if `x<1` then `1/(1+x)+(2x)/(1+x^2)+(4x^3)/(1+x^4)+..............oo`

If f(x)=|[x-2, (x-1)^2, x^3] , [(x-1), x^2, (x+1)^3] , [x,(x+1)^2, (x+2)^3]| then coefficient of x in f(x) is

Solve the following : (i) |x-2|=(x-2) " (ii) " |x+3| = -x-3 (iii) |x^(2)-x|=x^(2)-x " (iv) " |x^(2)-x-2| =2+x-x^(2)

If f(x)= |1x x+1 2x x(x-1)(x+1) x3 x(x-1) x(x-1)(x-2) (x+1)x(x-1)| then f(500) is equal to 0 b. 1 c. 500 d. -500

If f(x)= |{:(1,,x,,x+1),(2x,,x(x-1),,(x+1)x),(3x(x-1),,x(x-1)(x-2),,(x+1)x(x-1)):}| then the value of f(500) "_____"

If f(x)= ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ​ 1 2x 3x(x−1) ​ x x(x−1) x(x−1)(x−2) ​ x+1 (x+1)x (x+1)x(x−1) ​ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ​ Then f(100) is equal to

f(x) = |{:(x+c_(1),,x+a,,x+a),(x+b,,x+c_(2),,x+a),(x+b,,x+b,,x+c_(3)):}| " and " g(x)= (C_(1) -x)(c_(3)-x) Coefficient of x in f(x) is

If |[1,x,x^2] , [x,x^2,1] , [x^2,1,x]|=3 then find the value of |[x^3-1, 0, x-x^4] , [0,x-x^4, x^3-1] , [x-x^4, x^3-1, 0]|

"Let "f(x)=|{:(cos(x+x^(2)),sin (x+x^(2)),-cos(x+x^(2))),(sin (x-x^(2)),cos (x-x^(2)),sin (x-x^(2))),(sin 2x, 0, sin (2x^(2))):}|. Find the value of f'(0).

Find x if gof (x) = fog(x), given f (x) = 3x + 1 and g(x) = x + 3.

f(x)=x^x , x in (0,oo) and let g(x) be inverse of f(x) , then g(x)' must be