Basic Identities

Basic Identities of Algebra
Closure Property of Addition
Sum (or difference) of 2 real numbers equals a real number

Additive Identity
a + 0 = a

Additive Inverse
a + (-a) = 0

Associative of Addition
(a + b) + c = a + (b + c)

Commutative of Addition
a + b = b + a

Definition of Subtraction
a - b = a + (-b)

Closure Property of Multiplication 
Product (or quotient if denominator 0) of 2 reals equals a real number

Multiplicative Identity
a * 1 = a

Multiplicative Inverse
a * (1/a) = 1 (a 0)

(Multiplication times 0)
a * 0 = 0

Associative of Multiplication
(a * b) * c = a * (b * c)

Commutative of Multiplication
a * b = b * a

Distributive Law
a(b + c) = ab + ac

Definition of Division
a / b = a(1/b)

The "Vertical Line Test"

The "Vertical Line Test"

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Vertical line test is relation between two terms of X and Y. And this test used to determine If a relation is a function. If that shows no vertical lines and that intersect the graph at more than one point, the relation for each element of the domain are corresponding to exactly one element of the range.

Types of lines
Line:
The infinitely long and thin of the straight geometrical object is called line. Types of lines can be classified into following types:

• Straight Line
• Vertical line
• Horizontal line

Looking at this function stuff graphically, what if we had the relation that consists of a set containing just two points: {(2, 3), (2, –2)}? We already know that this is not a function, since x = 2 goes to each of y = 3 and y = –2.This characteristic of non-functions was noticed by I-don't-know-who, and was codified in "The Vertical Line Test": Given the graph of a relation, if you can draw a vertical line that crosses the graph in more than one place, then the relation is not a function. Here are a couple examples:

Types Of Vectors

Vector: A quantity that has magnitude as well as direction is called a vector.
Different types of vectors are,
a)  Zero or Null Vector: A vector whose initial and terminal points coincide, is called a zero vector (or null vector).
b) Unit vector: vector whose magnitude is unity (i.e., 1 unit) is called a unit vector.
c) Collinear Vectors: Two or more vectors are said to be collinear if they are parallel to the same line, irrespective of their magnitudes and directions.
d) Co-initial vectors:Two or more vectors having the same initial point are called coinitial vectors.

e) Negative of a Vectors: A vector whose magnitude is the same as that of a given vector, but direction is opposite to that of it, is called negative of the given vector.

f) Equal Vectors: Two vectors  are said to be equal, if they have the same
magnitude and direction regardless of the positions of their initial points,

Linear functions and Inequalities

Introduction to linear functions and inequalities:

Linear Functions:


In mathematics, the term linear function can refer to either of two different but related concepts:

• a first-degree polynomial function of one variable;
• a map between two vector spaces that preserves vector addition and scalar multiplication.

Inequalities
In mathematics, an inequality is a statement about the relative size or order of two objects or about whether they are the same or not

• The notation a <>less than b.
• The notation a > b means that a is greater than b.
• The notation a ≠ b means that a is not equal to b, but does not say that one is greater than the other or even that they can be compared in size.

linear functions example:

Solve the expression

5(-3y - 2) - (y - 3) = - 4(4y + 5) + 13

Solution:

• Given the equation

5(-3y - 2) - (y - 3) = -4(4y + 5) + 13

• Multiply factors.

-15y - 10 - y + 3 = -16y - 20 +13

• Group like terms.

-16y - 7 = -16y - 7

• Add 16y + 7 to both sides and write the expression as follows

0 = 0

• The above statement is true for all values of y and therefore all real numbers are solutions to the given equation.
Inequalities - Example :
Solve 4x + 3 <>
Solution:
4x + 3 <>
4x +3 – 3 <>
4x <>
4x – 8x <>
- 4x <>
x > – 1
The solution set is {0, 1, 2, 3…}

Polynomials

Introduction Of polynomials

A polynomial is a mathematical expression and it consist of a addition of terms. Each term may contains of a variables and variables with raised powers. The example of a polynomial is as follows:

AX ³ + BX ² + CX, AX ³ + BY²
The operations that are performed on the polynomials are,

• Adding Polynomials,
• Subtracting Polynomials,
• Multiplying Polynomials, and
• Dividing Polynomials.

Addition of Polynomials:

We add two polynomials adding the coefficients of the like powers.

Example :

Find the sum of Polynomials - 2x⁴ – 4x² + 6x + 3 and 4x + 6x³ – 6x² – 1.

Solution:

(-2x⁴ – 4x² + 6x + 3) + (6x³ – 6x² + 4x – 1) = - 2x⁴ + 6x³ – 4x² – 6x² + 6x + 4x + 3 – 1

(By using distributive and associative property)

= - 2x⁴ + 6x³ – (4 + 6) x² + (6 + 4) x + 2 (By arranging the coeeficient of same exponent)

= - 2x⁴ + 6x³ – 10x² + 10x + 2 (Adding the coefficient of coresponding exponent)
Answer of the given polynomial: - 2x⁴ + 6x³ – 10x² + 10x + 2

Hexagon, Cylinder Volume – Geometry Problems

GEOMETRY:
QUESTION:11
Find the volume of hexagon and cylinder. The cylinder has 7cm as height and radius is 4cm.
The hexagon exactly fits inside the cylinder.

 
Solution:
The volume of cylinder is = pie r^2 h
Here, r = 4cm
And h = 7cm
So, the volume of cylinder is = pie * 4*4*7 cm^3
= 112 pie cm^3

 
Now,

The volume of hexagon exactly fits inside the cylinder:
The volume of hexagon is base area * height.
Base area:
using sine law we can find base area. The regular hexagon
has 120 degree as interior angle. So, the triangle we drawn
through the center of the circle is equilateral triangle. The
radius is 4cm.
using. Sine law


 
sin 90 sin 30
------ = -------
4 x

 
1 0.5
-- = ---
4 x

 
X = 2cm
the length of the side of the hexagon is 2+2cm = 4cm.
the base area of the hexagon is:

 
3 root 3
--------- side^2
2

 
3 root 3
---------- * 4*4cm^2
2

 
=24 root 3.

 
The volume of the hexagon is:

 
= base area * height

 
= 24 root 3 cm^2 * 7 cm

 
= 168 root 3 cm^3

 

 

 

 

 

 
GEOMETRY:
QUESTION:12
Find the sides of a square if AM = 8cm

 
Solution:

 
According to the properties of a square, the diagonals bisect each
other. So,
AM = MC

 
AC = AM + MC

 
AC = 8cm + 8cm

 
AC = 16cm.

 
Using pythagoras theorem,

 
Take AB = x

 
x^2 + x^2 = 16^2

 
2x^2 = 256

 
x^2 = 128

 
x = 11.314.

 
Side of the square is 11.314cm

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Algebra word problems – part 2

ALGEBRA:
QUESTION:13
Find the Area of a rectangle the length of a rectangle is 3 more and 3 times the width of the
Rectangle.

 

 
Solution:
Take the Width as x cm
And the Length is three times and 3 more than
The Width. So, Length is 3x+3.

 
The Area of a Rectangle is:

 
Length * Width

 
(3x+3) ( x)

 
Expand the terms,

 
3x^2 +3x <------------- Answer.

 

 

 
AlGEBRA:
QUESTION: 14

 
Find (k^9)^-5 when k = 0.

 
Solution:

 
Using Exponent Law we can simplify the expression ,

 
K^9*-5 = k ^ -45 = 1/ k^45

 
If k =0 the value is undefined.