10 questions · Form 4 Additional Mathematics Bab 4: Indices, Surds and Logarithms
Rationalise the denominator of 6 / √3.
Prefer reading to quizzing? All 10 questions are listed below with the answer and explanation under each one.
1. Rationalise the denominator of 6 / √3.
Answer: A
Multiply numerator and denominator by √3: 6 × √3√3 × √3 = 6√33 = 2√3.
2. Simplify √20 + √45 - √5.
Answer: A
Simplify each surd: √20 = 2√5, √45 = 3√5. Then 2√5 + 3√5 - 1√5 = (2 + 3 - 1)√5 = 4√5.
3. 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.
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. Which of the following numbers is an example of a surd?
Answer: C
√16 = 4, √(94) = 32, and ∛27 = 3 are all rational numbers. √7 is irrational and cannot be expressed as a ratio of integers, making it a surd.
7. Simplify (3√2 + √5)(3√2 - √5).
Answer: A
Use difference of squares (a + b)(a - b) = a² - b²: (3√2)² - (√5)² = (9 × 2) - 5 = 18 - 5 = 13.
8. 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.
9. Given log₁₀ 2 = p and log₁₀ 3 = q, express log₁₀ 18 in terms of p and q.
Answer: A
18 = 2 × 3² = 2 × 9. log₁₀ (2 × 3²) = log₁₀ 2 + log₁₀ (3²) = log₁₀ 2 + 2 log₁₀ 3 = p + 2q.
10. 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.