5. The performance, p, of a machine is measured experimentally for four values of a design parameter, a (Table 2). α -1 1 2 4 р -0.75 0.75 0 3 Table 2: Experimental results of performance against design parameter Assume that the performance is related to the design parameter by a third- order polynomial function. (i) Derive four equations in terms of unknown coefficients. Obtain the matrix representation of the equations. (ii) (iii) (iv) Determine whether a unique solution exists. If it does, calculate the solution and find p when α = 3.

Write out the design matrix 1. Polynomial regression : 2. Data transformation: X and find its rank for the following Y₁ = Bot fixi + B₂X²³² + ... + B₂-, X; log7; = B₁ + B₁ (-log D₁) + E; P-1 > +E; " i=1.- `n models. n>p. i=1...n

Q.1 Suppose a simple economy with only an agriculture industry and a steel industry with the following technology matrix. Find the gross production of each industry if surpluses of 15 units of agriculture products and 35 units of steel are desired. A [0.3 0.2] Agricukue A = 0.1 0.4 Sed Q. 2 (a) find the derivatives for the following functions. (i) у %3D 5х - 6х3 (ii) (x*+5) / (х+3) (iї) у %3D (4х-1) (x? +6х - 3) (iv) у %3 (3x² + 2x + 1) Q.2 (b) if the profit function for a product is P (x) = 5600x + 85 x² - x' 200,000 dollars, selling how many items, x, will produce maximum profit? Find the maximum profit. Q.3 (a) for the following quadratic equation, determine (i) which way parabola open (ii) the coordinates of vertex (iii) the roots of the function P 90х — 200 -х? Q.3 (b) If the company has total costs C(x) = 15,000 + 35x + 0.1x and total revenue function given by R(x) = 385x – 0.9x?, find the break-even points. %3D %3D Q.4 Solve using Cramer's rule and inverse matrix method for the…

2. (a) Find the condition number of the coefficient matrix in the system X1 (1461) (²2) = (2²6) X2 as a function of d > 0. (b) Find the error magnification factor for the approximate root xa = (-1,3+6).