The correct option is C 3,−4
It is given that the angle between →a and →c is cos−1(14).
So, →a⋅→c=|→a||→c|cos(cos−1(14))
⇒→a⋅→c=14 ⋯(1)
→b−2→c=λ→a
Taking dot products with →a, we have
→a⋅→b−2(→a⋅→c)=λ(→a⋅→a)
⇒→a⋅→b−12=λ
⇒→a⋅→b=12+λ
Similarly, taking dot products with →b and →c
→b⋅→c=8−λ22−λ4 ⋯(2)
and →b⋅→c−2=λ(→a⋅→c) ⋯(3)
From equations (1),(2) and (3), we get
8−λ22−λ4−2=λ(14)
⇒λ=3,−4