X said to Y, “At the time of your birth I was twice as old as you are at present.” If the present age of X is 42 years, then consider the following statements:

1. 8 years ago, the age of X was five times the age of Y.

2. After 14 years, the age of X would be two times the age of Y.

Which of the above statements is/are correct?

The correct option is 2 only.

Let's solve the problem step-by-step to verify the given statements.

Step 1: Determine the present ages of X and Y.
Let the present age of Y be $y$ years.
We are given that the present age of X is 42 years.

According to the statement made by X: "At the time of your birth I was twice as old as you are at present."
Y was born $y$ years ago.
Therefore, the age of X at the time of Y's birth was:

$42-y$

According to the condition given in the question, this age was equal to twice Y's present age ($2y$):

$42-y=2y$

Solving for $y$:

$3y=42$

$y=14$

So, the present age of Y is 14 years, and the present age of X is 42 years.

Step 2: Evaluate Statement 1.
Statement 1 says: "8 years ago, the age of X was five times the age of Y."
Let's find their ages 8 years ago:

Age of X 8 years ago = $42-8=34$ years

Age of Y 8 years ago = $14-8=6$ years

Now, check if X's age was five times Y's age:

$5\times 6=30$

Since 34 ≠ 30, Statement 1 is incorrect.

Step 3: Evaluate Statement 2.
Statement 2 says: "After 14 years, the age of X would be two times the age of Y."
Let's find their ages after 14 years:

Age of X after 14 years = $42+14=56$ years

Age of Y after 14 years = $14+14=28$ years

Now, check if X's age will be two times Y's age:

$2\times 28=56$

Since 56 = 56, Statement 2 is correct.

Therefore, only statement 2 is correct.