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

(8 marks)

- 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)

- 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)

- 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)

- 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