GATE Electrical 2021 | GA | Question: 10

Question

Seven cars $\text{P, Q. R, S, T, U and V}$ are parked in a row not necessarily in that order. The cars $T$ and $U$ should be parked next to each other. The cars $S$ and $V$ also should be parked next to each other, whereas $P$ and $Q$ cannot be parked next to each other. $Q$ and $S$ must be parked next to each other. $R$ is parked to the immediate right of $V$. $T$ is parked to the left of $U$.

Based on the above statements, the only $\text{INCORRECT}$ option given below is:

• A. There are two cars parked in between $Q$ and $V$.
• B. $Q$ and $R$ are not parked together.
• C. $V$ is the only car parked in between $S$ and $R$.
• D. Car $P$ is parked at the extreme end.

Answer 1

1. The cars $T$ and $U$ should be parked next to each other.
   This means we can park these two cars in two ways, either $TU$ or $UT.$
2. The cars $S$ and $V$ also should be parked next to each other. 
   This means we can park these cars also in two ways, either $SV$ or $VS.$
3. The cars $P$ and $Q$ cannot be parked next to each other.
   This means we can not park the cars as $PQ$ or $QP.$
4. The cars $Q$ and $S$ must be parked next to each other. 
   This means we can park these two cars in two ways, either $QS$ or $SQ.$
5. $R$ is parked to the immediate right of $V$.
   This means we can park these two cars in only one way $VR.$
6. $T$ is parked to the left of $U$.
   This is not “immediately left” but combining with statement $1,$ “immediately left” follows. So, we can park these two cars in only one way $TU.$

Now we can combine the statements $(2)\;\& (4)$ and get two possible ways for cars $Q,S$ and $V.$

• $QSV$
• $VSQ$

But $VSQ$ is not possible because it violates statement $5.$

Now, using statement $5,$ we get the possible car parking sequence $QSVR.$

Now, we can check each and every option.

• A. There are two cars parked in between $Q$ and $V$, it is incorrect because only $S$ is in between $Q$ and $V.$
• B. $Q$ and $R$ are not parked together, it is correct.
• C. $V$ is the only car parked in between $S$ and $R,$ it is correct.
• D. Car $P$ is parked at the extreme end.
  For this option, we can consider all possible combinations of car parking which are only four as given below. $(P$ cannot come near $Q$ due to statement $3)$
  • $QSVR\; {\color{DarkBlue} {\color{Red} {P}}\;  {TU}}$
  • $QSVR\; {\color{DarkBlue} \;  {TU}}\;{\color{Red} {P}}$
  • ${\color{DarkBlue} {TU}}\; QSVR\; {\color{Red} {P}}$
  • ${\color{Red} {P}}\;{\color{DarkBlue} {TU}}\; QSVR\; $
  Here, we can see that car $P$ can be parked at the extreme (either left of right) end in three out of the four possible combinations. That is, though this is not a guarantee, it is still a possibility. 

So, the correct answer is $(A).$

Comments

ok, i got option A, but why not option D?

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@RISC

where will u park  car P then?

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@Lakshman Bhaiya

D. Car $P$ is parked at the extreme end.

possibilities as you have mentioned.

1. $𝑄𝑆𝑉𝑅𝑃𝑇𝑈$
2. $𝑄𝑆𝑉𝑅𝑇𝑈𝑃$
3. $𝑇𝑈𝑄𝑆𝑉𝑅𝑃$
4. $𝑃𝑇𝑈𝑄𝑆𝑉𝑅$

Here, we can see that "car $P$ can be parked at the extreme end" is true but option D should is incorrect because to be false, option D should be false in atleast one possibility and here option D is false in exactly one possibility. so, option D should also be incorrect.