}}{2}$
E) $\frac{1}{2}$
Answer & Solution
Answer: C
Using the angle addition formula: $\sin(75°) = \sin(45° + 30°) = \sin(45°)\cos(30°) + \cos(45°)\sin(30°) = \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2} = \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4} = \frac{\sqrt{6} + \sqrt{2}}{4}$.
24.
Tags: Complex Numbers · Hard · source: Original (AMC-style)
What is $i^{100}$?
A) $-1$
B) $0$
C) $1$
D) $i$
E) $-i$
Answer & Solution
Answer: C
The powers of $i$ cycle as: $i^1 = i$, $i^2 = -1$, $i^3 = -i$, $i^4 = 1$. Since $100 = 25 \cdot 4$, we have $i^{100} = (i^4)^{25} = 1^{25} = 1$.
25.
Tags: Sequences · Hard · source: Original (AMC-style)
The sequence $a_1, a_2, a_3, \ldots$ is defined by $a_1 = 1$ and $a_{n+1} = 2a_n + 1$ for $n \geq 1$. What is $a_5$?
A) $15$
B) $31$
C) $63$
D) $127$
E) $255$
Answer & Solution
Answer: B
We have $a_{n+1} + 1 = 2(a_n + 1)$. Let $b_n = a_n + 1$. Then $b_{n+1} = 2b_n$ with $b_1 = 2$. So $b_n = 2^n$, giving $a_n = 2^n - 1$. Therefore $a_5 = 2^5 - 1 = 32 - 1 = 31$.
Answer Key
| # | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Ans | A | B | B | A | A | B | B | B | C | B | B | C | D | C | B | C | B | C | A | D | B | A | C | C | B |