Properties of Greatest Integer Function `(i)[-n]=-[n] (ii)[x+k]=[x]+k (iii) [-x]=-[x]-1 (iv) [x]+[-x]={-1 ; xnotinZ ; 0 ; xepsilonZ} (iv) [x]-[-x]={2[x]+1 ; if xnotinZ ; 2[x] ; if xepsilonZ`

Properties of smallest integer function property (i)[-n]=-[n]where n epsilon N (ii) [-x]=-[x]+1; where xepsilon R-Z (iii) [x+n]=[x]+n where x epsilon R-Z and n epsilon Z (iv) [x]+[-x]={1; x notinZ ; 0 ; xinZ} (v) [x]-[-x]={2[x]-1; xnotinZ ; 2[x] ; if xinZ

If ((1+i)/(1-i))^(x)=1 AA n in N is (i) x = 2n+1 (ii) x =4n (iii) x=2n (iv) x=4n+1

If [x]^2- 5[x]+6= 0 , [x] denote the greatest integer function, then x ∈ [ 3 , 4 ] (ii)x ∈ ( 2 , 3 ](iii)x ∈ [ 2 , 4 )(iv)x ∈ [ 2 , 4 ]

if f(x)=(a-x^n)^(1/n), where a > 0 and n is a positive integer, then f(f(x))= (i) x^3 (ii) x^2 (iii) x (iv) -x

Find the domain of functions : (i) f(x)= 1/sqrt(1-cosx) (ii) (x^3-x+3)/(x^2-1) (iii) (3x)/(2x-8) (iv) x|x|

Find the derivative of the following functions from first principle.(i) x^3-27 (ii) (x - 1) (x - 2) (iii) 1/(x^2) (iv) (x+1)/(x-1)

Which of the following function is/are periodic? (a)f(x)={1,x is rational 0,x is irrational (b)f(x)={x-[x];2n

Prove the each of the following function is differentiable at x=0. (i) f(x) = cos|x| (ii) f(x) = x|x| (iii) f(x)=|x^(3)|(iv) f(x) =(x)/(1+|x|)

Solve for x (where [*] denotes greatest integer function and {*} represent fractional part function) (i) [2x]=1 (ii) {x}^(2)+[x]=2 (iii) 6{x}^(2)-5{x}+1=0 (iv) 6{x}^(2)-5{x}-1=0

sin^(2) x+2sin x cos x-3cos^(2) x = 0 if (i) tan x = 3 (ii) tan x = -1 (iii) x = n pi+(pi)/(4) , n in I (iv) x = n pi+tan^(-1) (3), n in I