Let f( ) = vmatrix & 2 1 & vmatrix . If S is the set of all values of [0, 2 ] satisfying f( ) + = 0 , then the sum of all elements in S is

Step-by-step solution

First, expand the determinant to find f( ) : f( ) = - 2 Substitute this into the given equation: - 2 + = 0 ( + 1) = 2 Expressing in terms of sine and cosine: ( + 1) = 2 Multiply both sides by (noting that 0 for to be defined): ( + 1) = 2 ^2 ( + 1 - 2 ^2 ) = 0 Using ^2 = 1 - ^2 , the term in the parentheses becomes: + 1 - 2(1 - ^2 ) = 2 ^2 + - 1 So the equation is: (2 ^2 + - 1) = 0 (2 - 1)( + 1) = 0 This gives three cases: 1) = 0 = 2 , 3 2 2) = 1 2 = 3 , 5 3 3) = -1 = However, we must check the domain. The function