GCE Further Mathematics
Past Questions

AI Explanation

By De Morgan's Law, the complement of a union equals the intersection of the complements: (A ∪ B)' = A' ∩ B'. Option A is De Morgan's Law for the complement of an intersection. Options C and D are not complements of A ∪ B.

AI Explanation

Substituting p = 2 and q = 3: 2 ⊕ 3 = 2² + 3² − (2)(3) = 4 + 9 − 6 = 7. Option B results from computing 4 + 9 − 2 = 11, option C from 4 + 9 = 13 without subtracting pq, and option D from adding pq instead of subtracting.

AI Explanation

The intersection P ∩ Q contains elements that belong to both P and Q. The elements common to both sets are 2 and 4, so P ∩ Q = {2, 4}. Option A is the union, option C is the symmetric difference, and option D contains elements only in Q.

AI Explanation

Substituting a = 3 and b = 2 into the operation: 3 * 2 = 3 + 2 − 2(3)(2) = 5 − 12 = −7. Common errors include forgetting to multiply by 2 in the last term (giving 5 − 6 = −1) or computing 2ab as 2(3 + 2) = 10 instead of 2 × 3 × 2 = 12.

AI Explanation

For the identity element e, we need a * e = a, so (a + e)/(1 + ae) = a. This gives a + e = a(1 + ae) = a + a²e, so e = a²e, meaning e(1 − a²) = 0 for all a, which holds when e = 0. Verifying: a * 0 = (a + 0)/(1 + 0) = a. ✓ The identity element is 0.

AI Explanation

Using the inclusion-exclusion principle: n(M ∪ P) = n(M) + n(P) − n(M ∩ P) = 25 + 20 − 10 = 35. The number of students studying neither subject is 40 − 35 = 5. Option B incorrectly uses only those studying both, while option C and D result from arithmetic errors.

AI Explanation

The inverse of 3 is the element x such that 3 Δ x = 0 (the identity). We need (3 + x) mod 5 = 0, so 3 + x must be a multiple of 5. With x = 2: (3 + 2) mod 5 = 5 mod 5 = 0. ✓ Thus the inverse of 3 is 2. Option D (x = 4) gives (3 + 4) mod 5 = 2, not 0.

AI Explanation

Since 15 sell neither, those selling at least one item = 100 − 15 = 85. By inclusion-exclusion: n(R ∪ B) = n(R) + n(B) − n(R ∩ B), so 85 = 60 + 45 − n(R ∩ B), giving n(R ∩ B) = 105 − 85 = 20. Options A, B, and D result from errors in applying the inclusion-exclusion formula.

AI Explanation

The complement A' consists of all elements in the universal set U that are NOT in A. Removing {2, 4, 6, 8} from U = {1, 2, 3, 4, 5, 6, 7, 8} leaves {1, 3, 5, 7}. Option B is A itself, option C is U, and option D is the empty set.

AI Explanation

Factor the numerator: 3^n(3 − 1) = 3^n × 2. Dividing by 3^(n−1) gives 3^n × 2 / 3^(n−1) = 3^(n−(n−1)) × 2 = 3¹ × 2 = 6. The key step is factoring 3^n from the numerator before applying index laws.

AI Explanation

√75 = 5√3, √48 = 4√3, √27 = 3√3. So 5√3 + 4√3 − 3√3 = 6√3. Each surd is simplified by extracting perfect square factors before combining like terms.

AI Explanation

log₂ 8 = 3 and log₂ 4 = 2, so log₂ 8 + log₂ 4 = log₂(8 × 4) = log₂ 32. Therefore x = 32. The product rule of logarithms states log_a m + log_a n = log_a(mn).

AI Explanation

log₃(x² − 2x) = 1 means x² − 2x = 3¹ = 3. Rearranging: x² − 2x − 3 = 0, which factors as (x − 3)(x + 1) = 0, giving x = 3 or x = −1. Both values make x² − 2x = 3 > 0, so both are valid solutions.

AI Explanation

Multiply numerator and denominator by the conjugate (3 + √2): numerator becomes 5(3 + √2) = 15 + 5√2, denominator becomes 3² − (√2)² = 9 − 2 = 7. The result is (15 + 5√2)/7.

AI Explanation

log_x 64 = 3 means x³ = 64. Since 4³ = 64, x = 4. The definition of logarithm converts the equation directly into an exponential form for easy solution.

AI Explanation

Let u = 2^x, so u² − 5u + 4 = 0, giving (u − 1)(u − 4) = 0. Thus u = 1 or u = 4, meaning 2^x = 1 = 2⁰ so x = 0, or 2^x = 4 = 2² so x = 2. Wait — u=4 gives x=2, so the answer should be x=0 or x=2, which is option A.

AI Explanation

72 = 8 × 9 = 2³ × 3². So log 72 = 3 log 2 + 2 log 3 = 3(0.3010) + 2(0.4771) = 0.9030 + 0.9542 = 1.8572 ≈ 1.8573. Expressing 72 as a product of prime factors is the essential first step.

AI Explanation

Multiply numerator and denominator by the conjugate (√5 + √3): numerator = (√5 + √3)² = 5 + 2√15 + 3 = 8 + 2√15; denominator = 5 − 3 = 2. Dividing gives (8 + 2√15)/2 = 4 + √15.

AI Explanation

f(1) = 1 - 2 + a + b = 4 gives a + b = 5. f(-1) = -1 - 2 - a + b = -2 gives -a + b = 1. Solving simultaneously: adding gives 2b = 6, so b = 3... wait, let me recheck: a+b=5 and -a+b=1, adding: 2b=6, b=3, a=2. The correct answer is B. [Correction: answer should be B]

AI Explanation

Let (5x-1)/[(x+1)(x-2)] = A/(x+1) + B/(x-2). Multiplying through: 5x - 1 = A(x-2) + B(x+1). Setting x = 2: 9 = 3B, so B = 3. Setting x = -1: -6 = -3A, so A = 2. The partial fractions are 2/(x+1) + 3/(x-2).

AI Explanation

Since (x - 2) is a factor, f(2) = 0. So 8 + 4k - 8 + 4 = 0, giving 4k + 4 = 0, hence k = -1. Substituting back confirms f(2) = 8 - 4 - 8 + 4 = 0.

AI Explanation

Testing x = 3: p(3) = 27 - 54 + 33 - 6 = 0, so (x - 3) is a factor by the Factor Theorem. Testing the other options gives non-zero values, confirming they are not factors.

AI Explanation

Substituting x = 2: f(2) = 2(8) - 3(4) + 2 - 5 = 16 - 12 + 2 - 5 = 1. The correct answer is 1. Common errors include sign mistakes or incorrect order of operations.

AI Explanation

By the Remainder Theorem, the remainder is f(-1) = 2(1) - 3(-1) + 1(-1)² - 5(-1) + 2 = 2 + 3 + 1 + 5 + 2 = 13. The answer is 13.

AI Explanation

Let (3x²+x+2)/[(x²+1)(x-1)] = (Ax+B)/(x²+1) + C/(x-1). Multiplying through: 3x²+x+2 = (Ax+B)(x-1) + C(x²+1). Setting x=1: 6 = 2C, so C=3. Expanding and comparing coefficients of x²: A+C=3, so A=0... Comparing constants: -B+C=2, so B=1. Comparing x terms: A-B=1, so A=2. Thus partial fractions = (2x-1)/(x²+1) + 3/(x-1).