The following questions are accompanied by two statements (A), (B). You have to determine which statements(s) is/are sufficient/ necessary to answer the questions.
What is the value of a two-digit number?
A. The sum of its digits is 12 and the difference of the squares of its digits is 48.
B. On reversing the digits of the original number, new number obtained is 36 less than the original number.

Options

A. Statement A alone is sufficient to answer the question but statement B alone is not sufficient to answer the questions.
B. Statement B alone is sufficient to answer the question but statement A alone is not sufficient to answer the question.
C. Both the statements taken together are necessary to answer the questions.
D. Either statement A or statement B by itself is sufficient to answer the question.

Answer

To solve the problem of determining the value of a two-digit number based on the provided statements, we can break down the solution into clear steps.

### Step 1: Define the Variables
Let the two-digit number be represented as \(10x + y\), where \(x\) is the digit in the tens place and \(y\) is the digit in the units place.

### Step 2: Analyze Statement A
From Statement A, we have:
1. The sum of the digits is 12:
\[ x + y = 12 \quad (1) \]
2. The difference of the squares of the digits is 48:
\[ x^2 - y^2 = 48 \quad (2) \]
This can be factored using the difference of squares:
\[ (x - y)(x + y) = 48 \]
From equation (1), we know \(x + y = 12\). Substituting this into the factored equation gives:
\[ (x - y)(12) = 48 \]
Simplifying this, we find:
\[ x - y = 4 \quad (3) \]

### Step 3: Solve the System of Equations
Now, we have two equations:
1. \(x + y = 12\) (from Statement A)
2. \(x - y = 4\) (derived from Statement A)

We can solve these equations simultaneously:
- Adding equations (1) and (3):
\[ (x + y) + (x - y) = 12 + 4 \implies 2x = 16 \implies x = 8 \]
- Substituting \(x = 8\) back into equation (1):
\[ 8 + y = 12 \implies y = 4 \]

### Step 4: Determine the Two-Digit Number
Now that we have \(x = 8\) and \(y = 4\), the two-digit number is:
\[ 10x + y = 10(8) + 4 = 84 \]

### Step 5: Analyze Statement B
From Statement B, we have:
- When the digits are reversed, the new number is 36 less than the original number:
\[ 10y + x = (10x + y) - 36 \]
Rearranging gives:
\[ 10y + x + 36 = 10x + y \]
This simplifies to:
\[ 9y - 9x = -36 \implies y - x = -4 \quad (4) \]

### Step 6: Combine Both Statements
From Statement A, we found \(x + y = 12\) and \(x - y = 4\). From Statement B, we found \(y - x = -4\).

Both statements lead to the same equations, confirming that the values of \(x\) and \(y\) are consistent.

### Conclusion
Both statements A and B together are necessary to determine the value of the two-digit number, which is 84.