Let S be the set of x -coordinates of the points of intersection of the curves C₁: y = x + x and C₂: y = 2 x in the interval [0, 4 ] . If the sum of all elements in S is k , then the value of k is

Step-by-step solution

The x -coordinates of the points of intersection are given by equating the two curves: x + x = 2 x Converting to sine and cosine, we get: x + 1 x = 2 x Cross-multiplying yields: x + 1 = 2 ^2 x Using the identity ^2 x = 1 - ^2 x , we have: x + 1 = 2(1 - ^2 x) 2 ^2 x + x - 1 = 0 Factorizing the quadratic equation: (2 x - 1)( x + 1) = 0 This gives x = 1 2 or x = -1 . If x = -1 , then x = 0 . However, at x = 0 , the original functions x and x are undefined. Thus, roots corresponding to x = -1 (which are x = 3 2 , 7 2 )