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.
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.
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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