8th Class

  Mensuration   His chapter deals with the concept of finding the surface area and volume of the regular figures. By regular figures we mean to say the figures whose parameters are known to us. Previously we have learnt to find the perimeter and area of the rectilinear figures, but now onwards we will learn to find the area of some polygons and also discuss the surface area and volume of some solid shapes namely cuboid cube, cone and cylinder.                                                  Area of a Polygon                                           Area of a given polygon can be found by dividing the given polygon into non-over lapping rectilinear figures. The area of the polygon will be equal to the sum of the areas of non-overlapping figures.  

• Example:

Find the area of the polygon given below if AP = 10 cm, BP = 20 cm, CP = 50 more...

  Data Handling   Statistics is the formal science of making effective use of numerical data relating to group of individuals or experiments. It deals with all aspects, including the collection, analysis and interpretation of data and also the planning of the collection of data in terms of the design of surveys and experiments. A statistician is someone who is particularly versed in the ways of thinking necessary for the successful application of statistical analysis. Often such people have gained this experience after starting work in number of fields. This is also a discipline called Mathematical Statistics, which is concerned with the theoretical basis of the subject.   Types of Data The data may be in the form of raw or grouped. The data which is not arranged in any form is known as the raw data and data which is arranged in a definite pattern more...

  Crop Production and Microorganisms   Cultivation Cultivation is the act of raising plants.     Land preparation and cultivation methodology is among the basic factors which affect the yield of crops. For getting better yield, it is important to prepare land thoroughly, so that the weeds are destroyed and water absorption capacity of the soil is increased.   Crops When the same kind of plants are grown in the fields on a large scale to obtain foods like cereals, pulses, vegetables and fruits, etc., it is called a crop.   Agriculture The growing of plants (or crops) in the fields for obtaining food is called agriculture. Crop production involves various agricultural practices such as:

• Preparation of soil- It includes tilling, levelling and maturing of the fields.

 

• Sowing - good quality seeds are sown more...

  Material   Material is a physical or chemical substance of which things can be made.   Fibres A fibre is a piece of fabric that is long, thin and flexible. Plant fibres are the basis of fabric such as cotton, silk and wool fibres come from animals. Many artificial fibres have also been invented such as rayon, nylon, polyester, acrylic etc.  

• Nylon - is entirely made of chemicals. It is very strong, elastic, light and water-resistant fibre. It is lustrous in appearance. It is used in making ropes, tents, fishing nets and parachutes.

 

• Polyester - is made from 'petroleum'. It is very strong, crease resistant, light, elastic and absorbs very little water. It is used in making pants, shirts, suits, jackets, etc.

 

• Rayon - is also known as "artificial silk7. Cellulose which is obtained from wood pulp more...

DATA HANDLING   FUNDAMENTALS The word 'statistics' is derived from the Latin word 'status' which means political state. Political states had to collect information about their citizens to facilitate governance and plan for development. Then, in course of time, statistics came to mean a branch of mathematics which deals with collection, classification, presentation and analysis of numerical data.   In this chapter, we shall learn about the classification of data, i.e., grouped data and ungrouped data, measures of central tendency, and their uses.   Data The word 'data' means, information in the form of numerical figures or a set of given facts. For example, the percentage of marks scored by 10 students of a class in a test are: 36, 80, 65, 75, 94, 48, 12, 64, 88 and 98.   Statistics is basically the study of numerical data. It includes methods of collection, more...

PERCENTAGE, PROFIT & LOSS   FUNDAMENTALS

• Gain = Selling price (S.P.) - Cost Price (C.P.)
• Loss \[=C.P.-S.\text{ }P.\]
• \[Gain%\text{ =}\frac{Gain}{C.P.}\times 100%;\text{ }S.P.\]

=\[C.P.+Gain=C.P.+C.P.\times \frac{Gain(in%)}{100}\] \[=C.P.\left[ 1+\frac{gain(in%)}{100} \right]\].

• In case of loss, \[S.\text{ }P.=C.P.\left[ 1-\frac{loss%}{100} \right]\]
• \[CP=\left( \frac{100}{100+gain%} \right)\times S.P.=\left( \frac{100}{100-loss%} \right)\times S.P.\]

Elementary Question:

Price of a book was decreased by 10% and then increased by 10%. If the original price of book is Rs. 100, what is its current price?

Decreased by 10%

Ans.     \[Rs.100\xrightarrow{decreased\,by\,10%}10%\,Rs.100\,means\] \[100-100\times \frac{10}{100}=100-10=90\]

\[Increased\,by\,10%\,Rs.90\xrightarrow{Increased\,by\,10%}\]

\[=90+90\times \frac{10}{100}=90+9=99\] Example: Gurpreet sells two watches forRs.1980/- each, gaining 10% on one and losing 10% on the other. Find her gain or loss per cent in the whole transaction.             Solution: SP of the first watch = Rs 1980/-; Gain = 10% \[\therefore \]CP of the first watch \[=\left\{ \frac{100}{(100+gain%)}\times SP \right\}=\left\{ \frac{100}{(100+10)}\times 1980 \right\}\] \[=Rs\left( \frac{100}{110}\times 1980 \right)=Rs1800/-\] more...

LOGARITHMS   FUNDAMENTALS

• Logarithm:- Let a be a positive real number other than 1 and \[{{a}^{x}}=m\], then x is called the logarithm of into the base and written as \[{{\log }_{a}}m\].

Example 1:- \[{{10}^{4}}=10000\] \[\Rightarrow \text{lo}{{\text{g}}_{10}}10000=4\] Example 2:- If \[{{3}^{-3}}=\frac{1}{27}\] \[\Rightarrow {{\log }_{3}}\frac{1}{27}=-3\]

• \[(I)\,\,\text{lo}{{\text{g}}_{a}}(mn)={{\log }_{a}}m+\text{lo}{{\text{g}}_{a}}n\]
• \[(II)\,\,\text{lo}{{\text{g}}_{a}}\frac{m}{n}=\text{lo}{{\text{g}}_{a}}m-\text{lo}{{\text{g}}_{a}}n\]
• \[(III)\,\,{{\log }_{a}}a=1\]
• \[(IV)\,\,{{\log }_{a}}1=0\]
• \[(V)\,\,\text{lo}{{\text{g}}_{a}}m\,({{m}^{p}})=p(\text{lo}{{\text{g}}_{a}}m)\]
• \[(VI)\,\,\text{lo}{{\text{g}}_{a}}m=\frac{1}{{{\log }_{m}}a}\]
• \[(VII)\,\text{lo}{{\text{g}}_{a}}m=\frac{{{\log }_{b}}m}{{{\log }_{b}}a}=\frac{\log m}{\log a}\]

 

SOLID SHAPES   FUNDAMENTALS

• Description of Some basic shapes:

(a) Square It has four sides and four comers. All its sides are of the same length.   (b) Rectangle. It has four sides and four comers. The opposite sides of a rectangle are parallel and of the same length. Every interior angle is a right \[\angle \]le.   (c) Triangle It has three sides and three vertices.   (d) Cuboid It has 6 flat faces, 12 straight edges and 8 vertices.   (e) Cube It has 6 flat faces, 8 vertices and 12 straight edges.   (f) Cylinder more...

NUMBER SYSTEM   FUNDAMENTALS  

• A number r is called a rational number if it can be written in the form \[\frac{p}{q}\], where p and q are integers and \[q\ne 0.\]

Example:- \[\frac{1}{2},\frac{1}{3},\frac{2}{5}\] etc.

• Representation of Rational Number as Decimals.

• Case I :- When remainder becomes zero \[\frac{1}{2}=0.5,\frac{1}{4}=0.25,\frac{1}{8}=0.125\]

It is a terminating Decimal expansion.

• Case II :- When Remainder never becomes zero.

Example:- \[\frac{1}{3}=.3333,\frac{2}{3}=.6666,\]it is a non - terminating Decimal expansion.

• There are infinitely large rational numbers between any two given rational numbers.

 

• Irrational Number:- The number which cannot be expressed in form of \[\frac{p}{q}\]and neither it is terminating nor recurring, is known as irrational number.

Examples:- \[\sqrt{2},\sqrt{3}\] etc.   Rationalization :- more...

RATIONAL NUMBERS   FUNDAMENTALS Rational Number:-

• A number which can be expressed as\[\frac{x}{y}\], where x and y are Integers and \[y\ne 0\] is called a rational number.

e.g., \[\frac{1}{2},\frac{2}{2},\frac{-1}{2},0,\frac{3}{-\,2}\] etc.

• Set of rational number is denoted by Z.
• A Rational number may be positive, zero or negative
• If \[\frac{x}{y}\] is a rational number and \[\frac{x}{y}>0\], then\[\frac{x}{y}\] is called a positive Rational Number.

e.g., \[\frac{1}{2},\frac{2}{5},\frac{-3}{-2},-\left( -\frac{1}{2} \right)\]etc.   Negative Rational Numbers:-

• If \[\frac{x}{y}\] is a rational number and \[\frac{x}{y}<0\], then \[\frac{x}{y}\]is called a Negative Rational Number.

e.g., \[\frac{-1}{2}.\frac{3}{-2},\frac{-7}{11}......\]etc.   Standard form of Rational Number:-

• A Rational number \[\frac{x}{y}\] is said to be m standard form, if x and y are integers having no common divisor other than one, where \[y\ne 0\].

            e.g., \[\frac{-1}{2},\frac{5}{6},\frac{8}{11}\]……etc. Note:- There are infinite rational numbers between any two rational numbers.   Property more...