Statement 1: Cubes of all odd natural numbers are odd.

Statement 2: Cubes of negative integers are positive or negative integers.

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.

An odd number can be represented as 2n + 1, where n is an integer.

Cube of odd number = (2n + 1)³

⇒ (2n)³ + 1³ + 3 x 2n x 1 x (2n + 1)

⇒ 8n³ + 1 + 6n(2n + 1)

⇒ 8n³ + 1 + 12n² + 6n

⇒ 8n³ + 12n² + 6n + 1 ……..(1)

If any no. odd or even is multiplied by an even number it becomes an even number.

Since, 8, 12 and 6 are even numbers so first three terms of equation (1) are even, and adding 1 at the end ensures the result is odd.

So, statement 1 is true.

Let's take some negative number, -2 and -3.

Cube of -2 = (-2)³ = -8

Cube of -3 = (-3)³ = -27

The cube of a negative integer is always a negative integer.

So, statement 2 is false.

Hence, Option 3 is the correct option.