Q1. [20 pts] Are the following statements true or false? Explain your reasons in one or two sentences. (a) [5 pts] Let A = and b= OT CO NO 2 Multiplying the second row of A by 2 and adding the first row to the third row gives the new matrix Then, Ar = b and Ar = b have the same solution. (Hint: derive the solutions if you are not sure.) (b) [5 pts] If a linear system has 2 equations and 3 unknowns, and the rank of the coefficient matrix is 2, then there are infinitely many solutions. (c) [5 pts] f(x) = (x] is continuously differentiable. (d) [5 pts] If two vectors in R' are linearly independent, then the span of them is a plane. Q2. [10 pts] Invert the matrix A in Q1(a). Q3. [25 pts] Let the demand function be F(p) = 2-4p/3 and cost function be C(x) = 23. (a) [5 pts] Derive the profit function II(r). (b) [5 pts] Derive the marginal revenue MR(x). (c) [5 pts] Derive the marginal cost MC(x). (d) [5 pts] Calculate output a* such that MR(r*) = MC(x*). (e) [5 pts] Is a* a local maximum? Is a* a global maximum? Why? Q4. [15 pts] Sketch the graph of f (x) = (22 -3)er.Q5. [15 pts] Find a nonparameterized equation of the plane through point (1,2,3) , (2, 1,2), and (5, 2,1) using cross products. Q6. [15 pts] Consider the production function F (K , L) = 2VKL. The unitr price of capital is 5 and the wage rate per unit of labor is 2. If X = 20 and L = 20, approximate the increase in production after a 0.1 increase in capital and 0.2 increase in labor. In what proportion should we increase K and L to increase production most rapidly