Evaluation G - 12 | 1.10 Solutions (Proof Methods)

1.10.1 Which method works by assuming the opposite and deriving a logical impossibility? (A) Deduction (B) Exhaustion (C) Contradiction (D) Induction (E) Counterexample.

Proof by contradiction assumes the statement is false and reaches a contradiction → Answer: C

1.10.2 Prove there is no largest even integer. Most appropriate method? (A) Exhaustion (B) Contradiction (C) Worked example (D) Counterexample (E) Diagram.

Assume there is a largest even integer N = 2k, then 2k+2 is larger and even → contradiction → Answer: B

1.10.3 To prove √2 irrational by contradiction, correct first assumption? (A) √2 = p/q in lowest terms (B) √2 is irrational (C) 2 is prime (D) p and q are both even (E) √2 = 2.

Assume the opposite of what we want to prove: √2 is rational = p/q in lowest terms → Answer: A

1.10.4 Proof by exhaustion is valid when: (A) infinite cases manageable (B) only prime numbers (C) every possible case is finite and individually verified (D) negation leads to contradiction (E) general algebraic argument.

Exhaustion requires checking every case, so the domain must be finite → Answer: C

1.10.5 Proving n² + n even by writing n(n+1) and noting one of n or n+1 is even. This is: (A) Exhaustion (B) Contradiction (C) Counterexample (D) Deduction (E) Induction.

General logical reasoning from known facts to conclusion → deduction → Answer: D

1.10.6 Disprove "All prime numbers are odd" with: (A) Deduction (B) Counterexample p=2 (C) Contradiction (D) Exhaustion under 100 (E) odd×odd=odd.

A single counterexample disproves a universal statement → Answer: B

1.10.7 In √3 irrational proof, 3q² = p² implies what about p? (A) p odd (B) multiple of 9 (C) multiple of 3 (D) p even (E) p = q.

If p² is divisible by 3, then p is divisible by 3 (since 3 is prime) → Answer: C

1.10.8 Proof by exhaustion that n³ − n divisible by 6 for n = 1 to 6. After checking all six cases, proof is: (A) Incomplete (B) Invalid (C) Complete and valid for 1 to 6 (D) Valid for all integers (E) Invalid.

Exhaustion only proves the statement for the cases checked, not for all integers → Answer: C

1.10.9 Which is correct about proof by deduction? (A) Tests every case (B) Assumes conclusion (C) Uses logical steps from accepted facts (D) Only for finite domains (E) Requires negation.

Deduction starts from known true statements and uses logical inference → Answer: C

1.10.10 Prove odd n → odd n². If n = 2k+1, then n² = 4k²+4k+1. Which conclusion completes proof? (A) even (B) = 2(2k²+2k)+1, odd (C) perfect square, odd (D) 4k²+4k odd (E) n² = (2k)²+1.

4k²+4k = 2(2k²+2k) is even, so 4k²+4k+1 is odd → Answer: B

1.10.11 Showing a=3, b=4, c=5 satisfies a²+b²=c² is: (A) Contradiction (B) Exhaustion (C) Disproof (D) Deduction (E) Specific example (existence proof).

One example proves existence → Answer: E

1.10.12 In proof by contradiction, after contradiction is reached, we conclude: (A) Original assumption modified (B) Incomplete (C) Original assumption false, statement true (D) True only for special cases (E) Assumption correct.

Contradiction shows the assumption is false, so the original statement is true → Answer: C

1.10.13 Counterexample to "For all real x, x² > x". Which value? (A) 2 (B) -3 (C) 1/2 (D) 10 (E) 5.

For x = 1/2, x² = 1/4, which is NOT > 1/2 → Answer: C

1.10.14 Valid proof by exhaustion that product of two consecutive integers is even: (A) One must be even (B) Check n=1,2,3 only (C) Assume odd, derive contradiction (D) Check both cases: n even gives even; n odd gives n+1 even → product even (E) Use formula n²+n.

Option D exhaustively checks both parity cases → Answer: D

1.10.15 p rational, q irrational. Which is always true about p+q? (A) rational (B) irrational, provable by contradiction (C) could be either (D) integer (E) irrational only if p≠0.

Assume p+q = rational, then q = (p+q) − p = rational − rational = rational → contradiction. So p+q is irrational → Answer: B

1.10.16 Prove "if n² even, then n even" by contradiction. What is assumed? (A) n even, n² odd (B) n² odd (C) n odd and n² even (D) n=0 (E) n² perfect square.

Assume the premise true (n² even) and conclusion false (n odd) → derive contradiction → Answer: C

1.10.17 Famous proof of infinitely many primes is by: (A) Exhaustion (B) Deduction only (C) Contradiction (Euclid's proof) (D) Counterexample (E) Specific example.

Euclid's proof assumes finitely many primes and constructs a new prime → contradiction → Answer: C

1.10.18 Best distinction between deduction and exhaustion: (A) Deduction uses examples (B) Deduction applies general logical argument; exhaustion checks every case (C) Exhaustion more rigorous (D) Deduction only for geometry (E) Exhaustion requires negation.

Deduction uses general reasoning; exhaustion checks each case individually → Answer: B

1.10.19 Proof that no perfect square ends in 3 by checking last digits 0-9 is: (A) Invalid (B) Contradiction (C) Exhaustion over all possible last digits (D) Deduction (E) Counterexample.

All possible last digits are exhausted (0-9) → Answer: C

1.10.20 Logical structure of proof by contradiction for statement P: (A) Assume P true, show P follows (B) Check all cases (C) Assume P false, derive a false statement, conclude P true (D) Find example where P holds (E) Assume P false, derive P true.

Proof by contradiction: assume ¬P, derive a contradiction, therefore P is true → Answer: C

Evaluation G - 12 | 2.1 Solutions (Sampling)

2.1.1 A researcher numbers all 800 students from 1 to 800, then uses a random number generator to select 80 students. This is: (A) Stratified (B) Systematic (C) Simple random sampling (D) Cluster (E) Convenience.

Each student has an equal chance of being selected using random numbers → simple random sampling → Answer: C

2.1.2 A school has 400 boys and 600 girls. A stratified sample of 100 students is taken. How many girls should be in the sample? (A) 40 (B) 50 (C) 60 (D) 70 (E) 80.

Girls proportion = 600/1000 = 0.6. Sample girls = 0.6 × 100 = 60 → Answer: C

2.1.3 A journalist interviews the first 20 people who walk past her office. This is: (A) Simple random (B) Systematic (C) Stratified (D) Convenience sampling (E) Quota.

Choosing the most accessible individuals → convenience sampling → Answer: D

2.1.4 From a population of 500, every 10th person on an alphabetical list is selected after a random start. This is: (A) Simple random (B) Cluster (C) Systematic sampling (D) Stratified (E) Opportunity.

Selecting items at regular intervals after a random start → systematic sampling → Answer: C

2.1.5 A city is divided into 50 residential blocks. Researchers randomly select 5 blocks and survey every household in those blocks. This is: (A) Stratified (B) Systematic (C) Simple random (D) Cluster sampling (E) Quota.

Randomly selecting entire clusters (blocks) and sampling all within them → cluster sampling → Answer: D

2.1.6 A factory produces 2000 items per day. A quality inspector tests every 40th item starting from item 17. How many items are tested per day? (A) 40 (B) 50 (C) 17 (D) 25 (E) 45.

Sampling interval = 40, so number sampled = 2000 ÷ 40 = 50 → Answer: B

2.1.7 A sample is taken by selecting all students in three randomly chosen classes. If classes are streamed by ability, which bias is most likely? (A) Response bias (B) Leading question (C) Non-response (D) Cluster bias causing under- or over-representation (E) Sampling frame bias.

Streamed classes mean some ability groups may be over- or under-represented → cluster bias → Answer: D

2.1.8 A survey asks: "Don't you agree that the new stadium is a waste of taxpayers' money?" Which problem makes this data unreliable? (A) Sample size too small (B) Leading question introducing response bias (C) Systematic sampling not used (D) Population not stratified (E) Data is qualitative.

The wording pushes the respondent toward an agreement → leading question → response bias → Answer: B

2.1.9 A company employs 120 part-time and 280 full-time workers. A stratified sample of 50 workers is needed. How many part-time workers? (A) 12 (B) 15 (C) 20 (D) 25 (E) 30.

Part-time proportion = 120/400 = 0.3. Sample part-time = 0.3 × 50 = 15 → Answer: B

2.1.10 Which is a census rather than a sample? (A) Testing 50 of 1000 lightbulbs (B) Every 5th voter (C) Recording age of every citizen in Nigeria (D) 3 random classes (E) 200 randomly chosen patients.

A census includes every member of the population → recording every citizen → Answer: C

2.1.11 A researcher collects data only from people who voluntarily respond to an online survey. Main disadvantage: (A) Sample too large (B) More expensive (C) Volunteers share characteristics → unrepresentative (D) Cannot collect quantitative data (E) Sampling frame is complete.

Volunteer/self-selection bias leads to an unrepresentative sample → Answer: C

2.1.12 A stratified sample of 200 people from a town. Strata: under 18 (500), 18-60 (1500), over 60 (1000). How many aged 18-60? (A) 50 (B) 75 (C) 100 (D) 150 (E) 120.

Total population = 500+1500+1000 = 3000. Proportion of 18-60 = 1500/3000 = 0.5. Sample = 0.5 × 200 = 100 → Answer: C

2.1.13 A dataset shows mean monthly income of 30 sampled workers is ₦85,000. A politician claims mean income of all Lagos workers is exactly ₦85,000. What is wrong? (A) Mean not appropriate (B) Sample statistic used as certain population parameter, ignoring sampling variability (C) Sample size too large (D) Income cannot be sampled (E) Mode should be used.

Sample statistics vary; they rarely equal the population parameter exactly → Answer: B

2.1.14 A systematic sample of size 40 is taken from a population of 1200. Random start = 11. What is the 3rd value selected? (A) 41 (B) 51 (C) 61 (D) 71 (E) 81.

Sampling interval = 1200 ÷ 40 = 30. Values: 11, 41, 71... 3rd = 71 → Answer: D

2.1.15 Which best describes a sampling frame? (A) Data collection method (B) Complete list of all population members from which a sample is drawn (C) Set of questions (D) Sample size formula (E) Diagram of strata.

A sampling frame is the list of all individuals in the population → Answer: B

2.1.16 A health study records blood pressure of patients attending a clinic on a Tuesday morning. Which limitation most affects appropriateness for all patients? (A) Blood pressure hard to measure (B) Sample is time- and location-specific, may not represent all patients (C) Should have used questionnaire (D) Blood pressure data is qualitative (E) Sample size too small.

Convenience sample at a specific time/place may not represent the whole patient population → Answer: B

2.1.17 Researcher A uses stratified random sampling on 500 households; Researcher B surveys 500 outside one supermarket. Whose data is more appropriate for national conclusions? (A) B, more variety (B) Both equally valid (C) A, stratified random sampling better represents diversity (D) B, reduces non-response (E) Neither, 500 too small.

Stratified random sampling ensures representation of subgroups; supermarket sample is biased → Answer: C

2.1.18 A school of 900 students wants a systematic sample of 45. What is the sampling interval? (A) 15 (B) 20 (C) 25 (D) 40 (E) 45.

Sampling interval = population ÷ sample size = 900 ÷ 45 = 20 → Answer: B

2.1.19 A survey on study habits asks only students in the school library at lunchtime. Why is this data not appropriate for the whole school? (A) Library students read more (B) Sample biased toward library users, who may have different habits (C) Lunchtime invalid (D) Sample size too large (E) Study habits need national tests.

Convenience sample from library excludes non-library users → bias → Answer: B

2.1.20 A population of 3000 employees is split: Administration (600), Operations (1800), Sales (600). A stratified sample of 150 is taken. How many from Operations? (A) 50 (B) 60 (C) 75 (D) 90 (E) 100.

Operations proportion = 1800/3000 = 0.6. Sample = 0.6 × 150 = 90 → Answer: D

Evaluation G - 12 | 2.2 Solutions (Data Representation)

2.2.1 Class 20≤x<30: freq 40, width 10 → freq density = 40/10 = 4. Class 30≤x<45: freq 60, width 15 → freq density = 60/15 = 4. Both have equal density → Answer: C

2.2.2 Frequency = height × width = 3.5 × 8 = 28 → Answer: B

2.2.3 For 200 data values, median = (200/2) = 100th value → read at cumulative frequency 100 → Answer: B

2.2.4 Lower quartile Q1 = (120/4) = 30th value → cumulative frequency 30 → Answer: B

2.2.5 IQR = Q3 − Q1 = 41 − 22 = 19 → Answer: C

2.2.6 IQR = 46 − 28 = 18. Upper outlier boundary = Q3 + 1.5×IQR = 46 + 27 = 73 → Answer: A

2.2.7 Smaller IQR = more consistent; higher median = generally higher scores → Class A more consistent and higher → Answer: A

2.2.8 Total frequency = (6×5) + (10×5) + (4×10) = 30 + 50 + 40 = 120 → Answer: C

2.2.9 Steeper slope on cumulative frequency curve means more data concentrated in that interval → Answer: B

2.2.10 60th percentile = 60% of 80 = 48th value → cumulative frequency 48 → Answer: B

2.2.11 Median (40) is equidistant from Q1 (25) and Q3 (55): both 15 away → symmetrical → Answer: C

2.2.12 Frequency density = frequency ÷ class width = 48 ÷ 20 = 2.4 → Answer: C

2.2.13 Median closer to Q3 (5.8 vs 7.1 difference 1.3, vs 5.8−3.2=2.6) → closer to Q3 indicates negative skew → Answer: B

2.2.14 Class frequencies: (5×4)=20, (8×5)=40, (3×10)=30. Total = 90. Fraction in second class = 40/90 = 4/9 → Answer: D

2.2.15 Box plot shows min, Q1, median, Q3, max. Mean cannot be read directly → Answer: D

2.2.16 Semi-interquartile range = (Q3 − Q1)/2 = (54 − 33)/2 = 21/2 = 10.5 → Answer: C

2.2.17 Drivers in 0.3≤t<0.5 (9) + 0.5≤t<0.9 (12) = 21 → Answer: A

2.2.18 Company X: higher median (₦180,000 vs ₦165,000) and smaller IQR (₦40,000 vs ₦95,000) → higher average and more consistent → Answer: B

2.2.19 Students scoring 65 or above = 200 − 140 = 60. Percentage = 60/200 × 100% = 30% → Answer: C

2.2.20 Modal class is the one with highest frequency density: 2.5, 8, 6, 1.5 → highest is 8 for class 20≤l<25 → Answer: B