**Given information:**
- The man's initial speed is 3 km/hr.
- The man's increased speed is 6 km/hr.
- The raindrops are falling vertically downwards at the man's initial speed.
- When the man increases his speed, the raindrops make an angle of 30 degrees with the vertical.

**To find:**
- The speed of the raindrops.

**Solution:**

Let's assume the speed of the raindrops to be 'v' km/hr.

When the man is walking at a speed of 3 km/hr, the raindrops are falling vertically downwards. This means the horizontal component of the raindrop's speed is 0 km/hr.

When the man increases his speed to 6 km/hr, the raindrops make an angle of 30 degrees with the vertical. This means the horizontal component of the raindrop's speed is v * cos(30).

Now, let's analyze the situation using the concept of relative velocity.

**Relative velocity:**

The relative velocity between the man and the raindrops is the difference between their velocities.

- When the man is walking at a speed of 3 km/hr, the relative velocity between the man and the raindrops is 0 km/hr (since the raindrops are falling vertically downwards).
- When the man increases his speed to 6 km/hr, the relative velocity between the man and the raindrops is v * cos(30) - 6 km/hr.

Since the relative velocity between the man and the raindrops is 0 km/hr in both cases, we can equate the two expressions to find the value of 'v' as follows:

0 = v * cos(30) - 6

Solving this equation, we get:

v * cos(30) = 6

v = 6 / cos(30)

Using the value of cos(30) = √3 / 2, we can calculate the speed of the raindrops as follows:

v = 6 / (√3 / 2)

v = (6 * 2) / √3

v = 12 / √3

Rationalizing the denominator, we get:

v = (12 / √3) * (√3 / √3)

v = (12√3) / 3

v = 4√3

Therefore, the speed of the raindrops is 4√3 km/hr.