Category: Maths Notes PPT

The first question that comes to your mind is what is a Pareto chart. It is even known as a Pareto analysis or Pareto diagram. There are different variations of the chart, like comparative Pareto charts and weighted Pareto charts. It’s a kind of bar graph wherein the length of the bars denotes cost, quantity, time, and frequency. The bars are arranged in such a way that the longest ones remain on the left while the shortest is on the right. The chart depicts more significant situations visually and is used as cause analysis equipment. It’s a graph indicating the occurrence of defects along with their collective impact. These charts are handy in finding flaws for prioritizing in perfect order to detect the greatest complete improvement.

Pareto Chart Analysis

Pareto chart analysis involves learning the right way of using the chart for varied applications. The process goes like this:

• First of all, you will have to determine the different classifications that you will use for arranging the items in this chart.

• Next, you need to choose the right measurement like time, cost, frequency, and quantity.

• The next step is deciding on the period for the Pareto chart for which the bar graph will be made, such as for a year, a month, or a week.

• Gather new data along with already existing data according to time and category.

• Take different measurements for specific categories.

• Select the right scale for measurements assembled. The maximum value will be the total made in the step above.

• For specific categories, bars need to be constructed and even labelled. Longest bars will be at the left while the shortest ones at the right.

These are the steps included in the Pareto chart analysis. Now coming to Pareto chart analysis example, you will find it below:

Pareto Chart Analysis Example

You need to come up with a chart portraying defects in a manufactured shirt. The defects need to be presented in a checklist of five points. Here, the five points are:

• Collar defect

• Sleeve defect

• Button defect

• Pocket defect

• Cuff defect

This will be the data table for the Pareto chart, which will help you in understanding the cumulative percentage:

Solved Example

  

 

Considering the data in this table or the Pareto chart analysis example, the bar graph for the same shall be created.

Interpreting a Pareto Diagram

So, you have now got a Pareto chart example with all its different elements. Here another trickier problem is interpreting the results of the chart. Considering the various discussions on Pareto chart analysis, the table above clearly shows all the shirt defects or problem areas that the manufacturing company needs to focus on.

The Pareto chart example clearly shows that the manufacturing company should take care of pocket defects and sleeve defects in the shirts it is producing.

Use of Pareto Chart

There are many differences in the use of Pareto charts and when to use Pareto charts. Pareto charts are best used for carrying out an analysis of bulk data. Let us have a look at the use of Pareto charts in different business industries.

• For analysing growth revenue of organisations in relation to the time

• For choosing specific data from different data sets and working on the same

• For explaining data sets to other individuals

• For analysing growth in population in a city or country or throughout the world

• For checking universal issues and focusing on resolving the most important ones

• For checking major complaints from the public and resolving the same according to priority

Pareto diagrams, more commonly known as 80/20 Pareto rule, are quite useful for different management officials. A Pareto diagram helps managers in determining issues in the workflow procedure. It helps them in understanding which 20% of problems in the company’s procedures are causing 80% of the issues. They can take care of these issues and ensure that the business procedures run smoothly overall.

The word perimeter is introduced from two Greek words ‘peri’ which defines around and ‘metron’; which defines measure. In Geometry, the perimeter of a two-dimensional shape is defined as the path or boundary that encloses the shape. if the given figure is a polygon such as a triangle, square, rectangle, etc, then the perimeter is the sum of the length of all the sides of a polygon. For example, a triangle with side length 4 cm has a perimeter 4 + 4 + 4 = 12.

Some geometrical shapes do not have a finite number of sides, so calculating the perimeter of such shapes is a bit uncomplicated. For example, the perimeter of a circle is stated as the circumference of a circle which can be calculated by using formula 2r, where r is the radius of the circle and the value is 3.14. Let us learn the formulas to calculate the perimeter of different two-dimensional shapes.

Perimeter Formulas of Two-Dimensional Shapes

Let us know learn the perimeter formulas of different two-dimensional shapes:

Perimeter of a Square

The perimeter of a square is the sum of the length of all its four sides. As we know, all the sides of a square are equal, hence the perimeter of a square will be 4 times its side i.e  4 × side.

The perimeter of a Square Formula (P) –  4 × Sides.

Perimeter of a Triangle

The perimeter of a triangle is the sum of the length of all its three sides. For example, if x, y, and z  are the three sides of a triangle, then the perimeter of that triangle will be x + y + z.

The perimeter of a Triangle Formulas (P) – Sum of the length of all its three sides.

Perimeter of Equilateral Triangle

The equilateral triangle is the triangle where all the sides and angles are equal. Hence, the perimeter of the equilateral triangle will be calculated by using the formula 3a, where ‘a’ is the side of the triangle.

The perimeter of Equilateral Triangle Formula  (P) = 3a or 3 × side.

Perimeter of Rhombus

The rhombus is often known as a diamond or diamond-shaped object. The total distance traveled along the boundary is termed the perimeter of a rhombus.

The perimeter of a Rhombus Formula (P) =  4a,

In the above perimeter of a rhombus formula, the variable ‘a’  is the length of the side of the rhombus.

Perimeter of Cube

The perimeter of a cube relies on the number of edges it has and the length of the edges of a cube. As the cube has 12 edges and all the edges are similar in length. Therefore the perimeter of a cube is calculated by using the formula 12l.

The perimeter of Cube Formula (P) =  12l

Where l is stated as the length of the edge of the cube.

Perimeter of Rectangle

The perimeter of a rectangle is defined as the sum of the length of all its 4 sides. As we know, opposite sides of a rectangle are equal, and accordingly, the perimeter of a rectangle will be twice the length of a rectangle plus twice the breadth of the rectangle and it is represented by the alphabet p.

The perimeter of a Rectangle Formula (P) =  2(L + B)units.

Perimeter of Circle

The perimeter of a circle is the measurement of the boundary of a circle. The perimeter of a circle is also defined as the circumference of a circle. The perimeter or circumference of a circle can be calculated using the formulas given below.

Perimeter of Circle Formula

If the radius of a circle is given, the perimeter or circumference of a circle will be calculated by using the formula 2πr, where r is the radius of the circle and the value of is 3.142 approx. 

If the diameter of a circle is given, the perimeter or circumference of a circle will be calculated by using the formula πd, where d is the diameter of the circle and the value of π is 3.142 approx.

Perimeter of Semicircle

A semicircle is a half-circle that is obtained either by dividing the whole circle into two halves or by dividing the circumference of a circle by 2. The perimeter of a semicircle is half of the circumference of an original circle C, plus the diameter d. The perimeter of the semicircle can be calculated by using the formula given below.

Perimeter of Semicircle Formula (P) =  ½ (2πr) + d, or  P = πr + d.

In the above perimeter of the semicircle formula, variables ‘d‘ and ‘r’ are the diameter and the radius of a circle respectively.

Perimeter of Parallelogram

A parallelogram is a two-dimensional geometric shape enclosed by its four sides. The perimeter of a parallelogram is the sum of all the lengths of all its four sides. We know the opposite sides of a parallelogram are equal and parallel to each other. Accordingly, the perimeter of a parallelogram is calculated by using the formula given below.

Perimeter of Parallelogram Formula (P) = 2(a + b) units.

Solved Examples

1. Calculate the perimeter of a square, if the length of the sides of a square is 10cm.

Solution: As we know, the perimeter of a square is 4 × side

Accordingly, the perimeter of a square will be 4 × 10 = 40 cm.

Hence, the perimeter of a given square is 40 cm.

2. The length and breadth of a rectangular garden is 150 m and 110 m respectively. Find the perimeter of a rectangular garden.

Solution: As we know, the perimeter of a rectangle is 2( length + breadth).

Accordingly, the perimeter of a rectangular garden is 2(150 + 110) = 2(260) = 520 meters.

Hence, the perimeter of a rectangular garden is 44 meters.

3. Find the perimeter of an equilateral triangle whose side is 8 cm

Solution: As we know, the perimeter of an equilateral triangle is 3 × side.

Accordingly, the perimeter of an equilateral triangle is 3 × 8 = 24 cm.

Hence, the perimeter of a given equilateral triangle is 24 cm.

Fun Facts

• Different rectangles with similar perimeter can have different areas.

• The perimeter of the square and rectangle is always smaller than their area whereas the perimeter of the triangle can be more than its area.

What Is a Pictograph?

The pictograph meaning can be explained as the way to represent data by depicting it through images is known as a pictograph. Every image is representing a number of something or units of the data. The representation of the data can be in the form of symbols or pictures.

This is a basic pictograph example.

This is an example which shows the data of a number of red boxes sold on each day of the week.

Some examples of Pictograph in Maths can be: 

Example 1: 

The data shows the number of apples sold individually in the months of April and May.

We can represent this data in the following manner by using a pictograph: 

Here one represents 5 apples. 

Example 2: 

The data shows the number of ice creams sold by shops A, B and C. 

We can represent this data in the following manner by using a pictograph: 

Here, one represents 6 ice creams. 

Example 3: 

We can represent this data in the following manner by using a pictograph: 

Here, represents 1000 cars.

Pictographs and problem sums 

Pictographs can also be a part of certain problem sums.  

Example 4: 

We have data on the flavour of ice-creams sold by Shop ABC in one day. 

Let us make a pictograph to understand this situation and answer the questions beneath: 

Let us try to interpret the information given in the data by answering some questions: 

1. Which is the best-selling flavour of ice cream at Shop ABC? 

Ans) Chocolate is the best-selling flavour of ice cream at Shop ABC.

2. Which is the least-selling flavour of ice cream at Shop ABC? 

Ans) Strawberry is the least-selling flavour of ice cream at Shop ABC.

3. What is the total number of ice creams sold in a day? 

Ans) the total number of ice creams sold is 33.

Different methods of pictorial representation

There are various other methods for representing data through pictorial ways. They are bar graphs, line charts or line graphs and pie charts.

1. Bar Graph 

A bar graph is a very simple graph which is used to compare different entities.  It is very easy to show the changes in some properties of a subject over time using a bar graph. They are widely used in the industry today for presentations and reports. They allow identifying different trends and patterns of the data from their bar graphs.

Let us take an example to understand the bar graph: 

This data can be plotted in a bar graph as follows: 

2. Line Graph 

It is also known as a line chart and is used to understand the change in the value of something over a period of time. The data is plotted a
s data points in the (x, y) format. It has two axes. The horizontal axis and the vertical axis. It is a straight line connected by data points. It is commonly used to make forecasts and predictions in the industry today. 

This is an example of a line chart or a line graph

3. Pie chart 

A pie chart is a circular representation of data depicting a pie. It is divided into slices to represent the numbers and their proportions in the whole data. Generally, they are used for representing percentage data. They can summarize a large amount of data and depict it in a visual manner easily.

Example: It is a depiction of the number of animals of each type on a farm.

Do You Know

1. Pictographs were used by many ancient cultures for their writing systems. As a matter of fact, many languages have a direct line of descent from pictographs. 

2. Pictographs are widely used as ideograms and memory aids. 

The numbers that are expressed in the form of a+ib where ‘i’ is an imaginary number called iota and has the value of (√-1) are known as complex numbers. Let’s take, for example, 2 + 3i is a complex number, where 2 is known to be a real number and 3i is an imaginary number. Therefore, the combination of both the real number and the imaginary number is known as a complex number.

Examples of Complex Numbers and Real Numbers

Examples of real numbers – 2, -13, 0.89,√5, etc.

Examples of imaginary numbers are -4i, 1.2i, (√2)i, 3i/2, etc.

An imaginary number is usually represented by ‘i’ or ‘j’, which is equal to √-1. Therefore, the square of an imaginary number gives a negative value.

Complex numbers are used to represent periodic motions such as, alternating current, water waves, light waves, etc., which rely on the cosine or sine waves, etc.

Polar Form of a Complex Number

We can also represent any given complex number in its polar form. The form z equals a + ib is called the rectangular coordinate form of a complex number.

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The horizontal axis is said to be the real axis whereas the vertical axis is said to be the imaginary axis. We find the real components and complex components in terms of r and θ.

where r = length of the vector, θ = angle made with the real axis.

What is the Polar Form of any Complex Number?

The polar form of a complex number is one way to represent a complex number apart from the rectangular form. Usually, complex numbers can be represented, in the form of z equals x + iy where ‘i’  equals the imaginary number.

But in polar form, we represent complex numbers as the combination of modulus and argument.

What is Absolute Value?

The modulus of a complex number is also known as the absolute value. This polar form can be represented with the help of polar coordinates of real as well as imaginary numbers in the coordinate system.

Polar Form Formula of Complex Numbers

Let us consider the coordinates (x, y) as the coordinates of complex numbers x+iy. We  can represent it in a cartesian plane, as given below:

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Here in the above diagram, the horizontal axis denotes the real axis, and the vertical axis denotes the imaginary axis. The real components and complex components of coordinates are found in terms of r and θ where r is the length of the vector, and θ is the angle made with the real axis.

Using the Pythagorean Theorem, we can write;

r² equals x² + y²

From trigonometric ratios, we know that;

Cos θ equals Adjacent side of the angle θ / Hypotenuse, can be written as Cos θ = x/r

Also, sin θ = Opposite side of the angle θ /Hypotenuse, can be written as Sin θ =y/r

Multiplying each side by r :

r cosθ = x and r sinθ = y

The rectangular form of any given complex number can be denoted by:

z equals x + iy

Now, substitute the values of variables x and y.

z equals x + iy = r (cosθ + i rsinθ)

In the case of any given complex number,

r signifies the absolute value or the modulus,  angle θ as the argument of the complex number.

Absolute Value of a Complex Number

Given, z equals x + yi, a complex number, the absolute value of z can be defined as |z|= x² + y².

It is known to be the distance from the origin to the point (x, y).

Notice that the absolute value of a real number gives us the distance of the number from 0 which is the origin, while the absolute value of a complex number gives the distance of the number from the origin, with coordinates (0, 0).

Adding Complex Numbers in Polar Form

Suppose we have any two given two complex numbers, one in a rectangular form and one in polar form. Now, we need to add these two numbers and represent them in the polar form again.

Let 7∠50°, 3 + 5i are the two complex numbers.

First, we will convert 7∠50° into a rectangular form.

7∠50° equals x + iy

Hence, x = 7 cos 50° = 4.5

y = 7 sin 50° equals 5.36

So, 7∠50° equals 4.5 + i 5.36

Therefore, when we add any two given complex numbers, we get;

(3 + i5)+ (4.5 + i 5.36) = 7.5 + I 10.36

Again, to convert the resulting complex number in polar form, we need to find the modulus as well as the argument of the number. Hence,

Modulus is equal to;

r equals |z|=√(x² + y²)

r equals √(7.52 + 10.362)

r equals 12.79

And the argument is equal to;

θ = tan^-1(y/x)

θ = tan^-1(10.36/7.5)

θ = 54.1°

Therefore, the required complex number is 12.79∠54.1°.

Solved Questions

Question 1) Add the complex numbers (5 + 7i) + (2.0 + 2.36i).

Solution) On adding the two complex numbers (5 + 2.0) + (7 + 2.36)i

7.0 + 9.36i.

Question 2) Add the complex numbers (2 + 5i) + (2 + 5i).

Solution) On adding the two complex numbers (2 + 2) + (5 + 5)i

4.0 + 10i.

The power of ten in mathematics is defined as any of the integer powers of a number multiplied by ten. To put it another way, we add ten to itself, a specific number of times (when the power is a positive integer). Also, the number 1 is a power of ten (the zeroth power) in its definition. In this article, you will understand the power of ten, its facts, converting numbers in the power of ten, & Scientific Notation Regarding the Power of 10. So, let us start by understanding the power of ten in the coming section.

Power of Ten

In Mathematics, the power of 10 is any whole-valued (integer) power of the number 10. In other words, the power of 10 states that the 10 multiplied to itself n number of times (when the power is any positive integer). Hence, the 10 power in long-form is the number 1 followed by n zeroes where n is the number that is greater than 0. For example, 10⁷ is written as 1,00,00,000.  

When n is the number that is smaller than 0, the 10 power is found by multiplying the base value  10 ‘n’ times in the denominator and placing 1 in the numerator. For example, 10⁻³ is written as

[frac{1}{10*10*10}] = 0.001

When n is equal to 0, the power of 10 is equal to 1. For example, 10⁰ = 1.

Read below to know the power of 10 Maths in detail.

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How to Convert Numbers in Power of 10 Maths?

To convert any number into the power of ten Math, two basic rules are followed.

1. If the number is given in the decimal notation, move the decimal point to the right side of its original position and place the decimal point after the first non zero digits. The power ten will be the number of places the original decimal point was moved and it will be negative as it was moved towards the right side.

Example:  0.0000732 = 7.32 x 10⁻⁵

2. If the whole number greater than 10 is to be changed into the power of 10 Math, then move the decimal point to the left side of its original position and place the decimal point after the first digit. The power of 10 will be the number of places the original decimal point was moved and it will be positive as it was moved towards the left side.

Example: 145,000 = 1.45 x 10⁵

Multiplying and Dividing By Positive Power of 10 Maths

1. When multiplying the number by the power of 10, we move the decimal points to the right side for each power of 10.

Example:

62.54 x 10¹ = 625.4

Here, the decimal point is shifted by one place to the right side.

62.54 x 10² = 6254.

Here, the decimal point is shifted by one place to the right side.

2. When dividing the number by the power of 10, we move the decimal points to the left side for each power of 10.

62.54 ÷ 10¹ = 6.254

Here, the decimal point is shifted one place to the left side.

62.54 ÷ 10² = 0. 6254.

Here, the decimal point is shifted two places to the left side.

Multiplying By the Negative Power of 10

Negative power tells how many times to divide the base number. When multiplying the number by the negative power of 10, we move the decimal points to the left side for each power of 10.

Example: 

6 x 10⁻³ = 6 x 1/10 x 1/10 x 1/10 = 6/1000 = 6 x 0.001 =  0.006

6.1 x 10⁻³ = 6.1 x 1/10 x 1/10 x 1/10 = 6.1/1000 = 6.1 x 0.001 = 0.0061

Scientific Notation Regarding Power of 10 Maths

The scientific notation, also known as the standard form, was given its name because it was first used by scientists to represent extremely small and large numbers. Exponents refer to the power of 10 multiplied by another number. Moreover, we can find them in both positive and negative forms.

Additionally, the positive form denotes multiplication, while the negative form represents division. The index of ten indicates how many places the decimal points should be moved to the right in the notation. 

In scientific notation, the numbers are represented in the form of a x 10ⁿ, where the variable a is the decimal with 1 ≤ a < 10ⁿ, and n is the integer.

To understand this, consider multiplying 1.35 by 10 to the fourth power. Alternatively, 1.35×10⁴.

You can then calculate it by 1.35 x  (10 x 10 x 10 x 10), or 1.35 x 10,000, to get the answer 13,500. Now, if we shift the decimal place to 1.35 over four places, we get 13,500.

Example:

Avogadro’s number in scientific notation is approximately written as 6.022141793 x 10²³. Here a is the decimal 6.022141793 and n is the exponent 23. 

Facts to Remember

• A power 10 with a positive exponent such that 10⁴, means that the decimal point is shifted towards the left.

• A power 10 with a negative exponent such that 10⁴, means that the decimal point is shifted towards the right.

Solved Example

1. What is 2.35 x 10⁴?

Solution:

2.35 x 10⁴ can be calculated as 2.35 x (10 x 10 x 10 x 10) = 2.35 x 10000 

When multiplying the number by the power of 10, we move the decimal points to the right side for each power of 10

Accordingly, 

2.35 x 10000 = 2,35,000

2. How Do You Write 0.0002 in Scientific Notation?

Solution:

According to the rule, to convert 0.0002 in scientific notation,  we will move the decimal point to the right side of its original position and place the decimal point after the first non-zero digit. The power ten will be the number of places that will be negative as it was moved towards the right side.

Therefore, the scientific notation for 0.0002 is 2 x 10⁻⁴

3. Can You Help Sam to Write 9.56 x 10¹¹ in Standard Notation?

Solution:

Here 9.56 is 956. Now, Sam will move the decimal point 11 places to the right side and add trailing zeros accordingly. 

Therefore, the standard notation for  9.56 x 10¹¹ is 956,000,000, 000.

4. What is the notation form of 3,00,00,00,000 or 300 crores?

a. 3 ×10⁹

b. 3 × 10⁸

c. 3 × 10¹⁰

d. 3 × 10¹¹

Solution: The answer is option a- 3 ×10⁹  

Because, 3 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 = 3,00,00,00,000.

5. How to express 10 to the power of 10?

Solution: To find 10 to the power of 10, we can write it in the exponent form as 1010, where 10 is the base and 10 is the power as well.

It means 10 is multiplied 10 times. 

So, 1010 = 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 = 10,000,000,000.

Hence, 10 to the power of 10 ca
n be expressed as 1010 = 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 = 10,000,000,000.

Conclusion

The idea of continuous and comprehensive growth is mighty simple, better learning. What helps in that, you may ask? Practising mathematical sums regularly helps remember facts and inculcate a better learning habit.

PMF or probability mass function is a simple concept in mathematics. It is a part of statistics. When you are learning about pmf you will find it very interesting and informative. It’s an informative and useful concept. Probability Mass Function is otherwise referred to as Probability Function or frequency function. PMF characterizes the distribution of a discrete variable which is unplanned or random. Example of a discrete random variable:

Let Y be the random variable of a function, and this is its probability mass function:

Py (y) = P (Y-y), for all y belongs to the range of Y.

Here are two conditions on which the probability function should fall upon:

• P y (y) ≥ 0

• ∑ yϵRange (y) P y (y) = 1

Probability Mass Function Definition

The definition of Probability Mass Function is that it’s all the values of R, where it takes into argument any real number. There are two times when the cost doesn’t belong to Y. First, when the case is equal to zero. The second time is when the value is negative, the value of the probability function is always positive.

Another name of PMF is the Probability Discrete Function (PDF). It’s given because when you are drawing the variable, it produces distinct outcomes or results. Two places where the discrete probability function is used is computer programming and statistical modelling.

 Probability Mass Function

The simple meaning of Probability Mass Function is the function relating to the probability of those events taking place or occurring. The word ”mass” is used to denote the expectations of discrete events.

Finding The Probability Mass Function 

It’s effortless to find the PMF for a variable. Given below are the steps that you need to follow to find the PMF of a variable:

Step 1: Start solving the question by fulfilling the first condition of the PMF. ( mentioned above)

Step 2: Take all the values of P ( X- x) and add it up. There will be a whole number ( 0, 1, 2), numbers with variables ( 1y, 2y 3y) and numbers which are squared ( 2 y2, 3 y2 ).

Step 3: Start using simultaneous equations to solve the sum.

Step 4: As you start using simultaneous equations, you will get two answers in the end. 

Step 5: You need to check which of the answers fulfils these two conditions:

(i) The value of the variable is never negative. 

(ii) The amount of the variable does not equal zero.

Step 6: The answer to the question is the one that follows both the conditions which are mentioned above. 

Probability Mass Function Applications

There are many places where the probability mass function is used and applied. Here are some of the places where there’s an application of PMF:

• One of the sections where PMF is used is statistics. It plays a vital and essential role in the study of statistics. Probability Function shows the various probabilities of the discrete variable data.

• PMF combines the variable for the random number that is identical or equal to the expectation for the random variable.

• Many people use PMF to calculate two main concepts in statistics- mean and discrete distribution.

• Another place where PMF is binomial and Poisson distribution is to find the value of the variables which are distinct and random. 

There are mainly two differences between PDF and PMF. Here are the two dissimilarities between them:

Distinction Between PDF and PMF:

Probability Mass Function Solved Example

Here is a probability mass function example which will help you get a better understanding of the concept of how to find probability mass function. 

Solved Example 1:

Let X be a random variable, and P (X=x) is the PMF given below;

1. Find the value of k.

2. Find the pmf probability of 

(i) P (X ≤ 6 )

Solution:

Given: ∑P(xi)=1

1. So, 

0+k+2k+2k+3k+ k2+ 2 k2+k  = 1 

9k 10 k2 = 1  

9k 10 k2 – 1 = 0

10k 2 + 10k – k – 1 = 0

10 k ( k + 1) – 1 (k – 1) = 0

(10k – 1 ) ( k +1 ) = 0

Therefore, 

10k – 1 = 0 and k + 1 = 0

Hence, 

k= 1/ 10, and k = -1 

K= -1 is not the desired answer because the pmf probability value lies between 0 and 1 ( it’s one of the conditions which is mentioned  above) 

Conclusion: The value of k is 1/10.

2.  (i)  P (X ≤ 6 )

So, 

P (X ≤ 6 ) = 1 – P ( x > 6)

= 1 – 7 k2 + k

= 1 – ( 7 ( 1/10) 2 + (1 / 10)) 

= 1 – ( 7/ 100 + 1 / 10) 

=  1 – ( 17/ 100)

= ( 100 – 17 )/ 100

= 83/ 100

Conclusion : P (X ≤ 6 ) = 83/ 100.