📘 Precalculus Mini Mock 25
Recommended: 75–90 minutes. No calculator.
Problems
1.
Tags: Basic Trig · Easy · source: Original (AMC-style)
What is sin(30°)?
A) $\frac{1}{2}$ B) $\frac{\sqrt{2}}{2}$ C) $\frac{\sqrt{3}}{2}$ D) $1$ E) $\frac{1}{\sqrt{2}}$
Answer & Solution
Answer: A
sin(30°) = 1/2.
2.
Tags: Cosine Values · Easy · source: Original (AMC-style)
What is cos(60°)?
A) $\frac{1}{2}$ B) $\frac{\sqrt{2}}{2}$ C) $\frac{\sqrt{3}}{2}$ D) $1$ E) $\frac{1}{\sqrt{2}}$
Answer & Solution
Answer: A
cos(60°) = 1/2.
3.
Tags: Tangent Values · Easy · 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.
4.
Tags: Special Angles · Easy · source: Original (AMC-style)
What is sin(90°)?
A) $0$ B) $\frac{1}{2}$ C) $\frac{\sqrt{2}}{2}$ D) $1$ E) $\frac{\sqrt{3}}{2}$
Answer & Solution
Answer: D
sin(90°) = 1.
5.
Tags: Trig Identities · Easy · source: Original (AMC-style)
What is sin²(30°) + cos²(30°)?
A) $0$ B) $\frac{1}{2}$ C) $1$ D) $\frac{3}{2}$ E) $2$
Answer & Solution
Answer: C
By the Pythagorean identity: sin²(θ) + cos²(θ) = 1 for any angle θ.
6.
Tags: Complex Numbers · Easy · source: Original (AMC-style)
What is (2 + 3i) + (1 + 2i)?
A) $3 + 5i$ B) $1 + i$ C) $3 + i$ D) $1 + 5i$ E) $2 + 5i$
Answer & Solution
Answer: A
Add real and imaginary parts separately: (2+1) + (3+2)i = 3 + 5i.
7.
Tags: Imaginary Unit · Easy · source: Original (AMC-style)
What is i²?
A) $1$ B) $-1$ C) $i$ D) $-i$ E) $0$
Answer & Solution
Answer: B
By definition, i² = -1.
8.
Tags: Complex Modulus · Easy · source: Original (AMC-style)
What is |3 + 4i|?
A) $3$ B) $4$ C) $5$ D) $7$ E) $12$
Answer & Solution
Answer: C
|a + bi| = √(a² + b²), so |3 + 4i| = √(9 + 16) = √25 = 5.
9.
Tags: Logarithm Basics · Easy · source: Original (AMC-style)
What is log₂(8)?
A) $2$ B) $3$ C) $4$ D) $8$ E) $16$
Answer & Solution
Answer: B
log₂(8) = 3 because 2³ = 8.
10.
Tags: Exponential Basics · Easy · source: Original (AMC-style)
What is 2³?
A) $4$ B) $6$ C) $8$ D) $9$ E) $12$
Answer & Solution
Answer: C
2³ = 2 × 2 × 2 = 8.
11.
Tags: Double Angle · Medium · source: Original (AMC-style)
What is sin(2 × 30°)?
A) $\frac{1}{2}$ B) $\frac{\sqrt{2}}{2}$ C) $\frac{\sqrt{3}}{2}$ D) $1$ E) $\frac{1}{\sqrt{2}}$
Answer & Solution
Answer: C
sin(60°) = √3/2.
12.
Tags: Complex Multiplication · Medium · 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
Using difference of squares: (1+i)(1-i) = 1² - i² = 1 - (-1) = 2.
13.
Tags: Trig Period · Medium · source: Original (AMC-style)
What is the period of sin(x)?
A) $\pi$ B) $2\pi$ C) $4\pi$ D) $\frac{\pi}{2}$ E) $\frac{\pi}{4}$
Answer & Solution
Answer: B
The period of sin(x) is 2π.
14.
Tags: Logarithm Properties · Medium · source: Original (AMC-style)
What is log₃(27) + log₃(3)?
A) $2$ B) $3$ C) $4$ D) $5$ E) $6$
Answer & Solution
Answer: C
log₃(27) + log₃(3) = log₃(27 × 3) = log₃(81) = 4.
15.
Tags: Exponential Growth · Medium · source: Original (AMC-style)
If 2ˣ = 8, what is x?
A) $2$ B) $3$ C) $4$ D) $5$ E) $6$
Answer & Solution
Answer: B
2ˣ = 8 = 2³, so x = 3.
16.
Tags: Complex Roots · Medium · source: Original (AMC-style)
What is √(-9)?
A) $3$ B) $-3$ C) $3i$ D) $-3i$ E) $9i$
Answer & Solution
Answer: C
√(-9) = √(9 × -1) = √9 × √(-1) = 3i.
17.
Tags: Trig Amplitude · Medium · source: Original (AMC-style)
What is the amplitude of 4sin(x)?
A) $1$ B) $2$ C) $4$ D) $8$ E) $16$
Answer & Solution
Answer: C
The amplitude of 4sin(x) is 4.
18.
Tags: Logarithm Change of Base · Medium · source: Original (AMC-style)
What is log₈(64)?
A) $1$ B) $2$ C) $3$ D) $4$ E) $8$
Answer & Solution
Answer: B
log₈(64) = 2 because 8² = 64.
19.
Tags: Complex Conjugate · Medium · source: Original (AMC-style)
What is the conjugate of 2 + 5i?
A) $2 - 5i$ B) $-2 + 5i$ C) $-2 - 5i$ D) $5 + 2i$ E) $5 - 2i$
Answer & Solution
Answer: A
The conjugate of a + bi is a - bi, so the conjugate of 2 + 5i is 2 - 5i.
20.
Tags: Trig Equations · Medium · source: Original (AMC-style)
What is the general solution to cos(x) = 0?
A) $x = 0$ B) $x = \pi$ C) $x = \frac{\pi}{2} + n\pi$ for integer n D) $x = n\pi$ for integer n E) $x = 2n\pi$ for integer n
Answer & Solution
Answer: C
cos(x) = 0 when x = π/2 + nπ for any integer n.
21.
Tags: Advanced Trig · Hard · source: Original (AMC-style)
What is sin(15°)?
A) $\frac{\sqrt{6} - \sqrt{2}}{4}$ B) $\frac{\sqrt{6} + \sqrt{2}}{4}$ C) $\frac{\sqrt{3} - 1}{2}$ D) $\frac{\sqrt{3} + 1}{2}$ E) $\frac{1}{2}$
Answer & Solution
Answer: A
Using the angle difference formula: sin(15°) = sin(45° - 30°) = (√6 - √2)/4.
22.
Tags: Complex Powers · Hard · source: Original (AMC-style)
What is (1 + i)⁴?
A) $1$ B) $-1$ C) $i$ D) $-i$ E) $-4$
Answer & Solution
Answer: E
(1 + i)² = 1 + 2i + i² = 1 + 2i - 1 = 2i. So (1 + i)⁴ = (2i)² = 4i² = 4(-1) = -4.
23.
Tags: Advanced Logarithms · Hard · source: Original (AMC-style)
What is log₂(3) + log₃(2)?
A) $1$ B) $2$ C) $\log_6(6)$ D) $\log_6(12)$ E) Cannot be simplified
Answer & Solution
Answer: E
This expression cannot be simplified using standard logarithm properties.
24.
Tags: Complex Equations · Hard · source: Original (AMC-style)
What is the solution to z² + 1 = 0?
A) $z = 1$ B) $z = -1$ C) $z = i$ D) $z = ±i$ E) $z = ±1$
Answer & Solution
Answer: D
z² + 1 = 0 gives z² = -1, so z = ±i.
25.
Tags: Advanced Trig Identity · Hard · source: Original (AMC-style)
What is sin(75°)cos(15°)?
A) $\frac{1}{4}$ B) $\frac{1}{2}$ C) $\frac{\sqrt{2}}{4}$ D) $\frac{\sqrt{3}}{4}$ E) $\frac{1}{8}$
Answer & Solution
Answer: A
Using product-to-sum: sin(75°)cos(15°) = (1/2)[sin(90°) + sin(60°)] = (1/2)[1 + √3/2] = 1/4.
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 | A | C | D | C | A | B | C | B | C | C | C | B | C | B | C | C | B | A | C | A | E | E | D | A |