Prove that: `|[x+y,x,x],[5x+4y,4x,2x],[10x+8y,8x,3x]|=x^3`.

Without expanding the following determinant, show that : |[[3x+y, 2x, x],[4x+3y, 3x, 3x],[5x+6y,4x,6x]]|=x^3

Prove that: {:|(x,y,x+y),(y,x+y,x),(x+y,x,y)| = -2(x^3+y^3)

Prove that the three st. lines : 2x - 3y = 7, 3x -4y = 13 and 8x-11y = 33 meet in a point.

Prove that the circles x^(2) +y^(2) - 4x + 6y + 8 = 0 and x^(2) + y^(2) - 10x - 6y + 14 = 0 touch at the point (3,-1)

If y = [ x + 2 + sqrt(x^2 + 4x + 10)]^n then prove that (x^2 + 4x + 10) y_2 + (x + 2) y_1 - n^2y = 0 .

Find the values of 'x' and 'y' when : {:[(x+2y, 3y),(4x,2)] = [(0,-3),(8,2)]