If `f(x)=|((1+x)^a,(1+2x)^b,1),(1,(1+x)^a,(1+2x)^b), ((1+2x)^b,1,(1+x)^a)|` then find constant term and coefficient of x

Consider the polynomial fucntion f(x) = |{:((1+x)^(2),,(1+2x)^(b),,1),(1,,(1+x)^(a),,(1+2x)^(b)),((1+2x)^(b),,1,,(1+x)^(a)):}| a,b being positive integers. Which of the following is true ?

If f(x)=log((1+x)/(1-x)) , then f((2x)/(1+x^2)) equal to:

if f(x)=(x)/((1+x))+(x)/((1+x)(1+2x))+(x)/((1+2x)(1+3x))+. . . infinite terms , Discuss continuity at x=0

If f.R rarr R,f(x)=(x^(2)+bx+1)/(x^(2)+2x+b),(b gt 1) " and " f(x), 1/f(x) have the same bounded set as their range, the value of b is

If x*a=0, x*b=1, [x a b]=1 and a*b ne 0 , then find x in terms of a and b.

If a^2+b^2+c^2=-2 , and f(x)= |{:(1+a^2x,(1+b^2)x,(1+c^2)x),((1+a^2)x,1+b^2x,(1+c^2)x),((1+a^2)x,(1+b^2)x,1+c^2x):}| , then f(x) is a polynomial of degree

Suppose f(x) is a function satisfying the folowing conditions: (i)f(0)=2,f(1)=1 (ii) f(x) has a minimum value at x=(5)/(2) (iii) for all xf'(x)= |{:(2ax,2ax-1,2ax+b+1),(b,b+1,-1),(2(ax+b),2ax+2b+1,2ax+b):}| The number of solutions of the equation f(x) +1=0 is

If f(x)=|((x-a)^(4),(x-a)^(3),1),((x-b)^(4),(x-b)^(3),1),((x-c)^(4),(x-c)^(3),1)|" then "f'(x)=lambda.|((x-a)^(4),(x-a)^(2),1),((x-b)^(4),(x-b)^(2),1),((x-c)^(4),(x-c)^(2),1)| . The value of lambda is

If f(x)=sqrt(x^(2)+1),g(x)=(x+1)/(x^(2)+1) and h(x)=2x-3 , then find f'[h'{g'(x)}] .