Mathematics II

Frontmatter

1. Areas and Volumes I

Abstract

After working through this chapter you should be able to

(1)

Calculate the surface areas and volumes of pyramids, cones and spheres.

 

(2)

Define frustum and calculate the volumes and surface areas of frusta of pyramids and cones.

 

(3)

Calculate the total surface areas and volumes of composite figures.

 

(4)

Draw a graph to represent a given irregular area or volume to a suitable scale.

 

(5)

List the methods of measuring an irregular area.

 

(6)

Calculate an irregular area using the following rules (a) the mid-ordinate rule, (b) the trapezoidal rule, (c) Simpson’s rule.

 

(7)

Calculate an irregular volume using the above rules.

 

(8)

Give the results to an accuracy consistent with the data used.

 

H. G. Davies, G. A. Hicks

2. Trigonometry I

Abstract

After working through this chapter you should be able to

(1)

Determine the values of the trigonometric ratios for angles from 0° to 360° inclusive.

 

(2)

Derive the relationships tan θ = sin θ/cos θ and sin² θ + cos² θ = 1 for a right-angled triangle.

 

(3)

Plot the graphs of sin θ, cos θ and tan θ for angles from 0° to 360° inclusive.

 

(4)

State the sine and cosine rules.

 

(5)

State the conditions under which these rules may be used.

 

(6)

Use these rules to calculate unknown sides and angles in any triangle.

 

(7)

Solve practical problems using these rules.

 

(8)

Calculate the area of any triangle using the formulae ½ab sin C and √ [s (s − a) (s −b) (s − c)].

 

H. G. Davies, G. A. Hicks

3. Graphs I

Abstract

working through this chapter you should be able to

(1)

Decide on scales for Cartesian axes.

 

(2)

Plot straight-line graphs, and determine the intercept and gradient.

 

(3)

Plot straight-line graphs of experimental data.

 

(4)

Reduce the non-linear graphs

to linear form.

 

H. G. Davies, G. A. Hicks

4. Statistics I

Abstract

After working through this chapter you should be able to

(1)

List the usual methods of measuring central tendency and dispersion of a set of numerical data.

 

(2)

Use the range to group data into equal class intervals.

 

(3)

Define arithmetic mean and calculate it for ungrouped data and for grouped data with equal intervals.

 

(4)

Define mode and determine it for a set of numerical data.

 

(5)

Define median and determine it by arranging the data in rank order.

 

(6)

Determine median and quartiles from cumulative frequency curve.

 

(7)

Define standard deviation and determine it for a set of grouped and ungrouped data.

 

(8)

Compare two distributions with the same mean but different standard deviations.

 

H. G. Davies, G. A. Hicks

5. Algebra I

Abstract

After working through this chapter you should be able to

(1)

Evaluate expressions.

 

(2)

Estimate errors.

 

(3)

Transpose formulae.

 

(4)

Factorise quadratic expressions.

 

(5)

Solve quadratic equations (1) by factorisation, (2) using the formula.

 

(6)

Identify the sum and product of roots of a quadratic equation, in terms of the coefficients a, b and c.

 

(7)

Construct and solve quadratic equations occurring in practical situations.

 

(8)

Solve simultaneous equations (1) both linear (2) one linear, one quadratic.

 

H. G. Davies, G. A. Hicks

6. Graphs II

Abstract

After working through this chapter you should be able to

(1)

Recognise direct and inverse proportionality.

 

(2)

Plot graphs of quadratic equations, and solve the equation graphically.

 

(3)

Define maximum and minimum points.

 

(4)

Determine the quadratic equation from data.

 

(5)

Solve simultaneous equations graphically.

 

H. G. Davies, G. A. Hicks

7. Differentiation and Integration

Abstract

After working through this chapter you should be able to

(1)

Calculate the gradient of a chord to a curve.

 

(2)

Recognise incremental values.

 

(3)

Identify the differential coefficient with the gradient of the tangent, and differentiate simple expressions.

 

(4)

Define indefinite integration, and evaluate the indefinite integral of simple expressions.

 

(5)

Recognise the necessity for the constant of integration.

 

(6)

Define the definite integral, and evaluate the definite integral of simple expressions.

 

H. G. Davies, G. A. Hicks

8. Trigonometry II

Abstract

After working through this chapter you should be able to

1.

Define radian and convert radians to degrees and vice versa.

 

2.

Use radian measure to calculate lengths of arcs.

 

3.

Calculate areas of sectors and segments.

 

4.

Calculate projections of lengths and areas.

 

5.

Calculate lengths and areas of plane figures.

 

6.

Identify angle between (i) a line and a plane and (ii) between two planes.

 

7.

Calculate lengths and angles for solid figures.

 

H. G. Davies, G. A. Hicks

9. Mensuration II

Abstract

After working through this chapter you should be able to

1.

Define centroid and state its position for a rectangle, triangle, circle and semicircle.

 

2.

Calculate surface areas and volumes of frusta of cones, pyramids and spheres.

 

3.

Calculate the area and perimeter of an ellipse using given formulae.

 

4.

Calculate an irregular volume using the prismoidal rule.

 

5.

State and use Pappus’ theorem to find the volume of a solid of revolution.

 

H. G. Davies, G. A. Hicks

10. Algebra II

Abstract

After working through this chapter you should be able to

1.

Arrange fractions as continued fractions and calculate the error in the convergents.

 

2.

Use continued fractions to solve practical problems.

 

3.

Define a logarithm and deduce the laws of logarithms.

 

4.

Use these laws to simplify expressions and to solve equations.

 

5.

Use logarithms to plot a straight-line graph to verify the law y = ax ⁿ for experimental data and to determine the constants a and n.

 

6.

Draw graphs of exponential functions.

 

7.

Solve practical problems involving exponential functions.

 

H. G. Davies, G. A. Hicks

Backmatter