|
|
|
|
|
-
-
-
-
-
-
-
-
|
|
|
5. Completing the square.
|
-
|
|
|
6. Solution of quadratic equations.
|
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
|
|
|
41. Knowledge and use of tanθ = sinθ/cosθ
|
-
-
-
-
-
|
|
|
Further Pure 1
|
-
|
|
|
I. Algebra and Graphs
|
-
|
|
|
88. Graphs of rational functions of the form (ax + b)/(cx +d) , (ax + b)/(cx^2 + dx + e) or (x^2 + ax + b)/(x^2 + cx + d)
|
-
|
|
|
89. Graphs of parabolas, ellipses and hyperbolas with equations y^2 = 4ax, x^2/a^2 + y^2/b^2 = 1, x^2/a^2 - y^2/b^2 = 1 and xy = c^2.
|
-
|
|
|
II. Complex Numbers
|
-
|
|
|
90. Non-real roots of quadratic
equations.
|
-
|
|
|
91. Sum, difference and product
of complex numbers in the
form x + i y .
|
-
|
|
|
92. Comparing real and
imaginary parts.
|
-
|
|
|
III. Roots and coefficients of a
quadratic equation
|
-
|
|
|
93. Manipulating expressions
involving α +β and αβ .
|
-
|
|
|
IV. Series
|
-
|
|
|
94. Use of formulae for the sum
of the squares and the sum of
the cubes of the natural
numbers.
|
-
|
|
|
V. Calculus
|
-
|
|
|
95. Finding the gradient of the
tangent to a curve at a point,
by taking the limit as h tends
to zero of the gradient of a
chord joining two points
whose x-coordinates differ
by h.
|
-
|
|
|
96. Evaluation of simple improper
integrals.
|
-
|
|
|
VI. Numerical Methods
|
-
|
|
|
97. Finding roots of equations by
interval bisection, linear
interpolation and the
Newton-Raphson method.
|
-
|
|
|
98. Solving differential equations of the form dx/dy = f(x).
|
-
|
|
|
99. Reducing a relation to a
linear law.
|
-
|
|
|
VII. Trigonometry
|
-
|
|
|
100. General solutions of
trigonometric equations
including use of exact values
for the sine, cosine and
tangent of 6/π, 4/π, 3/π
|
-
|
|
|
VIII. Matrices and Transformations
|
-
|
|
|
101. 2 × 2 and 2 × 1 matrices;
addition and subtraction,
multiplication by a scalar.
Multiplying a 2 × 2 matrix by
a 2 × 2 matrix or by a 2 × 1
matrix.
|
-
|
|
|
102. The identity matrix I for a
2 × 2 matrix.
|
-
|
|
|
103. Transformations of points in
the x − y plane represented
by 2 × 2 matrices.
|
-
|
|
|
Mechanics 1
|
-
|
|
|
I. Mathematical Modelling
|
-
|
|
|
237. Use of assumptions in
simplifying reality.
Candidates are expected to use mathematical models to solve
problems.
Mathematical analysis of
models.
|
-
|
|
|
238. Interpretation and validity of
models.
|
-
|
|
|
239. Refinement and extension of
models.
|
-
|
|
|
II. Kinematics in One and Two
Dimensions
|
-
|
|
|
240. Displacement, speed, velocity,
acceleration.
|
-
|
|
|
241. Sketching and interpreting
kinematics graphs.
|
-
|
|
|
242. Use of constant acceleration
equations.
|
-
|
|
|
243. Vertical motion under gravity.
|
-
|
|
|
244. Average speed and average
velocity.
|
-
|
|
|
245. Application of vectors in two
dimensions to represent
position, velocity or
acceleration.
|
-
|
|
|
246. Use of unit vectors i and j.
|
-
|
|
|
247. Magnitude and direction of
quantities represented by a
vector.
|
-
|
|
|
248. Finding position, velocity,
speed and acceleration of a
particle moving in two
dimensions with constant
acceleration.
|
-
|
|
|
249. Problems involving resultant
velocities.
|
-
|
|
|
III. Statics and Forces
|
-
|
|
|
250. Drawing force diagrams, identifying forces present and clearly labelling diagrams.
Candidates should distinguish between forces and other quantities. Force of gravity. Friction, limiting friction, coefficient of friction, the
relationship of F µR
|
-
|
|
|
251. Normal reaction forces.
|
-
|
|
|
252. Tensions in strings and rods,
thrusts in rods.
Modelling forces as vectors.
|
-
|
|
|
253. Finding the resultant of a
number of forces acting at
a point
|
-
|
|
|
254. Finding the resultant force
acting on a particle.
|
-
|
|
|
255. Knowledge that the resultant
force is zero if a body is in
equilibrium
|
-
|
|
|
IV. Momentum
|
-
|
|
|
256. Concept of momentum
|
-
|
|
|
257. The principle of conservation
of momentum applied to two
particles.
|
-
|
|
|
V. Newton's Laws of Motion.
|
-
|
|
|
258. Newton's three laws of
motion.
|
-
|
|
|
259. Simple applications of the
above to the linear motion of
a particle of constant mass.
Including a particle moving up or down an inclined plane.
Use of F= μR as a model
for dynamic friction.
|
-
|
|
|
VI. Connected Particles
|
-
|
|
|
260. Connected particle problems.
|
-
|
|
|
VII. Projectiles
|
-
|
|
|
261. Motion of a particle under
gravity in two dimensions.
|
-
|
|
|
262. Calculate range, time of
flight and maximum height.
|
-
|
|
|
263. Modification of equations to
take account of the height of
release.
|
-
|
|
|
Decision 1
|
-
|
|
|
I. Simple Ideas of Algorithms
|
-
|
|
|
342. Correctness, finiteness and
generality. Stopping
conditions.
|
-
|
|
|
343. Bubble, shuttle, shell,
quicksort algorithms.
|
-
|
|
|
II. Graphs and Networks
|
-
|
|
|
344. Vertices, edges, edge weights,
paths, cycles, simple graphs.
|
-
|
|
|
345. Adjacency/distance matrices.
|
-
|
|
|
346. Connectedness.
|
-
|
|
|
347. Directed and undirected
graphs
|
-
|
|
|
348. Degree of a vertex, odd and
even vertices, Eulerian trails
and Hamiltonian cycles.
|
-
|
|
|
349. Trees.
|
-
|
|
|
350. Bipartite graphs.
|
-
|
|
|
III. Spanning Tree Problems
|
-
|
|
|
351. Prim's and Kruskal's
algorithms to find minimum
spanning trees. Relative
advantage of the two
algorithms.
|
-
|
|
|
352. Greediness.
|
-
|
|
|
IV. Matchings
|
-
|
|
|
353. Use of bipartite graphs.
|
-
|
|
|
354. Improvement of matching
using an algorithm.
|
-
|
|
|
V. Shortest Paths in Networks
|
-
|
|
|
355. Dijkstra's algorithm
|
-
|
|
|
VI. Route Inspection Problem
|
-
|
|
|
356. Chinese Postman problem.
|
-
|
|
|
VII. Travelling Salesperson
Problem
|
-
|
|
|
357. Conversion of a practical
problem into the classical
problem of finding a
Hamiltonian cycle.
|
-
|
|
|
358. Determination of upper
bounds by nearest neighbour
algorithm.
|
-
|
|
|
359. Determination of lower
bounds on route lengths
using minimum spanning
trees.
|
-
|
|
|
VIII. Linear Programming
|
-
|
|
|
360. Graphical solution of twovariable
problems.
|
-
|
|
|
IX. Mathematical modelling
|
-
|
|
|
361. The application of
mathematical modelling to
situations that relate to the
topics covered in this module.
|
-
-
-
-
|
|
|
Further Pure 2
|
-
|
|
|
I. Roots of Polynomials
|
-
|
|
|
104. The relations between the
roots and the coefficients of
a polynomial equation; the
occurrence of the non-real
roots in conjugate pairs when
the coefficients of the
polynomial are real.
|
-
|
|
|
II. Complex Numbers
|
-
|
|
|
105. The Cartesian and polar coordinate
forms of a complex
number, its modulus,
argument and conjugate.
The sum, difference, product
and quotient of two complex
numbers.
|
-
|
|
|
106. The representation of a
complex number by a point
on an Argand diagram;
geometrical illustrations.
Simple loci in the complex
plane.
|
-
|
|
|
III. De Moivre's Theorem
|
-
|
|
|
107. De Moivre's theorem for
integral n.
|
-
|
|
|
108. De Moivre's theorem; the nth
roots of unity, the
exponential form of a
complex number.
|
-
|
|
|
109. Solutions of equations of the form z^n = a + ib.
|
-
|
|
|
IV. Proof by Induction
|
-
|
|
|
110. Applications to sequences
and series, and other
problems.
|
-
|
|
|
V. Finite Series
|
-
|
|
|
111. Summation of a finite series
by any method such as
induction, partial fractions or
differencing.
|
-
|
|
|
VI. The calculus of inverse
trigonometrical functions
|
-
|
|
|
112. Use the derivatives of the inverse trigonometric functions as given in the formulae booklet. To include the use of the standard integrals.
|
-
|
|
|
VII. Hyperbolic Functions
|
-
|
|
|
113. Hyperbolic and inverse
hyperbolic functions and
their derivatives; applications
to integration.
|
-
|
|
|
VIII. Arc length and Area of
surface of revolution about
the x-axis
|
-
|
|
|
114. Calculation of the arc length
of a curve and the area of a
surface of revolution using
Cartesian or parametric
coordinates.
|
-
|
|
|
Further Pure 3
|
-
|
|
|
I. Series and Limits
|
-
|
|
|
115. Maclaurin series
|
-
|
|
|
116. Expansions of ex, ln(1+ x) ,
cos x and sin x, and (1+ x)n
for rational values of n.
|
-
|
|
|
Further Pure 4
|
-
|
|
|
I. Series and Limits
|
-
|
|
|
117. Knowledge and use, for
k > 0, of limxke−x as x
tends to infinity and
limxk ln x as x tends to zero.
Improper integrals.
|
-
|
|
|
I. Vectors and Three- Dimensional Coordinate Geometry
|
-
|
|
|
129. Definition and properties of
the vector product.
Calculation of vector
products.
|
-
|
|
|
130. Calculation of scalar triple
products.
|
-
|
|
|
131. Applications of vectors to
two- and three-dimensional
geometry, involving points,
lines and planes.
|
-
|
|
|
132. Cartesian coordinate
geometry of lines and planes.
Direction ratios and direction
cosines.
|
-
|
|
|
II. Matrix Algebra
|
-
|
|
|
133. Matrix algebra of up to 3×3
matrices, including the
inverse of a 2×2 or 3×3
matrix.
|
-
|
|
|
134. The identity matrix I for
2×2 and 3×3 matrices.
|
-
|
|
|
135. Matrix transformations in
two dimensions: shears.
|
-
|
|
|
136. Rotations, reflections and
enlargements in three
dimensions, and
combinations of these.
|
-
|
|
|
137. Invariant points and invariant
lines.
|
-
|
|
|
138. Eigenvalues and eigenvectors
of 2 × 2 and 3 × 3 matrices.
|
-
|
|
|
139. Diagonalisation of 2 × 2 and
3 × 3 matrices.
|
-
|
|
|
III. Solution of Linear Equations
|
-
|
|
|
140. Consideration of up to three
linear equations in up to
three unknowns. Their
geometrical interpretation
and solution.
|
-
|
|
|
IV. Determinants
|
-
|
|
|
141. Second order and third order
determinants, and their
manipulation.
|
-
|
|
|
142. Factorisation of
determinants.
|
-
|
|
|
143. Calculation of area and
volume scale factors for
transformation representing
enlargements in two and
three dimensions.
|
-
|
|
|
V. Linear Independence
|
-
|
|
|
144. Linear independence and
dependence of vectors.
|
-
|
|
|
Further Pure 5
|
-
|
|
|
I. Series and Limits
|
-
|
|
|
118. Use of series expansion to
find limits.
|
-
|
|
|
Further Pure 6
|
-
|
|
|
II. Polar Coordinates
|
-
|
|
|
119. Relationship between polar
and Cartesian coordinates.
|
-
|
|
|
Further Pure 7
|
-
|
|
|
II. Polar Coordinates
|
-
|
|
|
120. Use of the formula area = integral from α to β of 1/2*r^2dθ
|
-
|
|
|
Further Pure 8
|
-
|
|
|
III. Differential Equations
|
-
|
|
|
121. The concept of a differential
equation and its order.
|
-
|
|
|
Further Pure 9
|
-
|
|
|
III. Differential Equations
|
-
|
|
|
122. Boundary values and initial
conditions, general solutions
and particular solutions.
|
-
|
|
|
Further Pure 10
|
-
|
|
|
IV. Differential Equations-First Order
|
-
|
|
|
123. Analytical solution of first order linear differential equations of the form dy/dx + Py = Q where P and Q are functions of x.
|
-
|
|
|
Further Pure 11
|
-
|
|
|
IV. Differential Equations-First Order
|
-
|
|
|
124. Numerical methods for the
solution of differential
equations of the form
dy/dx= f(x,y )
|
-
|
|
|
Further Pure 12
|
-
|
|
|
IV. Differential Equations-First Order
|
-
|
|
|
125. Euler's formula and
extensions to second order
methods for this first order
differential equation.
|
-
|
|
|
Further Pure 13
|
-
|
|
|
V. Differential Equations - Second Order
|
-
|
|
|
126. Solution of differential equations of the form a*(d^2y)/(dx^2) + b*dy/dx + cy = 0 , where a, band c are integers, by using an auxiliary equation whose roots may be real or
complex.
|
-
|
|
|
Further Pure 14
|
-
|
|
|
V. Differential Equations - Second Order
|
-
|
|
|
127. Solution of equations of the form a*(d^2y)/(dx^2) + b*dy/dx + cy = f(x) where a, band c are integers by finding the complementary function and a particular integral
|
-
|
|
|
Further Pure 15
|
-
|
|
|
V. Differential Equations - Second Order
|
-
|
|
|
128. Solution of differential equations of the form: (d^2y)/(dx^2) + P*dy/dx + Qy = R where P, Q, R are functions of x. A substitution will always be given which reduces the differential equation to a form which can be solved using the other methods.
|
-
|
|
|
Statistics 1
|
-
|
|
|
I. Numerical Measures
|
-
|
|
|
145. Standard deviation and
variance calculated on
ungrouped and grouped data.
|
-
|
|
|
146. Linear scaling.
|
-
|
|
|
147. Choice of numerical
measures.
|
-
|
|
|
II. Probability
|
-
|
|
|
148. Elementary probability; the
concept of a random event
and its probability.
|
-
|
|
|
149. Addition law of probability.
Mutually exclusive events.
|
-
|
|
|
150. Multiplication law of
probability and conditional
probability.
Independent events.
|
-
|
|
|
151. Application of probability
laws.
|
-
|
|
|
III. Binomial Distribution
|
-
|
|
|
152. Discrete random variables.
|
-
|
|
|
153. Conditions for application of
a binomial distribution.
|
-
|
|
|
154. Calculation of probabilities
using formula.
|
-
|
|
|
155. Calculation of probabilities
using tables.
|
-
|
|
|
156. Mean, variance and standard
deviation of a binomial
distribution.
|
-
|
|
|
IV. Normal Distribution
|
-
|
|
|
157. Continuous random vari
|
-
|
|
|
158. Properties of normal
distributions.
|
-
|
|
|
159. Calculation of probabilities.
|
-
|
|
|
160. Mean, variance and standard
deviation of a normal
distribution.
|
-
|
|
|
V. Estimation
|
-
|
|
|
161. Population and sample.
|
-
|
|
|
162. Unbiased estimates of a
population mean and
variance.
|
-
|
|
|
163. The sampling distribution of
the mean of a random sample
from a normal distribution.
|
-
|
|
|
164. A normal distribution as an
approximation to the
sampling distribution of the
mean of a large sample from
any distribution.
|
-
|
|
|
165. Confidence intervals for the
mean of a normal distribution
with known variance.
|
-
|
|
|
166. Confidence intervals for the
mean of a distribution using
a normal approximation.
|
-
|
|
|
167. Inferences from confidence
intervals.
|
-
|
|
|
VI. Correlation and Regression
|
-
|
|
|
168. Calculation and
interpretation of the product
moment correlation
coefficient.
|
-
|
|
|
169. Identification of response
(dependent) and explanatory
(independent) variables in
regression.
|
-
|
|
|
170. Calculation of least squares
regression lines with one
explanatory variable. Scatter
diagrams and drawing a
regression line thereon.
|
-
|
|
|
171. Calculation of residuals.
|
-
|
|
|
172. Linear scaling.
|
-
|
|
|
Statistics 2
|
-
|
|
|
I. Discrete Random Variables
|
-
|
|
|
173. Discrete random variables
and their associated
probability distributions.
|
-
|
|
|
174. Mean, variance and standard
deviation.
|
-
|
|
|
175. Mean, variance and standard
deviation of a simple function
of a discrete random variable.
|
-
|
|
|
II. Poisson Distribution
|
-
|
|
|
176. Conditions for application of
a Poisson distribution.
|
-
|
|
|
177. Calculation of probabilities
using formula.
|
-
|
|
|
178. Mean, variance and standard
deviation of a Poisson
distribution.
|
-
|
|
|
179. Distribution of sum of
independent Poisson
distributions.
|
-
|
|
|
III. Continuous Random Variables
|
-
|
|
|
180. Differences from discrete
random variables.
|
-
|
|
|
181. Probability density functions,
distribution functions and
their relationship.
|
-
|
|
|
182. The probability of an
observation lying in a
specified interval.
|
-
|
|
|
183. Median, quartiles and
percentiles.
|
-
|
|
|
184. Mean, variance and standard
deviation.
|
-
|
|
|
185. Mean, variance and standard
deviation of a simple function
of a continuous random
variable.
|
-
|
|
|
186. Rectangular distribution.
|
-
|
|
|
IV. Estimation
|
-
|
|
|
187. Confidence intervals for the
mean of a normal distribution
with unknown variance.
|
-
|
|
|
V. Hypothesis Testing
|
-
|
|
|
188. Null and alternative
hypotheses.
|
-
|
|
|
189. One tailed and two tailed
tests, significance level,
critical value, critical region,
acceptance region, test
statistic, Type I and
Type II errors.
|
-
|
|
|
190. Tests for the mean of a
normal distribution with
known variance.
|
-
|
|
|
191. Tests for the mean of a
normal distribution with
unknown variance.
|
-
|
|
|
192. Tests for the mean of a
distribution using a normal
approximation.
|
-
|
|
|
VI. Chi-Squared (χ2) Contingency
Table Tests
|
-
|
|
|
193. Introduction to χ2
distribution.
|
-
|
|
|
194. Use of sum (Oi - Ei)^2/Ei as an approximate χ^2-statistic.
|
-
|
|
|
195. Conditions for approximation
to be valid.
|
-
|
|
|
196. Test for independence in
contingency tables.
|
-
|
|
|
Statistics 3
|
-
|
|
|
I. Further Probability
|
-
|
|
|
197. Bayes' Theorem.
|
-
|
|
|
II. Linear Combinations of Random
Variables
|
-
|
|
|
198. Mean, variance and standard
deviation of a linear
combination of two (discrete or
continuous) random variables.
|
-
|
|
|
199. Mean, variance and standard
deviation of a linear
combination of independent
(discrete or continuous) random
variables.
|
-
|
|
|
200. Linear combinations of
independent normal random
variables.
|
-
|
|
|
III. Distributional Approximations
|
-
|
|
|
201. Mean, variance and standard
deviation of binomial and
Poisson distributions.
|
-
|
|
|
202. A Poisson distribution as an
approximation to a binomial
distribution.
|
-
|
|
|
203. A normal distribution as an
approximation to a binomial
distribution.
|
-
|
|
|
204. A normal distribution as an
approximation to a Poisson
distribution.
|
-
|
|
|
IV. Estimation
|
-
|
|
|
205. Estimation of sample sizes
necessary to achieve confidence
intervals of a required width
with a given level of
confidence.
|
-
|
|
|
206. Confidence intervals for the
difference between the means
of two independent normal
distributions with known
variances.
|
-
|
|
|
207. Confidence intervals for the
difference between the means
of two independent
distributions using normal
approximations.
|
-
|
|
|
208. The mean, variance and
standard deviation of a sample
proportion.
|
-
|
|
|
209. Unbiased estimate of a
population proportion.
|
-
|
|
|
210. A normal distribution as an
approximation to the sampling
distribution of a sample
proportion based on a large
sample.
|
-
|
|
|
211. Approximate confidence
intervals for a population
proportion and for the mean of
a Poisson distribution.
|
-
|
|
|
212. Approximate confidence
intervals for the difference
between two population
proportions and for the
difference between the means
of two Poisson distributio
|
-
|
|
|
V. Hypothesis Testing
|
-
|
|
|
213. The notion of the power of a
test.
|
-
|
|
|
214. Tests for the difference
between the means of two
independent normal
distributions with known
variances.
|
-
|
|
|
215. Tests for the difference
between the means of two
independent distributions using
normal approximations.
|
-
|
|
|
216. Tests for a population
proportion and for the mean of
a Poisson distribution.
|
-
|
|
|
217. Tests for the difference
between two population
proportions and for the
difference between the means
of two Poisson distributions.
|
-
|
|
|
218. Use of the supplied tables to test H0:ρ =0 for a bivariate normal population.
|
-
|
|
|
Statistics 4
|
-
|
|
|
I. Geometric and Exponential
Distributions
|
-
|
|
|
219. Conditions for application of
a geometric distribution.
|
-
|
|
|
220. Calculation of probabilities
for a geometric distribution
using formula.
|
-
|
|
|
221. Mean, variance and standard
deviation of a geometric
distribution.
|
-
|
|
|
222. Conditions for application of
an exponential distribution.
|
-
|
|
|
223. Calculation of probabilities
for an exponential
distribution.
|
-
|
|
|
224. Mean, variance and standard
deviation of an exponential
distribution.
|
-
|
|
|
II. Estimators
|
-
|
|
|
225. Review of the concepts of a
sample statistic and its
sampling distribution, and of
a population parameter.
|
-
|
|
|
226. Estimators and estimates.
|
-
|
|
|
227. Properties of estimators.
|
-
|
|
|
III. Estimation
|
-
|
|
|
228. Confidence intervals for the
difference between the
means of two normal
distributions with unknown
variances.
|
-
|
|
|
229. Confidence intervals for a
normal population variance
(or standard deviation) based
on a random sample.
|
-
|
|
|
230. Confidence intervals for the
ratio of two normal
population variances (or
standard deviations) based on
independent random samples.
|
-
|
|
|
IV. Hypothesis Testing
|
-
|
|
|
231. Tests for the difference
between the means of two
normal distributions with
unknown variances.
|
-
|
|
|
232. Tests for a normal population
variance (or standard
deviation) based on a random
sample.
|
-
|
|
|
233. Tests for the ratio of two
normal population variances
(or standard deviations)
based on independent
random samples.
|
-
|
|
|
V. Chi-Squared (χ2) Goodness of
Fit Tests
|
-
|
|
|
234. Use of sum (Oi - Ei)^2/Ei as an approximate χ^2 -statistic.
|
-
|
|
|
235. Conditions for approximation
to be valid.
|
-
|
|
|
236. Goodness of fit tests.
|
-
|
|
|
Mechanics 2
|
-
|
|
|
I. Mathematical Modelling
|
-
|
|
|
264. The application of
mathematical modelling to
situations that relate to the
topics covered in this module.
|
-
|
|
|
II. Moments and Centres of
Mass
|
-
|
|
|
265. Finding the moment of a
force about a given poin
|
-
|
|
|
266. Determining the forces acting
on a rigid body when in
equilibrium.
|
-
|
|
|
267. Centres of Mass.
|
-
|
|
|
268. Finding centres of mass by
symmetry (e.g. for circle,
rectangle).
|
-
|
|
|
269. Finding the centre of mass of
a system of particles.
|
-
|
|
|
270. Finding the centre of mass of
a composite body.
|
-
|
|
|
271. Finding the position of a body
when suspended from a given
point and in equilibrium.
|
-
|
|
|
III. Kinematics
|
-
|
|
|
272. Relationship between
position, velocity and
acceleration in one, two or
three dimensions, involving
variable acceleration.
|
-
|
|
|
273. Finding position, velocity and
acceleration vectors, by the
differentiation or integration
of f (t)i+g(t)j+h(t)k , with
respect to t.
|
-
|
|
|
IV. Newton's Laws of Motion
|
-
|
|
|
274. Application of Newton's laws
to situations, with variable
acceleration.
|
-
|
|
|
V. Application of Differential Equations
|
-
|
|
|
275. One-dimensional problems
where simple differential
equations are formed as a
result of the application of
Newtons second law.
|
-
|
|
|
VI. Uniform Circular Motion
|
-
|
|
|
276. Motion of a particle in a
circle with constant speed.
|
-
|
|
|
277. Knowledge and use of the
relationships v = rω, a = rω^2 = a = v^2/r
|
-
|
|
|
278. Angular speed in radians s-1
converted from other units
such as revolutions per
minute or time for one
revolution.
|
-
|
|
|
279. Position, velocity and
acceleration vectors in terms
of i and j.
|
-
|
|
|
280. Conical pendulum.
|
-
|
|
|
VII. Work and Energy
|
-
|
|
|
281. Work done by a constant
force.
|
-
|
|
|
282. Gravitational potential
energy.
|
-
|
|
|
283. Kinetic energy.
|
-
|
|
|
284. The work-energy principle.
|
-
|
|
|
285. Conservation of mechanical
energy.
|
-
|
|
|
286. Work done by a variable
force.
|
-
|
|
|
287. Hooke's law.
|
-
|
|
|
288. Elastic potential energy for
strings and springs.
|
-
|
|
|
289. Power, as the rate at which a
force does work, and the
relationship P = Fv.
|
-
|
|
|
VIII. Vertical Circular Motion
|
-
|
|
|
290. Circular motion in a vertical
plane.
|
-
|
|
|
II. Dimensional Analysis
|
-
|
|
|
292. Finding dimensions of
quantities.
Prediction of formulae.
Checks on working, using
dimensional consistency.
|
-
|
|
|
III. Collisions in one dimension
|
-
|
|
|
293. Momentum.
Impulse as change of
momentum.
Impulse as Force × Time.
Impulse as ∫ F dt
|
-
|
|
|
294. Conservation of momentum.
Newton's Experimental Law.
Coefficient of restitution.
|
-
|
|
|
IV. Collisions in two dimensions
|
-
|
|
|
295. Momentum as a vector.
|
-
|
|
|
296. Impulse as a vector.
|
-
|
|
|
297. Conservation of momentum
in two dimensions.
|
-
|
|
|
298. Coefficient of restitution and
Newton's experimental law.
|
-
|
|
|
299. Impacts with a fixed surface.
|
-
|
|
|
300. Oblique Collisions
|
-
|
|
|
V. Further Projectiles
|
-
|
|
|
301. Elimination of time from
equations to derive the
equation of the trajectory of
a projectile
|
-
|
|
|
VI. Projectiles on Inclined Planes
|
-
|
|
|
302. Projectiles launched onto
inclined planes.
|
-
|
|
|
Mechanics 3
|
-
|
|
|
I. Relative Motion
|
-
|
|
|
291. Relative velocity.
Use of relative velocity and
initial conditions to find
relative displacement.
Interception and closest
approach.
|
-
|
|
|
Mechanics 4
|
-
|
|
|
I. Moments
|
-
|
|
|
303. Couples.
|
-
|
|
|
304. Reduction of systems of
coplanar forces.
|
-
|
|
|
305. Conditions for sliding and
toppling.
|
-
|
|
|
II. Frameworks
|
-
|
|
|
306. Finding unknown forces
acting on a framework.
Finding the forces in the
members of a light, smoothly
jointed framework.
Determining whether rods are
in tension or compression.
|
-
|
|
|
III. Vector Product and Moments
|
-
|
|
|
307. The vector product
i×i = 0, i×j = k, j×i = - k,
etc
|
-
|
|
|
308. The result |a×b|=|a||b|sinθ
|
-
|
|
|
309. The moment of a force as
r×F.
|
-
|
|
|
310. Vector methods for resultant
force and moment.
|
-
|
|
|
311. Application to simple
problems.
|
-
|
|
|
IV. Centres of mass by Integration
for Uniform Bodies
|
-
|
|
|
312. Centre of mass of a uniform
lamina by integration.
|
-
|
|
|
313. Centre of mass of a uniform
solid formed by rotating a
region about the x-axis.
|
-
|
|
|
V. Moments of Inertia
|
-
|
|
|
314. Moments of inertia for a
system of particles
|
-
|
|
|
315. Moments of inertia for
uniform bodies by
integration.
|
-
|
|
|
316. Moments of inertia of
composite bodies.
|
-
|
|
|
317. Parallel and perpendicular
axis theorems.
|
-
|
|
|
VI. Motion of a rigid body about
a smooth fixed axis.
|
-
|
|
|
318. Angular velocity and
acceleration of a rigid body.
|
-
|
|
|
319. Motion of a rigid body about
a fixed horizontal or vertical
axis.
|
-
|
|
|
320. Rotational kinetic energy and
the principle of conservation
of energy.
|
-
|
|
|
321. Moment of momentum
(angular momentum).
|
-
|
|
|
322. The principle of conservation
of angular momentum.
|
-
|
|
|
323. Forces acting on the axis of
rotation
|
-
|
|
|
Mechanics 5
|
-
|
|
|
I. Simple Harmonic Motion
|
-
|
|
|
324. Knowledge of the definition
of simple harmonic motion.
|
-
|
|
|
325. Finding frequency, period and
amplitude
|
-
|
|
|
326. Knowledge and use of the
formula v^2 = ω^2(a^2−x^2)
|
-
|
|
|
327. Formation of simple second
order differential equations
to show that simple harmonic
motion takes place.
|
-
|
|
|
328. Solution of second order differential equations of the form (d^2x)/(dt^2) = −ω^2*x
|
-
|
|
|
329. Simple Pendulum.
|
-
|
|
|
II. Forced and Damped
Harmonic Motion
|
-
|
|
|
330. Understanding the terms
forcing and damping and
solution of problems
involving them.
|
-
|
|
|
331. Candidates should be able to
set up and solve differential
equations in situations
involving damping and
forcing.
|
-
|
|
|
332. Light, critical and heavy
damping.
|
-
|
|
|
333. Resonance.
|
-
|
|
|
334. Application to spring/mass
systems.
|
-
|
|
|
III. Stability
|
-
|
|
|
335. Finding and determining
whether positions of
equilibrium are stable or
unstable.
|
-
|
|
|
IV. Variable Mass Problems
|
-
|
|
|
336. Equation of motion for
variable mass
|
-
|
|
|
V. Motion in a Plane using Polar
Coordinates
|
-
|
|
|
337. Polar coordinates
|
-
|
|
|
338. Transverse and radial
components of velocity in
polar form.
|
-
|
|
|
339. Transverse and radial
components of acceleration
in polar form.
|
-
|
|
|
340. Application of polar form of
velocity and acceleration.
|
-
|
|
|
341. Application to simple central
forces.
|
-
|
|
|
Decision 2
|
-
|
|
|
I. Critical Path Analysis
|
-
|
|
|
362. Representation of compound
projects by activity networks,
algorithm to find the critical
path(s); cascade (or Gantt)
diagrams; resource
histograms and resource
levelling.
|
-
|
|
|
II. Allocation
|
-
|
|
|
363. The Hungarian algorithm.
|
-
|
|
|
III. Dynamic Programming
|
-
|
|
|
364. The ability to cope with
negative edge lengths.
|
-
|
|
|
365. Application to production
planning.
|
-
|
|
|
366. Finding minimum or
maximum path through a
network.
|
-
|
|
|
367. Solving maximin and
minimax problems.
|
-
|
|
|
IV. Network Flows
|
-
|
|
|
368. Maximum flow/minimum cut
theorem.
|
-
|
|
|
369. Labelling procedure.
|
-
|
|
|
V. Linear Programming
|
-
|
|
|
370. The Simplex method and the
Simplex tableau.
|
-
|
|
|
VI. Game Theory for Zero Sum
Games
|
-
|
|
|
371. Pay-off matrix, play-safe
strategies and saddle points.
|
-
|
|
|
372. Optimal mixed strategies for
the graphical method.
|
-
|
|
|
VII. Mathematical modelling
|
-
|
|
|
373. The application of
mathematical modelling to
situations that relate to the
topics covered in this module.
|
|
|
|
|
|
|
|
|
|
|
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