Question Solved1 Answer 4. One important use of matrices is to represent linear operators (maps). A linear operator F maps vectors v1, v2 and v3 into vectors W1, W2 and w3, respectively, where: --[1- - -1 2 2 -2 -3 -3 -3 2 1 and 3 0 5 -5 W1 = W2 5 -3 -2 1 2 W3 2 4 -3 0 -5 (a) Find the dimensions of the domain space and of the range space, respectively, of the operator F (b) Find the operator F.

QUESTION 1 Fill in the Boolean Algebra axiom or theorem (e.g. 1a, 1b, 2a, etc.) for each step of the following Boolean algebra minimization (x+ylyxx* xy + xly + yly (x+y)(x+ly) = x + xy + xly + yly (x+y)(x+ly) = x + xy + xly + 0 (x+y)(x+ly) = x + xy + xly (x+y)(x+ly) = x(1 + y + y) (x+y)(x + y) = x(1 + 1) (x+y)(x+ly) = x(1) (x+y)(x + y) = x

Answer the following in 100 words(each). “Because debt interest lowers a company’s tax bill, companies should always try to borrow as much as possible”. Say whether you agree or disagree and explain why. “The cost of debt for a company is always lower than the cost of equity. For this reason, a company should always finance new projects with debt”. Say whether you agree or disagree and explain why. Explain why standard methods of calculating the NPV of a project do not capture the value of ‘real options’. “Companies whose assets have low betas have low costs of capital and this gives them a competitive advantage in takeover battles since they will place a higher value on the assets of takeover targets than companies with high beta assets”. Say whether you agree or disagree and explain why. “A company’s lenders and its shareholders can sometimes disagree about the investments that the company should make. Keeping leveraging low helps to keep these disagreements to a minimum”. Say whether you agree or disagree and explain why.

Question 4.9. Locate the centroid y of the beam's cross-sectional area and then determine the moments of inertia of this area and the product of inertia with respect to u and v axes. The axes have their origin at the centroid C. (Use Mohr's Circle) Solution 200 mm 25 mm 25 mm 60⁰ 75 mm 75 mm 25 mm y Question 4.9. Locate the centroid y of the beam's cross-sectional area and then determine the moments of inertia of this area and the product of inertia with respect to u and v axes. The axes have their origin at the centroid C. (Use Mohr's Circle) Solution 200 mm 25 mm 25 mm 60⁰ 75 mm 75 mm 25 mm y

A machine produces electrical components. 99.7% of the components have lengths between 1.176 cm and 1.224 cm. Assuming this data is normally distributed, what are the mean and standard deviation? Another machine produces electrical components; however for this machine there are 68% of the components have lengths between 1.176 cm and 1.224 cm. Assuming this data is normally distributed, what are the mean and standard deviation?

2) Create a maths problem and model solution corresponding to the following question: "Find the directional derivative of the following two-variable function for the supplied point and unit vector" Evaluate the directional derivative at the point (2, 5). Ensure that your supplied function uses both x and y, has 3 terms, and that at least one of those terms contains vx . You may choose the vector representing the direction you will evaluate the directional derivative in, and the derivative must be a multiple of 2. 3) Create a maths problem and model solution corresponding to the following question: "Determine the derivative with respect to x for the following function" Your problem should provide a function that requires use of the chain rule and the product rule to differentiate, contains tanh(2x), and produces a derivative that includes In(3) 4) Create a maths problem and model solution corresponding to the following question: "Evaluate the following integral using trigonometric substitution" The integral should make use of the substitution x = atan, and also require a second substitution to solve. The square root component should be multiplied by a polynomial.

4) Create a maths problem and model solution corresponding to the following question: "Evaluate the following integral using trigonometric substitution" The integral should make use of the substitution x = atane, and also require a second substitution to solve. The square root component should be multiplied by a polynomial.