(a)

The polar equation is  and .

The graph of is spiral , so it is either or .

Since , there are revolutions.

When , .

When , .

.

Therefore, distance from origin to point at which first revolution that intersect the polar axis is

three times distance from origin to point at which last revolution that intersect the polar axis.

All the conditions are satisfied in the option graph.

Thus, option is correct choice.

(b)

The polar equation is  and .

The graph of is spiral , so it is either or .

Since , there are revolutions.

When , .

When , .

.

Therefore, distance from origin to point at which first revolution that intersect the polar axis is

  times distance from origin to point at which last revolution that intersect the polar axis.

All the conditions are satisfied in the option graph.

Thus, option is correct choice.

(c)

The polar equation is  .

When , .

When , .

When , .

When , .

Graph is the only option that satisfies all the four conditions.

(d)

The polar equation is .

When , .

When , .

When , .

Graph is the only option that satisfies all the four conditions.

(e)

The polar equation is .

When , .

When , .

When , .

Graph is the only option that satisfies all the four conditions.

(f)

The polar equation is .

When , .

When , .

When , .

When , .

Thus, graph of never touches positive -axis at any point.

Graph is the only option that satisfies all the four conditions.

(a) .

(b) .

(c) .

(d) .

(e) .

(f) .