In Figure, it is given that `R T=T S ,\ /_1=2/_2\ a n d\ /_4=2/_3` . Prove that ` R B T\ ~=\ S A T`

In Fig. 4.100, if (Q T)/(P R)=(Q R)/(Q S) and /_1=/_2 . Prove that P Q S ~ T Q R . (FIGURE)

In an AP , it s being given that T_(4)/T_(7) = 2/3 . " Find " T_(7)/T_(10)

In Fig. 4.238, S and T are points on the sides P Q and P R respectively of P Q R such that P T=2c m , T R=4c m and S T is parallel to Q R . Find the ratio of the areas of P S T and P Q R . (FIGURE)

In a harmonic progression t_(1), t_(2), t_(3),……………., it is given that t_(5)=20 and t_(6)=50 . If S_(n) denotes the sum of first n terms of this, then the value of n for which S_(n) is maximum is

If t_r is the rth term is the expansion of (1+a)^n , in ascending power of a , prove that r (r +1) t_(r+2) = (n - r + 1) (n - r) a^2 t_r

Find the shortest distance between the following (6 -7) lines whose vector equations are : (i) vec(r) = (1 -t) hati + (t - 2) hatj + (3 - 2t) hatk and vec(r) = (s + 1) hat(i) + (2s - 1) hatj - (2s + 1) hatk (ii) vec(r) = (3 -t) hati + (4 + 2t) hatj + (t - 2) hatk and vec(r) = (1 + s) hati + (3s - 7 ) hatj + (2s -2) hatk. where t and s are scalars.

The activity of a radioactive substance is R_(0) at t = 0 , R_(1) at t = t and R_(2) at t = 2t . The decay constant of this species is/are given by:

Find the Cartesian equation of the following planes : a. vec(r). (hati + hatj - hatk ) = 2 b. vec(r). (2 hati + 3 hatj - 4 hatk ) = 1 (c ) vec(r). [ (s - 2t) hati + (3 - t ) hatj + (2 s + t ) hatk] = 15