Find the eigen values of the following matrix and discuss the applications of eigen values in engineering disciplines.

Calculus by Geoffrey

Calculus

Q1. a. Find the eigen values of the following matrix and discuss the applications of eigen values in engineering disciplines.

(8 marks)

  1. Temperature of a disk brake plate at any point (x, y) varies is represented by the T(x,y)=100/(1+x3+y3 ) where T measure in °C and x, y in meters. Find the rate of change of temperature with respect to x direction and y direction and also the rate at a point (2,1). (12 marks)

Q2. a. In an automobile testing the relationship between the displacement s, velocity v and acceleration a of a piston is given by the following set of linear simultaneous equations:

Use Gauss-Jordon elimination method to determine the values of s, v and a.

(15 marks)

  1. The results obtained during helical spring loading test are as follows:

Force (Newton) Time (Seconds)

11.4 0.56

18.7 0.35

11.7 0.55

12.3 0.52

14.7 0.43

18.8 0.34

19.6 0.31

⦁ Determine the equation of the regression line of time on force.

⦁ Find the equation for the regression line of force on time.

⦁ Draw the scatter diagram. (10 marks)

Q3 a. In an oil rig a thermodynamic system, K = A , where R, K and A are constants

Find , (16 marks)

  1. Find the stationary point of the function y = x2 − 2x + 3 and hence determine the nature of this point. (14 marks)

Q4. a. Solve the linear equation using MATLAB

5x = 3 y – 2 z + 10

8 y + 4 z = 3 x + 20

2 x + 4 y – 9 z = 0

(5 marks)

  1. Consider the two matrices A= and B= using MATLAB, determine the following

⦁ A + B

⦁ AB

⦁ A2

⦁ AT

⦁ B-1

⦁ BT AT

⦁ A2 + B 2 + AB

⦁ Determinant of AB (20 marks)

Marking Scheme

Question Description Marks

Question 1 ⦁ Steps of eigen value determination 8

⦁ Determination of the rate of change of temperature with respect to x direction 4

Determination of the rate of change of temperature with respect to y direction 4

Determination of the rate at a point 4

Question 2 ⦁ Steps of elimination method 10

Values determination 5

⦁ Determination of the equation of the regression line of time on force. 6

Equation for the regression line of

force on time. 2

Scatter diagram. 2

Question 3 ⦁ ∂k/∂T 8

∂A/∂T 8

⦁ Calculation of stationary point 7

Nature of this point determination 7

Question 4 ⦁ Solution by MATLAB 5

⦁ Determination of

⦁ A + B

⦁ AB

⦁ A2

⦁ AT

⦁ B-1

⦁ BT AT

⦁ A2 + B2 + 2AB

⦁ Determinant of AB 8×2.5