Explanation: (E) Let \(D_Z\) be the days Zakir takes to produce x chairs, so \(D_K = D_Z + 2\). Work Rate: \(R_Z = \frac{x}{D_Z}\) chairs/day, \(R_K = \frac{x}{D_K} = \frac{x}{D_Z + 2}\) chairs/day. Combined rate: \(R_{Z+K} = \frac{24x}{6} = 4x\) chairs/day. \(R_{Z+K} = R_Z + R_K \implies 4x = \frac{x}{D_Z} + \frac{x}{D_Z + 2}\). Divide by x: \(4 = \frac{1}{D_Z} + \frac{1}{D_Z + 2} \implies 4 = \frac{D_Z + 2 + D_Z}{D_Z(D_Z + 2)} = \frac{2D_Z + 2}{D_Z^2 + 2D_Z}\). \(4(D_Z^2 + 2D_Z) = 2D_Z + 2 \implies 4D_Z^2 + 8D_Z = 2D_Z + 2 \implies 4D_Z^2 + 6D_Z - 2 = 0 \implies 2D_Z^2 + 3D_Z - 1 = 0\). Solving this quadratic equation for \(D_Z\) yields an irrational number. \(D_K = D_Z + 2\) is also irrational. Days for Kalam to produce 2x chairs: \(D_{K(2x)} = D_K \cdot 2 = 2(D_Z + 2)\), which is an irrational number. Since the options are integers, the answer is none of these.
