10 questions · Form 4 Additional Mathematics Bab 4: Indices, Surds and Logarithms
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Prefer reading to quizzing? All 10 questions are listed below with the answer and explanation under each one.
1. If 3^(x + 2) = 27^(x - 1), find the value of x.
Answer: A
Express base as 3: 3^(x + 2) = (3³)^(x - 1) => 3^(x + 2) = 33x - 3) => x + 2 = 3x - 3 => 2x = 5 => x = 52.
2. Rationalise the denominator of 6 / √3.
Answer: A
Multiply numerator and denominator by √3: 6 × √3√3 × √3 = 6√33 = 2√3.
3. Convert the logarithmic equation log₃ 81 = 4 into index form.
Answer: B
Using log_a N = x ≤> a^x = N, log₃ 81 = 4 converts directly to 3⁴ = 81.
4. Simplify the surd √72 into its simplest form a√b.
Answer: C
Find the largest perfect square factor: √72 = √36 × 2 = √36 × √2 = 6√2.
5. Solve 5^x = 20 correct to 3 decimal places.
Answer: A
Take log₁₀ on both sides: log₁₀ (5^x) = log₁₀ 20 => x log₁₀ 5 = log₁₀ 20 => x = log₁₀ 20 / log₁₀ 5 ≈ 1.301030.69897 ≈ 1.861.
6. Express 12 + √3 with a rational denominator.
Answer: A
Multiply by conjugate (2 - √3): 1 × (2 - √3)(2 + √3)(2 - √3) = 2 - √34 - 3 = 2 - √3.
7. Given log₂ y = 3 - 2 log₂ x, express y in terms of x.
Answer: A
log₂ y + 2 log₂ x = 3 => log₂ y + log₂ (x²) = 3 => log₂ (y x²) = 3 => y x² = 2³ = 8 => y = 8x².
8. What is the value of log_x x - log_x 1?
Answer: A
Since log_x x = 1 and log_x 1 = 0, we have 1 - 0 = 1.
9. Solve the logarithmic equation log₂ (x + 2) + log₂ (x - 2) = 3.
Answer: A
Combine logs: log₂ [(x+2)(x-2)] = 3 => log₂ (x² - 4) = 3 => x² - 4 = 2³ = 8 => x² = 12 => x = √12 = 2√3. (Reject x = -2√3 since log domain x > 2).
10. Find the conjugate surd of (3 - √5) and simplify their product.
Answer: A
The conjugate of (3 - √5) is (3 + √5). Product: (3 - √5)(3 + √5) = 3² - (√5)² = 9 - 5 = 4.