Calculus
Q1. a. Find the eigen values of the following matrix and discuss the applications of eigen values in engineering disciplines.
(8 marks)
b. 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)
b. 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)
b. 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)
b. 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
