Step 1: Understanding the Question:
This question belongs to the Number System topic, focusing on the properties of perfect squares. A perfect square is an integer that can be written as the square of another integer. In a multiple-choice format, the goal is often to eliminate incorrect options using mathematical shortcuts rather than calculating the square root of every number. This tests a candidate's familiarity with digit patterns and number theory basics.
Step 2: Key Formulas and approach:
We use "Unit Digit Elimination" and "Trailing Zero Rules" as our primary approach.
1. A perfect square never ends in the digits 2, 3, 7, or 8.
2. If a number ends in zero, it must have an even number of zeros at the end (00, 0000, etc.) to be a perfect square.
3. Once we narrow down the options, we use "Estimation" by finding the squares of nearby multiples of 10 (like 80 squared or 90 squared).
Step 3: Detailed Explanation:
We examine Option B (2362). It ends in 2. According to our rules, no perfect square can end in 2. So, we eliminate it.
We examine Option D (5688). It ends in 8. No perfect square ends in 8. So, we eliminate it.
We examine Option E (1253). It ends in 3. No perfect square ends in 3. So, we eliminate it.
Now we check Option A (4410). It ends in a single zero. A perfect square must have an even number of trailing zeros. Therefore, 4410 is not a perfect square (though 441 is $21^2$).
This leaves only Option C (7921). Let's verify it using estimation.
We know $80^2 = 6400$ and $90^2 = 8100$. Since 7921 is between 6400 and 8100, its root must be between 80 and 90.
The number ends in 1. A number's square ends in 1 only if the number itself ends in 1 or 9. Since 7921 is very close to 8100, we test 89.
$89 \times 89 = (90 - 1)^2 = 8100 - 180 + 1 = 7921$.
This confirms Option C is the perfect square of 89.
Step 4: Final Answer:
The perfect square among the options is 7921.