vertex of parabola

In the previous post we have discussed about Permutation and Combination and In today's session we are going to discuss about vertex of parabola.


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In mathematics, parabolas are basically graphs which have equations of the form:

                                                F(x) = ax² + bx + c                             where a≠ 0,

In parabolas, there are both highest and lowest points depending upon where they are going to open up or down. These points are called vertex. Parabolas are basically used to justify many real life situations like height above ground of an object thrown upward, to check period of time after some time, to check the area of rectangle with a particular width, to check the maximum height of any object etc. In all these situations we need to find some co-ordinates which we get with the help of parabolas. In parabolas there is vertical line of symmetry which passes through its vertex. This line of symmetry helps us to find out co-ordinates. To understand the vertex of a parabola we take an example where we need to examine line of symmetry also. So we have an equation which is

                                                                F(x) = x² – 4x - 5

And we have to find out the vertex of parabola. Now to get the roots of 'x' we use the quadratic formula which is:

                                                                X = [-b ±√(b² – 4ac)]/ 2a,

Where the values of a, b and c are 1, -4 and 5 in the equation. Now we put these values in above equation. So

                                                                X= - ( - 4)±√((-4)² – 4(1) (-5) ) / 2(1),

After solving further we get         x = (4 ± 6) / 2 = 2 ± 3,

These value is also called as x- intercept. Above point shows that it starts from 2 and add 3 to get one intercept, and when we subtract 3 from 2 we get another intercept. So 2 is the midpoint of two intercepts. Finally, we have intercept points which is 5 and -1. Now if we find 'y' value which passes through 'x' value then we put 'x' value in the given equation so

                                                                F(2) = (2)² – 4(2) – 5 = -9,

So the vertex of parabola is (2, -9).

Next we will discuss How to Calculate Molar Mass.

Cbse sample papers 2013 are available online.

Permutation and Combination

 In the previous post we have discussed about parametric equation and In today's session we are going to discuss about Permutation and Combination.

Collection of different objects and symbols in a particular sequence or particular order is known as permutation.

Collection of different objects is called as combination here order doesn't matters. In other words it is an unordered collection of a unique size.

Now we will talk about the formula for finding the Permutation and Combination. 

Formula to find the permutation is given as:

Permutation = ⁿp_r = n! / (n – r)!,

And formula to find the combination is given as:

Combination = ⁿC_r = ⁿp_r / r!;

Here, value of ‘n’ and ‘r’ indicate the non- negative integers and also r ≤ n value.

Value of ‘r’ indicates the size of each permutation.

Value of ‘n’ indicates the size of set from which element are permuted.

‘!’ indicate the factorial operator.

Now we will see example of combination and permutation.

Example: Calculate the number of permutation and combination where value of ‘n’ is 7 and the value of ‘r’ is 4?

Solution: We know that formula for permutation and combination is:

Permutation = ⁿp_r = n! / (n – r)!,

Combination = ⁿC_r = ⁿp_r / r!;

Given, n = 7 and r = 4.

First we will find the factorial of 7. The factorial of 8 is = 7 * 6 * 5 * 4 * 3 * 2 * 1 = 5040.

Now find the factorial of (7 – 4);

The factorial of (7 – 4) is = (7 – 4)! = 3!,

So the factorial of 3 is = 3 * 2 * 1 = 6;

Now divide 5040 by 6;

Permutation = 5040 / 6 = 840;

Now find the factorial of 4.

The factorial of 4 is = 4 * 3 * 2 * 1 = 24;

Now divide 840 by 24.

Combination = 840 / 24 = 35.

We will see How to Find the Volume of a Cube in the next session.

Cbse class 12 board papers can be downloaded from CBSE board website.

parametric equation

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Parametric equation is used to define a relation with help of parameters. Parameter equation is given as: x = f (c) and y = g (c). Now we will understand how to find the second derivative of this equation. Second derivative of a parametric equation is given as:

⇒ d²y / dx² = d / dx (dy / dx), it can also be written as:

⇒ d²y / dx² = d / dc (dy / dx).dc / dx, we can also write it as:

⇒ d²y / dx² = d / dc (y / x) 1 / x;

⇒ d²y / dx² = xy – yx / x³;

Now we will understand this derivative with help of an example:

Suppose we have x (c) = 8c² and y (c) = 5c, then first derivative of this equation is given as:

⇒ [(dy / dc) / (dx / dc)] = y (c) / x (c),

Here the value of x (c) indicates the derivative of ‘x’ with respect to ‘c’. If derivative of an equation is given in this way then we will use chain rule for calculating the equation. So on moving further we get:

⇒ dy / dx = [dy / dc . dc / dx] by the definition of chain rule:

⇒ dy / dc = dy / dx . dx / dc,

Differentiate both given function with respect to’ c’ and we get:

⇒ dx / dc = 16 c and dy / dp = 5,  if we put these values in the equation then we get:

⇒ dy / dx = y / x = 5 / 16c, which is the required solution.

Median of a Triangle can be defined as a line segment which joins a vertex to the midpoint of the opposing side.

Icse 2013 solved papers can be downloaded from different websites. In the next session we will discuss about Permutation and Combination. 

Definition of complex number

In the previous post we have discussed about 11th Grade Math and In today's session we are going to discuss about  Definition of complex number.

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Here we are going to discuss an important concept that is complex number. Complex number can be defined as any number that can be written in the form of p + iq, here value of ‘p’ and ‘q’ stands for real number and ‘i’ stands for imaginary unit. Here value of ‘i’ is given as √ -1. In above given form ‘p’ is denoted as real part and ‘q’ is denoted as imaginary part of complex number. Complex number p + iq can also be described using the point (p, q). If its real part is given as zero then it is called as pure imaginary and if its imaginary part is zero then it is known as real number. Now we will discuss some rules of complex number.

 i = √ -1, the value of 'i' is represented as √ -1. Here if we use iota symbol in the equations then we can write the square root of negative numbers. For example: (√ x (√ y)), we can also write this given value as √ xy.

Let’s discuss some rules that are used for solve imaginary numbers. As we discussed above value of 'i' is given as √ -1, on squaring the iota we get:

= i² = √ -1 * √ -1, we get the value of i² = -1.

In case of i³ it can also write this equation in squaring form:

= i³ = i² * i; (we know that value of i² is -1).

= -1 * i = -i.

In case of i⁴, first write it in the squaring form,

= i⁴ = i² * i², put the value of i² we get:

= -1 * -1 = 1. This is all about complex number definition.

There are different types of Multiplication Properties which are identity, commutative and so on.

Iit jee syllabus can be downloaded online.

11th Grade Math

Mathematics is an interesting subject, which needs innovation and practice to prove yourself. When we look at the contents taught to students till class 10, we observe that they are quite different from what child studies in grade 11. Now when we talk about topics of grade 10th, it includes the basic concepts of mathematics which includes arithmetic, geometry at basic stage, trigonometric basics too.

But moving ahead, we observe that 11th Grade Math includes the topics which are quite new for the students and are of different interest. When we learn about the integration and differentiation in beginning it looks as it we are learning some research topics. The high level mathematics helps the child to relate some of the topics of mathematics with physics. Even learning Physics becomes easy for the students of mathematics in grade 11, as many of the proofs and solving equations are based on the concepts which the child is learning in math class. Besides this we have the topic of mathematical induction, permutation and combination along with the higher level of probability, which helps the child to develop the logical skills and the reasoning concepts.

Learning mathematics in grade 11 has its own thrill, but the child need to be in the regular practice of the subject. We often observe that the science student who has opted for the mathematics in grade 11 is busy in solving the mathematical problems in about 45 – 60 % of his academics studies, and the left over subject is distributed evenly among all other subjects. Mathematics needs involvement and interest of the child to perform well in the subject.

Ordinary Differential Equations is one of the topic taught in class 11. Icse Syllabus For Class 8 is also available online and papers of different subjects can be downloaded. 

 

What is graphing exponential functions

As we all are very well aware about the concept of exponent that describe a value to the power of any number through which any number can multiplied by itself. In the same aspect exponential function can also be describe as a function which can be represented as the power of x of exponential function. This can be represented as a^x. In the section, the discussion held on the topic of graphing exponential functions. To understand the methodology of plotting a exponential function on to a graph we first need to understand why we use exponential function. The basic reason behind using a exponential function is that it is capable of modeling or representing the rate of change between the independent and dependent variable.
The exponential function is a kind of function which can be represented in the form of f (x) = a^x. In this exponential function ‘a’ is a variable which include any positive value but greater than 1. The concept of graphing an exponential function can be categorized into two categories on the basis of their shapes. When the value of variable ‘a’ of exponential function is positive but less than one then it generate a curve from top left side to right down side. On other hand when value of variable ‘a’ of exponential function is greater than one then it generate a curve of graph left down side to right top side. This function have their application in various field of real world like estimating population growth, compound interest, bacterial calculation and radioactive growth. Some time the concept of exponential function also be describe as xy^a where x is a positive value and y is also a positive value but not equal to 1.
In chemistry the concept of Properties of Acids can be describe as characteristics which is able to define the feature of acids. Indian certificate of secondary education having the syllabus of all subjects for any particular class, which is popularly known as icse syllabus 2013.

solve using matrices

In the previous post we have discussed about 11th Grade Math and In today's session we are going to discuss about solve using matrices. Matrices are defined as the set of objects or set of numbers or may be a set of characters in the form of a particular format of rows and columns. These rows and columns are defined for all matrices and a matrix is also known as two dimensional arrays. Various operations can be performed over matrices that are mathematical operations such as addition, subtraction, multiplication and inverse of a matrix can also be determined. Matrices are defined in the order “m x n” where m is the number of rows and n is the number of columns. An equation can be represented in the form of matrix, as left hand side of equals to symbol is represented in the matrix form to the left side only and similarly the right side is represented to the right side in the matrix form. Let us consider an example with the equations x + y = 5 and 3x – y = -1. These equations can be represented in the form of matrix and we will solve using matrices. Representing these equations in the form of matrix we get coefficients of x and y in one matrix and x and y in other matrix. And the constants will be represented to the right hand side in the form of matrix.

Potential energy is defined as the energy which is in the form of stored energy. Potential energy formula is given as potential energy is equals to the product of mass of object to the acceleration of gravity to the height of object. In a mathematical form it is represented as P.E= m x g x h where m is the mass of object, g is the acceleration due to gravity and h is the height of object. Icse board is similar to cbse board which runs all over India. It appears to give more practical knowledge and icse 2013 board papers can be easily discovered on net. It provides help to the students by giving an idea of questions.