If `1/(1+l)=v`, prove that
`v+v^(2)+v^(3)+………+v^(n)=(1-v^(n))/l`

int (1-v) (dx)) / (1 + v + v ^ (2))

For any two vectors vec ua n d vec v prove that ( vec udot vec v)^2+| vec uxx vec v|^2=| vec u|^2| vec v|^2a n d ( vec1+| vec u|^2)( vec1+| vec v|^2)=(1- vec udot vec v)^2+| vec u+ vec v+( vec uxx vec v)|^2

For any two vectors vec ua n d vec v prove that ( vec udot vec v)^2+| vec uxx vec v|^2=| vec u|^2| vec v|^2a n d ( vec1+| vec u|^2)( vec1+| vec v|^2)=(1- vec udot vec v)^2+| vec u+ vec v+( vec uxx vec v)|^2

If V = n_v V_g then show that X = G (n_v -1) .

The threshold frequencies for photoemission for three metals numbered 1,2,3 are respectively v_(1), v_(2), v_(3) and v_(1) gt v_(2)gt v_(3) . An incident radiation of frequency v_(0)gtv_(2) …………………cause the photoemission from 3 but…………….cause photoemission from 1.

Find 3 -dimensional vectors vec v_ (1), vec v_ (2), vec v_ (3) satisfying vec v_ (1) * vec v_ (1) = 4, vec v_ (1) * vec v_ (2) = - 2, vec v_ (1) * vec v_ (3) = 6, vec v_ (2) * vec v_ (2) = 2, vec v_ (2) * vec v_ (3) = - 5, vec v_ (3) * vec v_ (3)

If u_(n)=sin(n theta) sec^(n) theta, v_(n)=cos(n theta) sec^(n) theta , n in N, n ne 1," then " (v_(n)-v_(n-1))/(u_(n-1))+(1)/(n)(u_(n))/(v_(n)) =

(9v + 1) + (3v + 1)