Two functions f and g are defined as follows:

Determine, with reasons, whether fg and gf exist. If the composite function exists, give the rule, domain and range in exact form.

Relative to the origin *O*, three distinct fixed points *A, B* and C have position vectors a, **b **and **c** respectively. It is known that b is a unit vector, |**a|**=3, |**c**| = 2 and the angle *AOC* is 60º.

(i) State the geometrical interpretation of |**b⋅c**|.

It is further given that .

(ii) Find the ratio of the area of triangle *AOB* to the area of triangle *BOC*.

(iii) Show that where 2**a** + **c** = *k* **b** where *k*∈ℜ, *k*≠0 ,.

By considering (2**a** + **c**)**⋅**(2**a **+** c**) , find the exact values of *k*.

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**Key Concept of solving this question**

This question is testing on the concept of intersection between two lines. Students are required to deduce that if two lines intersect each other, there should be a unique solution for the parameters of lines. With this information, students should be able to solve for the unknown a, then proceed to find the acute angle between the lines by using the dot product.

**Handwritten Solution**