If `g(x)=(f(x))/((x-a)(x-b)(x-c)),w h e r ef(x)` is a polynomial of degree < 3, then prove that `(dg(x))/(dx)=|{:(1,a, f(a)(x-a)^(-2)),(1,b,f(b)(x-b)^(-2)), (1,c,f(c)(x-c)^(-2)):}|/|{:(a^2,a,1),(b^2,b,1),(c^2,c,1):}|`

If f(x) g(x) and h(x) are three polynomials of degree 2 and Delta = |( f(x), g(x), h(x)), (f'(x), g'(x), h'(x)), (f''(x), g''(x), h''(x))| then Delta(x) is a polynomial of degree (dashes denote the differentiation).

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

If a^2+b^2+c^2=-2a n df(x)= |1+a^2x(1+b^2)x(1+c^2)x(1+a^2)x1+b^2x(1+c^2)x(1+a^2)x(1+b^2)x1+c^2x| , then f(x) is a polynomial of degree 0 b. 1 c. 2 d. 3

If (d(f(x)))/(dx) = e^(-x) f(x) + e^(x) f(-x) , then f(x) is, (given f(0) = 0)

If f(x), h(x) are polynomials of degree 4 and |(f(x), g(x),h(x)),(a, b, c),(p,q,r)| =mx^4+nx^3+rx^2+sx+r be an identity in x, then |(f'''(0) - f''(0),g'''(0) - g''(0),h'''(0) -h''(0)),(a,b,c),(p,q,r)| is

If f(x) is continuous such that abs(f(x)) le 1, forall x in R " and " g(x)=(e^(f(x))-e^(-abs(f(x))))/(e^(f(x))+e^(-abs(f(x)))), then range of g(x) is

A certain polynomial P(x)x in R when divided by k x-a ,x-ba n dx-c leaves remainders a , b ,a n dc , resepectively. Then find remainder when P(x) is divided by (x-a)(x-b)(x-c)w h e r eab, c are distinct.

If f(x) is a polynomial of degree n(gt2) and f(x)=f(alpha-x) , (where alpha is a fixed real number ), then the degree of f'(x) is

f(x)= x, g(x)= (1)/(x) and h(x)= f(x) g(x). If h(x) = 1 then…….

Let f:R->R and g:R->R be two one-one and onto functions such that they are mirror images of each other about the line y=a . If h(x)=f(x)+g(x) , then h(x) is (A) one-one onto (B) one-one into (D) many-one into (C) many-one onto