True or False :
`a + ar + ar^2 + ar^3+…….`to`infty = (a)/(1-r)` for all `r ne1`.

True or False : d/(dx) (logx) = 1/x for all x ne 0 .

True or False : d/(dx) (log|x|) = 1/x for all x ne 0

True or False : If A is a square matrix and |A| ne 0, then (A^3)^-1 = (A^-1)^3)

Statement -1 For all natural numbers n , 0.5+0.55+0.555+...... upto n terms =(5)/(9){n-(1)/(9)(1-(1)/(10^n))} , Statement-2 a+ar+ar^2+....+ar^(n-1)=(a(1-r^n))/((1-r)) , for 0lt r lt 1 .

Prove the following by using the principle of mathematical induction for all n in N :- a + ar + ar^2+...+ ar^(n-1)=(a(r^n-1))/(r-1) .

True/False If AD is the median of triangleABC and P is a point on AC such that ar(triangleADP): ar (triangleABD) = 2:3 , then ar(trianglePDC) : ar(triangleABC) =1:3

True or False : The function f(x) = |sinx| is differetiable at all x in R .

Let a, r, s, t be non-zero real numbers. Let P(at^(2),2at),Q(ar^(2),2ar)andS(as^(2),2as) be distinct points on the parabola y^(2)=4ax . Suppose that PQ is the focal chord and lines QR and PK are parallel, where K the point (2a,0). The value of r is

Show that the products of the corresponding terms of the sequences a, ar, ar^2 ,... ar^(n-1) and A, AR, AR^2 ,.... AR^(n-1) form a G.P, and find the common ratio.