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ENGSCI 211FC Mathematical Modelling 2, 2012

WORKSHEET 1 – B ASIC SKILLS TEST PRACTICE 

Logarithms

1. Simplify

   to the nicest possible result.

2. Evaluate . (If you don’t have a calculator, just determine an appropriate formula.)

Polynomial & rational functions

3. Divide        by .

4. Factorise  (Hint: Use the preceding result.)

5. Write down the roots of .

6. Write down a trial partial fraction expansion (PFE) for         .

7. Determine the coefficients in the PFE of  and write down the fully determined partial fractions expansion.

8. Write down a trial PFE for

  

.

9. Determine the coefficients in your expansion & write down the fully determined PFE for .

Other Functions10. What are the domain and range of   .

11. Sketch    over its domain.

12. Sketch  over the interval  13. Express   in its nicest possible form, given    and     .

14. Find a general expression for the roots

 of

. (Hint: Use a trig’ identity to convert this

into a simpler polynomial equation.)

Complex numbers15. Evaluate the modulus  of .

16. Evaluate the argument  of .

17. Evaluate the conjugate  of .

18. Evaluate the quotient .

Differentiation

19. Find & classify the stationary point(s) of     .20. Differentiate   .

21. Differentiate      .

22. Determine the tangent slope at    if     .

Integration

23. Evaluate     .

24. Determine the integral .

25. Determine the integral

.

26. Determine the integral .

27. Determine the integral   .

28. Determine the integral   .

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ENGSCI 211 – Mathematical Modelling 2

2 Initial version: 28 February 2012

Last updated: 29 February 2012

Differential Equations29. Classify the following ODEs as fully as possible:

a.        

b.      

c.        30. Solve    with    and     .

31. Find the general solution  of     .

32. Find an integrating factor for   .

33. Use your integrating factor to find the general solution for   . [Please note typo] 

34. Write down the characteristic polynomial associated with .

35. Find the roots of the characteristic polynomial, & hence write down the general solution.

Vectors

36. Evaluate the norm   of   .37. Evaluate the dot product between   and   .

38. Use your preceding answers to evaluate the cosine of the angle  between   and .

39. Evaluate the cross product  between   and .

40. Evaluate the sine of the angle between   and .

Matrices

41. Evaluate the determinant  of    .

42. Evaluate the product

of

 

 and

 .

43. Evaluate .

44. Write down the augmented matrix associated with the following system of equations:  

45. Reduce the augmented system to uppertriangular form.

46. Solve the preceding system of equations for , , and .

Probability

47. Determine the expectation  where  is uniformly distributed over .

48. Evaluate the mean of the sample  .

49. Evaluate the median of .

50. Evaluate the mode of .

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