Statement 1: When the sum of a two - digit number and the number obtained by reversing its digit is divided by 11, the quotient is equal to the sum of two digits.

Statement 2: When the sum of a two digit number and the number obtained by reversing its digit is divided by the sum of the two digit, the quotient is always 11.

Which of the following options is correct ?

1. Both the statements are true.

2. Both the statements are false.

3. Statement 1 is true, and statement 2 is false.

4. Statement 1 is false, and statement 2 is true.

Let the digit at ten's place be x and at unit's place be y.

The two-digit number = 10 × x + y = 10x + y

Number obtained by reversing its digits = 10 × y + x = 10y + x

Sum of the number and reversed numbers= (10x + y) + (10y + x)

= 10x + x + y + 10y

= 11x + 11y

= 11(x + y)

On dividing by 11, we get :

$\dfrac{\text{11(x + y)}}{11}$ = (x + y)

The quotient is equal to the sum of the digits.

So, statement 1 is true.

On dividing the sum of numbers by sum of the digits = $\dfrac{11(x + y)}{x + y}$ = 11

The quotient is equal to 11.

So, statement 2 is true.

Hence, option 1 is the correct option.