`f_(1)(x)=x+a,f_(2)(x)=x^(2)+bx+c` and `Delta=|[1,1,1],[f_(1)(x_(1)),f_(1)(x_(2)),f_(1)(x_(3))],[f_(2)(x_(1)),f_(2)(x_(2)),f_(2)(x_(3))]|`, then `Delta` is independent of

If f_(n)(x),g_(n)(x),h_(n)(x),n=1, 2, 3 are polynomials in x such that f_(n)(a)=g_(n)(a)=h_(n)(a),n=1,2,3 and F(x)=|{:(f_(1)(x),f_(2)(x),f_(3)(x)),(g_(1)(x),g_(2)(x),g_(3)(x)),(h_(1)(x),h_(2)(x),h_(3)(x)):}| . Then, F' (a) is equal to

If f(x_(1)-x_(2)),f(x_(1))f(x_(2)) & f(x_(1)+x_(2)) are in A.P. for all x_(1),x_(2) and f(0)!=0 then

If 2f(x^(2))+3f((1)/(x^(2)))=x^(2)-1, then f(x^(2)) is

If f_(r)(x), g_(r)(x), h_(r) (x), r=1, 2, 3 are polynomials in x such that f_(r)(a) = g_(r)(a) = h_(r) (a), r=1, 2, 3 and " "F(x) =|{:(f_(1)(x)" "f_(2)(x)" "f_(3)(x)),(g_(1)(x)" "g_(2)(x)" "g_(3)(x)),(h_(1)(x)" "h_(2)(x)" "h_(3)(x)):}| then F'(x) at x = a is ..... .

If f(x)=log((1+x)/(1-x))f(2(x)/(1+x^(2)))=2f(x)

Let f_(1) (x) and f_(2) (x) be twice differentiable functions where F(x)= f_(1) (x) + f_(2) (x) and G(x) = f_(1)(x) - f_(2)(x), AA x in R, f_(1) (0) = 2 and f_(2) (0) = 1. "If" f'_(1)(x) = f_(2) (x) and f'_(2) (x) = f_(1) (x) , AA x in R . then the number of solutions of the equation (F(x))^(2) =(9x^(4))/(G(x)) is...... .

Consider the fucntion ,f_(1)(x)=sqrt(|x|)f_(2)(x)=sqrt(x);f_(3)(x)=x^((1)/(3))f_(4)(x)=(x^(2))^((1)/(3))

If f(x)=(1+x)(2+x^(2))^((1)/(2))(3+|x^(3)|)^((1)/(3)) then f'(-1) is