Introduction to solving discrete variables

Introduction to solving discrete variables:

      In our article let us discuss the solving discrete variables. Solving discrete variables is the important one in math and it expailns how to solve discrete variable. Discrete variables are mainly  based on two types of variables and we will study about solving discrete variables with its example.
Example for discrete variable: In an event of the nearest millisecond to the time.

Solving Discrete Variable:

Description about solving discrete variable:

Solving discrete variable:

    Solving discrete variable is the important one in mathematics. The discrete variable only takes the values under in its boundary itself. The limits of discrete value are completed. The best example for discrete variable is 'Real world quantities approximation'.

Example for solving discrete variable:

Solution:

            If the student flips the coin, he counts how much possible number of heads as outcomes in an  event. The possible number of head values is present between 0 and other values such as x=1, 2, 3 like that.

             The above examples are used to solve the discrete variable with its discrete probability function.

Solving Discrete Variable Probability Distribution:-

DISCRETE PROBABILITY DISTRIBUTION:-

Definition:

If the variable is said to be discrete variable then its probability distribution of the variable is known as discrete probability distribution.

The following example clearly explains the concept of discretr random variable with its probability function.

For example:

solution:

If a coin flips in twice, we get  4 possible values of outcomes such as HH, HT, TH, and TT.

The random variable X represents number of heads in an event.

                                  No. of heads                                     Probability

                                        0                                                        `1/4`

                                        1                                                        `1/2`

                                        2                                                        `1/2`

Another example:

An event of flipping a coin three times then find the probability of possible number of tails as outcomes and x=1,2,3 ?

soluton:

If you flip the coin 3 times, we shall get possible values of 8 outcomes such as HHH, HHT, HTH, HTT, THH, THT, TTH and TTT

Here the possible number of values for the above event is showed below.

                                  Number of tails                                  Probability

                                             0                                              `1/8`

                                             1                                               `7/8`

                                             2                                               `5/8`

                                             3                                              `1/8`

so the above examples clears the concept of solving discrete variable.

Definition of intersecting lines

Definition of intersecting lines:

Consider the equations

a₁x+b₁y+c₁ = 0

a₂x+b₂y+c₂ =0

These equations represent two straight lines. If a point P lies either in  any equation then P satisfies the equation

(a₁x+b₁y+c₁ = 0) (a₂x+b₂y+c₂) =0→ (3)

Conversely if P is a point satisfying the equation (3), then P satisfies either (1) or (2) and hence p lies either (1) or (2). Thus equation (3) represents both the lines (1) and (2).By expanding the equation (3) it can be reduced to the form ax²+2hxy+by²+2gx+2fy+c=0 a second degree linear equation in x and y.

            Consequently the combined equation of the two lines a a₁x+b₁y=0, a₂x+b₂y=0 which passes through the origin is (a₁x+b₁y)(a₂x+b₂y) =0 and it is of the form ax²+2hxy+by² = 0, a second degree homogeneous equation in x and y.

1)If h²≥ ab, then ax²+2hxy+by² = 0 represents a pair of straight lines passing through the origin.

Note:- If h²< ab then ax²+2hxy+by² = 0 represents two imaginary lines having real  point of intersection, the origin.

Note 2:- If h²= ab then ax²+2hxy+by² = 0 represents coincident lines.

Note 3:- h²> ab then ax²+2hxy+by² = 0 represents two real and different lines.

Intersecting lines:- If the two pair of lines are intersects at a point is called the pair of intersecting lines.

Angle between a pair of lines:- If ө is an angle between the lines represented by ax²+2hxy+by² = 0 then cos ө = (a+b)/√((a-b)²+4h²

1)If ө be the acute angle between the lines represented by ax²+2hxy+by² = 0, then cos ө =( mod of a+b)/ √((a-b)²+4h²

2) If ө be the acute angle between the lines represented by ax²+2hxy+by² = 0, then Tan ө = 2√(h²-ab)

3) If ө=0, then the two lines will be coincident and tanө = 0 which implies h²-ab =0  which implies h²=ab

3)If ө= ∏ /2, then the two lines will be perpendicular and cos ө = 0 which implies a+b =0 which implies co efficient of x²+coefficient of y²

More on Definition of Intersecting Lines 

Angle bisectors:- The equations of bisectors of angle between the lines a₁x+b₁y+c₁ = 0, a₂x+b₂y+c₂ =0 are

(a₁x+b₁y+c₁)/ √(a₁²+b₁²) (+ or -) (a₂x+b₂y+c₂ )/ √(a₂²+b₂²) =0 If c₁c₂>0 then the line bisection the angle containing the origin between the lines is

(a₁x+b₁y+c₁)/ √(a₁²+b₁²)  - (a₂x+b₂y+c₂ )/ √(a₂²+b₂²) =0 and the line bisecting  the other angle is

(a₁x+b₁y+c₁)/ √(a₁²+b₁²) + (a₂x+b₂y+c₂ )/ √(a₂²+b₂²) =0

Pair of angle bisectors:- The equation to the pair of bisectors between the pair of  lines ax²+2hxy+by² = 0 is h(x²-y²) = (a-b)xy

DEFINITION:-A pair of lines L₁L₂ = 0 is said to be equally inclined to a line L =0 if the lines L₁ =0, and L₂=0 subtend the same angle with the line L =0

1)The equation to the pair of lines passing through the origin and forming an equilateral triangle with the line ax+by+c=0 is (ax+by)²- 3(bx-ay)²=0

Solved Problems on Definition of Intersecting Lines

Q 1: Find the equations of bisectors of the angle between the lines x+y-2 =0 and x-7y+5 =0 and distinguish them.

Sol :- Equations of the bisectors of the angle between the given lines are   (x+y-2) / √(1+1) (+ or -) (x-7y+5)/ √(1+49) =0

          (x+y-2) / √2 (+ or -) (x-7y+5)/ 5√2 = 0

        5(x+y-2)(+ or -) 1(x-7y+5) = 0

5x+5y-10+x-7y+5 = 0 or 5x+5y-10-x+7y-5 = 0

6x-2y-5 =0 or 4x+12y-15 = 0

If ө is the acute angle between x+y-2 =0, 6x-2y-5 =0 then

            Cos ө = mod of (6-2)/( √(1+1) √(36+4) = 4/4√5 = 1/√5<1/√2

                                                                                         = cos (∏ /4)

                                                                                          = ө >∏ /4

Therefore 6x-2y-5 =0 is the obtuse angle bisector and hence 4x+12y-15 =0 is the acute angle bisector.

Q 2:  Find the equation to the pair of lines passing through the origin and forming an equilateral triangle with the line 2x+3y+5 =0 Also find the area.

Sol :- The equation to the required parallel lines is

       (2x+3y)²-3(3x-2y)² =0

   (4x²+9y²+12xy)-3(9x²+4y²-12xy) = 0

   -23x²+48xy-3y² = 0

   23x²-48xy+3y² = 0

Therefore the length of altitude p = 5/ √(4+9) = 5/√(13)

Area of the triangle = P²/√3 = 25/13√3 sq. unit

Math Problems Help

Introduction to solving factoring calculator:

In this article we are discussing about solving factoring calculator. The decomposition of the elements is called as factors of that element. We can get the elements by multiplying the some of its factors. For example the number 12 can be factored into 3 x 4 and x² - 16 can be factored into (x-4) (x+4).The factors can be positive elements. The devices that use to get an output by giving proper input are known as calculators. By using the solving factoring calculator we can solve for factors of the number.

Solving Factoring Calculator - Example Problems:
Example 1:
Factor the given expression, 2x² + 4x
Solution:
= 2x² + 4x
Taking the common terms outside,
= 2x(x + 2)
The answer is 2x (x+2).
Example 2:
Factor the given expression, 5x² + 155x = 0
Solution:
5x² + 155x = 0
Taking the common terms outside,
5x(x + 31) = 0
5x = 0
Divide 5 on both sides,

x = 0
x + 31 = 0
Subtract 31 on both sides,
x + 31 – 31 = 0 – 31
x = -31
The answer is 0,-31.

WORKING WITH CIRCLES

The study of Circles in an important topic under Geometry. Circles are 2 dimensional figures. They are defined as a geometrical figure which is formed by a set of equidistant points from a particular point. It is also given as the locus of a particular point. A circle has different characteristics to study about like:

• The Circumference
• The Radius
• The Diameter
• The Circle
• The Area of a circle
• The Chord
• The Segment
• The Arc

The students need to understand each character in detail to solve problems based on them. Each of them is given by a formula for computation of it's value. The students need to be familiar with the formulas and the steps to be followed while computing the values.
Students can now-a-days get ample support online. They get exhaustive amount of solved examples observing which students can learn solving problems on their own. If they encounter any hassles or doubts, they can approach the online Geometry tutors who are expert in their subjects.

Gain free Math Knowledge

Dividing square roots is easy ,..

• Problem 1:

Do you know the square root of 1962?

Answer:

The concluding answer for square root of 196 is 3√218.

• Problem 2:

What is the square root of 1965?

To learn more on what is a square root

Answer:

The concluding answer for the square root of 1965 is 44.328321.

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Familiarization with math

Introduction to different types of lines:-

A line is the gathering of points, along a straight line path and going to opposite direction, and also a line segment is part of a line, it’s having two end points. A ray also a part of line, one side is end and other side goes to one direction.

A line which do not have starting point imagine it is endless from both directions. A continuous level of the length contains more than 2 points. just math tutoring Put the little arrows on both sides for understanding.

In the following three different types of lines:

1. Parallel line,
2. Perpendicular line, and
3. Intersecting line.
4. Math at your fingertips
   

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Introduction to classifying Quadrilaterals:

A quadrilateral is a figure which is closed drawn by joining in order four non collinear points existing in a plane. A quadrilateral is any four-sided geometric figure

A parallelogram is a 4-sided a quadrilateral with both pairs of opposite sides parallel .Every quadrilateral has two diagonal. They intersect each other at a single point.

A quadrilateral has 4-sides, four angles and two diagonals. The sum of interior angles of a quadrilateral = 4-right angles or 360 degree.

Do you know what is a quadrilateral ....?

The Classification are :

Trapezium ,Parallelogram , Rhombus ,Square & a rectangle.

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