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 of Rational Number
Let x and y are two rational number and y > x, then the rational number between x and y is\[\frac{1}{2}\left( x+y \right).\]
e.g., find 2 rational number between \[\frac{1}{3}\]and \[\frac{1}{2}\]
Solution:- Let \[x=\frac{1}{3}\] and \[y=\frac{1}{3}\] and y > x.
Then, Rational no. between\[\frac{1}{3}\]and\[\frac{1}{2}\]is
\[\frac{1}{2}\left( \frac{1}{3}+\frac{1}{2} \right)=\frac{1}{2}\left( \frac{2+3}{6} \right)=\frac{5}{12}\]
Again Let \[x=\frac{5}{12}\] and \[y=\frac{1}{2}\] and y > x. then
Rational no. between \[\frac{5}{12}\] and \[\frac{1}{2}\] is
\[\frac{1}{2}\left( \frac{5}{12}+\frac{1}{2} \right)=\frac{1}{2}\left( \frac{5+6}{12} \right)=\frac{1}{2}\times \frac{11}{12}=\frac{11}{24}\]
Hence the Rational Numbers between \[\frac{1}{3}\] and \[\frac{1}{2}\] are \[\frac{5}{12}\] and \[\frac{11}{24}\].
Let x and y are two rational number and y > x. Consider to find n rational numbers between x and y. Let d = \[\frac{y-x}{n+1}\]
Then 'n' rational number lying between x and y are \[\left( x+d \right),\left( x+2d \right),\left( x+3d \right),\_\_\_\left( x+nd \right).\]
Example:- Find 9 rational number between 2 and 3.
Solution:- Let x = 2 and y = 3 then y > x
Now \[\mathbf{d}=\frac{y-x}{n+1}=\frac{3-2}{9+1}=\frac{1}{10}\]
Then, rational number are, 2 + 0.1, 2 + 0.2, 2 + 0.3, 2 + 0.4, 2 + 0.5, 2 + 0.6, 2 + 0.7, 2 + 0.8, 2 + 0.9 = 2.1, 2.2, 2.3, 2.4, 2.5, 2.6, 2.7, 2.8 and 2.9.
Representation of Rational Number on the Number line
To represent - on the number line first we draw a number line
Let O represent 0 (zero) and A represent 1. So divide OA into 4 equal parts, each point in the middle representing P, Q and R. Point R represent\[\frac{3}{4}\].
Operations on Rational Numbers
Addition of Rational Numbers:
Example: Find the sum of the rational numbers \[\frac{-4}{9},\frac{15}{12}\] and \[\frac{-7}{18}\].
Solution: \[\frac{-4}{9}+\frac{15}{12}+\frac{-7}{18}=\frac{-16+45-14}{36}=\frac{15}{36}=\frac{5}{12}\]
Properties of Addition of Rational Number
Closure Property:- If a and b are two rational numbers, then a + b is always a rational number.
E.g., Let \[a=3\], \[b=-2,\] then \[a+b=3+\left( -2 \right)-1\]
Commutative Property:- If a and b are two rational number then a + b = b + a.
E.g., Let \[a=\frac{1}{2}\]and \[b=\frac{1}{3}\] then more...
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 \[{{3}^{-3}}=\frac{1}{27}\]
\[\Rightarrow {{\log }_{3}}\frac{1}{27}=-3\]
(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
It has 3 faces \[\to \] 1 curved face and 2 flat faces.
(g) Cone
It has 2 faces \[\to \] 1 curved face and 1 flat face.
It has 1 curved edge.
Two- dimensional shapes have only length and breadth.
Three dimensional shapes have length, breadth and height or depth.
Three-dimensional (or 3-D) shapes can be visualized on a two dimensional (or 2-D) surface.
A net is a skeleton-outline in 2-D, which when folded results in a 3-D shape. The same solid can have several types of nets.
Dice are cubes with dots on each face: Opposite faces of a die always have a total of seven dots on them. Generally, dice have number, 1 to 6 on their faces.
A solid can be sketched in two ways.
(a) An oblique sketch which does not have proportional faces but gives a realistic feel of the 3-D solid.
e.g., Oblique sketch of a cuboid.
(b) An isometric sketch, drawn on an isometric dot paper, which has proportional measurements of the solid.
Different sections of a solid can be viewed in many ways:
(a) Slicing the shape results in the cross - section of the solid.
(b) Observing a 2 - D shadow of a 3 - D shape.
(c) Looking at the shape from different angles, i.e., the front-view, the side-view and the top-view.
Front and top view of a cylinder
Trigonometry is the study of relationship between the sides and angles of a triangle.
Trigonometrical ratio
Trigonometric ratio of angle in a right angled AABC are defined as follows:
\[\sin \theta =\frac{AB}{AC}=\frac{P}{h}\]
\[Cos\theta =\frac{AB}{AC}=\frac{b}{h}\]
\[\tan \theta =\frac{AB}{AC}=\frac{p}{b}\]
The ratio \[\text{cosec}\theta ,\,\text{sec}\theta \] and \[\cot \theta \] are respectively the reciprocals of the \[sin\theta ,cos\theta \]and \[tan\theta .\]
i.e., \[\text{sin}\,\theta =\frac{1}{\text{cosec}\,\theta },\text{cos}\theta =\frac{1}{\sec \theta }\text{and}\,\,\text{tan}\,\theta =\frac{1}{\cot \theta }\]
Trigonometric ratio of some specific angles
Cuboid:- A cuboid is a solid bounded by the rectangular plane regions. A cuboid has six faces, 12 edges and 8 vertices.
Total surface Area of the cuboid \[=2\left( lb+bh+hl \right)\] sq. units.
Volume of the cuboid \[={{l}^{2}}\times b\times h\]
Diagonal of the cuboid \[=\sqrt{{{l}^{2}}+{{b}^{2}}+{{h}^{2}}}\]
Cube:- A cuboid whose length, breadth and height are equal is called a cube.
If length of each edge of a cube is a.
Then, volume of the cube \[={{a}^{3}}\]
Total surface area of the cube \[=6{{a}^{2}}\]
Diagonal of the cube\[=\sqrt{3a}.\]
Cylinder:- It is formed by rotating one side of a rectangle about its opposite side.
Volume of the cylinder \[=\pi {{r}^{2}}h\]
Area of the base \[=\pi {{r}^{2}}\]
Area of the curved surface \[=2\pi rh\]
Total surface Area
\[=27\pi rh+2{{\pi }^{2}}h=2\pi r(h+r)\]
Right Circular Cone:- A right circular cone is a solid generated by rotating a right angled triangle around its height.
Radius = r, Height = h
Slant height = l
Volume of the cone \[\frac{1}{3}=\pi {{r}^{2}}h\]
Area of the Base \[=\pi {{r}^{2}}\]
Area of the curved surface \[=\pi r\sqrt{{{h}^{2}}+{{r}^{2}}}=\pi rl\]
Sphere:- The set of all points in the space which are equidistant from fixed point is called a sphere.
Radius = r
Volume of a sphere\[~=\frac{4}{3}\pi {{r}^{3}}\]
Surface Area of a sphere \[=4\pi {{r}^{2}}\]
Hemisphere:- A plane through the centre of the sphere divides the sphere into two equal parts each of which is called a hemisphere.
Radius \[=Ox=r\]
Volume of a Hemisphere\[\frac{2}{3}\pi r3\]
Curved surface area of a Hemisphere \[=27\pi {{r}^{2}}\]
Total surface area of a Hemisphere \[=37\pi {{r}^{2}}\]
The first power of a number is the number itself. i.e., \[{{\mathbf{a}}^{\mathbf{1}}}=\mathbf{a}\]
The second power is called square. E.g., square of ‘5’ is \[{{5}^{2}}\]
The third power is called cube. E.g., cube of y is \[{{y}^{3}}\]
Number in Standard form:-
A number written as \[(x\times {{10}^{y}})\] is said to be in standard form if x is a decimal number such that \[1\le x\le 10\] and y is either a positive or a negative integer.
Expressing large numbers in standard form
E.g., Express 246784 in standard form
Solution:- \[246784=2.46784\times 100000\]
\[=\left( 2.4678\times {{10}^{5}} \right)\]
Expressing very small numbers is standard form
E.g., Express 0.00003 in standard form.
Direct & Inverse Proportional (Time & Work)
FUNDAMENTALS
Let 3 pens cost Rs. 9, then 6 pens will cost Rs. is
Clearly. More pens will cost more.
Again, if 2 women can do a piece of work in 7 hours, then 1 woman alone can do it in 14 hours.
Thus, less people at work, more will be the time taken to finish it.
Thus, change in one quantity brings a change in the other.
Variation: If two quantities depend upon each other in a way such that the change in one results in a corresponding change in the other, then the two quantities are said to be in variation. This variation may be direct (i.e. increase in one quantity leads to increase in other quantity) as illustrated in the example of "cost of pens" above. Variation may also be indirect (i.e. increase in one quantity leads to decrease in other quantity) as illustrated in the example of "work done by women" above.
There are many situations in our daily life where the variation in one quantity brings a variation in the other.
ILLUSTRATIONS:
More no. of articles will cost more (Direct Variation)
More is the number of workers at a work, less is the time taken to complete the work. (Indirect Variation)
For a given amount of money deposited in a bank, more is the rate of interest, more is the interest earned upon it in a fixed time period. (Direct Variation)
More is the distance covered by train, more will be electricity consumed by it. (Direct Variation)
Faster is the speed of train, lesser will be the time taken to cover a given distance. (Indirect Variation)
Direct Proportionality: Two quantities x and y are said to be in direct proportion if increase / decrease in the value of one variable x, leads to increase / decrease in the value of y, in such a way that the ratio \[\frac{x}{y}\] remains constant.
Hence, if x and y are directly proportional, then\[\frac{x}{y}=k\], where k is a constant. As x takes the values \[({{x}_{1}}={{x}_{2}}={{x}_{3}})\]and y. takes the values. \[({{y}_{1}},{{y}_{2}},{{y}_{3}})\]then,
\[\frac{{{x}_{1}}}{{{y}_{1}}}=\frac{{{x}_{2}}}{{{y}_{2}}}=\frac{{{x}_{3}}}{{{y}_{3}}}=....=K\]
Examples (1), (3) and (4) given above are the cases of direct proportion.
Consider a gas in a closed cylinder such that its volume (V) is kept constant. If you increase the temperature (T) of the gas, then its Pressure (P), which can be measured through a manometer, also increases.
If V= constant; \[\frac{P}{T}=k;\]
(in the example above, just try to understand \[\frac{{{x}_{1}}}{{{y}_{1}}}=\frac{{{x}_{2}}}{{{y}_{2}}}=\frac{{{x}_{3}}}{{{y}_{3}}}=....=k\] where the analogy for x is Pressure (P) and the analogy for y is temperature (T).
See the table on the right side. Column II represent temperature (T) whereas Column III represents pressure (P).
Divide each data in Column III by each data in Column II.
FACTORIZATION
FUNDAMENTALS
FACTORS: When an algebraic quantity can be expressed as the product of two or more algebraic quantities, then each of these quantities is called a factor of the given algebraic quantity and the process of finding factors, is called FACTORIZATION.
Remarks: Factorization is the opposite process of multiplication,
EXAMPLE Look at the examples given below:
It is advisable that students memorize squares of numbers from 1 to 20. E.g. here, \[\mathbf{1}{{\mathbf{5}}^{\mathbf{2}}}=\mathbf{225}\] and \[\mathbf{1}{{\mathbf{7}}^{\mathbf{2}}}=\mathbf{289}\] are readily used.
Factorization when a Common Monomial Factor Occurs in Each Term.
METHOD: Step 1. Find the HCF of all the terms.
Step2. Divide each term by this HCF.
Step3. Write the given expression= HCF \[\times \] quotient obtained in step 2.
Conceptual Framework / Idea behind above steps: HCF itself is one of the factors. Hence,
\[\text{other}\,\text{factor}\,\text{will}\,\text{be}\,\text{equal}\,\text{to}\frac{\text{Given}\,\text{pression}}{\text{HCF}}\]
EXAMPLE 1. Factorize (i.e. break into factors) each of the following:
(1) \[13n+117\]
(2) \[{{n}^{3}}+2n+{{n}^{2}}\]
(3) \[15{{x}^{2}}{{y}^{2}}{{z}^{2}}+5x{{y}^{2}}z+5xyz\]
(4) \[6ab-9bc\]
(1) \[13n+117=13\left( n+9 \right)\]
(2) \[{{n}^{3}}+2n+{{n}^{2}}=n({{n}^{2}}+2+n)\]
(3) \[15{{x}^{2}}{{y}^{2}}{{z}^{2}}+5x{{y}^{2}}z+5xyz=5xyz(3xyz+y+1)\]
Factorization when one or more Binomial is Common
METHOD: Step 1. Find the common binomial by intelligent thinking or by trial & error.
Step 2. Divide each term by this common binomial.
Step 3. Write the given expression = this binomial \[\times \] quotient obtained in Step 2
EXAMPLES. Factorize:
(1) \[6x\left( 3a-4b \right)+10y\left( 3a-4b \right)\]
(2) \[6\left( 16x-23y \right)-22{{\left( 16x-23y \right)}^{2}}\]
(3) \[mn{{\left( ax-2by \right)}^{2}}+m{{n}^{2}}(ax-2by)\]
We have,
(1) \[6x\left( 3a-4b \right)+10y\left( 3a-4b \right)\]
\[=\left( 6x+10y \right)\left( 3a-4b \right)\]
(2) \[6\left( 16x-23y \right)-22{{\left( 16x-23y \right)}^{2}}\]
\[=\left( 16x-23y \right)\left[ 6-22\left( 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, classification, presentation, analysis of data and inferences from data. Data as can be qualitative or quantitative in nature. If one speaks of honesty- beauty, colour, etc., the data is qualitative, while height, weight, distance, marks, etc., are quantitative. Data can also be classified as: raw data, and grouped data.
Raw Data
Data obtained from direct observation is called raw data.
The marks obtained by 100 students in a monthly test are an example of raw data or ungrouped data.
In fact, little can be inferred from this data. However, arranging the marks in ascending order in the above example is a step towards making raw data more meaningful.
Grouped Data
To present the data in a more meaningful way, we condense the data into convenient number of classes or groups, generally not exceeding 10 and not less than 5. This helps us in perceiving at a glance, certain salient features of data.
Some Basic Definitions
Before getting into the details of tabular representation of data, let us review some basic definitions:
Observation: Each numerical figure in a data is called an observation.
Frequency: The number of times a particular observation occurs is called its frequency.
Tabulation or Presentation of Data
A systematical arrangement of the data in a tabular form is called 'tabulation' or 'presentation' of the data. This grouping results in a table called ''frequency table' that indicates the number of scores within each group.
Many conclusion about the characteristics of the data, the behavior of variable, etc., can be drawn from this table.
The quantitative data that is to be analyzed statistically can be divided into three categories:
Individual Series: Any raw data that is collected forms an individual series.
Example:
The weights of 10 students.
36, 35,32,40,65,48,54,71, 62 and 33
Percentage of marks scored by 10 students in a test:
46, 66, 96, 99, 90, 36, 81, 73, 59, 48
Discrete Series: A discrete series is formulated from raw data. Here, the frequency of the observations is taken into consideration.
Example: Given below is the data showing the more...
\[=90+90\times \frac{10}{100}=90+9=99\]
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
\[=\left\{ \frac{100}{(100+gain%)}\times SP \right\}=\left\{ \frac{100}{(100+10)}\times 1980 \right\}\] \[=Rs\left( \frac{100}{110}\times 1980 \right)=Rs1800/-\]
SP of the second watch = Rs 1980/-; Loss% = 10%; CP of the second watch
Total CP of the two watches
\[=Rs\left( 1800+2200 \right)=Rs4000/-\]
\[\therefore \]Total SP of the two watches
\[=Rs\left( 1980\times 20 \right)=Rs\,3960/-\]
Since \[\left( SP \right)<\left( CP \right),\] there is less in the whole transaction.
Loss \[=Rs\left( 3960-4000 \right)=Rs40/-;\]
\[~\therefore Loss%=\frac{40}{4000}\times 100%=1%\,loss\]
Hence, Gurpreet loses 1 % in the whole transaction.
Further concepts on percentage:
A number can be split into two parts such that one part is P% of the other. Then the two parts are:
If the circumference of a circle is increased (or) decreased by P%, then the radius of a circle increases (or) decreases by P%.
Elementary question 1: The circumference of a circle is 88 cm, if the circumference is increased by 50%, find percentage increase in radius.
Answer: \[{{C}_{1}}=88\,cm\,{{C}_{2}}={{C}_{1}}+\frac{50}{100}\times {{C}_{1}}=88+44=132\] \[{{r}_{1}}=\frac{88}{2\pi }=14\,\,and\,\,{{r}_{2}}=\frac{132}{2\pi }=21\]
\[\therefore \]% age increase in radius
\[\frac{{{r}_{2}}-{{r}_{1}}}{{{r}_{1}}}\times 100%=\frac{21-14}{14}\times 100%=50%\]
OVERHEADS: Suppose you go to market and purchase a bed set and a TV. Naturally, you need to pay transportation charges and also labour cost and sometimes repairing charges, if there is damage. These extra expenses are called overheads. For calculating the total cost price, we add overheads to the purchase price.
Elementary question 2: Mohan Kumar purchased and old bike for Rs.20000 and spent Rs. 4000 on its overhauling. Then, he sold it to his friend Jatin for Rs. 21000. Find his loss/gain percentage.
Solution: Cost price = Rs. 20000, overheads = Rs. 21000;
Total cost price
\[=Rs\left( 20000+4000 \right)=24000;\]
Selling price = Rs. 21000.
Since\[\left( SP \right)<\left( CP \right)\], Mohan Kumar makes a loss.
\[\text{Loss}=Rs(24000-21000)=Rs.3000;\]
\[\text{Loss}%=\frac{3000}{24000}\times 100%=12\frac{1}{2}%\]
Elementary question 3: By selling 36 m of cloth, a cloth-maker loses an amount equal to the selling price of 4m of cloth. Find her gain or loss percent.
Solution: Let C. P. of 1 m cloth \[=x\Rightarrow CP\] of 36 m cloth = 36x
Let her sell at p% loss \[\Rightarrow SP=36x\left( 1-\frac{p}{1000} \right)\]
\[\therefore Loss=\frac{36xp}{100}\]But this equal S. P. of 4m cloth
\[\Rightarrow more...