GATE Chemical 2018 | GA | Question: 8

Question

To pass a test, a candidate needs to answer at least $2$ out of $3$ questions correctly. A total of $6,30,000$ candidates appeared for the test. Question $A$ was correctly answered by $3,30,000$ candidates. Question $B$ was answered correctly by $2,50,000$ candidates. Question $C$ was answered correctly by $2,60,000$ candidates. Both questions $A$ and $B$ were answered correctly by $1,00,000$ candidates. Both questions $B$ and $C$ were answered correctly by $90,000$ candidates. Both questions $A$ and $C$ were answered correctly by $80,000$ candidates. If the number of students answering all questions correctly is the same as the number answering none, how many candidates failed to clear the test?

• A. $30,000$
• B. $2,70,000$
• C. $3,90,000$
• D. $4,20,000$

Answer 1

Answer D)

Say there are x students who answered all correctly

So, according to condition there are x student who answered none

Equation using inclusion exclusion principle

$630000-x=330000+250000+260000-100000-90000-80000+x$

$x=30000$

Now, it is given that to pass u need to answer at least 2 out of 3 questions

So, those who answered one answer correctly also considered to be fail

Only A answered correctly$330000-100000-80000+30000=180000$

Only B answered correctly $250000-100000-90000+30000=90000$

Only C answered correctly $260000-80000-90000+30000=120000$

So, total failed student $180000+90000+120000+30000=420000$

Comments

@srestha

why 60,000 divide by 2 ???

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Exactly I want to ask, why 2x is taken? Because (aubuc) =630000

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@Chidesh1397 @mohan123   This is because LHS indicates the number of students who have answered at least one question correctly, which is equal to total number of students  $-$  number of students who have answered all questions correctly , which is equal to  total number of students  $-$  number of students who have not answered not a single question correctly(using the condition mentioned in the question).

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For ease of calculation, we can remove four zeroes from each number.

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Yes, that’s for the exam but here she has to write it, otherwise would look incomplete.

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Hi, shouldn't LHS be equal to "total students - students who did not attempted any questions correct" instead of no of students who answered all questions correct. As students who answered all correctly have answered at least one question correctly.

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Only A answered correctly330000−100000−80000+ 30000 =180000

@Deepak Poonia

@Lakshman Bhaiya

why we are adding 30000 herein above answer

 ?

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@MSG1999

Think with the perspective of Venn diagram. When subtracting $A \cap B$ and $A \cap C$ from $A$, we remove the intersection of $A \cap B \cap C$ 2 times. to account for this are then adding the intersection.

Answer 2

Answer D: 4,20,000

(Note: Figures used such that 100,000 is taken as 100 and 90,000 is taken as 90 for simplification in the solution picture. Multiply all figures by 1000 to get final answer).