QUESTION 1. (a) A gas is being compressed in a closed cylinder

QUESTION 1.

(a)

A gas is being compressed in a closed cylinder and the values of pressures and corresponding volumes at constant temperature are as shown:

Pressure (kPa)

Volume (m3)

160

0.034

180

0.036

200

0.030

220

0.027

240

0.024

260

0.025

280

0.020

Find the coefficient of correlation for these values. (10 marks)

(b)

The mean mass of bricks for building produced by a brick manufacturer is 5 kg and the standard deviation of the masses is 0.036 kg. This manufacturer produced 10,000 bricks, find how many bricks are likely to have masses of:

between 4.88 and 4.92 kg

between 4.92 and 5.04 kg, and

more than 5.04 kg.

Assuming the masses are normally distributed. (10 marks)

QUESTION 2

Solve the following 2nd order differential equations:

(8 marks)

(12 marks)

QUESTION 3

(a)

If find the total differential,

(8 marks)

(b)

The rate of flow, Q, of a liquid through a pipe of diameter, d and length L is given by

where is the pressure difference between the ends, and is the coefficient of viscosity.

If can be measured to within 0.5%, to within 1.1%, to within 0.3% and to within 0.1%, calculate the maximum percentage error in the value of .

(12 marks)

QUESTION 4

(a)

Find

2A + 3B

A x B

B x C

(6 marks)

(b)

Use Matrix method to solve the simultaneous equations:

(14 marks)

Question 5.

Given that:

Determine, at the point (1, 2, 1),

grad f

div F, and

curl F (20 marks)

Formulas

Trapezoidal rule:

Simpson’s rule:

Newton-Raphson method:

Euler’s method:

Functions of Two or more Variables

and — stationary point is a minimum

and — stationary point is a maximum

— stationary point is a saddle point

Statistics formulas and Tables

Hyperbolic Functions

Calculus

Product Rule: If , then

Quotient Rule: If , then

Chain Rule:

Integration by parts:

Volume of Revolution:

Table of Standard Differential Coefficients

f(x)

nxn-1

1/x (x 0)

sin x

cos x

sinh x

cosh x

cosx

-sin x

cosh x

sinh x

tan x

tanh x

cosec x

-cosec x cot x

cosech x

-cosech x coth x

sec x

sec x tan x

sech x

-sech x tanh x

cot x

–

coth x

Table of Standard Integrals

f(x)

sin x

-cos x

sinh x

cosh x

cosx

sin x

cosh x

sinh x

tan x

tanh x

cosec x

cosech x

sec x

sech x

cot x

coth x

Algebra

The Binomial Coefficients:

This result is valid for all x if n is a positive integer and for for all other values of n.

Quadratic equations:

If then

Logarithms:

Matrices

If , then , and

Sine and Cosine rule

Cosine rule: ,

Sine rule:

Area of triangle:

Some formulas and Tables of Laplace Transforms

S/No.