`vec(a),vec(b)` and `vec(c )` are unit vectors such that `|vec(a)+vec(b)+3vec(c )|=4` . Angle between `vec(a)` and `vec(b)` is `theta_(1)` , between `vec(b)` and `vec(c )` is `theta_(2)` and between `vec(a)` and `vec(c )` varies `[pi//6, 2pi//3]` . Then the maximum value of `cos theta_(1)+3cos theta_(2)` is

Options

A. `3`
B. `4`
C. `2sqrt(2)`
D. `6`

Answer

B