Which of the following options is the only INCORRECT combination ? (A) (II) (iii) (P) (B) (II) (iv)

Q: Does 'incorrect combination' mean that at least one part is false, or must all parts be false?

In the context of this specific JEE problem, we are looking for the combination where the components are false. Always look at the provided options and the truth values of the components to see how they align. If the combination is (III), (i), (R), and all are false, that is clearly the intended incorrect combination.

Q: Why do we need to check all options? Can't we just find the first incorrect one?

While it might be tempting to stop at the first one you find, in a high-stakes exam like JEE, verifying your answer is crucial. By checking all options, you confirm that the others are indeed correct, which validates your initial truth table. It is a safety net against silly mistakes.

Q: Is there a trick to these matching questions?

The 'trick' is simply systematic organization. These questions are designed to test your patience and attention to detail. By creating a clear truth table first, you turn a complex-looking problem into a simple matching exercise. Stay calm, be methodical, and trust your data.

Question

Which of the following options is the only INCORRECT combination ?

Options:

(A) (II) (iii) (P)

(B) (II) (iv) (Q)

(C) (I) (iii) (P)

(D) (III) (i) (R)

Sigma Percentile · Accepted Answer

### The Detective's Mindset: Unraveling the Logic Puzzle

Imagine you are standing in the middle of a high-stakes investigation. You have a set of clues, and your job is to find the one piece of evidence that doesn't fit.

In the world of JEE Advanced, this is exactly what a logic-based matching problem is. It is not just about math; it is about the art of deduction. Today, we are going to walk through this problem, not as a chore, but as a detective story.

We are looking for the one **INCORRECT** combination. This means we are hunting for the 'False' among the 'True'.

### Phase 1: Mapping the Truth

Before we even look at the options, we must establish our baseline. Think of this as gathering our intelligence. We have three columns, and each contains statements that are either $True$ or $False$.

Let us lay them out clearly:

*   Column 1: $(I) \rightarrow True$, $(II) \rightarrow True$, and $(III) \rightarrow False$.
*   Column 2: $(i) \rightarrow False$, $(ii) \rightarrow True$, $(iii) \rightarrow True$, and $(iv) \rightarrow True$.
*   Column 3: $(P) \rightarrow True$, $(Q) \rightarrow True$, $(R) \rightarrow False$, and $(S) \rightarrow True$.

This is our map. Without this map, we are wandering in the dark. With it, the path becomes clear.

### Phase 2: The Systematic Investigation

Now that we have our truth values, we can evaluate the options. This is where the detective work begins. We need to check each combination to see if it holds up.

Option 1 is $(II), (iii), (P)$. Substituting our values, we get $(True, True, True)$. This is a perfectly correct combination.

Option 2 is $(II), (iv), (Q)$. Again, we substitute: $(True, True, True)$. Another correct combination.

Option 3 is $(I), (iii), (P)$. Substituting: $(True, True, True)$. It is also correct. We are looking for the one that fails, the one that is entirely incorrect.

### Phase 3: The Final Breakthrough

Finally, we arrive at Option 4: $(III), (i), (R)$. Let us look at the values we mapped earlier.

$(III)$ is $False$. $(i)$ is $False$. And $(R)$ is $False$.

When we put them together, we get $(False, False, False)$. This is the combination we have been hunting for! It is the only one where all components are false.

In the high-pressure environment of the JEE, it is easy to rush and make a mistake. But notice how, by being systematic and patient, we turned a potentially confusing problem into a clear, logical conclusion.

We verified the others to be sure, and we found our target. Remember, the key to these problems is not speed; it is precision. Keep your truth table clean, check your values twice, and you will never fall into the trap.

Animated Solution for Mathematics - Sets and Relations: Which of the following options is the only INCORRECT combination ?

Select Answer:

Visualized Solution

Objective

Column 1 Truth Values

Column 2 Truth Values

Column 3 Truth Values

Evaluating Option 1

Evaluating Option 2

Evaluating Option 3

Evaluating Option 4

Final Conclusion

The Sigma Insight: Cartesian Product of Sets

The Detective's Mindset

Phase 1

Phase 2

Phase 3

Similar Questions

Comprehension Passage

If $n (X) =^{m} C_{6}$ , then the value of $m$ is ________.

If the value of $n (Y) + n (Z)$ is $k^{2}$ , then $∣ k ∣$ is ________.

Let $A = {1, 3, 4, 6, 9}$ and $B = {2, 4, 5, 8, 10}$ . Let $R$ be a relation defined on $A \times B$ such that $R = {((a_{1}, b_{1}), (a_{2}, b_{2})) : a_{1} \leq b_{2} and b_{1} \leq a_{2}}$ . Then the number of elements in the set $R$ is

Let A and B two sets containing 2 elements and 4 elements respectively. The number of subsets of $A \times B$ having 3 or more elements is

Let the number of elements in sets $A$ and $B$ be five and two respectively. Then the number of subsets of $A \times B$ each having at least 3 and at most 6 elements is

Let $A = {2, 3, 4}$ and $B = {8, 9, 12}$ . Then the number of elements in the relation $R = {((a_{1}, b_{1}), (a_{2}, b_{2})) \in (A \times B, A \times B) : a_{1} divides b_{2} and a_{2} divides b_{1}}$ is

Let $Z$ be the set of integers. If $A = {x \in Z : 2^{(x + 2) (x^{2} - 5 x + 6)} = 1}$ and $B = {x \in Z : - 3 < 2 x - 1 < 9}$ , then the number of subsets of the set $A \times B$ , is :