A Maths  30 days revision plan  2 hours a day
 
Content
Exercises
Day 1
Set language and notation
 
use set language and notation , and Venn diagrams to describe sets and represent relationships between sets as follows :

A = { x : x is a natural number }
B = { ( x , y ) : y = m x  + c  }
C = { x : a   <   x   <  b }
D = { a , b , c , ... }  

understand and use the following notation :

Union of A and B
A B
Intersection of A and B
A ∩ B
Number of elements in set  A
n(A)
"... is an element of ..."
"... is not an element of ..."
Complement of set A
A'
The empty set
Universal set
U
A is a subset of B
A B
A is a proper subset of B
A ⊂ B
A is not a subset of B
A B
A is not a proper subset of B
A ⊄ B


Day 2
Functions
 
understand the terms :

function
domain
range (image set)
one-one function
inverse function
composition of functions

use the notation  f ( x ) = sin x , f : x |-->  l g  x ,  ( x > 0 ) ,
f  – 1 ( x )  and  f 2 ( x ) [ = f ( f ( x ) ) ]

understand the relationship between y = f ( x ) and 
y = | f ( x ) | , where f ( x ) may be linear , quadratic or trigonometric


Q1

Q2

Q3

Q6
Day 3 explain in words why a given function is a function or why it does not have an inverse

find the inverse of a one-one function and form composite functions

use sketch graphs to show the relationship between a function and its inverse


Q7

Q8

Q9
Day 4
Quadratic functions
 
find the maximum or minimum value of the quadratic function f : x |--> a x 2 + b x + c  by any method

use the maximum or minimum value of f ( x ) to sketch the graph or determine the range for a given domain


quadratic formula
Q1

quadratic functions
Q1

Q2

Q3

Day 5 know the conditions for f ( x ) = 0  to have

(i) two real roots
(ii) two equal roots
(iii) no real roots

and the related conditions for a given line to

(i) intersect a given curve
(ii) be a tangent to a given curve
(iii) not intersect a given curve

solve quadratic equations for real roots and find the solution set for quadratic inequalities


quadratic functions
Q4

Q5

Q6
Day 6
Indices and surds
 
perform simple operations with indices and with surds, including rationalising the denominator


indices
Q1

Q2

surds
Q1

Q2

Q3
Day 7
Factors of polynomials
 
know and use the remainder and factor theorems

 
Day 8 find factors of polynomials

solve cubic equations

 
Day 9
Simultaneous equations
 
solve simultaneous equations in two unknowns with at least one linear equation

 
Day 10
Logarithmic and exponential functions
 
know simple properties and graphs of the logarithmic and exponential functions including  l n x  and  e  x 

 
Day 11 know and use the laws of logarithms (including change of base of logarithms)
solve equations of the form ax= b
 
Day 12
Straight line graphs
 
interpret the equation of a straight line graph in the form
y = mx +c
transform given relationships, including y = axn and y = Abx, to straight line form and hence determine unknown constants by calculating the gradient or intercept of the transformed graph
 
Day 13 solve questions involving mid-point and length of a line
know and use the condition for two lines to be parallel or perpendicular
 
Day 14
Circular measure
 
solve problems involving the arc length and sector area of a circle  
Day 15 knowledge and use of radian measure  
Day 16
Trigonometry
 
know the six trigonometric functions of angles of any magnitude (sine, cosine, tangent, secant, cosecant, cotangent)
understand amplitude and periodicity and the relationship between graphs of e.g. sin x and sin 2x
 
Day 17 draw and use the graphs of y = a sin(bx) + c,
y = a cos(bx) + c, y = a tan(bx) + c, where a, b are positive integers and c is an integer
 
Day 18 use the relationships AAcossin= tan A, AAsincos= cot A,
sin2 A + cos2 A = 1, sec2 A = 1 + tan2 A,
cosec2 A = 1 + cot2 A, and solve simple trigonometric equations involving the six trigonometric functions and the above relationships (not including general solution of trigonometric equations)
prove simple trigonometric identities
 
Day 19
Permutations and combinations
 
recognise and distinguish between a permutation case and a combination case
know and use the notation n!, (with 0! = 1), and the expressions for permutations and combinations of n items taken r at a time
answer simple problems on arrangement and selection (cases with repetition of objects, or with objects arranged in a circle or involving both permutations and combinations, are excluded)
 
Day 20
Binomial expansions
 
use the Binomial Theorem for expansion of (a + b)n for positive integral n
use the general term 

 a


rnn –
r
br, 0
r
n
 
Day 21
Vectors in 2 dimensions
 
use vectors in any form, e.g. , AB , p, ai + bj
know and use position vectors and unit vectors
find the magnitude of a vector. Add and subtract vectors and multiply vectors by scalars
compose and resolve velocities
use relative velocity including solving problems on interception (but not closest approach)
 
Day 22
Matrices
 
display information in the form of a matrix of any order and interpret the data in a given matrix;
• solve problems involving the calculation of the sum and product (where appropriate) of two matrices and interpret the results;
• calculate the product of a scalar quantity and a matrix;
• use the algebra of 2 x 2 matrices (including the zero and identity matrix);
• calculate the determinant and inverse of a non-singular 2 x 2 matrix and solve simultaneous linear equations
Day 23
Differentiation and integration
 
understand the idea of a derived function;
• use the notations f ˈ (x), f "(x),;dddddd,dd22=xyxxyxy
• use the derivatives of the standard functions xn (for any rational n), sinx, cosx, tanx, ex, lnx, together with constant multiples, sums and composite functions of these
 
Day 24 differentiate products and quotients of functions;
• apply differentiation to gradients, tangents and normals, stationary points, connected rates of change, small increments and approximations and practical maxima and minima problems;
• discriminate between maxima and minima by any method;
• understand integration as the reverse process of differentiation;
• integrate sums of terms in powers of x excluding x1
 
Day 25 integrate functions of the form
(ax + b)n (excluding n = –1), eax+b, sin(ax + b),
cos(ax + b);
• evaluate definite integrals and apply integration to the evaluation of plane areas;
 
Day 26 • apply differentiation and integration to kinematics problems that involve displacement, velocity and acceleration of a particle moving in a straight line with variable or constant acceleration, and the use of x-t and v-t graphs  
Day 27 Test 1 (must pass)  
Day 28 Test 2 (must pass)  
Day 29 Test 3 (should pass)  
Day 30 Test 4 (good to pass)