Coordinate Geometry

Coordinate geometry is a branch of geometry. In which two numbers are called coordinates, Coordinate geometry is used to specify the position of a point in a plane and use of algebraic methods in the learning of geometric figures.

The equation of a curve represents the essential property of dissimilar points on the curve, which we locate by means of real numbers. The coordinate geometry is also called as plane Cartesian geometry. Coordinate geometry problems deals with the lines, planes and equations involving in it. These types of problems are easy to solve. Coordinate geometry problems are simple and can be understood easily.

Below are is one example of Coordinate Geometry:

Find the equation of straight line passing through the given points (1, 3) and (3, − 4).

Solution:

We use point slope formula to solve this problem on coordinate geometry

The equation of a straight line is =
Here (x₁, y₁) = (1, 3) and (x₂, y₂) = (3, − 4).
Substituting the above, the required line is =
y − 3/7 = x − 1/ (− 2) ⇒ =
⇒ 2(y − 3) = − 7 (x − 1) ⇒ 2y −6 = − 7x + 7
⇒ 7x + 3y = 13.

Postulates

A statement whose validity is accepted without proof is called a postulate. In addition to point, line plane etc, it is also necessary to start with certain other basic statements that are accepted without proof. In geometry these are called postulates.

A postulate, though itself is an unproved statement, can be cited as a reason to support a step in a proof. Postulates are just like axioms in arithmetic and algebra, that they are accepted without proof.

Some of the postulates we use often are:
* The line containing any two points in a plane lies wholly in that plane.
* An angle has only one and only one bisector.
* Through any point outside a line, one and only one perpendicular can be drawn to the given line.
* A segment has one and only one mid point.
* Linear pair postulate: If a ray stands on a line, then the sum of the two adjacent angles so formed is 180o.

Matrices and Determinants

Matrices : A rectangular array of entries is called a Matrix. The entries may be real, complex or functions. The entries are also called as the elements of the matrix. The rectangular array of entries are enclosed in an ordinary bracket or in square bracket.

Determinants : Let A = [a_ij] be a square matrix. We can associate with the square matrix A, a determinant which is formed by exactly the same array of elements of the matrix A. A determinant formed by the same array of elements of the square matrix A is called the determinant of the square matrix A and is denoted by the symbol det.A or |A|.

Binomial Distribution

Binomial Distribution is a statistical experiment which means the number of successes in n repeated trials of a binomial experiment. It is also called as Bernoulli distribution or Bernoulli trial.
For example:
For a clinical trial, a patient may live or die. Here the researcher faces the number of survivors and not how much time the patient lives after treatment.

Properties of Binomial Distribution.
Listed below are some properties of Binomial Distribution.

• The experiment has n repeated trials.
• Each trial can have two possible outcomes. One is success and another one is failure.
• Here the trials are independent.
• Mean = n * P.
• Variance = n * P * (1 – P).
• Standard Deviation = sqrt[ n * P * ( 1 – P ) ].

Binomial distribution Formula.
Below is the formula for Binomial Distribution:

b(x; n, P) = _nC_x * P^x * (1 - P)^{n – x}
Here the Notation are,
B(x; n, P) = Binomial Probability.
X = successes
N = number of trials
P = Probability of success
nCx = Number of combinations of n trials, x is success

Algebra Problems

Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures. Together with geometry, analysis, topology, combinatorics, and number theory, algebra is one of the main branches of pure mathematics.

Question 1)

Solve the following equation:

3(x-1)=8

Answer:

3(x-1)=8

3x-3=8

3x=8+3

3x=11

Therefore x = 11/3


Question 2)

Solve the following equation.

6y-5/2y = 7/9

Answer:

6y-5/2y = 7/9

(6y-5) * 9 = 14y (by cross multiplication)

54y-45 = 14y

54y-14y = 45

40y = 45

Y = 45/40

Therefore Y = 9/8

For more examples on Algebra Problems click here

Real Valued Functions

A function whose range is within the real numbers be assumed to be a real function, moreover called a real-valued function. During math, a real-valued function is a function to associates near each part of the domain a real number within the image. f be a function as of set A toward a set B but all element x within A be able to be related through a unique element within B. It can be written as, -->ƒ : A → B

Operation on Real Functions:

The following are the Operation on Real Functions: Sum Function, Difference Function, Product Function, Quotient Function, Scalar Multiplication Function, Composite Functions, Inverse Functions.

Limits

Left Hand Limit: Let f(x) tend to a limit l₁ as x tends to a through values less than 'a', then l₁ is called the left hand limit.

Right Hand Limit: Let f(x) tend to a limit l₂ as x tends to 'a' through values greater than 'a', then l₂ is called the right hand limit.

We say that limit of f(x) exists at x = a, if l₁ and l₂ are both finite and equal.

Bar Graphs - Advantages and Disadvantages.

A bar graph or chart is a graphical representation of data in which typically bars of uniform width are drawn with equal spacing between them on one axis (say, the x-axis), depicting the variable. The values of the variable are shown on the other axis (say, the y-axis) and the heights of the bars depend on the values of the variable. Bar charts are in drawn in the shape of rectangle and it is used to represent discrete values.

Steps to build a bar graph and charts.

Following are the steps to build a bar graph

• The initial step is to Title of the graph or chart, Label for each axis and Scale for each axis from the table

• Decide the frequency axis and whether bar charts will go horizontally or vertically

• Draw a set of axes that is to use to build the chart.

• Use the data from the given table to draw in the bars on the graph.

Advantages and disadvantages of bar graph or charts

Advantages of bar charts or graph:

• Easy to prepare
• Easily understood by all parties
• It shows the total plan in impact form.
• Good communication tool
• Comparison can be made easy and it will save time of the user to make quick comparison of large data.

Disadvantages of bar charts:

• It does not show inter relationships between actions

• Managing/ supervising projects becomes difficult without those relationships between activities
• It is difficult to evaluate the impact of an unexpected event on the rest of the construction process

Sequence and Series

A set of numbers arranged in a definite order according to some definite rule is called a sequence. A sequence is a function whose domain is the set N of natural numbers.

Indicated sum of the terms in a sequence is called a series. The result of performing the additions is the sum of the series.


General Properties of Sequence and Series in Mathematics

Properties of sequence and series in mathematics:

1) The convergence or divergence of an infinite series remains unaffected by the addition or removal of finite numbers.

2) If a series in which all terms are positive is convergent, the series remains convergent even when some or all its terms are negative

3) The convergence or divergence of an infinite series remains unaffected by multiplying each term by a finite number.

Some Facts of Sequence and Series in Mathematics

Convergence, Divergence and Oscillation of a series:

Consider the infinite series Σun=u1+u2+.....+un+......∞

and let the sum of the first n terms be Sn=u1+u2+u3+.....+un

Clearly Sn is a function of n and as increases indefinitely three possibilities arise:

1)If Sn tends to finite limit as n→∞,Σun is convergent

2)If Sntends to ±∞ asn→∞,then Σun is divergent

3)If Sndoes not tend to unique limit as n→∞ then Σunis oscillatory.

Congruence

Two figures are congruent if they have the same shape and size. More formally, two sets of points are called congruent if, and only if, one can be transformed into the other by an isometry, i.e., a combination of translations, rotations and reflections.

Properties of congruence of triangles:

Reflexive Property --> -->

Every triangles is congruent to itself.

Symmetric property:

If ▲ABC is congruent to ▲DEF, then ▲DEF is congruent to ▲ABC

Transitive Property:

If ▲ABC is congruent to▲DEF, ▲DEF is congruent to ▲LMN, Then ▲ABC is congruent to▲LMN

The above properties prove the congruence of triangles

Determination of congruence:

Angle Angle Angle (AAA)

When three angles of the triangles are equal we can say that two triangles are similar triangles. That is corresponding angles are having equal measurement.

Side Side Side (SSS)

When three corresponding sides of the triangles are equal we can say that the triangles are similar triangles.

Side Angle Side (SAS)

When two sides in one triangle are in the same ratio to the corresponding sides of the other triangle, and the included angles are equal , we can say that both are similar triangle.

Angle Angle Side (AAS)

When two pairs of angles of two triangles are equal in measurement, and pair of corresponding sides is equal in length , then the triangles are called as congruent triangle.

Right Angle Hypotenuse (RHS)

When two triangles have their hypotenuse equal in length, and the shorter sides of two right triangles is equal in length, then the triangles are called as congruent triangle.

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