Using intermediate value theorem, prove that there exists a number `x` such that `x^(2005)+1/(1+sin^2x)=2005.`

Using mean value theorem, prove that tanx > x for all x(0,pi/2)dot

By using binomial theorem, expand: (1+x+x^2)^3

Using binomial theorem expand (1+x/2-2/x)^4,\ x!=0.

Using Rolle's theorem prove that the equation 3x^2 + px - 1 = 0 has the least one it's interval (-1,1).

Using binomial theorem, write down the expansions of the following: (1-2x+3x^2)^3

Using binomial theorem, write down the expansions of the following: \ (1+2x-3x^2)^5

Using binomial theorem expand (x^(2)+(1)/(x^(2)))^(4)

int(cosec^2x-2005)/cos^[2005]x.dx

Let f(x) = (1-x)^(2) sin^(2)x+ x^(2) for all x in IR and let g(x) = int_(1)^(x)((2(t-1))/(t+1)-lnt) f(t) dt for all x in (1,oo) . Consider the statements : P : There exists some x in IR such that f(x) + 2x = 2 (1+x^(2)) Q : There exist some x in IR such that 2f(x) + 1 = 2x(1+x) Then

Verify mean value theorem for the function f(x) = x^(3)-2x^(2)-x+3 in [0,1]