If a,b,c ar enon-coplanar vectors and `lamda` is a real number, then the vectors `a+2b+3c,lamdab+4c and (2lamda-1)c` are non-coplanar for

Options

A. no value of `lambda`
B. all except one value of `lambda`
C. all except two value of `lambda`
D. all values of `lambda`

Answer

C

Explanation

Let `alpha=a+2b+3c, beta=lambdab+4c`
and `gamma (2lambda-1)c`
Then, `[ alpha, beta, gamma] = | (1, 2 , 3),(0 ,lambda, 4),(0,0,(2lambda-1))|[ "a b c"]`
`(alpha, beta, gamma] = lambda (2 lambda -1) [abc] `
Now, consider `lambda (2 lambda -1) =0`
`lambda =0,(1)/(2) " " [ because [abc] ne 0]`
Hence, `alpah, beta and gamma ` are non-coplanar for all value of `lambda` expcept two values `0 and (1)/(2)`.