The post What are matrices and their types? appeared first on Goseeko blog.

]]>A matrix is a rectangular arrangement of the numbers. We call these numbers inside the matrix as elements of the matrix.

HereThe expression of a matrix ‘A’ is-

The vertical elements are columns and the horizontal elements are rows of the matrix.

The order of matrix A is m by n or (m× n)

We denote a matrix ‘A’ is as-

Where, i = 1, 2, …….,m and j = 1,2,3,…….n

Here ‘i’ denotes row and ‘j’ denotes column.

**1. Rectangular matrix-**

A matrix in which the number of rows is not equal to the number of columns, then we say that the given matrix is a rectangular matrix.

Example:

The order of matrix A is 2×3, hence, it has two rows and three columns.

Matrix A is a rectangular matrix.

**2. Square matrix-**

A matrix with equal number of rows and columns, is a square matrix.

Example:

The order of matrix A is 3 ×3 , that means it has three rows and three columns.

Matrix A is a square matrix.

**3. Row matrix-**

A matrix with a single row and any number of columns is called row matrix.

Example:

**4. Column matrix-**

A matrix with a single column and any number of rows is called row matrix.

Example:

**5. Null matrix (Zero matrix)-**

A matrix in which each element is zero, then we call it a null matrix or zero matrix, therefore, we denote it by O.

Example:

**6. Diagonal matrix-**

A matrix is said to be diagonal matrix if all the elements except principal diagonal are zero

therefore, The diagonal matrix always follows-

So that, for Example:

**7. Scalar matrix-**

A diagonal matrix in which all the diagonal elements are equal to a scalar, hence, we call it a scalar matrix.

Example-

**8. Identity matrix-**

A diagonal matrix is said to be an identity matrix if its each element of diagonal is unity or 1.

Hence, we denote it by – ‘I’

**9. Triangular matrix-**

If every element above or below the leading diagonal of a square matrix is zero. therefore, we call it a triangular matrix.

Hence, there are two types of triangular matrices-

**(a) Lower triangular matrix-**

If all the elements below the leading diagonal of a square matrix are zero, therefore, we call it a lower triangular matrix.

Example:

**(b) Upper triangular matrix-**

If all the elements above the leading diagonal of a square matrix are zero, hence, we call it a lower triangular matrix.

Example –

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]]>The post What is graph theory? appeared first on Goseeko blog.

]]>It also has uses in social sciences, chemical sciences, information retrieval systems, linguistics even in economics also.

Graphs are discrete structures consisting of vertices and edges that connects these vertices.

There are several different types of graphs that differ with respect to the kind and number of edges that can a connect a pair of vertices.

Graph theory is a relatively new area of mathematics. Graph is the form of representing of descriptive data in the terms of verticals and edges. hence, we use Graph theory in various fields like computer science, information technology, genetics, telecommunication etc.

A graph is a collection of vertices and edges in which each edge is assigned to pair of points, we call them terminal.

Hence, we can say that a graph is a network of dots connect by lines.

Mathematically we can define a graph as-

A graph is a pair of set (V, E) where-

1. V is a non-empty set, we call its elements vertices.

2. E is collection of two-element subset of V, we call it edges.

**Terminology: **A graph G is an order pair (V, E) where V is a non-empty set and E is the set of edges in which each element of E is assign to a unique unorder pair of elements of V.

We denote an element of a set E as- e = (v, su) where u, v ∈ V.

hence, U and v are the end vertices of edge e.

**Note- In any graph the number of vertices with odd degree must be even.**

**Loop: **If both the end vertices of an edge are same then we call the edge as a loop.

**Parallel edge: **If two or more edges have same terminal vertices, then we call these edges as parallel edges.

**Interested in learning about similar topics? Here are a few hand-picked blogs for you!**

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]]>The post What is a ring? appeared first on Goseeko blog.

]]>(1) R is an additive Abelian group,

(2) R is an multiplicative semigroup

(3) The two distributive laws hold good, viz

a. (b + c) =a.b + a.c. (left distributive law)

(a + b). c = a.c + b.c, (right distributive law) for all a, b, c R

1. In order to be a ring, there must be a non-empty set equipped with two binary operations, viz, + and . , satisfying the postulates mentioned above. Later on we will simply write ‘‘Let R be a ring’’. It should be borne in mind that there are two binary operations, viz, + and , on R.

2. It should also be borne in mind that the binary operations + and . may not be own usual addition and multiplication.

3. Since R is an additive Abelian group, the additive identity, denoted by 0, belongs to R. We call it the zero element of R. Here 0 is a symbol.

4. If a , bR, then – bR ( R is an additive group). a+ (– b) is, generally, denoted by a – b.

5. a.b is, generally, written as ab.

**Commutative Ring: **R is commutative if ab = ba, for all a, bR.

**R with unity **: R is said to be a ring with unity (or identity) if there exists an element e R such that

ae = a = ea, for all a R.

**Note**: A ring may or may not have a unity.

2. If R is a ring with unity, unity may be same as the zero element. For example, {0} is a commutative ring with unity. Here 0 is the additive as well as the multiplicative identity. It is known as the zero ring.

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]]>The post What are the definite and indefinite integrals? Explained with examples appeared first on Goseeko blog.

]]>The limits of calculus are also used to define the integral. In this post, we will learn the definition and formulas of the definite and indefinite integrals along with a lot of examples.

The definite integral is a type of integral in which the upper and the lower limits are applied to integrate the functions. In this type of integral, the interval of initial and final terms of a graph are used. The interval (u, v) is also known as the boundaries of the function.

The equation used to define the definite integral is given below.

**· In**

** and v are the lower and upper boundaries respectively and it is known as the integral notation of definite integral.**

**·**** ****f(y) is the integrand function.**

**·**** ****dy is the variable of integral.**

**·**** ****F(b) – F(a) is the fundamental rule of calculus to apply the boundaries.**

**·**** ****M is the numerical value of the definite integral.**

**On the other hand, the indefinite integral is the other type of integral used to find the volume and area under the graph. In this type of integral, the upper and the lower limits are not applied. The simple integral notation is used in the indefinite integral.**

**ʃ f(y) dy = F(y) + C**

**·**** **** ʃ it is the integral notation.**

**·**** ****f(y) is the integrand function.**

**·**** ****dy is the variable of integration.**

**·**** ****F(y) is the result of the function after integrating the function.**

**·**** ****C is the integral constant.**

**These types of integral are frequently used to find the area, volume, and central points. **

**Below are some well-known rules of integral.**

Rule name | Rules |

The sum rule | ʃ (f(y) + g(y)) dy = ʃ f(y) dy + ʃ g(y) dy |

The constant rule | ʃ C dy = Cy, where c is any constant |

The difference rule | ʃ (f(y) – g(y)) dy = ʃ f(y) dy – ʃ g(y) dy |

The constant function rule | ʃ (f(y) dy = C ʃ (f(y) dy |

The power rule | ʃ [(f(y)]^{n} = [f(y)]^{n+1} / n + 1 |

**By using the rules of integration, the problems of the definite and indefinite integral can be solved easily. Below are some examples of these types of integral.**

**Example 1: For the definite integral**

**Integrate 5x**^{3}** + 12sin(x) – 5x**^{2}**y**^{5}** + 11x**^{2}** + 5 with respect to x have boundary values [1, 4].**

**Solution**

**Step 1: ****Use the integral notation of the definite integral and write the given function along with the variable of integration.**

**Step 2: ****Now use the sum rule and the difference rule of integral and apply the definite integral notation separately to each term. **

**Step 3: ****Use the constant and constant function rule of integral on the above expression.**

**Step 4: ****Now integrate the above expression by using the power and trigonometric rules of integral. **

Step 5: Use the fundamental theorem of calculus ’

To avoid such a large number of steps to get the result, you can simply use an integral calculator. Follow the below step to integrate the function with the steps.

**Step 1:** Select the type of integral i.e., definite or indefinite.

**Step 2:** Input the integrand.

**Step 3:** choose the integrating variable.

**Step 4:** Enter the upper and lower limit in case of the definite integral.

**Step 5:** Press the calculate button to get the result.

**Step 6:** The solution with steps will show below the calculate button.

**Example 2: For indefinite integral**

I**ntegrate x ^{5} – sin(x) – 5y^{5} + 13x^{3} + 5x with respect to x.**

**Solution**

**Step 1:** Use the integral notation of the indefinite integral and write the given function along with the variable of integration.

ʃ (x^{5} – sin(x) – 5y^{5} + 13x^{3} + 5x) dx

**Step 2:** Now use the sum rule and the difference rule of integral and apply the indefinite integral notation separately to each term.

ʃ (x^{5} – sin(x) – 5y^{5} + 13x^{3} + 5x) dx = ʃ (x^{5}) dx – ʃ sin(x) dx – ʃ 5y^{5} dx + ʃ 13x^{3} dx – ʃ 5x dx

**Step 3:** Use the constant and constant function rule of integral on the above expression.

ʃ (x^{5} – sin(x) – 5y^{5} + 13x^{3} + 5x) dx = ʃ (x^{5}) dx – ʃ sin(x) dx – 5y^{5} ʃ dx + 13ʃ x^{3} dx – 5ʃ x dx

**Step 4:** Now integrate the above expression by using the power and trigonometric rules of integral.

ʃ (x^{5} – sin(x) – 5y^{5} + 13x^{3} + 5x) dx = x^{5+1 }/ 5 + 1 – (-cos(x)) – 5y^{5} (x) + 13 (x^{3+1} / 3 + 1) – 5 (x^{1+1}_{ }/ 1 + 1) + C

ʃ (x^{5} – sin(x) – 5y^{5} + 13x^{3} + 5x) dx = x^{6 }/ 6 – (-cos(x)) – 5y^{5} (x) + 13 (x^{4} / 4) – 5 (x^{2}_{ }/ 2) + C

ʃ (x^{5} – sin(x) – 5y^{5} + 13x^{3} + 5x) dx = x^{6}/6 + cos(x) – 5xy^{5} + 13x^{4}/4 – 5x^{2}/2 + C

Now you are witnessed that the types of the integral can be solved easily without any difficulty. In this post, we’ve learned about the basics of the types of integrals along with examples and also learned how to solve definite and indefinite integrals using an antiderivative calculator. Now you can solve any problem either by definite integral or indefinite easily.

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]]>The post What is a bar diagram? appeared first on Goseeko blog.

]]>Shape of a bar is like a Rectangle. we call them one dimensional diagrams because only length of the bar matters and not the width.

The following are the different types of bar diagrams:

(i) Simple Bar Diagram

(ii) Subdivided Bar Diagram

(iii) Multiple Bar Diagram

(iv) Percentage Bar Diagram

(v) Deviation Bar Diagram

(vi) Broken Bar Diagram

Here we will discuss about subdivided bar diagram and Percentage Bar Diagram

If there are various components of a variable to represent in a single diagram then we use subdivided bar diagrams in this condition.

We represent a bar in the order of magnitude from the largest component at the base of the bar to the smallest at the end of the bar, but we keep the order of various components in each bar in the same order.

We use Different shades or colours to distinguish between different components. To explain such differences, we should use the index in the bar diagram.

We can represent Subdivided bar diagrams vertically or horizontally. If the number of components are more than 10 or 12, then we do not use the subdivide bar diagrams because in that case, here will be overloaded information in the diagram and we cannot compare of understand it easily.

**Example: Represent the following data by subdivided bar diagram:**

**Sol:**

First of all we calculate the cumulative cost on the basis of the given amounts:

subdivided bar diagram is given below:

When we draw the Subdivided bar diagram on the basis of the percentage of the total, we call them a percentage bar diagram.

When we draw such diagram, we keep the length of all the bars equal to 100 and we form the segments in these bars to represent the components on the basis of percentage of the aggregate.

First of all we assume the total of the given variable equal to 100. Then we calculate the percentage for each and every component of the variable. After then the cumulative percentage are calculated for every component. Finally the bars are subdivided into the cumulative percentage and presented like subdivided bar diagram.

**Example:**

**Draw a percentage bar diagram for the following data:**

**Solution: **

percentage bar diagram is-

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]]>The post What is simplex method? appeared first on Goseeko blog.

]]>Linear programming functions are in standard form when trying to maximize the objective function. Subject to constraints,

Here and after adding the Slack variable, the system of corresponding constraint equations Where,

S1, s2…………sm is called a slack variable. These are non-negative numbers that are added to remove the inequality from the equation.

**Example: Maximize-**

**Subject to-**

**By using simplex method.**

**Solution**

Here we have

And the inequalities-

By adding s1 and s2 we get-

Putting decision variables equals to zero, we get-

Since the coefficient (–3) of x3 is most negative, so *x*3 is an entering variable.

The smallest ratio is 1 in the *s*2-row, therefore *s*2 is the departing variable. The pivot entry (2) is at the intersection of *x*3-column and *s*2-row.

To make pivot entry (1), we multiply third row by ½, we get-

Applying, we get

Since, Z row of the table has non-negative entries in the column of variables, therefore, this is the case of optimal solution. From the last column of the table we have x1 = 0, x2 = 0 and x3 = 1 and the maximum value of Z = 3.

Solve a product-mix problem for which the values of ?1 and ?2 that

Maximize: ? = 3?1 +2?2

Subject to: −?1 + 2?2 ≤ 4

3?1 + 2?2 ≤ 14

?1 − ?2 ≤ 3

?1 ≥ 0 , ?2 ≥ 0

Sol:

The solution process to maximize first introduce the slack variables ?1 , ?2 ??? ?3 . Hence

????????: ? = 3?1 + 2?2 +0. ?1 +0. ?2 + 0. ?3

We get

?1 + 2?2 +?1 +0. ?2 + 0. ?3 = 4

3?1 +2?2 + 0. ?1 +?2 +0. ?3 = 14

?1 − ?2 + 0. ?1 + 0. ?2 + ?3 = 3

?1 , ?2 , ?1 , ?2 , ?3 ≥ 0

From the table the pivot element is 1 hence the pivot column is the first column and the pivot row is the third row because of the most negative number on that column and least positive ratio respectively.

Here again the second column is the pivot column and the second row is also pivot row.

If the value of ?? − (contribution of unit price to the profit) are all positive or zero, the current basic feasible solution is optimal. But there values one or more negative choose the variable corresponding to which the value of ?? − ?? is least i.e most negative.

Hence the optimal solution will be-

And Max. Z = 14

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]]>The post What is maxima and minima? appeared first on Goseeko blog.

]]>As we know that the value of a function at maximum point is called maximum value of a function. Similarly the value of a function at minimum point is called minimum value of a function.

The maxima and minima of a function is an extreme biggest and extreme smallest point of a function in a given range (interval) or entire region. Pierre de Fermat was the first mathematician to discover a general method for calculating maxima and minima of a function. The maxima and minima complement each other.

If f(x) is a single valued function defined in a region R then

Maxima is a maximum point if and only if

Minima is a minimum point if and only if .

Let f(x, y) be a defined function of two independent variables.

Then the point x = a and y = b is said to be a maximum point of f(x, y) if

For all positive and negative values of h and k.

Similarly the point x = a and y = b is said to be a minimum point of f(x, y) if

For all positive and negative values of h and k.

Critical points of a function of two variables are those points at which both partial derivatives of the function are zero. A critical point of a function of a single variable is either a local maximum, a local minimum, or neither. With functions of two variables there is a fourth possibility – a saddle point.

1. Maximum and minimum values of a function occur alternatively

2. Function may have several maximum and minimum values in an interval

3. At some point the maximum value may be less than the minimum value

4. The points at which a function has maximum or minimum value are called turning points and the maximum and minimum values are known as extreme values, or extremum or turning values.

5. The values of x for which f(x) = 0 are often called critical values

For a function y = f(x) to attain a maximum point at x = a,

For minimum point-

**Necessary Condition**– If a function f(x) is maximum or minimum at a point x = b and if f’(b) exists then f’ (b) = 0.

**Sufficient Condition-** If b is a point in an interval where f(x) is defined and if f ‘(b) = 0 and f’’(b) is not equal to 0, then f(b) is maximum if f’’(b) <0 and is minimum if f’’(b) > 0. (The proof is not shown at present).

**Working Rule:**

To find the maximum or minimum point of a curve y = f(x).

Find f ‘ (x) and equate it to zero. From the equation f ‘(x) = 0, find the value of x, say a and b.

Here the number of roots of f ‘(x) = 0 will be equal to the number of degree of f ‘(x) = 0.

Then find f ‘(a – h) and f ‘(a + h), then note the change of sign if any (here h is very small).

If the change is from positive to negative, f(x) will be maximum at x = a. If again the change of sign is from negative to positive, f(x) will be maximum at x = a.

Similarly for x = b.

First we find the first derivative of y = f(x) i.e dy/dx and make it zero.

From the equation dy/dx = 0 find the value of x say a and b.

The again we find the second derivative of y or

Put x = a in , if at x = a is negative then the function is maximum at x = a and maximum value will be f(a).

If the value of at x = a is positive, then the function is minimum and the minimum value will be f(a)

Similarly we take for x = b.

**Example: Examine for maximum and minimum for the function **

Sol.

Here the first derivative is-

So that, we get-

Now we will get to know that the function is maximum or minimum at these values of x.

For x = 3

Let us assign to x, the values of 3 – h and 3 + h (here h is very small) and put these values at f(x).

Then-

which is negative for h is very small

which is positive.

Thus f’(x) changes sign from negative to positive as it passes through x = 3.

So that f(x) is minimum at x = 3 and the minimum value is-

And f(x) is maximum at x = -3.

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]]>The post What is the sample space and types of events?￼ appeared first on Goseeko blog.

]]>A sample space is the set of all possible outcomes of a random experiment. We denote it by S, and the total number of elements in the sample is the size of the sample space and we denote it by n(S).

Discrete sample space- Sample space in which sample points are finite or countably infinite.

For example-

If we throw a dice, then the spl space is

S = {1, 2, 3, 4, 5, 6} and n(S) = 6.

If we toss a coin twice or simultaneously then-

S = {HH, HT, TH, TT}

If toss a coin 4 times or toss four coins simultaneously then-

S = {HHHH, HHHT, HHTH, HTHH, THHH, HHTT, HTHT, HTTH, THHT,

THTH, TTHH, HTTT, THTT, TTHT, TTTH, TTTT} and n(S) = 16.

Note-If a random experiment with x possible outcomes is performed n times, then the total number of elements in the sample is x^n

1. Exhaustive Events or Sample Space: The set of all possible outcomes of a single performance of an experiment is exhaustive events or sample space. Each outcome is a sample point.

In case of tossing a coin once, S = (H, T) is the sample space. Two outcomes – Head and Tail

– constitute an exhaustive event because no other outcome is possible.

2. Trial and Event: Performing a random experiment is called a trial and outcome is termed an event. Tossing of a coin is a trial and the turning up of the head or tail is an event.

3. Equally likely events: Two events are said to be ‘equally likely’, if one of them cannot be expected in preference to the other. For instance, if we draw a card from a well-shuffled pack, we may get any card. Then the 52 different cases are equally likely.

4. Independent events: Two events may be independent, when the actual happening of one does not influence in any way the probability of the happening of the other.

5. Mutually Exclusive events: Two events are known as mutually exclusive, when the occurrence of one of them excludes the occurrence of the other. For example, on tossing a coin, either we get head or tail, but not both.

6. Compound Event: When two or more events occur in composition with each other, the simultaneous occurrence is called a compound event. When a die is thrown, getting a 5 or 6 is a compound event.

7. Favorable Events: The events, which ensure the required happening, are said to be favorable events. For example, in throwing a die, to have the even numbers, 2, 4 and 6 are favorable cases.

Odds in favor of an event and odds against an event-

If the number of favorable cases are ‘m’ and the number or not favorable cases are ‘n’.

Then-

1. Odds in favor of the event = m/n

2. Odds against the event = n/m

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]]>The post What is the Pigeonhole principle? appeared first on Goseeko blog.

]]>This principle can be applied to many problems where we want to show that a given situation can occur.

**Example-1: Suppose a class contains 13 students, then two of the students (pigeons) were born in the same months (pigeonholes)**

**Example-2: **Suppose a department contains 13 professors, then two of the professors (pigeons) were born in the same month (pigeonholes).

**Example-2: **Find the minimum number of elements that one needs to take from the set *S *= {1*, *2*, *3*, . . . , *9} to be sure that two of the numbers add up to 10.

Here the pigeonholes are the five sets {1, 9}, {2, 8}, {3, 7}, {4, 6}, {5}. Thus any choice of six elements (pigeons) of *S *will guarantee that two of the numbers add up to ten.

**Example-2: Show that if seven numbers from 1 to 12 are chosen, then two of them will add up to 13.**

Sol.

First we will construct six different sets, each containing two numbers that add up to 13 as follows-

Each of the seven numbers belong to one of these six sets.

Since there are six sets, then according to the pigeonhole principle that two of the numbers chosen belong to the same set. These numbers add upto 13.

Example-3: Suppose a bag contains 10 red balls, 10 white balls and 10 blue balls. what is the minimum number of balls we have to choose randomly for the bag to ensure that we 4 balls of same color?

Solution:

By applying Pigeonhole principle-

number of colors(pigeonholes) n = 3

number of balls (pugeons) k + 1 = 4

hence the minimum number of balls required = kn + 1

by simplifying we obtain kn + 1 = 10.

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]]>The post What is a frequency polygon? appeared first on Goseeko blog.

]]>We can define as follows-

If we plot the points with the upper limits of the classes as abscissae and the cumulative frequencies corresponding to the values less then the upper limits as ordinates and join the points so plotted by line segments, the curve thus obtained is nothing but known as “less than cumulative frequency curve” or “less than ogive”.

If we plot the points with the lower limits of the classes as abscissa and the cumulative frequencies corresponding to the values more than the lower limits as ordinates and join the points so plotted by line segments, the curve thus obtained is nothing but known as “more than cumulative frequency curve” or “more than Ogive”.

**Note-**

**Median may also the obtained by drawing dotted vertical line through the point of inter section of both the ogives, when drawn on a single figure.**

**Draw less than type and more than type ogive of the following data-**

Weekly wages | 0-10 | 10-20 | 20-30 | 30-40 | 40-50 |

workers | 45 | 55 | 70 | 40 | 10 |

**Sol:**

We prepare the table for both less than type and more than type:

Weekly wages | Workers | Less than cumulativeWages less than workers | more than cumulativeWages more than workers |

0-10 | 45 | 10 45 | 0 220 |

10-20 | 55 | 20 100 | 10 175 |

20-30 | 70 | 30 170 | 20 120 |

30-40 | 40 | 40 210 | 30 50 |

40-50 | 10 | 50 220 | 40 10 |

From above data, we construct both the ogives as shown in Figure below

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]]>