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.
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