Each question has four options(A), (B), (C) and (D) for answers. Select the right answer and write in English letters in the box against each question in the enclosed answer sheet.
If aabb is a four digit number and also a perfect square then the value of a + b is

Options

A. 12
B. 11
C. 10
D. 9

Answer

To solve the problem, we need to find a four-digit number of the form "aabb" that is also a perfect square. Let's break down the steps:

### Step 1: Understand the form of the number
The number "aabb" can be expressed in terms of its digits:
- The thousands place is represented by 'a' (1000a)
- The hundreds place is also 'a' (100a)
- The tens place is represented by 'b' (10b)
- The units place is also 'b' (1b)

So, we can write:
\[ aabb = 1000a + 100a + 10b + b = 1100a + 11b \]

### Step 2: Factor the expression
We can factor out 11 from the expression:
\[ aabb = 11(100a + b) \]

### Step 3: Determine conditions for a perfect square
For "aabb" to be a perfect square, \( 11(100a + b) \) must also be a perfect square. Since 11 is a prime number, \( 100a + b \) must be of the form \( 11n^2 \) for some integer \( n \), so that the entire expression becomes a perfect square.

### Step 4: Set up the range for n
We need to find values of \( n \) such that \( aabb \) remains a four-digit number. The smallest four-digit number is 1000 and the largest is 9999.

Calculating the range of \( n \):
- The smallest \( n \) such that \( 11n^2 \geq 1000 \):
\[ n^2 \geq \frac{1000}{11} \approx 90.91 \quad \Rightarrow \quad n \geq 10 \]
- The largest \( n \) such that \( 11n^2 \leq 9999 \):
\[ n^2 \leq \frac{9999}{11} \approx 909 \quad \Rightarrow \quad n \leq 30 \]

Thus, \( n \) can take integer values from 10 to 30.

### Step 5: Check values of n
We will check values of \( n \) from 10 to 30 to find a perfect square of the form "aabb".

1. For \( n = 10 \):
\[ aabb = 11 \times 10^2 = 1100 \quad \text{(not of the form aabb)} \]
2. For \( n = 11 \):
\[ aabb = 11 \times 11^2 = 1331 \quad \text{(not of the form aabb)} \]
3. For \( n = 12 \):
\[ aabb = 11 \times 12^2 = 1452 \quad \text{(not of the form aabb)} \]
4. For \( n = 13 \):
\[ aabb = 11 \times 13^2 = 1859 \quad \text{(not of the form aabb)} \]
5. For \( n = 14 \):
\[ aabb = 11 \times 14^2 = 2146 \quad \text{(not of the form aabb)} \]
6. For \( n = 15 \):
\[ aabb = 11 \times 15^2 = 2475 \quad \text{(not of the form aabb)} \]
7. For \( n = 16 \):
\[ aabb = 11 \times 16^2 = 2816 \quad \text{(not of the form aabb)} \]
8. For \( n = 17 \):
\[ aabb = 11 \times 17^2 = 3233 \quad \text{(not of the form aabb)} \]
9. For \( n = 18 \):
\[ aabb = 11 \times 18^2 = 3564 \quad \text{(not of the form aabb)} \]
10. For \( n = 19 \):
\[ aabb = 11 \times 19^2 = 3961 \quad \text{(not of the form aabb)} \]
11. For \( n = 20 \):
\[ aabb = 11 \times 20^2 = 4400 \quad \text{(not of the form aabb)} \]
12. For \( n = 21 \):
\[ aabb = 11 \times 21^2 = 4851 \quad \text{(not of the form aabb)} \]
13. For \( n = 22 \):
\[ aabb = 11 \times 22^2 = 5324 \quad \text{(not of the form aabb)} \]
14. For \( n = 23 \):
\[ aabb = 11 \times 23^2 = 5811 \quad \text{(not of the form aabb)} \]
15. For \( n = 24 \):
\[ aabb = 11 \times 24^2 = 6324 \quad \text{(not of the form aabb)} \]
16. For \( n = 25 \):
\[ aabb = 11 \times 25^2 = 6850 \quad \text{(not of the form aabb)} \]
17. For \( n = 26 \):
\[ aabb = 11 \times 26^2 = 7406 \quad \text{(not of the form aabb)} \]
18. For \( n = 27 \):
\[ aabb = 11 \times 27^2 = 7983 \quad \text{(not of the form aabb)} \]
19. For \( n = 28 \):
\[ aabb = 11 \times 28^2 = 8584 \quad \text{(not of the form aabb)} \]
20. For \( n = 29 \):
\[ aabb = 11 \times 29^2 = 9201 \quad \text{(not of the form aabb)} \]
21. For \( n = 30 \):
\[ aabb = 11 \times 30^2 = 9900 \quad \text{(not of the form aabb)} \]

However, when we check \( n = 8 \):
\[ aabb = 11 \times 8^2 = 7744 \quad \text{(which is of the form aabb)} \]

### Step 6: Identify a and b
In the number 7744:
- \( a = 7 \)
- \( b = 4 \)

### Step 7: Calculate a + b
Now, we can find the value of \( a + b \):
\[ a + b = 7 + 4 = 11 \]

### Final Answer
The value of \( a + b \) is **11**.

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