Without using a calculator, find the approximate value of (49.001)^2 and choose the nearest option.

Introduction / Context:
This question tests your ability to approximate the square of a number that is very close to an easy benchmark value. Estimation skills like this are important in many aptitude tests because they allow you to check the reasonableness of answers without doing long, exact calculations.

Given Data / Assumptions:

• We need to approximate (49.001)^2.
• We are given only coarse options: 2500, 2400, 2600 and 2300.
• We may use algebraic identities and basic mental arithmetic.
• We are looking for the closest approximate value, not an exact decimal expansion.

Concept / Approach:
Notice that 49.001 is extremely close to 49, whose square is easy to compute exactly as 49^2 = 2401. We can use the identity (a + b)^2 = a^2 + 2ab + b^2 with a = 49 and b = 0.001. Because b is very small compared to a, the extra terms 2ab and b^2 will be small corrections to 2401, so the square of 49.001 will be a little more than 2401, and very close to 2400 in the context of the given options.

Step-by-Step Solution:
Compute the benchmark square: 49^2 = (50 − 1)^2 = 2500 − 100 + 1 = 2401.Write 49.001 as 49 + 0.001. Using (a + b)^2, we have (49 + 0.001)^2 = 49^2 + 2 × 49 × 0.001 + (0.001)^2.Calculate the correction term: 2 × 49 × 0.001 = 0.098.The last term (0.001)^2 = 0.000001 is negligible at this scale.So (49.001)^2 ≈ 2401 + 0.098 ≈ 2401.098, which is slightly larger than 2401.

Verification / Alternative check:
Compare 2401.098 to each option. It is extremely far from 2600 and 2300, and about 100 units away from 2500.It is only about 1 unit away from 2400, which makes 2400 the best approximation among the choices.

Why Other Options Are Wrong:
2500 would correspond roughly to 50^2, but 49.001 is clearly smaller than 50, so its square cannot be as large as 2500.2600 is even farther and would require a much larger base number, so it is clearly too big.2300 is too small because even 48^2 is 2304, and our number is larger than 48, so its square must exceed 2304.

Common Pitfalls:
A common error is to round 49.001 up to 50 and then square, giving 2500, without recognizing that a one unit increase at this scale significantly changes the square.Another pitfall is to ignore that we only need an approximate answer and attempt an exact multiplication, which wastes time and can lead to mistakes.

Final Answer:
The square of 49.001 is slightly above 2401 and is best approximated by 2400 among the given options.