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 ,(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 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_(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_(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_r(x),g_r(x),h_r(x),r=1,2,3 are differentiable function and y=|(f_1(x), g_1(x), h_1(x)), (f_2(x), g_2(x), h_2(x)),(f_3(x), g_3(x), h_3(x))| then dy/dx= |(f\'_1(x), g\'_1(x), h\'_1(x)), (f_2(x), g_2(x), h_2(x)),(f_3(x), g_3(x), h_3(x))|+ |(f_1(x), g_1(x), h_1(x)), (f\'_2(x), g\'_2(x), h\'_2(x)),(f_3(x), g_3(x), h_3(x))|+|(f_1(x), g_1(x), h_1(x)), (f_2(x), g_2(x), h_2(x)),(f\'_3(x), g\'_3(x), h\'_3(x))| On the basis of above information, answer the following question: Let f(x)=|(x^4, cosx, sinx),(24, 0, 1),(a, a^2, a^3)| , where a is a constant Then at x= pi/2, d^4/dx^4{f(x)} is (A) 0 (B) a (C) a+a^3 (D) a+a^4

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

f(x)=x^(2)-2x,x in R, and g(x)=f(f(x)-1)+f(5-f(x)) Show that g(x)>=0AA x in R

If f(x), g(x) and h(x) are second degree polynomials. Prove that sum_(r=1)^n Delta_r is independent of x where Delta_r=|[2r+1, 6n(n+2),f(x) ],[2^(r-1), 3*2^(n+1)-6, g(x)],[ r(n-r+1), n(n+1)(n+2),h(x)]|

If :R rarr R^(+) be a differentiable function and g(x)=e^(2x)(2f(x)-3(f(x))^(2)+2(f(x))^(3))f or x in R ,then

If f,g,a n dh are differentiable functions of xa n d(x)=|fgh(xf)'(xg)'(x h)'(x^(f2)f)' '(x^2g)' '(x^2h)' '| prove that ^(prime)=|fgff'g' h '(x^3f' ')'(x^3g' ')'(x^3h ' ')'|

f:R rarr R,f(x)=x^(3)+3x+2 and g:R rarr R such that f(g(x))=x. Then value of int_(2)^(6)g(x)dx is