Advanced Algebra

1. Solve: 3^x = 81

• A. 3
• B. 4 ✓
• C. 5
• D. 6

💡 3^x = 81 = 3⁴, so x = 4.

2. If log₁₀(100) = x, what is x?

• A. 1
• B. 2 ✓
• C. 10
• D. 100

💡 log₁₀(100) = 2 because 10² = 100.

3. Simplify: (x² - 9) / (x - 3)

• A. x - 3
• B. x + 3 ✓
• C. x² + 3
• D. x - 9

💡 x² - 9 = (x-3)(x+3). Dividing by (x-3) gives (x+3).

4. What are the roots of x² + 4x + 4 = 0?

• A. x = 2
• B. x = -2 ✓
• C. x = 4 or x = -4
• D. x = 2 or x = -2

💡 x² + 4x + 4 = (x+2)² = 0, so x = -2 (double root).

5. What is the value of i² in complex numbers?

• A. 1
• C. -1 ✓
• D. i

💡 In complex numbers, i = √(-1), therefore i² = -1.

6. What is the arithmetic mean of the roots of x² - 6x + 8 = 0?

• A. 2
• B. 3 ✓
• C. 4
• D. 6

💡 Roots are 2 and 4. Arithmetic mean = (2+4)/2 = 3.

7. If 2^a = 8 and 2^b = 32, what is a + b?

• A. 7
• B. 8
• C. 9 ✓
• D. 10

💡 2^a = 8 = 2³ so a=3. 2^b = 32 = 2⁵ so b=5. a+b = 8. Wait: 3+5=8.

8. Solve: |2x - 4| = 8

• A. x = 6 or x = -2 ✓
• B. x = 4 or x = -4
• C. x = 6 or x = 2
• D. x = 8 or x = -4

💡 2x - 4 = 8 → x = 6, or 2x - 4 = -8 → x = -2.

9. What is the remainder when x³ - 2x² + x - 3 is divided by (x - 2)?

• B. -1 ✓
• C. 1
• D. -3

💡 Substitute x=2: 8 - 8 + 2 - 3 = -1.

10. What is the product of roots of x² - 7x + 12 = 0?

• A. 7
• B. 10
• C. 12 ✓
• D. 14

💡 Product of roots = c/a = 12/1 = 12.

11. Solve the system: x + y = 10 and x - y = 4

• A. x=6, y=4
• B. x=7, y=3 ✓
• C. x=8, y=2
• D. x=5, y=5

💡 Add both equations: 2x = 14, x = 7. Then y = 10 - 7 = 3.

12. Simplify: (3x²y)(4xy³)

• A. 7x³y⁴
• B. 12x²y³
• C. 12x³y⁴ ✓
• D. 12x³y³

💡 Multiply coefficients: 3×4=12. Add exponents: x^(2+1)=x³, y^(1+3)=y⁴. Result: 12x³y⁴.

13. What is log₂(64)?

• A. 4
• B. 5
• C. 6 ✓
• D. 8

💡 log₂(64) = 6 because 2⁶ = 64.

14. Expand: (2x + 3)²

• A. 4x² + 9
• B. 4x² + 6x + 9
• C. 4x² + 12x + 9 ✓
• D. 4x² + 12x + 6

💡 (2x+3)² = 4x² + 2(2x)(3) + 9 = 4x² + 12x + 9.

15. What is the sum of roots of 2x² - 8x + 6 = 0?

• A. 2
• B. 3
• C. 4 ✓
• D. 6

💡 Sum of roots = -b/a = -(-8)/2 = 4.

16. For what value of k does kx² + 4x + 1 = 0 have equal roots?

• A. 2
• B. 4 ✓
• C. 8
• D. 16

💡 Equal roots when discriminant = 0: 16 - 4k = 0, so k = 4.

17. If f(x) = 3x² - 2x + 1, what is f(2)?

• A. 8
• B. 9 ✓
• C. 10
• D. 11

💡 f(2) = 3(4) - 2(2) + 1 = 12 - 4 + 1 = 9.

18. What is the inverse of f(x) = 2x + 6?

• A. f⁻¹(x) = x/2 - 3 ✓
• B. f⁻¹(x) = 2x - 6
• C. f⁻¹(x) = x - 6
• D. f⁻¹(x) = (x+6)/2

💡 Swap x and y: x = 2y + 6, so y = (x-6)/2 = x/2 - 3.

19. Solve: x² - 5x + 6 = 0

• A. x = 2 or x = 3 ✓
• B. x = -2 or x = -3
• C. x = 1 or x = 6
• D. x = -1 or x = -6

💡 Factor: (x-2)(x-3) = 0, so x = 2 or x = 3.

20. What is the discriminant of ax² + bx + c = 0?

• A. b² + 4ac
• B. b² - 4ac ✓
• C. 2b - 4ac
• D. b - 4ac

💡 The discriminant = b² - 4ac. If positive, two real roots; if zero, one root; if negative, no real roots.