Question 2

We will check each statement based on mathematical definitions.

Step 1 — Hexagons and Polygons

Let's think about what a polygon is. A polygon is a closed shape. It has straight sides. A hexagon is a shape. It has six straight sides. It is also a closed shape. So, a hexagon fits the definition of a polygon.

Step 2 — Polygons and Pentagons

A polygon is a general name. It means any closed shape with straight sides. A pentagon is a specific type of polygon. It has five straight sides. Since pentagons are polygons, some polygons are indeed pentagons.

Step 3 — Even Numbers and Divisibility

Let's recall the definition of an even number. An even number is any integer. It must be perfectly divisible by 2. This is the definition itself. So, every single even number is divisible by 2. The statement says "not all" are. This contradicts the definition.

Step 4 — Real Numbers and Irrational Numbers

Real numbers are a big group. This group includes rational numbers. It also includes irrational numbers. Irrational numbers cannot be written as a simple fraction. Examples are $\sqrt{2}$ or $\pi$. Since irrational numbers exist, some real numbers are irrational.

Step 5 — Real Numbers and Rational Numbers

Real numbers are made of two types. They are rational numbers and irrational numbers. If all real numbers were rational, then irrational numbers would not exist. But we know irrational numbers exist. So, it is true that not all real numbers are rational. Some real numbers are irrational.

Answer

(i) True (ii) True (iii) False (iv) True (v) True