## Character Sketch of Vera in the Open Window

In the story ” The Open window” Framton Nuttel was suffering from a nerve disorder. The doctor advised him to spend a few days in the village. They thought that it would give him peace of Mind. Framton Nuttel’s sister had lived in the village four years ago. She knew many people there. She sent him to the village with letters of introduction.

## The Open Window Characters

Mr. Framton Nuttel came to Mrs. Sappleton’s home because he had a letter of introduction for her from his sister. Mrs. Sappleton’s was busy in some work upstairs. So Vera gave him company. The French window of the drawing-room was open.

### Write a Character Sketch of Vera

Vera told a fake story about her aunt. She said that her husband and two brothers went for hunting three years ago but they never returned. it was said that they were lost in the marshes while they were hunting Birds. Their bodies were not recovered so far. Vera said that the death of her husband and two brothers made her aunt insane. Her out hoped that one day they would return through the window. so she always kept the window open.

### Mrs. Sappleton Character Sketch

At this Mrs. Sappleton’s arrived and began to talk about her husband and brothers. She said that her husband and two brothers went for hunting in the morning. They would return soon through the open window.

### Character Sketch of Framton Nuttel and why he confused?

Mr. Framton Nuttel was very confused at the contrast between their stories. Soon he saw three people with a dog through the open window. Their clothes were muddy. Mrs. Sappleton’s shouted with joy that her husband and brothers had arrived. Mr.Nuttel was extremely horrified. He took them for their ghosts. So he ran away.

### Nuttel Run Away

Mrs. Sappleton’s asked why the man had run away to see them. Vera made up another story she said that Mr. Nuttel was afraid of dogs. Once he was attacked by a dog in India. He had to hide in the grave and the dogs were barking the whole night on the grave. She said that Mr. Nuttel had run away to see the dog. we can say that vera is responsible for all confusion which upsets Framton Nuttel.

### Who Was Framton Nuttel in The Open Window?

Mr.Nuttel was an elderly mature man who came to the countryside in search of mental peace, He was a sensitive man and suffered from mental tension. He wanted to spend a few days in the village and had a desire to enjoy its calm and serenity.

He was a stranger in the village, His sister lived in the village for four years. He had a letter of introduction for Mrs. Sappleton from his sister.

### who is vera in the open window

Mrs. Sappleton was upstairs while her niece Vera received Mr. Nuttel. She was a girl of fifteen and was very imaginative and self-possessed. Mr.Nuttel told Vera, the purpose of his visit to the village. He told Vera that he was a complete stranger in the village.

He did not know anything about her aunt except her address and name. He was a patient of nerves and wanted to enjoy the solitude of the village in order to relax his nerves. He was greatly disturbed by the noise and excitement of the city.

### Why was the window kept open?

Vera told Mr.Nuttel that her aunt was a very miserable woman. Her great tragedy happened just three years ago. It was connected one way or other with the French window that opened onto a lawn. She told Mr.Nuttel that her husband and her two brothers with their dog went off for their day’s shooting.

They never came back. They were engulfed in a treacherous marshy piece of land. Their bodies were never recovered That was the dreadful part of it. She told Mr.Nuttel that her aunt kept the window open every evening till it was dusk. She expected that they would come back from this window.

She told Mr.Nuttel how her aunt said farewell to her husband and her brothers. Her husband with his white waterproof coat on his arm and her youngest brother singing a sweet song looked very charming at the time of departure. In this way, Vera impressed Mr.Nuttel a lot with the narration of her aunt’s tragedy.

### Character Sketch of Mrs. Sappleton in the Open Window

Mrs. Sappleton was an eldełly, and respectful woman. She was Vera’s aunt. She was dignified in appearance and behavior. She was very polite and kind to Mr. Frampton Nuttel who came to live with her for a few days. She was upstairs when he came to see her. When she made her appearance in the room, she was very apologetic for being late. She was greatly concerned about the feelings of others.

#### A Truthful Woman

She was a truthful and plain-speaking lady. She told Mr. Frampton the actual reason for Keeping the window of her room open on an October evening. She did not know that Vera had told him a concocted story earlier. She told him that her husband and her two brothers had gone out hunting on that morning. They were expected to come back from that window.

#### A Talkative Woman

She was very talkative. She rattled on cheerfully about the shooting and the scarcity of birds. Mr. Frampton wanted to tell her about his illness but she did not provide him a chance to do so. In the end, she told her husband that Mr. Nuttel was an extraordinary man. He was an ailing person and left the room without making an apology. In the beat of conversation, she could hot study properly the mind of her guest

#### A Faithful Wife

Mrs. Sappleton was a faithful wife and loving sister to her brothers. She was very much concerned about the safe return of her husband and her two brothers who had gone out hunting. At their safe arrival, her face beamed up with pleasure. This shows that she had a loving nature.

#### Not Very Intelligent Wife

Her lack of understanding of human nature was reflected in her relationship with her niece. She was no doubt very kind and soft to Vera, but she failed to understand her natural desires and longings. She could not provide to Vera suitable girl-friends for her companionship. She did not provide her with some form of amusement or recreation to satisfy her suppressed feelings. Her lack of attention made Vera a liar and she invented false stories to satisfy her inner self.

## Explain Counting Technique with Permutation and Combination

In this topic of Counting Techniques, we will read and compare permutations and combinations on counting and types of counting and discuss their formulas and numerical.

## Define Counting Technique in statistic?

In some cases, the sample or event space is so large that it isn’t feasible to write it out. Mathematical tools can be useful to count the size of the sample space and event space in that case. Such tools are called counting techniques.

### Types of Counting Techniques in Mathematical Method

There are two types of counting techniques.

• Permutation Technique
• Combination Technique

### What is Meant by the Permutation Technique?

In the mathematical method, the number of ways to arrange things is called a permutation.

#### For Example:

Arranging people, digits number of anything, letters, and color are examples of permutation.

### What is called Combination?

In the mathematical method, the number of ways to choose things is called a Combination.

#### For Example:

An example of a combination is any cricket team.

For example, if a team coach has 16 team members, then he selects 11 out of these 16 members. This process of choosing is called a combination.

### Concept Of Permutation and Combination

The simple concept of permutation and combination is given in the form of graphs below.

### Cases of Permutaion

There are three types of cases of permutation.

• Repetition is allowed
• Repetition is not allowed
• Distinct Permutation (when they are not all difference)

#### Define Repetition is Allowed

Repetition is allowed: when a single word, or a group of words, is repeated for effect.

#### for example:

when tossing two coins. and the result is

HH, HT, TH, TT

#### Define Repetition is not allowed

Repetition is not allowed: when a single word, or a group of words, is not repeated for effect.

#### For Examples

when tossing two coins. and the result is

HH, HT, TH, TT

## Define Rest in Physics with Examples?

Define Rest in Physics: When a body doesn't change its position with respect to its surrounding, it means that the body is in Rest. Surrounding means the locations in the neighborhood of that frame in which a few are extra (diverse) items are present.

### Define Motion and Explain with Example?

 On the other aspect a frame or Body is called in Motion, if it modifications its role with recognize to its surrounding. It method that the body isn't always in relaxation as it adjustments its position.

### What are the five Examples of Motion in Physics?

If a body does now not exchange its function, it is said to be on relaxation, and if it adjustments its role it’s far called movement. The examples of movement encompass: running, biking, jumping, swimming, consuming, ingesting, gambling, writing, typing, shifting motors, throwing balls.

### Explain Rest and MotionWith Example in Physics

Someone sitting in a moving vehicle is at rest due to the fact she or he is not changing his or her position with admiration to different men and women who are also sitting in the car. But to an observer outdoor the car, the man or woman’s insides of the automobile are in movement.

### Define Rest and Motion with Examples in Physicsand Types of Motion

A few objects circulate alongside a straight line, a few circulate in some different methods.. In step with this concept, there are three types of motion which might be:

• Translatory movement (linear , random and circular)
• Rotatory motion
• Vibratory motion

### What is meant by Translatory Motion?

In simple words:
In translational motion, a frame or body movements along a line. objects move without any rotation. This line could be straight. It can also be curved.

#### Which of these is an example of Translatory Motion?

The motion of a Ferris wheel is a common instance of translator motion. A strolling girl, a stone falling immediately, and a moving vehicle bus, etc..

#### What are the three types of Translatory Motion?

• ⇒linear motion
• ⇒random motion
• ⇒circular motion

### Define Linear motion in Physics?

Whilst an object moves in an instant direction, its motion is stated to be linear movement.

#### Examples of linear Motion

The movement of an airplane is a pleasant example of linear movement because of its actions in a straight direction.

### What is Linear Motion in General form

In nature, a body could be said to be in motion if it adjustments its position with appreciation to a reference factor and time. Even as describing the linear motion we require the best one coordinate axis at the side of time to explain the motion of a particle then it is stated to be in a linear movement or rectilinear movement. In linear motion, the particles will pass from one point to some other factor both in an instant line or a curved route. Relying upon the direction of movement linear movement is further subdivided divided as:

• Rectilinear Movement – it’s miles a route of motion in a directly line.
• Curvilinear Motion – it’s far a path of movement in a curved line.

### Define Random motion in Physics?

While a body 0r any object moves in an abnormal manner, its motion is stated to be a random movement. It way that the body’s movement is disordered.

#### Examples of Random Motion

The motion of molecules of oxygen, movement of birds flying in the sky. Its simple instance could be determined in our environment i. E., the motion of a kite inside the sky and movement of clouds, and many others.

### What is Circular Motion?and Example

When a frame(body) or any item actions in a round path, its motion is said to be round. In simple phrases, the motion or movement of an item in a round direction is said to be around the motion of that body of the object.

#### Examples of circular motion

A toy education transferring on a circular track is the maximum commonplace example and planets revolving across the primary source of heat and mild (solar).

### Rotatory motionDefinitionwith example

In simple words:
The spinning motion of a body or an item around its axis is stated to be the rotatory motion of that frame or item. Axis means a particular line round which a body or object rotates.

#### Example of Rotatory motion

It is a very common example is the movement of a wheel approximately its axis and that of a steerage wheel.

### Vibratory motionDefinitionwith example

In simple words:
To and for a motion of an object or any frame body approximately its mean function is stated to be vibratory motion.

#### Example of vibratory motion

The motion of a toddler at the same time as sitting on a swing. The motion of the pendulum of a wall clock is likewise an essential example of vibratory motion. Its most simple example from our daily life is the vibration of a cellular smartphone.

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## History of Matrix Order

The idea of matrices was introduced by a famous mathematician Arther Cayley in 1857. Matrices are widely used in both physical and social science. In this article we will introduce a new mathematical form, called a matrix, that will enable us to represent a number of different qualities as a single unit.

### Details of Matrix:

Before going to know about the Matrix order we can briefly understand matrices. A matrix is a rectangular array of any number or function. Matrices are 2-dimensional .it is represented by rows and columns. Rows are denoted by m and columns are denoted by n. Now we move towards matrix order. The term matrix order is defined as multiply the number of rows and columns in that matrix. We write the order of the matrix in such a manner that the number of rows first show  and then the number of columns.

## What is Matrix order in math?

A matrix is a square or a rectangular array of numbers written within square brackets or parentheses in a definite order, in rows and columns. The term matrix order is defined as the number of rows and columns in that matrix. for example

Generally, the matrices (plural of the matrix) are denoted by capital letters… A, B, C………….etc. while the elements of a matrix are denoted by small letters a, b, c…………. and numbers 1, 2, 3,……..

### Easy Matrix Order Example:

A=\begin{bmatrix}3&5\\5&1\end{bmatrix} ,

B=\begin{bmatrix}2&3&2\\2&2&1\\4&3&4\end{bmatrix} ,

C=\begin{bmatrix}1\\2\end{bmatrix} ,

D=\begin{bmatrix}1&2\end{bmatrix}

We write the order of matrix in such a manner that number of row first show  and then the number of columns

Example

Let suppose a matrix A has m number of row and n number of the column then the order of the matrix is m*n

Let A be matrix

A=\begin{bmatrix}1&2&2\\1&2&0\end{bmatrix}

Here are 2 rows and 3 columns So order of matrix A is 2*3.

### Types of Matrix in Maths

Some important types of matrix order are as follow:

### Define Row matrix with Examples

The matrix having only one row is called a row matrix.

Examples:

Example No 1.

A=\begin{bmatrix}1&2&3\end{bmatrix}

The order of this matrix is 1×3

Example No 2.

B=\begin{bmatrix}1&2&3&4&5\end{bmatrix}

.Order of this matrix is 1×5

### Define Column matrix with Examples

The matrix having only one column is called a column matrix.

Examples:

Example No 1.

A=\begin{bmatrix}1\\2\\3\end{bmatrix}

Order of this matrix is 3×1

Example No 2.

B=\begin{bmatrix}1\\2\\3\\4\end{bmatrix}

Order of this matrix is 4×1

### Define Null Matrix with Examples

A matrix in which all entries are zero is called a null matrix.

Examples:

Example No 1.

A=\begin{bmatrix}0&0&0\\0&0&0\\0&0&0\end{bmatrix}

Order of this matrix is 3×3

Example No 2.

B=\begin{bmatrix}0&0&0\\0&0&0\end{bmatrix}

Order of this matrix is 2×3

### Explain Square matrix with Examples

A matrix in which a number of rows and number of columns are equal is called a square matrix.

Examples:

Example No 1.

A=\begin{bmatrix}1&2&3\\4&5&6\\7&8&9\end{bmatrix}

Order of this matrix is 3×3

Example No 2.

B=\begin{bmatrix}1&2\\3&4\end{bmatrix}

Order of this matrix is 2×2

### Definition of Diagonal matrix with Easy Examples

A square matrix in which diagonals entries are non-zero and off-diagonals entries are zero is called a diagonals matrix

Examples:

Example No 1.

A=\begin{bmatrix}1&0&0\\0&2&0\\0&0&2\end{bmatrix}

Order of this matrix is 3×3

Example no 2.

B=\begin{bmatrix}1&0\\0&2\end{bmatrix}

Order of this matrix is 2×2

### Define Upper triangular matrix with Examples in Math

A square matrix in which all entries below the main diagonals are zero is called upper triangular matrix.

Examples:

Example No 1.

A=\begin{bmatrix}1&2&2\\0&2&2\\0&0&5\end{bmatrix}

order of this matrix is 3-by-3

Example No 2.

B=\begin{bmatrix}1&1\\0&1\end{bmatrix}

Order of this matrix is 2-by-2

The upper triangular matrix must be a square matrix.

### Lower triangular matrix with Examples

A square matrix in which all items overhead the main diagonals are zero is called a lower triangular matrix.

Examples:

Example No 1.

A=\begin{bmatrix}1&0&0\\1&2&0\\2&2&2\end{bmatrix}

Order of this matrix is 3-by-3

B=\begin{bmatrix}1&0&0&\\4&6&0&0\\4&4&5&0\\2&5&4&1\end{bmatrix}

The order of this matrix is 4-by- 4

A lower triangular matrix is also must be a square matrix.

### What is Symmetric matrix in Math?

A square matrix in which its transpose is equal to itself is called a symmetric matrix

## Define Hyperbolic Function in Simple Way

• Sin Hyperbolic Function
• Cos Hyperbolic Function

### Simply Define sinh x hyperbolic function

Sinh x = \frac12\left(e^x-\;e^{-x}\right) is called hyperbolic sine function. Its domain and range are the set of all real numbers.

### Formula for Cosine hyperbolic function in Math with Detail

Cosh x = \frac12\left(e^x+\;e^{-x}\right) is called hyperbolic cosine function. its domain is the set of all real numbers and the range is the set of all numbers in the interval.

[1, +∞)

### Four Hyperbolic Functions and Formulas and Their Details

The remaining four hyperbolic functions are defined in the terms of the hyperbolic sine and the hyperbolic cosine function as follows:

### Four Hyperbolic Function Formulas are

The hyperbolic functions have the same properties that resemble to those of trigonometric functions.

## Speed Velocity and Acceleration Problems in Physics

Speed:

Objects travel a certain distance in a given time frame and this is called their speed.

Formula:

### Define Velocity:

Basically, velocity is just how fast a person’s displacement changes.

Formula:

### which explains the information needed to calculate speed and velocity in Physics?

Distance covered and time taken are required to calculate speed.

it is denoted by v and its formula is:

v\;=\;\frac st

Where S is Distance and t is time in Formula

It is necessary to take time into consideration when calculating the velocity of the moving body.

it is also denoted by v and its formula is:
v\;=\;\frac dt

### when is the average velocity of an object equal to the instantaneous velocity?

The average velocity of an object is equal to its instantaneous velocity if its acceleration is zero. There cannot be any change in speed or direction of an object if its acceleration is zero. It follows that if an object is moving in the same direction or at zero acceleration, its speed or direction can’t change.

### which pair of sentences is describing the same velocity?

Vehicles moving in opposite directions have the same velocity.

Essentially, acceleration is the rate at which velocity changes, while velocity is the rate at which position changes. Thus, velocity, acceleration, and position are connected.

## The main difference between speed and velocity involves

Speed is defined as:
The distance covered by an object in a unit of time is said to be its speed.

speed formula:

However, Velocity is defined as:
The rate of change of displacement of anybody is said to be velocity.

## which explains the information needed to calculate speed and velocity?

For calculation of speed, we require distance covered and taken time

it is denoted by v and its formula is:

v\;=\;\frac st

S is Distance and t is time.

However, For the calculation of velocity, we require displacement of that moving body and take time.

it is also denoted by v and its formula is:
v\;=\;\frac dt

### when is the average velocity of an object equal to the instantaneous velocity?

If the acceleration of an object is zero then the average Velocity is equal to instantaneous velocity. And if the acceleration of an object is zero there can be no change in speed or direction of that object.

### which pair of sentences is describing the same velocity?

A moving car and a truck have the same velocity.

Actually, acceleration is a rate of change of velocity and velocity is a rate of change of position. So the position, velocity, and acceleration are related to each other.

## What is an Asymptote in Calculus

In mathematical terms, an asymptote is a curve that approaches infinity, asymptotes are lines that appear parallel to it.

### Types of Asymptotes in Calculus

There are three types of Asymptote in math.

### How to find Horizontal Asymptote of Exponential Function?

Horizontal Asymptote: A horizontal asymptote occurs when A curve approaches some constant value b as x approaches infinity (or *infinity)..

### How to find Vertical Asymptote of Exponential Function

Vertical Asymptote: A horizontal asymptote occurs when a function of X approaches some constant, c (from the left or right) then the curve continues to infinity (or −infinity).

### What is Oblique Asymptote?

Oblique Asymptote: A Oblique Asymptote occur when, as x goes to infinity (or −infinity) the curve then becomes a line y=mx+b

### Asymptote for a Curve Definition in Math

Definition: A straight line l is called an asymptote for a curve C if the distance between l and C approaches zero as the distance moved along l (from some fixed point on l ) tends to infinity.

The curve C can approach asymptote l as one moves along l in one direction, or in the opposite direction, or in both directions.

### How to Find Equation of Horizontal Asymptote?

Suppose the equation y\;=\;f(x) of C is such that y is real and y\rightarrow a as X\rightarrow\infty or X\rightarrow-\infty then y\;=\;a is a Horizontal asymptote. For, the distance between the curve and the straight line y\;=\;a is y\;-\;a and this approaches zero as X\rightarrow\infty or X\rightarrow-\infty.

### How to Find Equation of Vertical Asymptote?

If the equation of C is such that y is real and Y\rightarrow\infty or Y\rightarrow-\infty as x\rightarrow a from one side then the straight line x = a is a vertical asymptote.

To see this, observe that (1) x – a is the distance between the curve and the straight line and that this distance is supposed to approach zero (2) Y\rightarrow\infty or Y\rightarrow-\infty as x\rightarrow a , so that

\lim_{y\rightarrow\pm\infty}\;(x-a)\;=\;\lim_{x\rightarrow a}\;(x-a)\;=\;0

Thus, to locate vertical asymptotes we have to find a number a such that \lim_{x\rightarrow a}\;y\;=\;\infty\;or\;-\infty

Similarly, if y\rightarrow mx\;+\;c as x\rightarrow\infty or x\rightarrow-\infty then y\;=\;mx\;+\;c is an asymptotes ( which is neither vertical nor horizontal).

Thus, we inquire for \lim_{x\rightarrow\pm\infty}\;y is studying asymptotes.

### Find Horizontal and Vertical asymptote for Algebraic equations

For Algebraic equations, we can find horizontal and vertical asymptotes as follows:

For horizontal asymptotes we write the given equation in the form x\;=\;\frac{\psi(y)}{\theta(y)} and consider those values of y for which \theta(y)\;=\;0.

Similarly, to find a verticle asyptote, we write the given equation in the form y\;=\frac{f(x)}{g(x)} and consider those value of x for which g(x) = 0.

### Working Rule for Asymptotes Parallel to the Axes

In an equation of a curve, the coefficient of the highest power of x (respectively of y) equated to zero gives asymptotes (if any) parallel to the x-axis (respectively y-axis)

## Define Homogeneous Function with Example

A function f defined by

u\;=\;f\;(x,\;y,\;z,\;…)

of any number of variables is said to be homogeneous of degree n in these variables if multiplication of these variables by any number t(\neq0) result in the multiplication of the function by t^n, i.e.,

f\;(tx,\;ty,\;tz,\;…) = t^n f\;(x,\;y,\;z,\;…) (1)

provided that (tx,\;ty,\;tz,\;…) is in the domain of f.

Taking t=\frac1x,(x\neq0) the equation (1) becomes

• f\left(1,\;\;\frac yx,\;\frac z{x\;}\right)\;=\;\frac1{x^n}f(x,\;y,\;z,\;…)\;
• \;f(x,\;y,\;z,\;…)\;\;=x^n\;f\left(1,\;\;\frac yx,\;\frac z{x\;}\right)\;
• =x^ng\left(\frac yx,\;\frac zx\;…\right)