If `f_r(x),g_r(x),h_r(x),r=1,2,3` are polynomials 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^(prime)(x) at x=a` is____________________

If f_r(x),g_r(x),h_r(x),r=1,2,3 are polynomials such that f_r(a)=g_r(a)=h_r(a),r=1,2,3a n d 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^(prime)(x)a tx=a 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),g(x) and h(x) are three polynomials of degree 2, then prove that phi(x) = |[f(x),g(x),h(x)],[f ^ (prime)(x),g^(prime)(x),h^(prime)(x)],[f^(primeprime)(x),g^(primeprime)(x),h^(primeprime)(x)]| is a constant polynomial.

If f(x),g(x),h(x) are polynomials in x of degree 2 and F(x)=|fghf'g' h 'f' 'g' ' h ' '| , then F^(prime)(x) is equal to 1 (b) 0 (c) -1 (d) f(x)dotg(x)doth(x)

If f(x) and g(x) are two polynomials such that the polynomial h(x)=xf(x^3)+x^2g(x^6) is divisible by x^2+x+1, then f(1)=g(1) (b) f(1)=1g(1) h(1)=0 (d) all of these

If f(x) = sqrtx , g(x) = root(3)(x+1) , and h(x) = root(4)(x+2), " then "f(g(h(2)))=

If f(x)=sqrt(x^2+1),\ \ g(x)=(x+1)/(x^2+1) and h(x)=2x-3 , then find f^(prime)(h^(prime)(g^(prime)(x))) .

If f, g : R to R such that f(x) = 3 x^(2) - 2, g (x) = sin (2x) the g of =

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...... .

f(x) : [0, 5] → R, F(x) = int_(0)^(x) x^2 g(x) , f(1) = 3 g(x) = int_(1)^(x) f(t) dt then correct choice is