Let D(x) = vmatrix x+ & x & x x & x+ & x x & x & x+ vmatrix . The graph of y = D(x) in the xy -plane is a straight line passing through the points (1, 15) and (-1, -5) . If + + = 6 , then the value of ^3 + ^3 + ^3 is equal to

Step-by-step solution

Expanding the given determinant, we apply the operations R₁ R₁ - R₂ and R₂ R₂ - R₃ : D(x) = vmatrix & - & 0 0 & & - x & x & x+ vmatrix Expanding along the first row: D(x) = [ (x+ ) - (- )(x)] - (- )[0 - (- )(x)] D(x) = ( x + + x) + ( x) D(x) = + x( + + ) This shows y = D(x) is a linear function. We are given that it passes through (1, 15) and (-1, -5) . Substituting x = 1 : + ( + + ) = 15 ... (i) Substituting x = -1 : - ( + + ) = -5 ... (ii) Adding (i) and (ii): 2 = 10 = 5 Subtracting (ii) from (i): 2( + + ) = 20 +