Prove that for all real values of `x and y ,x^2+2x y+3y^2-6x-2ygeq-11.`

Prove that the greatest value of x y is c^3/sqrt(2a b) , if a^2x^4+b^2y^4=c^6dot

Prove that graphs y=2x-3 and y=x^2-x never intersect.

Prove that the radii of circles x^2 + y^2 = 1, x^2 + y^2 - 2x -6y = 6 and x^2 + y^2 - 4x - 12y = 9 are in A.p.

Let (f(x+y)-f(x))/2=(f(y)-a)/2+x y for all real xa n dydot If f(x) is differentiable and f^(prime)(0) exists for all real permissible value of a and is equal to sqrt(5a-1-a^2)dot Then (a) f(x) is positive for all real x (b) f(x) is negative for all real x (c) f(x)=0 has real roots (d)Nothing can be said about the sign of f(x)

If the equation y^2+x^2-10x+21=0 is satisfied by real values of x and y, prove that x lies between 3 and 7 and y lies between (-2) and 2.

For all positive values of x and y the value of ((1+x+x^(2))(1+y+y^(2)))/(xy) is-

(a) If x : y = 3 : 4 , then find the value of the ratio (2x + y) : ( 9x - 2y) . (b) If x : y = 5 : 4 , then find the value of the ratio (x^(2) + xy) : (x^(2) - xy) .

Let y=sqrt(((x+1)(x-3))/((x-2))) . Find all the real values of x for which y takes real values.

The minimum value of the functon y=x^(2)-6x+11 is -

Group the like terms from the algebraic expressions given: 2x, y, 12xy, 13y^2 , -5x, 18y, -4xy, -2y^2 , 21x^2y , 3x, 3xy, -xy, -y, -6x^2 , -15x^2 .