Institution: PROBASHI KALLYAN BANK Exam: অফিসার (জেনারেল) (05-05-2023) || 2023
Topic: Arithmetic Mathematics (পাটিগণিত), Subject: Mathematics (সাধারণ গণিত)

Best Answer Explained in Below-:

To determine which option is smaller than \( a \) (where \( a = 0.67 \)), let's analyze each choice in relation to the value of \( a \).

1. **Option A: \( \sqrt{a} \)**
Since \( 0 < a < 1 \), the square root of \( a \) will be larger than \( a \). Specifically, \( \sqrt{0.67} \approx 0.82 \), which is greater than \( 0.67 \).

2. **Option B: \( 1/a \)**
The expression \( 1/a \) represents the reciprocal of \( a \). For \( a = 0.67 \), this yields \( 1/0.67 \approx 1.49 \), which is clearly greater than \( 0.67 \).

3. **Option C: \( a^2 \)**
Squaring a number between 0 and 1 results in a smaller number. Therefore, \( a^2 = (0.67)^2 \approx 0.4489 \), which is less than \( 0.67 \).

4. **Option D: \( 1/a^2 \)**
Similar to \( 1/a \), this expression calculates to \( 1/(0.67^2) \approx 2.23 \), which is also greater than \( 0.67 \).

From this analysis, we see that the only option smaller than \( a \) is **C: \( a^2 \)**, because squaring a fraction less than 1 results in a number that is also less than the original fraction. Therefore, the correct answer is indeed \( a^2 \).