If ` S = [{:( 1,1,1),( 1,a^(2) ,a),( 1,a,a^(2)):}]` where ` a=e^(2ipi//3) ,`find `absS `

The determinant of the matrix A=|(1,1,1),(1,omega,omega^2),(1,omega^2,omega)|, where omega=e(2pii)/3, is

if A[{:(1,3,2),(2,0,3),(1,-1,1):}], then find A^(3)-2A^(2)+A-I_(3).

If A=[{:(,2,-1),(,-1, 3):}]" evaluate "A^2-3A+3I , where I is a unit matrix of order 2.

A={:[(1,2,1),(3,-1,2),(1,0,3)]:},B={:[(1,1,1),(-1,1,1),(2,-2,2)]:} then find matrix X such that 2A+X=2B.

If S_(n) = 1 + (1)/(2) + (1)/(2^(2))+….+(1)/(2^(n-1)) , find the least value of n for which 2- S_(n) lt (1)/(100) .

Let S = {1, 2, 3} . Determine whether the functions f : S->S defined as below have inverses. Find f^(-1) , if it exists.(a) f = {(1, 1), (2, 2), (3, 3)} (b) f = {(1, 2), (2, 1), (3, 1)} (c) f =

Evaluate S=sum_(n=0)^n(2^n)/((a^(2^n)+1) (where a>1) .

Let S = {1, 2, 3} . Determine whether the functions f : S->S defined as below have inverses. Find f^(-1) , if it exists.(a) f = {(1, 1), (2, 2), (3, 3)} (b) f = {(1, 2), (2, 1), (3, 1)}

If S _(1), S _(2) , S _(3)……., S _(2n) are the sums of infinite geometric series whose first terms are respectively 1,2,3,…..,2n and common ratio are respectively, 1/2, 1/3, …….., (1)/(2n +1), find the value of , S_(1) ^(2) + S_(2) ^(2) +…....+ S _(2n -1) ^(2).

Consider two charged metallic spheres S_(1) , and S_(2) , of radii R_(1) , and R_(2) respectively. The electric fields E_(1) , (on S_(1) ,) and E_(2) , (on S_(2) ) their surfaces are such that E_(1)//E_(2) = R_(1)//R_(2) . Then the ratio V_(1) (on S_(1) )/ V_(2) (on S_(2) ) of the electrostatic potential on each sphere is