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Csec Mathematics Summary, Cheat Sheet of Mathematics

Mathematics

All Csec Mathematics topics are included in this pdf in a consized form.

Typology: Cheat Sheet

2022/2023

Uploaded on 05/11/2023

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MATHEMATICS CSEC SUMMARY 2022
Section 1 – Number Theory and Computation
Section 2 – Consumer Arithmetic
Section 3 – Sets
Section 4 – Measurements
Section 5 – Statistics
Section 6 – Algebra
Section 7 – Relations, Functions and Graphs
Section 8 – Geometry and Trigonometry
Section 9 – Vectors and Matrices
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MATHEMATICS CSEC SUMMARY 2022

Section 1 – Number Theory and Computation

Section 2 – Consumer Arithmetic

Section 3 – Sets

Section 4 – Measurements

Section 5 – Statistics

Section 6 – Algebra

Section 7 – Relations, Functions and Graphs

Section 8 – Geometry and Trigonometry

Section 9 – Vectors and Matrices

Section 1 – Number Theory and Computation

Sets of numbers :

Natural numbers, N = {1, 2, 3,….}

Whole numbers, W = {0, 1, 2, 3, …..}

Integers, Z = { …, - 2, - 1, 0, 1, 2, …}

Rational numbers, Q = {

!

"

, p and q are integers, q ≠ 0}

Irrational numbers, 𝑄

Real numbers, R = Q ∪ 𝑄

Significant figures rules :

  1. All non-zero numbers ARE significant
  2. Zeros between two non-zero digits ARE significant
  3. Leading zeros are NOT significant. .e.g. 0.0045 has 2 sig. fig.
  4. Trailing zeros to the right of the decimal ARE significant. e.g. 45.00 has 4 sig. fig.
  5. Trailing zeros in a whole number with decimal shown ARE significant.
  6. Trailing zeros in a whole number with no decimal shown are NOT significant.

Properties of numbers:

a) Closure: If a,b ∈ 𝑅 then a*b ∈ 𝑅.

Associative: (x + y) + z = x + (y + z)

c) Commutative: x + y = y + x and x. y = y. x. (d) Distributive: x. (y + z) = x. y. +

x. z

e) Additive Identity: x + 0 = 0 + x = x. (f) Multiplicative Identity: x. 1 = 1.

x = x

g) Additive Inverse: x + ( - x ) = ( - x ) + x = 0 (h) Multiplicative Inverse: x./

$

$

  1. x

Ratios:

A ratio of a : b : c implies that the fractions being shared are

Section 3 – Sets

  • n( A ∪ 𝐵 ) = 𝑛(𝐴) + 𝑛(𝐵) − 𝑛(𝐴 ∩ 𝐵)

Venn diagrams

Section 4 – Measurements

Length Mass

10 mm = 1 cm 1g = 1000mg

100 cm = 1 m 1kg = 1000g

1000 mm = 1 m 1kg = 2.2lbs

1000 m = 1 km 1lb = 16 ounces

Speed =

9-*%.'1+

%-7+

Units: ms

  • 1

or kmh

  • 1

Distance = speed x time

Time =

9-*%'.1+

*!++

To construct a cumulative frequency graph and read off the Quartiles we do the following:

Quartiles:

Lower Quartile, Q 1

<

th

term

Median, Q 2

=

th

term

Upper Quartile, Q 3

>

<

th

term

Inter Quartile Range = Q 3

– Q

1

Semi-Inter Quartile Range =

?

!

3?

"

=

Section 6 – Algebra

Basic Algebra rules:

i. x + x = 2x

ii. x – 2x = x ( 1 – 2) = - x

iii. x + y = x + y

Indices Rules

i. x

m

. x

n

= x

m+n

ii. x

m

÷ x

n

= x

m-n

iii. x

0

iv. (x

m

n

= x

mn

v. x

  • m

$

Simplifying:

When simplifying fractions:

Expanding brackets:

(a + b)(a + b ) = a

2

+ 2ab + b

2

Factorizing :

  1. ab + ca = a ( b + c)
  2. x

2

y + y

2

x = xy (x + y)

  1. px

2

  • qx + apx + aq = x(px + q) + a (px + q)

= (px + q)(x + a)

  1. a

2

  • b

2

= (a - b)(a + b)

Solving:

'

@

1

9

cross multiply to obtain ad = bc then solve for unknown.

  1. When solving simultaneously use elimination, if 2 linear equations, or substitution

method, if a linear and a quadratic.

  1. Direct variation : a 𝛼 b and Inverse variation : a 𝛼

@

a = kb a =

A

@

Sign rules

  • x - = +
  • x + = -

  • x - = -

  • x + = +

To factorize a quadratic:

ax

2

  • bx + c
  1. Obtain a, b and c
  2. Get two numbers that when

multiplied give ac and when added

gives b

  1. Replace b with those two numbers

then factorize

NB.

An expression as no equal sign [=],

but an equation has an equal sign

Composite Functions

Inverse of a function

Co-ordinate Geometry:

Distance between two points : L(𝑥

=

=

=

=

Mid-point: (

$

$

B$

"

=

C

$

BC

"

=

Gradient : m =

C

$

3 C

"

$

$

3 $

"

!'0'((+( (-.+* D'E+ +"F'( /0'9-+.%* [ 7

"

H 7

$

]

!+0!+.9-1F('0 (-.+,!0&9F1% &; /0'9-+.% +"F'( 3 #. [ 7

"

7

$

H 3 #]

functions of functions, substitute

one function into the next

eg

𝑓P𝑠(𝑥)R = 𝑓 /

𝑔P𝑓(𝑥)R = 𝑔( 2 𝑥 − 1 ) =

=

= 𝑓P𝑓

R

3 #

Steps

1)let y = f(x)

  1. interchange x and y

  2. Solve for y

3 #

3 #

Equation of a line : y = mx + c m – gradient

c – y-intercept (cuts the y-axis)

To find the equation of a line:

  • find gradient of line
  • obtain a point on the line
  • substitute in 𝑐 = 𝑦 − 𝑚𝑥

NB.

  • Solving equations simultaneously gives the points of intersection of the equations.

Quadratic:

General form : y = ax

2

  • bx + c [highest power of x is 2]

To complete the square : y = a(x + h)

2

  • k. where h =

@

='

and k = c – ah

2

To sketch a quadratic :

  • Shape: minimum , a > 0

Maximum, a < 0

  • Turning point : ( - h , k)
  • Maximum or minimum value is always k.
  • X-value which gives max or minimum value is – h.
  • X-intercepts: solve ax

2

  • bx +c = 0
  • Y-intercept: (0,c)

Inequalities:

  • Solve inequalities like equations, but
  • Change the inequality sign when ÷ by a negative
  • For < or ≤ : shade below the line
  • For > or ≥: shade above the line

< less than / fewer than

> greater than / more than

≤ at most / no more than

≥ at least / no less than

Transformations:

Section 9 – Vectors and Matrices

Vectors:

Position vector, 𝑂𝑃

kkkkk⃗

= /

kkkkk⃗

= − /

Addition: /

Subtraction: /

Multiplying Vectors

a) By a scalar

If 𝑂𝑃

kkkkk⃗

kkkkk⃗

= 2 /

b) Two vectors

If we have two vectors 𝑃

k⃗

0 and 𝑄

k⃗

0 then

P.Q = ad + bc

is called dot or scalar product

P (3,5)

L

=

=

Displacement Vectors

If 𝐴𝐵

kkkkk⃗

= p and

kkkkk⃗

= q

kkkkk⃗

= 𝐵𝐴

kkkkk⃗

  • 𝐴𝐶

kkkkk⃗

= −𝑝 + 𝑞

alternate route from B to C

parallel vector are multiples of each other a=kb

Collinear

B

A C

q

p

kkkkk⃗

= 𝐴𝐵

kkkkk⃗

  • 𝐵𝐶

kkkkk⃗

kkkkk⃗

kkkkk⃗

kkkkk⃗

C

B

A

To show collinear

  1. show

kkkkk⃗

∥ 𝐵𝐶

kkkkk⃗

  1. Show

kkkkk⃗

  • 𝐵𝐶

kkkkk⃗

= 𝐴𝐶

kkkkk⃗

Eg. 1) 𝐴 + 𝐵 = )

∴ not same size

We cannot add

    • )
  • = )
  • = )

Multiplication

(a) By a scalar

If we have 𝐴 = )

Then 𝐾𝐴 = 𝑘

Eg. If 𝐴 = )

  • then

(a) 2 𝐴 = 2

(b) 3 𝐴 = 3 )

  • = )

(c) Two Matrices

To multiply two matrices

(1) No. of columns in first matrix = no. of rows in second matrix

ie. 𝑎 × × × 𝑐

(2) Product is no. of rows in matrix one by no. of columns in matrix

two.

ie. 𝑎 × × 𝑐. Product = 𝑎 × 𝑐

2 × 1

1 × 2

2 × 1

2 × 1

(3) Always rows in matrix one × (by) columns in matrix 2.

(1) If 𝐴 = )

  • and 𝐵 = )

2 × × 2 ≡ 2 × 2

  • )

= D

( 2 × 10 ) + (− 1 )( 3 ) ( 2 )( 1 ) + (− 1 )(− 1 )

3 × 0

2 × 3

E

  • )

= D

0 × 2

1 × 3

0 × − 1

1 × 2

( 3 × 2 ) + (− 1 × 3 ) ( 3 × − 1 ) + ( 2 × − 1 )

E

Determinant

If 𝐴 = )

  • then the determinant, det 𝐴 𝑜𝑟 |𝐴|

Example 𝐴 = )

  • 𝐵 = )

2 × 2