Prove that if the equation `x^2+9y^2-4x+3=0` is satisfied for real values of `xa n dy ,t h e nx` must lie between 1 and 3 and`y` must lie between-1/3 and 1/3.

Prove that for real values of x ,(a x^2+3x-4)//(3x-4x^2+a) may have any value provided a lies between 1 and 7.

The real number k for which the equation, 2x^3+""3x""+""k""=""0 has two distinct real roots in [0, 1] (1) lies between 2 and 3 (2) lies between -1 and 0 (3) does not exist (4) lies between 1 and 2

Prove that for all real values of xa n dy ,x^2+2x y+3y^2-6x-2ygeq-11.

Prove that the values of the function (sinxcos3x)/(sin3xcosx) cannot lie between 1/3 and 3 for any real x

If f(x , y) satisfies the equation 1+4x-x^2=sqrt(9sec^2y+4cos e c^2y) then find the value of xa n dtan^2ydot

Find the greatest value of x^2 y^3 , where x and y lie in the first quadrant on the line 3x+4y=5 .

If t is a real number satisfying the equation 2t^3-9t^2+30-a=0, then find the values of the parameter a for which the equation x+1/x=t gives six real and distinct values of x .

Let xa n dy be two real variable such that x >0a n dx y=1 . Find the minimum value of x+y .

x y=(x+y)6na n d(dy)/(dx)=y/x t h e nn= 1 b. 2 c. 3 d. 4

Solve the equation (dy)/(dx)+(2xtan^(-1)y-x^3)(1+y^2)=0