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GATE 2023 EC GA Q.9: solution and answer

GATE 2023 · ECGA Q.9 · 2 marks · MCQ

Out of 1000 individuals in a town, 100 unidentified individuals are covid positive. Due to lack of adequate covid-testing kits, the health authorities of the town devised a strategy to identify these covid-positive individuals. The strategy is to:

(i) Collect saliva samples from all 1000 individuals and randomly group them into sets of 5.

(ii) Mix the samples within each set and test the mixed sample for covid.

(iii) If the test done in (ii) gives a negative result, then declare all the 5 individuals to be covid negative.

(iv) If the test done in (ii) gives a positive result, then all the 5 individuals are separately tested for covid.

Given this strategy, no more than ______ testing kits will be required to identify all the 100 covid positive individuals irrespective of how they are grouped.

  1. (A)

    700

  2. (B)

    600

  3. (C)

    800

  4. (D)

    1000

See the question in the paper
GATE 2023 EC GA question 9Tap to open full size

Answer and solution

ANSWER

No more than 700 testing kits are needed: 200 kits for the pooled tests, plus 5 kits for each of at most 100 positive sets. The answer is (A) 700.

Official answer: ASource: Official answer key, GATE 2023 EC (IIT Kanpur)

  1. Number of sets: . Step (ii) uses one kit on the mixed sample of every set, so kits are always used.

  2. A set needs more kits only if its mixed sample tests positive. Step (iv) then tests all members separately, which uses more kits.

  3. If sets test positive, the total is kits. So the worst case ("irrespective of how they are grouped") is the grouping with the most positive sets.

  4. A set tests positive only if it holds at least one positive person. There are positive people and sets, so at most sets can be positive. That happens when every positive person is in a different set, which is possible because .

  5. Extra kits in the worst case: .

  6. Most kits ever needed:

  7. Check: any grouping that puts two or more positives in one set has fewer than positive sets, so it uses fewer than kits. The bound of is therefore never exceeded, and it is actually reached, so the answer is , option (A).

KEY IDEA

Total kits = (number of sets) + 5 × (number of positive sets). The pooled tests always cost 200 kits, so the worst case is the grouping with the most positive sets. With 100 positives, that is 100 sets, one positive in each.

WATCH OUT FOR

One mistake is to count only the 100 positive people and answer 200 + 100 = 300, or to forget the 200 pooled tests and answer 500. Both miss that a positive pool means retesting all 5 members, and that the 200 pooled tests happen whatever the result. Another is to take the worst case as all 200 sets being positive (200 + 1000 = 1200), which is impossible because there are only 100 positive people.

From the GATE 2023 EC paper. Every question of this paper · Download the full paper · All 41 Reasoning questions, with notes

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