Ninth Grade Math

 In the previous post we have discussed about Circle Area and In today's session we are going to discuss about Ninth Grade Math.


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In Ninth Grade Math, we study polynomials, in which we will learn about the types of polynomials, their degrees and thus we can calculate the zeros of polynomials. This chapter also includes the study of Remainder theorem and Factor theorem.

When we write the polynomial of 1 degree, we write it as ax + b. In case of a quadratic polynomial, we get two values of zeros and similarly a cubic polynomial has three zeros Further we say that to find the zeros of the given polynomial, we will equate the given polynomial to zero and thus we get the value of the variable.

Let us look at the following example : If p(x) = 3x - 6, then to find the value of zero of the polynomial, we proceed as follows:

3x – 6 = 0,

Or 3x = 6,

Or x = 6 / 3 = 2,

Thus we say that on putting the value of x = 2, we will get the value of the polynomial as zero. Let us check the above statement:

3 * (2) – 6 = 0,

6 – 6 = 0,

Now we will take the quadratic polynomial. Standard form of quadratic polynomial is ax² + bx + c, where we have a, b, c as rational numbers and 'a' is not equals to zero.

Now we say that if 25x² – 9 is the quadratic polynomial, then to find its zero, we say:

25x² – 9 = 0,

(5x)²  - 3² = 0,

Now applying the formula a² – b² = (a + b) * (a – b), we get:

 (5x + 3) * (5x – 3) = 0,

So either 5x + 3 = 0 or 5x – 3 = 0,

So, either x = -3/5 or x = 3/5,

Question based on Lateral Area of a Cone, is often found in the icse 2013 board papers.

Circle Area

In the previous post we have discussed about Tenth Grade Math and In today's session we are going to discuss about Circle Area. Circle is studied under conic section it is a branch of mathematics and there are other conic sections like ellipse, hyperbola, etc. Circle is a kind of conic section or a kind of ellipse whose eccentricity is zero and its two foci are coincident.

We can also define circle as a simple shape in geometry having different points in the plane which are equidistant from a particular point which is called as center of circle and you should also know that distance between any of these points in the plane and center is known as radius.

For calculating area of circle we must know some of the terms related to circle:

Circumference of the circle is the distance around the circle. Diameter of the circle is the distance traveled across the circle through the center.

In calculation of area of circle we use a Greek letter called pi (π) and approx value of pi is 3.14.

Actually, if we understand the real meaning of area of circle then it is just the number of square units inside a circle. You just have to know the total number of squares in the circle and area of each square and only then by multiplying the area of Single Square and total number of squares, we can get area of circle.

But there is one easier formula for this:

A = π. R²,

Where 'A' is the area of circle and 'r' is the radius of circle. In calculations we use π = 3.14.

There is one fact which says that area of a circle is equals to area to the area of a triangle if the base of triangle is of the length of circle’s circumference and height is equal to the circle’s radius.

Mastering Chemistry Answers can be done by understanding the concepts of chemistry.

Icse sample papers 2013 helps in understanding the pattern of question papers.

Tenth Grade Math

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Tenth Grade Math comprises of various important theories like algebra, geometry & trigonometry, understanding number system, probability or data organization and measurements. In case of algebra we discuss concepts like rationalization, factorization, solving linear and quadratic equations, sketching graphs and understand the characteristics of different equations etc. Questions based on algebra can be solved by using general mathematical formulas.

In probability and data management we would study about problems regarding arrangement of statistical data, arithmetic and geometric means, discrete & continuous frequency distribution techniques etc. Measurements include theorems like Pythagoras theorem, construction of angle bisectors, perpendiculars bisectors, medians and measures of various angles using compass. In trigonometric math we study about the trigonometric functions like sine, cosine, tangent etc. and their uses in solving the problems related to heights and distances. Let us take some examples of 10th grade math.

Example 1: Factorize the quadratic equation 4 x² + 9 x + 5 = 0?

Solution: Method that we can use here for factorizing the quadratic polynomial is mid – term splitting. We do it as follows:

4 x² + 9 x + 5 = 0

4 x² + 4 x + 5 x + 5 = 0

4 x (x + 1) + 5 (x + 1) = 0

Or x = -1, -5 /4

Example 2: If the triangle ABC is a right angled triangle with side of the longest side equal to 5 units and that of base equal to 4 units then find base?

Solution: Here we make use of Pythagoras theorem:

5² = 4² + x²

Or x = 3 units.

Next we discuss about the concept of Latent heat of vaporization which can be defined as the amount of warmth that is freed or absorbed by any system or a substance when it alters its state without altering the temperature. For instance, melting of ice.

In icse 2013 solved papers these concepts have been detailed. In the next session we will discuss about Circle Area.

9th Grade Math

In 9th Grade Math, we have a chapter of a polynomial, in which we will learn about the concepts related to the polynomial. Firstly we will start with what all is a polynomial. We define a polynomial as the combination of the terms which are classified as the linear polynomial, quadratic polynomial or the cubic polynomial.
In case the polynomial is a linear polynomial, its standard form is ax + b, where a <> 0. On the other hand , we write quadratic polynomial in the form of ax>2 + bx + c , where a <> 0. Similarly cubic polynomial is written as ax>3 + bx>2 + cx , where a <> 0.
To find the zero of the given polynomial, we mean that the roots of the polynomial are to be calculated. So we will equate the given polynomial to 0 and get the value of x. Suppose we have a linear polynomial say P(x) =3x + 6. So we say that to find the zero of the polynomial we write P(x) = 0
So 3x + 6 = 0,
Or 3x = -6 ,
Or x = -6 /3 = -2 Ans.
Similarly we can find the zero of the other polynomials. We must remember that the number of zeros of the given polynomial is equal to the degree of the polynomials. So a quadratic polynomial has two zeros, a cubic polynomial has 3 zeros and so on….
To learn more about the different Organic Chemistry Practice Problems, we can take the help of online tutors and understand the concept more clearly. We can download the iit jee sample papers, from the internet which will surely help the students to understand the pattern of the question papers. If some of the series of such question papers are solved, it develops the confidence level of the child.  

Rational Exponents

Rational exponents in the field of mathematics can be defined as a fractional number play a role of exponents for nay number. To understand this concept we need to first understand the definition of rational number and definition of exponents. In mathematics, exponents can be define as a power of any number which shows no. of times a number can multiplied by itself. The mathematical representation of an exponent can be defined by following given notation that is a^b .

The above mathematical notation can be define as exponential representation where ‘a’ and ‘b’ are integer value and the value of a can be consider as base value and value of b can be consider as value of exponent. On the side Rational can be specified as a number which is able to represent two integer values in the form of quotient. Suppose x and y are two different integer value then they can be represented as x / y form.

Now we come to the point that is rational exponent, it can be define as a number where base value has a power of any fractional number. These are also able to for representing an integer and nth root value. Some of the popular rational exponents are square root exponents, cube root exponent and 4th root exponent. Suppose there is a number 5, now it can be represented as 5^1/2 = √5, 5^1/3 = ³√5,

5^1/4 = ⁴√5 and so on. In mathematics, generally these are defined as:

X^y/z = ^z√xy.

In the field of chemistry the Properties of Acids can be define as feature of acids with different chemical in different situations. In India, those school who affiliated from ICSE board then they have to follow icse syllabus 2013 for better result.  In the next session we will discuss about Geometric Progression. 

Geometric Progression

Geometric Progressions in mathematics are the successions which have a particular relation between the numbers in the series like other progressions. Every number is achieved by multiplying the previous number by a constant value. A definite proportion is maintained between the successive terms of the sequence. Suppose we have a geometric sequence given as follows: s, s * p, s * p², s * p³, s * p⁴ and so on. Where, s denotes the first term of the GP and p represents the definite ratio maintained between the terms.

For example, the sequence 8, 16, 32, 64 is a geometric progression with the first term as 8 and the fixed ratio as 16 /8 = 2.

We solve the geometric progression questions as follows: 1^st we have to write down the specified facts or figures given regarding the progression. It can be any combination of data that is available to us, it can either be the 1st term and the relation between the numbers or the 1st term and the succeeding term of the progression.

For instance, if the 1st term is 4 and the successive term is 20. The GP can be written as:

4, 4 * 5, 4 * 5², 4 * 5³, 4 * 5⁴ and so on.

Next we discuss another concept of maths that is related to the product differentiation of a function. The derivative of a function that contains the product can be solved as follows:

Suppose we have a function: y = f (x) * g (x),

D (y) / D (x) = g (x) D (f (x)) / D (x) + f (x) D (g (x)) / D (x).

We solve such problems by differentiating the two functions simultaneously. These concepts are important in maths and we can free download cbse books to refer these topics. In the next session we will discuss about Rational Exponents. 

properties of exponents

Exponents can be defined as the power to the base of any expression. Any mathematical expression is expressed in the form of base and exponents. If there is no exponent then it is said to have 1 as an exponent. There are different properties of exponents that are defined in this section. The first property of exponents is product of power property according to this property if two expressions are multiplied having same base and different exponents then in their resultant the powers are added having base as same. Second property is of Zero exponent according to this property, when two expressions of same base but one of them having 0 as exponent then on adding 0 to any other power will produce the same exponent in the resultant also. Third property is Negative Exponent property in which base are same of both the expressions but one power is negative and since it is negative therefore the exponent in resultant is obtained by subtracting the exponents.

Fourth property is quotient of powers property according to which when two expressions of same base having different exponents are divided then in the resultant their powers are subtracted.
Fifth property that is power of product property says that if two different base having same exponents are multiplied then their resultant can also be determined by first multiplying and then taking power of it. According to power of quotient property, if two expressions are divided having different base but same exponent then their power can be taken after dividing the terms.
Protein purification is termed as the process used for the extraction of single type of protein from a mixture or we can say complex mixture. It is used for the characterization of structure or function and also for the interactions of the proteins. CBSE is termed as Central Board Of Secondary Education which runs all over India in every state. Cbse syllabus for class 11 consists of serial pattern of topics for each and every subject and can be easily checked on internet also.

How to Make a Bar Graph

In mathematics, we will see different types of graph such as histogram graph, function graph. Here we will understand that How to Make a Bar Graph. Generally bar graph are used to categorized the different information, situations by taking bars in the graph. Bar graph is also called as bar charts. Here we need to follow some steps while constructing bar graph are given below:
Step 1: To plot a bar graph first of all we have to plot the axis in the graph. As we discuss above that two axis are there in a graph that is named as x – axis and y -  axis.
Step 2: Then after we have to select an appropriate scale and construct equal intervals.
Step 3: Now using the information of given data items label the horizontal axis.
Step 4: According to data items plot the bars.
Step 5: At last named the title of a graph.
Now we have to apply these steps in example.
 For example: Using the table mention below plot a bar chart.
Scores on the practice Test and the Test

Scores on the practice Test and the Test Student Practice Test Test
Hussen	60	70
Tomer	75	90
Marrine	55	55
Jiff	80

 Solution:
  First of all select or named the title from above mention table. As we see in the table that the title of above table is 'Score on practice Test and Test'. After that choose vertical bars. Here scores are different from one another mostly by 5, 10, 15 or 20. It means select a scale of 10.
 If we find any variation between scores then scores are like 1, 2, 3, 4, or 5, then it is better to select a scale of 1 or 2.
 After marking names on x – axis and scores on y – axis. Suppose if we decide to plot horizontal bars then put names on y – axis and scores on x – axis. At last plot the bars. Bar graph is shown below:

This is how we plot double bar graph using above steps. Now we will see Properties of Real Numbers.Commutative property of addition and commutative property of multiplication. for the prepration of 2013 board exam please prefer icse 2013 board papers.

equiangular triangle

In the previous post we have discussed about Define Equilateral Triangle and In today's session we are going to discuss about equiangular triangle. Hi friends, we will study different types of triangle such as equilateral triangle, isosceles triangle and so on. Here we will discuss one of the important triangle that is equiangular triangle. An equiangular triangle can be defined as a triangle such that the entire interior angles are of equal length and angle are of 60 degree.

The interior angles of an equiangular triangle always reach to 180 degree, and each angle of equilateral triangle is always third part of it. It means we can say that all the angles are of 60 degree. Let’s discuss some properties related to equiangular triangular.

Area – The area of equiangular triangle is given by:

Area = √3 / 4 s², here‘s’ denotes the length of any one side.

In case of equiangular triangle, the radius of a circle is just half of the radius of the circumference. Interior angles – 60 degree is the interior angles of an equiangular triangle. Perimeter – addition of the  entire sides of an equiangular triangle is perimeter.

Perimeter = s + t + u, here ‘s’, ‘t’, ‘u’ are the lengths of three sides of an equiangular triangle. Now we will discuss that how to find the area of equiangular triangle with the help of example: (know more about equiangular triangle, here)

Example: Find the area of equiangular triangle where the length of sides are 16, 16, 16 inch?

Solution: We know that the formula for finding the area of equiangular triangle is:

Area = √3 / 4 s²,

Given, length = 16 inch, put the value of all sides in a given formula:

 Area = √3 / 4 s²

Area = √3 / 4 (16)²,

Area = √3 / 4 (256),

Area = 256 √3 / 4,

So, the area of an equiangular triangle is 256 √3 / 4.

Same side Interior Angles can be defined as the angle pairs which are on inside of two lines and also on same side of traversal. icse guess papers 2013 is very helpful for exam point of view.

Define Equilateral Triangle

An equilateral triangle in the mathematics can be defined as the triangle which has all the 3 sides of the equal length. This is the general definition but in the traditional or we can say the Euclidean type of the geometry, the equilateral triangles are also those triangles which are equiangular which means that all the 3 internal angles of the triangle are congruent to each other like the 3 sides and each angle is equal to 60 degrees. The equilateral triangles are the regular polygons and thus they can also be known as the regular triangles.
According to the definition of the equilateral triangles we can derive many results. Suppose the length of each side of any equilateral triangle is l then we can derive various results for this triangle with the help of the Pythagorean theorem which may be given as follows. (know more about Equilateral Triangle, here)
The area of any triangle which is equilateral may be calculated by the formula A = ( √3/4 ) * l². The perimeter of any such triangle which is equilateral is equal to thrice the length of any side of the triangle that is P = 3l. Also the height or we can say the altitude from any side of the equilateral triangle is given by h = ( √3/2 ) * l. 
We can also derive some of the results for the inscribed and the circumscribed circles of the equilateral triangles. For example the radius of the circumscribed circle of any triangle which is equilateral may be given as R = ( √3/3 ) * l. Then the radius of the inscribed circle of any triangle which is equilateral may be given as r = ( √3/6 ) * l.
In order to get more help on the topics: Equilateral Triangle and Rutherford Atomic Theory you can visit our next article. CBSE Syllabus 2013 is designed in a manner to help students in learning important topics in a very simpler way.

Surface Area of Cube

In the previous post we have discussed about Volume of Cube and In today's session we are going to discuss about Surface Area of Cube. A solid object that has six equal square faces is said to be cube, its structure is just like a regular hexahedron. Let's see how to find the Surface Area of Cube formula? To find the surface area of a cube we need to follow some of the steps which are mention below:

Step 1: To find the Surface Area of Cube first it is necessary to have the length of one side of a cube. According to the definition of a cube all the sides of a cube are same, so the length of all the sides is also same. Suppose we have length of one side of a cube is 21 inch. Then find the surface area of a cube. The formula to find the surface area of a cube is given by:

surface area of a cube = 6a2, here 'a' denotes the side of a cube.

Step 2: if we put the value of the edges of a cube in the formula then we get the surface area of a cube. So on putting the value in the given formula.

surface area of a cube = 6a2, here the value of 'a' is 21 inch.

surface area of a cube = 6 (21)2,

surface area of a cube = 6 * 441, so the surface area of a cube is 2646 inch2.

Now we will discuss some parts of a cube. (know more about Cube, here)

Face – All the sides of a cube are said to be the faces of a cube because cube is has all the sides of equal length. A cube having six faces and all faces are square. All the faces of a cube have four equal sides and four equal interior angles.

Edge – A line segment that is obtained when two edges are meeting at a same point. 12 edges are present in a cube, as we know that cube has all the faces are congruent to each other and all the edges has same length.

Stoichiometry Problems is deals with the chemistry, it shows the quantitative relationship in between reactant and products. To prepare for the board exam please go through cbse latest sample papers.

Volume of Cube

In the previous post we have discussed about simplifying trig expressions and In today's session we are going to discuss about Volume of Cube.  Hi friends, there are different types of shapes like sphere, rectangle etc. Here we will discuss one of the important shape that is cube. Cube can be defined as a solid object or figure which has six square faces. All the angles and sides of a cube are same. Now we will see the Volume of Cube. Volume of a cube can be defined as the number of cubical units fill exactly in a cube. The formula to find the volume of a cube is given as:

Volume of a cube = s³, here 's' denotes the length of one edge of a cube.

Let's understand some steps to find the volume of a cube. (know more about Volume of Cube, here)

Step 1: To find the volume of a cube it is necessary to have the length of one edge of a cube. Suppose we have given the length of one edge of a cone is 7 inch, then find the volume of a cube.

Step 2: If we have the value of one edge of a cone then put the value in the formula to find the volume.

So, the volume of a cube = s³, given the value of s = 7 inch.

The volume of a cube = (7)³, we can also write it as:

Volume of a cube = 7 * 7 * 7;

Volume of a cube = 343 inch³.

So, here we get the volume of a cube is 343 inch³.

Now we will see some formulas based on the cube.

If we want to find the surface area of a cube then we use formula. To find the surface area of a cube the formula is given as:

Surface area of a cube = 6s², here 's' denotes the one side length of a cube.

Before entering in the examination hall please solve all the iit jee sample papers. It is very helpful for exam point of view. If we solve the Stoichiometry Practice Problems then we score good marks in exam.

simplifying trig expressions

In the previous post we have discussed about Measures of Central Tendency Definition and In today's session we are going to discuss about simplifying trig expressions. Trigonometry is deals with the branch of mathematics which is used to compare with the angles and sides of triangles. There are different types of expression and also there are different types of method to simplifying trig expressions. Here we need to follow some steps to solve the trigonometric expression.
Step 1: To solve the trigonometric expression first of all we have to take a trigonometry expression. Suppose we have a trigonometry expression as:
 Sec (⊼/ 2 – y) – tan (⊼/ 2 – y) sin (⊼/ 2 – y).
Step 2: Then we use corresponding identities of given trigonometry values. In this expression, in place of Sec (⊼/ 2 – y) we use cosec y, in place of tan (⊼/ 2 – y), we use cot y and in place of sin (⊼/ 2 – y). (know more about simplifying trig expressions, here)
⇨Sec (⊼/ 2 – y) – tan (⊼/ 2 – y) sin (⊼/ 2 – y),
So, using the above values we can write the trigonometry expression as:
= cosec y – cot y cos y, on moving ahead we can also write the expression as:
= cosec y – (cos y / sin y) cos y, (because cot y can also be written as cos y / sin y).
Step 3: Then multiply the value present in the parenthesis to the value present outside the parenthesis. So the above trigonometric expression we can write it as:
= cosec y – (cos y / sin y) cos y, on multiplying we get:
= cosec y – cos ² y / sin y,
We can also write the cosec y as 1 / sin y, so put this value in place of cosec y.
= 1 / sin y - cos ² y / sin y, here in this expression sin y is the LCM. So, taking LCM we get the next step of expression as:
= (1 - cos ² y) / sin y, we can also write the (1 - cos ² y) as sin ² y. So, we can write it as:
= sin ² y / sin y = sin y. This is how we can solve the trigonometric expression. There are different Types of Pollution in the environment. The iit entrance exam is applicable for iit colleges.

Measures of Central Tendency Definition

Central Tendency Measure: It is such type of measure which helps us in locating the middle of group of data .Its Common measure are mean, mode and median

Measures definition:
1.      Mean: It is the most commonly used measure where we sum all numbers from the group of data and divide it by the count of numbers in the data group. In simple terms we call it as average too.
2.      Mode: It is most frequently occurring value in the set of data .It can be unimodal or bimodal or multimodal. .Unimodal is that where group of data has one mode and bimodal it has two modes
3.      Median: This values is the numbers which lies in the middle of data set when the group of data is arranged in the series of either ascending or descending numbers .In the case of even count of numbers median is calculated by calculating the mean value of the two middle numbers .This value divides the data set exactly into half .We cannot calculate median for qualitative data set. It calculation is only possible after sorting of the data
Understand the measures of central tendency definition with help of this solved example
Example: Find the central tendency of the data
6, 1, 3, 2, 5, 6, 6, 6, 9?
Solution: First we will calculate the mean,
6+1+3+2+5+6+6+6+9/9,
= 4.88.
So mean is 4.88.
Now calculate median by arranging the data in ascending order
1 , 2 , 3 , 5 6 , 6 , 6 , 6 , 9
Now apply the formula as
= (n+1/2)th,
= 10/2th,
5th term.
5th term is 6 so median is 6. (know more about Measures of Central Tendency Definition, here)

Mode is the most frequently occurring term in the series,
Here ‘6’ is the most repetitive term so mode will be ‘6’.



Get all information on physics important topic like types of waves and get all cbse sample papers 12 class on various online educational portals and In the next session we will discuss about simplifying trig expressions. 

solving trigonometric equations

In mathematics, an equation that contains trigonometric functions is known as trigonometric equations. For example: Cos u = ½. The roots of trigonometric function are obtained by the inverse trigonometric functions. Now see how to solving trigonometric equations. Suppose we have sin u + sin 2u + sin 3u = 0. This trigonometric equation can be reduced to form 2 sin 2u cos u + sin 2u = 0. In other word we can also write as: Sin 2u (2 cos u + 1) = 0. When we solve this equation we get the value of sin 2u as 0.  Sin 2u (2 cos u + 1) = 0;
⇨ Sin 2u = 0. And the value of cos u is:
⇨ 2 cos u + 1 = 0;
⇨ 2 cos u = -1;
⇨ Cos u = -1/2. So the value of cos u is -1/2. This equation gives the result of the trigonometric equations. So u = ½ arcsin 0 = n ⊼/2;
u = arcos (-1/2),
u = 2/3 ⊼ (3n + ½), Where, the value of ‘n’ is either positive or negative integer. Let’s discuss how to solve the trigonometric equations. Let we have the trigonometric equation 4 tan³ u – tan u = 0, that lies in the interval [0, 2⊼]. Then it can be solved as shown below:
So the given trigonometric equation is: 4 tan³ u – tan u = 0. We can write the equation as:
⇨ 4 tan³ u – tan u = 0;
⇨ Tan u (4 tan² u – 1) = 0, so the value of tan ‘u’ is 0;
Or tan u = + 1/√3. For every value of u ∈ [0, 2⊼],
⇨ Tan u = 0; It means the value of ‘u’ is 0, ⊼ or 2⊼.
While
Tan u = 1/√3,
u = ⊼ / 6 or 7⊼/ 6,
Tan u = -1/√3,
u = 5⊼ / 6 or 11⊼/ 6. This is how we can solve the trigonometric equation. Now we will see the Equation for Velocity. Equation of velocity is given by: Equation of velocity (v) = displacement (D) / time (T). The students of class 1st follow the icse syllabus for class 1.

Complex Conjugate

In mathematics, a number written in the from of u + iv is said to be a complex number. Here 'u' and 'v' are real numbers and 'i' is is iota symbol. The value of iota is given as: i = √-1. In this given expression 'u' is real part and 'v' is imaginary part. Now we will discuss different Complex Conjugate rules. The complex number is represented by the symbol 'Z' that contain real part and imaginary part, so we can write it as:

z = Re (z) + i im (z) = u + iv. Let the value of Im z = 0, then the value of z = u that is a real number. If the value of Re (z) = 0 then we get z = iv that is pure imaginary number. The complex conjugate rules are mention below: (know more about Complex Conjugate, here) 

Rule 1: Addition and subtraction:

The addition and subtraction rule of complex number is given as:

z₁ + z₂ = (u₁ + u₂) + i (v₁ + v₂);

Rule 2: Multiplication:

Multiplication rule is given as:

z₁ X z₂ = (u₁ + v2) X (u₂ + i v₂) = (u₁ u₂ – v₁ v₂) + i (u₁ v₂ – u₂ v₁);

Rule 3: Division rule:

Division rule of complex number is given as:

z₁ / z₁ = (u₁ + i v₁) / (u₂ + i v₂) = (u₁ u₂ + v₁ v₂) + i (u₂ v_{1 –} u₁ v₂) / (u₂₂ + v₂₂);

Let's talk about some properties of complex conjugate.

1. (zu)' = s'u'

2. z' = z, it is applicable if and only if the value of 'z' is real.

3. Zn' = z'n for any integer number 'n'.

4. | z' | = | z |;

5. | z |² = zz' = z'z;

6. z'' = z, in which the conjugate of the conjugate of a complex number 'z' is again that number.

7. Z – 1 = z' / | z |², it is applicable if any only if the value of 'z' is not zero.

8. Exp (z') = (exp (z))';

Second Law of Motion is deal with the branch of physics. In this we have studies the different types of motions. Before entering in the 10^th board examination please go through the CBSE sample papers for class x and In the next session we will discuss about solving trigonometric equations. 

Scalar

We have heard about the term scalar many times but do we actually understand the real meaning of the term scalar. So here we are going to discuss about its definition, applications, etc.

If we see a scalar quantity in a very basic way then it is a quantity which only has magnitude or which does not depend upon directions and on the other hand a vector quantity is a quantity which magnitude as well as direction. And if we see in the context of linear algebra then we will know that scalar is nothing but a real number. And also there is one property or operation which relates the vectors, in the vector space with the scalars and that is the scalar multiplication. In this operation, we multiply a scalar or a real number with a vector in vector space to get another vector.

We have so many operations and properties related to scalars like scalar product, dot product, etc. But many of us get confused between the scalar multiplication and scalar product as we define the scalar product as when in a vector space, we multiply two vectors to produce a scalar or a number.   (know more about Scalar, here)

We can also define scalar as a quantity having only one component which does not change or is invariant under the rotations in the coordinate plane.

Some examples of scalar quantities are length, area, temperature, volume, speed, energy, mass, power, work, etc as they all do not have any directions or does not depend on it.

And some of the vector quantities which depend upon the direction are velocity, acceleration, force, weight, momentum, etc.

We have a matrix related with the term scalar called scalar matrix and is used to represent a matrix of the form KI where the alphabet K is a scalar and I is the identity matrix.

In order to get help in understanding the topics: scalar, simplifying expressions and Tamilnadu Board, you can just visit our next page and In the next session we will discuss about Complex Conjugate. 

How to Learn Pythagorean Theorem

In the previous post we have discussed about Cones and In today's session we are going to discuss about Pythagorean Theorem, The Pythagorean Theorem is a theorem of the mathematics which establishes a relationship among all 3 sides in a right angled triangle. So let us discuss the statement of this theorem.
The Pythagorean Theorem states that in such a triangle which is right angled the area of a square the side of which is the hypotenuse of the right angled triangle that is the side which is opposite to the ninety degree angle is equal to the addition of the areas of those squares the sides of which are the 2 legs of the triangle that is those 2 sides which make the ninety degree angle. The Pythagorean Theorem is also sometimes known as the Pythagoras theorem.
The Pythagoras theorem can also be framed in a different way as follows. In a triangle which is right angled the square of the length of the hypotenuse of the right triangle is equal to the sum of the squares of the lengths of the other 2 sides of the right triangle that is the sides which make the ninety degree angle.
This theorem can also be represented in the mathematical way. Suppose that p, q and r are the lengths of the 3 sides of a right angled triangle then according to the Pythagoras theorem it can be given that,
p² + q² = r²
where r is the length of the hypotenuse and p & q are the lengths of the 2 sides which make the ninety degree angle. (want to Learn more about Pythagorean Theorem, click here),
The Pythagoras theorem can be proved by many different ways. For example this theorem can have the algebraic type of proofs and the geometric types of the proofs.
Linear equations solver makes your calculations easy and fun and linear equation is also an important topic of mathematics. icse board is a reputed board in India and provide quality education to students.

Cones

In the previous post we have discussed about fundamental counting principle and In today's session we are going to discuss about Cones, Its a three dimensional surface in which circular base and one vertex is present is known as Cones. In other word, we can say that a cone has flat base, and one side is curved surface. Let’s discuss some formulas that are related to the cone. Suppose if we want to calculate the surface area of base then the formula is given by:

Surface area of base = ⊼ * r2,

Surface area of side = ⊼ * r * s,

Or Surface area of side = ⊼ * r * √ (r2 + h2),

Volume of a cone = ⊼ * r2 * (h / 3),

Here ‘H’ is the height of a cone, ‘R’ is the radius of a cone and ‘S’ denoted the side length.

In geometry, cone has a pointy end that is known as vertex or apex of a cone. The flat part of a cone is said to be the base of a cone. Its is having conical shape. If we rotate a triangle then a new shape is formed that is also known as cone. Similarly, if we rotate a triangle along the shortest side then it can be result as right angled triangle. Now, we will discuss the different types of properties of a cone. ((want to Learn more about Cones, click here),)

Properties of a cone are used to calculate the volume, surface area and the total surface area of a cone. Volume of a cone is given by: Volume of a cone = âŠ¼ / 3 * r2 * h, Here ‘r’ is the radius the cone’s base and ‘h’ is the height of a cone. Now surface area of a cone is given by:

SA = ⊼ * r * √ (r2 + h2), Now total surface area of a cone is given by:

TA = SA + ⊼r2,

Where ‘r’ is the radius of cone’s base and SA is the cone’s surface area of face. The Variance and Standard Deviation is used find the average form the mean values. To get more information about the standard deviation then follow the icse text books.

fundamental counting principle

fundamental counting principle can be referred as a process which is used for determining the total no. of possible outcomes in situation when two or more features can vary. In the simple definition we can define the fundamental counting principal as a way to demonstrate the total no. of events t t can occur in the form of figure. The concept of fundamental counting principal is differing from the concept of probability. In probability, we perform the task of calculating the no. event of events which might be occurring. In the fundamental counting principal, in case we have n no. of choices for first attempt and m no. of choices for second attempt then we can say that there are n * m no. of choices we have.
In mathematics, fundamental counting principle can be referred as a process which is used for determining the total no. of possible outcomes in situation when two or more features can vary. In the simple definition we can define the fundamental counting principal as a way to demonstrate the total no. of events t t can occur in the form of figure. The concept of fundamental counting principal is differing from the concept of probability. In probability, we perform the task of calculating the no. event of events which might be occurring. In the fundamental counting principal, in case we have n no. of choices for first attempt and m no. of choices for second attempt then we can say that there are n * m no. of choices we have.
Here we show you an example that helps to understanding the concept of fundamental counting principle:
Example: Suppose Ron have three shirts and 2 paints in a tour, then calculate the possibility of days he has for wearing these clothes in different ways?
Solution: As in the above given example there are 2 paints and 3 shirts then we can say that there are 6 possibilities to wear these clothes in different form in 6 days. These possibilities can also be obtained by performing multiplication of 2 * 3 that yield the same answer.
The above given example are performed or solve by simple mathematical calculation. This concept of fundamental counting principle can also be used by using another concept which is known as tree diagram. This is a most popular example to understand the concept of fundamental counting principal in a pictorial form. In the last we can say that calculating total no. of outcomes we need to perform the multiplication between the no. of possibilities of the first characteristic to the number of possibilities of the second characteristic. Sometime a question arises that How to Find the Perimeter of a Rectangle? Then the solution of this problem are given in mathematics by multiplying the addition of length and breath of rectangle by 2. In cbse board, cbse previous year board papers guide the students to make a plan of action to perform well in the board examinations.

Data Analysis

In the previous post we have discussed about Associative Property of Addition and In today's session we are going to discuss about Data Analysis. The data analysis as the name suggests is just the analysis of the data. But we will look into the broader meaning of the term data analysis. The data analysis is a type of method which is generally used for enquiring about the data, cleaning the data, for taking the data from one place to another and also for data modeling. The motive of the data analysis that is the need of analyzing any type of the data is to get some vital information, to provide the conclusions and also to help in the process of making any type of the decision. (know more about Data analysis, here)
Let us take an example of the data analysis now. The mining of the data is also a special type of process of the data analysis which generally emphasizes on the modeling and on the discovery of the information for a motive which is predictive and not for any type of the purpose which is completely descriptive.
The term data analysis is also sometimes used in place of the ‘modeling of the data’. The data analysis is just a method under which various types of the phases can be separated. The data analysis is very much related to the visualization of the data.
Now we have seen the various definitions of the data analysis. So let us move on and study some further details of the data analysis. Therefore we will discuss now about the cleaning of the data which is nothing but only a part of the process of the data analysis. The cleaning of the data is a critical process under which the data is being enquired to check whether it is correct or not and finally the data which is found out to be incorrect are corrected if it is required.
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Associative Property of Addition

In the previous post we have discussed about The Unit Circle and In today's session we are going to discuss about Associative Property of Addition, Different properties of addition can be checked for the numbers namely closure property, commutative property and associative property for addition.  By Associative Property Of Addition, we mean that if we have any three natural numbers say a, b and c then we say that the addition of these three numbers is associative, which indicates that even if the order of addition of the three numbers is changed, the sum of the three numbers remains same. So we say that the numbers a and b are added first and then number c is added to it, the sum we get will be the same as when we add the numbers b and c and then the number a is added to it.  Thus it can be mathematically expressed as follows:
 ( a + b )  + c = a + ( b + c )
 The associative property of addition also holds true for the whole numbers, integers and even for the fraction numbers.  If we have 3 fraction numbers say a1/b1, a2/ b2 and a3 / b3, then we say that the associative property of addition also holds true for the fraction numbers which can be indicated numerically as follows:  ( a1/b1 +  a2/ b2 ) + ( a3 / b3) =  ( a1/b1) + (  a2/ b2  + a3 / b3 )
 We also check the associative property of addition for the rational numbers and observe that the associative property of addition also holds true for the addition of the rational numbers too. (know more about Associative property, here)
To understand the concept of how to do fractions, without the help of the teacher, we can take the help of online math tutor and understand the concepts. We can also download CBSE Board Hindi Syllabus to know about the marks distribution of different topics and understand the pattern of the question paper.

The Unit Circle

In the previous post we have discussed about How to Find Altitude and In today's session we are going to discuss about The Unit Circle. The unit circle is a mathematical term used to prove many theorems plus has many other applications also. The unit circle is self defined as the word unit or unity is for 1, so the unit circle is nothing but a circle with radius one. According to Cartesian coordinate system, the unit circle is used quite frequently in trigonometry and according to which the unit circle is the circle with radius one and centered at the origin (0, 0) in the Euclidean plane. The unit circle is denoted by S¹. If we would generalize this term in higher dimensions, then we call it as the unit sphere.
Another simple way of defining a circle with a radius one is that its center is put on the Euclidean graph where both axes x axis and y axis intersect or cross. And it is a very simple and great way to learn and talk about angles and lengths.
As we all are aware with the term Pythagorean theorem, but for those who do not know the actual sense of the theorem, here is the explanation: if we consider a point (x, y) on the unit circle and let x and y are positive, then x and y will be the lengths of the legs of a right angle triangle whose hypotenuse has length 1. Hence x and y satisfy the equation given by the Pythagorean Theorem: which says the sum of the squares of the lengths of the other two sides of right triangle is equal to the square of hypotenuse length.
x² + y² = 1
The above equation says that the reflection of any point (x, y) on the unit circle about x or y axis is also on the unit circle as x² = (- x²) for all x.
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