State which of the following sets are finite and which infinite :
(i) `{x:x` is a positive integral root of `x^(2)-2x-15=0}`
(ii) `{x:x` is human being in the world}
(iii) `{x:x` is a multiple of `3}`
(iv) `{x:x` is a real number between `a and 2}.`

Answer

To determine whether the given sets are finite or infinite, we will analyze each one step by step.

### Step 1: Analyze the first set
**Set (i)**: `{x : x is a positive integral root of x^2 - 2x - 15 = 0}`

1. **Find the roots of the equation**: We will use the quadratic formula, which is given by:
\[ x = \frac{-b \pm \sqrt{D}}{2a} \]
where \(D = b^2 - 4ac\). Here, \(a = 1\), \(b = -2\), and \(c = -15\).

2. **Calculate the discriminant (D)**:
\[ D = (-2)^2 - 4 \cdot 1 \cdot (-15) = 4 + 60 = 64 \]
Since \(D > 0\), there are two distinct real roots.

3. **Calculate the roots**:
\[ x = \frac{2 \pm \sqrt{64}}{2 \cdot 1} = \frac{2 \pm 8}{2} \]
This gives us:
\[ x_1 = \frac{10}{2} = 5 \quad \text{and} \quad x_2 = \frac{-6}{2} = -3 \]
The positive integral root is \(5\).

4. **Conclusion**: The set has only one positive integral root, which is finite.

### Step 2: Analyze the second set
**Set (ii)**: `{x : x is a human being in the world}`

1. **Population count**: The number of human beings can be counted, and as of now, it is approximately 8 billion.

2. **Conclusion**: This set is finite because the population of humans is a definite number.

### Step 3: Analyze the third set
**Set (iii)**: `{x : x is a multiple of 3}`

1. **Identify the multiples**: The multiples of 3 are 3, 6, 9, 12, 15, ..., and so on.

2. **Conclusion**: There is no upper limit to the multiples of 3; they can go on indefinitely. Therefore, this set is infinite.

### Step 4: Analyze the fourth set
**Set (iv)**: `{x : x is a real number between a and 2}`

1. **Understanding real numbers**: The set of real numbers between any two distinct numbers (in this case, \(a\) and \(2\)) contains infinitely many values.

2. **Conclusion**: This set is infinite because there are infinitely many real numbers between any two distinct points.

### Final Summary
- Set (i): Finite (1 element)
- Set (ii): Finite (definite population)
- Set (iii): Infinite (multiples of 3)
- Set (iv): Infinite (real numbers between any two points)