GATE Mechanical 2017 Set 1 | GA | Question: 5

Question

$P$, $Q$ and $R$ talk about $S's$ car collection. $P$ states that $S$ has at least $3$ cars. $Q$ believes that $S$ has less than $3$ cars. $R$ indicates that to its knowledge, $S$ has at least one car. Only one of $P, Q$ and $R$ is right. The number of cars owned by $S$ is.

• A. $0$
• B. $1$
• C. $3$
• D. Cannot be determined.

Comments

i simply did this. 

 

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nice approach @rhl, i was overthinked and tried to do the exclusive OR between the statements .

Answer 1

There are $3$ possible cases :-

$(1)$ $P$ is right, $Q$ and $R$ are wrong

It means $P$'s statement "S has at least $3$ cars" is true and $R$'s statement "S has at least one car" is false which means $S$ has actually less than one car which is contradicting $P$'s statement that $S$ has at least $3$ cars. So, It is NOT a possible case.

$(2)$ $R$ is right, $P$ and $Q$ are wrong

It means $P$'s statement "S has at least $3$ cars" is false. So, $S$ actually has less than $3$ cars. Now, Q's statement "S has less than 3 cars" is also false which means $S$ has actually at least $3$ cars. Both statements are contradicting each other. So, It is NOT a possible case.

$(3)$ $Q$ is right, $P$ and $R$ are wrong

It means $Q$'s statement "S has less than $3$ cars" is true and $P$'s statement "S has at least $3$ cars" is false which means $S$ has actually less than $3$ cars. $R$'s statement "S has at least one car" is also false which means $S$ has actually less than $1$ car. All $3$ statements are not contradicting each other. So, It is a possible case. Since, all $3$ statements are true. So, we can conclude from all the $3$ statements that "$S$ has less than $1$ car" (or) "$S$ has no car".

So, Answer is $(A)$ 

Comments

I’ve done the mistake in this question because i’ve thought Q:less than 3 means it may not be equal to 0 .so i’ve chose option D but actually less than 3 means we have to consider 0 cars also

Answer 2

n = no. of cars S posses.

If n = 1 then both Q and R statements will be correct. So (B) is not the correct option.

Similarly, if n = 3 then both P and R statements will be correct. So (C) is also not the correct option.

If however n = 0 then only Q statement will be correct.

Therefore correct option is option (A).

Answer 3

Atleast 3 implies atleast 1 but reverse may not be true.

Comments

Atleast 3 implies atleast 1 is correct. Reverse need not be true for implication. If reverse is true it becomes a bi conditional statement. How can we apply implication to this question?

Answer 4

Let $n$ be the number of cars that $S$ has

According to $P, \: n\geq 3$, on the contrary according to $Q, \: n <3$. According to $R, \: n \geq 1$, this is proportional to $P$'s statement.

So answer is option A