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Simple Equations | Study
Mind Map Overal Idea Content Speed Notes Quick Coverage An equation isa condition ona variable suchthat two expressions in the variable should have equalvalue. Thevalue of thevariable for whichthe equation issatisfied is called the solution ofthe equation. An equation remains the same if the LHSand the RHSare interchanged. (Scroll down till end of the page)… readmore
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An equation isa condition ona variable suchthat two expressions in the variable should have equalvalue.
Thevalue of thevariable for whichthe equation issatisfied is called the solution ofthe equation.
An equation remains the same if the LHSand the RHSare interchanged. (Scroll down till end of the page)
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In case ofthe balanced equation, if we add the same number to both thesides, or subtract the same number from both the sides,
or
multiply both sidesby the same number, or divide both sidesby the samenumber, the balance remains un disturbed,
i.e.,the value of the LHS remains equal to the value of the RHS The above property gives a systematic method of solving an equation.
We carry out a series of identical mathematical operations on the two sides of the equation in such a waythat on oneof the sides we get justthe variable. Thelast step isthe solution of the equation.
Transposing means moving to the other side.
Transposition of a number has the same effect as adding same number to (or subtracting the same number from) both sides of the equation.
Whenyou transpose a number fromone side ofthe equation tothe other side, you change itssign.
For example, transposing +3 fromthe LHS tothe RHS in equation x + 3 = 8 gives x = 8 – 3 (= 5).
We can carry out the transposition of an expression in thesame way as the transposition of a number.
We havelearnt how to construct simple algebraic expressions corresponding to practical situations.
Wealso learnt how,using the technique of doing thesame mathematical operation (for example adding the samenumber) on bothsides, we could build an equation starting fromits solution.
Further, we also learnt that we could relate a given equation tosome appropriate problem/puzzlefrom the equation. practical situation and build a practical word.
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Advanced Tools For Study Assess Interact Acids, Bases and Salts | Assess
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COAL AND PETROLEUM | Assess
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Data Handling | Study
Mind Map Overal Idea Content Speed Notes Quick Coverage The collection, recording and presentation of data help us organiseour experiences and draw inferences from them. Before collecting data we need to know what we would use it for. The data that is collected needs to be organised in a propertable, so that it becomeseasy to… readmore
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The collection, recording and presentation of data help us organiseour experiences and draw inferences from them.
Before collecting data we need to know what we would use it for.
The data that is collected needs to be organised in a propertable, so that it becomeseasy to understand and interpret (Scroll down till end of the page)
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Average is a numberthat represents or shows the central tendencyof a group of observations or data.
Arithmetic mean is one of the representative values of data.
Mean = sum of all observations/ Number of observations.
Mode is another form of central tendency or representative value.
The mode of a set of observations is the observation that occurs most often.
If each of the value in a data is occurring one time, then all are mode.
Sometimes we also say that this data has no mode since none of them is occurring frequently.
Median is also a form of representative value.
It refers to the value which lies in the middle of the data with half of the observations above it and the other half below it.
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A bar graph is a representation of numbers using bars of uniform widths.
Double bar graphshelp to comparetwo collections of data at a glance.
Double bar graphshelp to comparetwo collections of data at a glance.
There are situations in our life, that are certain to happen, some that are impossible and some that may or may not happen.
The situation that may or may not happen has a chanceof happening.
Probability: A branch of mathematics that is capable of calculating the chance or likelihood of an event taking place (in percentage terms).
If you have 10 likelihoods and you want to calculate the probability of 1 event taking place,it is said that its probability is 1/10 or event has a 10% probability of taking place.
Events that have many possibilities can have probability between 0 and 1.
Important Formulae – Data Handling
1. A trial is anaction which results in one or several outcomes. 2. An experiment in whichthe result ofa trial cannot be predicted inadvance is called a random experiment.
3. An event associated to a random experiment is thecollection of someoutcomes of theexperiment.
4. An event associated witha random experiment is said tohappen if anyone of theoutcomes satisfying thedefinition of theevent is anoutcome of theexperiment when it is performed.
5. The Empirical probability ofhappening of an event E is defined as: P(E)= Number of trials in which the event happened/ Total number of trials.
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Advanced Tools For Study Assess Interact Heat | Study
Mind Map Overal Idea Content Speed Notes Quick Coverage Heat: It is a form of energy, which makes any object hot or cold. The materials which allow heat to pass through them easily are conductors of heat. The materials which do not allow heat to pass through them easily are called insulators. Temperature: The degree… readmore
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Heat: It is a form of energy, which makes any object hot or cold.
The materials which allow heat to pass through them easily are conductors of heat.
The materials which do not allow heat to pass through them easily are called insulators.
Temperature: The degree of hotness of an object is called temperature.
Heat is the cause of temperature.
Our sense of touch is not reliable to measure the temperature. (Scroll down till end of the page)
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Heat:
It is a form of energy, which makes any object hot or cold.
Temperature: The degree of hotness of an object is called temperature. Our sense of touch is not reliable to measure the temperature.
Thermometer is a device used for measuring temperatures.
Heat is the cause of temperature.
Clinical thermometer is used to measure our body temperature.
Transfer of Heat: Heat flows from a hotter object to a colder object until both objects reach the same temperature.
The heat flows from a body at a higher temperature to a body at a lower temperature.
There are three ways in which heat can flow from one object to another.
These are conduction, convection and radiation.
Conduction: It is the process by which heat is transferred from the hotter end to the colder and end of an object.
Convection: It is the flow of heat through a fluid from places of higher temperature to places of lower temperature by movement of the fluid itself.
Radiation: It is the mode of transfer of heat in which energy is directly transferred from one place to another.
It does not need any material medium.
Dark-coloured objects absorb radiation better than the light-coloured objects.
That is the reason we feel more comfortable in light-coloured clothes in the summer.
Woollen clothes keep us warm during winter.
It is so because wool is a poor conductor of heat and it has air trapped in between the fibres.
Thermometer:
Thermometer is a device used for measuring temperatures.
Clinical thermometer:
Clinical thermometer is used to measure our body temperature.
A thermometer used to measure the temperature of our body is called a clinical thermometer.
For other purposes, we use the laboratory thermometers.
The range of these thermometers is usually from –10°C to 110°C.
The normal temperature of the human body is 37°C.
It consists of a long, narrow, uniform glass tube with a bulb containing mercury at one end.
There is a kink near the bulb.
The range of clinical thermometer is from 35°C to 42°C. (Or from 94°F to 108°F).
Laboratory Thermometer:
Laboratory Thermometer: It is a thermometer used to measure the temperature of objects other than our body.
It consists of a column of mercury enclosed in a glass casing.
The column is continuous without any kink.
It measures a range of temperature from -10˚C to 110˚C.
Sea Breeze:
Sea Breeze: Durign the day, the land heats up faster than the sea.
Warm air above the land rises and cold air from sea takes its place.
Warm air from the land moves towards the sea to compele the cycle.
This produces a sea breeze from the sea to the land.
Land Breeze:
Land Breeze: At night the land cools faster than sea.
The warm air above the sea rises.
This warm air is replaced by colder air from the land producing a land breeze.
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Pair of Linear Equations in Two Variables | Study
Mind Map Overal Idea Content Speed Notes Quick Coverage Content : (Scroll down till end of the page) Study Tools Audio, Visual & Digital Content Content … Key Terms Topic Terminology Term Important Tables Table: . Assessments Test Your Learning readmore
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Advanced Tools For Study Assess Interact SURFACE AREAS AND VOLUMES | Study
Mind Map Overal Idea Content Speed Notes Quick Coverage Plane figure The figures which we can be drawn on a flat surface or that lie on a plane are called Plane Figure. Example – Circle, Square, Rectangle etc. Solid figures The 3D shapes which occupy some space are called Solid Figures. Example – Cube, Cuboid,… readmore
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Plane figure
The figures which we can be drawn on a flat surface or that lie on a plane are called Plane Figure.
Example – Circle, Square, Rectangle etc.
Solid figures
The 3D shapes which occupy some space are called Solid Figures.
Example – Cube, Cuboid, Sphere etc. (Scroll down the till the end of the page)
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Volume
Space occupied by any solid shape is the capacity or volume of that figure. The unit of volume is a cubic unit.
Surface Area
The area of all the faces of the solid shape is its total surface area. The unit of surface area is a square unit.
Lateral or Curved Surface Area
The surface area of the solid shape after leaving the top and bottom face of the figure is called the lateral surface of the shape. The unit of lateral surface area is a square unit.
Surface Area and Volume of a Cube
Cube is a solid shape having 6 equal square faces.
Lateral surface area of a cube 4s2 Total surface area of a cube 6s2 The volume of a cube s3 Diagonal √3 s, s = edge of the cube = side length of face of cube Surface Area and Volume of a Cube Example
What is the capacity of a cubical vessel having each side of 8 cm?
Solution
Given side = 8 cm So, Volume of the cubical vessel = l3 = (8)3 = 256 cm3.
Surface Area and volume of a Cuboid
Cuboid is a solid shape having 6 rectangular faces at a right angle.
Lateral surface area of a cuboid 2h(l + b) Total surface area of a cuboid 2(lb + bh + lh) Volume of a cuboid lbh Diagonal l = length, b = breadth, h = height Surface Area and volume of a Cuboid Example
What is the surface area of a cereal box whose length, breadth and height is 20 cm, 8 cm and 30 cm respectively?
Solution
Given, length = 20 cm, breadth = 8 cm, Height = 30 cm
Total surface area of the cereal box = 2(lb + bh + lh)
= 2(20 × 8 + 8 × 30 + 20 × 30)
= 2(160 + 240 + 600)
= 2(1000) = 2000 cm2.
Surface Area and Volume of a Right Circular Cylinder
If we fold a rectangular sheet with one side as its axis then it forms a cylinder. It is the curved surface of the cylinder. And if this curved surface is covered by two parallel circular bases then it forms a right circular cylinder.
Curved surface area of a Right circular cylinder 2πrh Total surface area of a Right circular cylinder 2πr2 + 2πrh = 2πr(r + h) The volume of a Right circular cylinder πr2h r = radius, h = height Surface Area and Volume of a Right Circular Cylinder Surface Area and Volume of a Hollow Right Circular Cylinder
If a right circular cylinder is hollow from inside then it has different curved surface and volume.
Curved surface area of a Right circular cylinder 2πh (R + r) Total surface area of a Right circular cylinder 2πh (R + r) + 2π(R2 – r2) R = outer radius, r = inner radius, h = height Surface Area and Volume of a Hollow Right Circular Cylinder Example
Find the Total surface area of a hollow cylinder whose length is 22 cm and the external radius is 7 cm with 1 cm thickness. (π = 22/7)
Solution
Given, h = 22 cm, R = 7 cm, r = 6 cm (thickness of the wall is 1 cm).
Total surface area of a hollow cylinder = 2πh(R + r) + 2π(R2 – r2)
= 2(π) (22) (7+6) + 2(π)(72 – 62)
= 572 π + 26 π = 598 π
= 1878.67 cm2
Surface Area and Volume of a Right Circular Cone
If we revolve a right-angled triangle about one of its sides by taking other as its axis then the solid shape formed is known as a Right Circular Cone.
Curved surface area of a Right Circular Cone πrl = πr[√(h2 + r2)] Total surface area of a Right Circular Cone πr2 + πrl = πr(r + l) The volume of Right Circular Cone (1/3) πr2h r = radius, h = height, l = slant height Surface Area and Volume of a Right Circular Cone Surface Area and Volume of a Sphere
A sphere is a solid shape which is completely round like a ball. It has the same curved and total surface area.
Curved or Lateral surface area of a Sphere 4πr2 Total surface area of a Sphere 4πr2 Volume of a Sphere (4/3) πr3 R = radius Surface Area and Volume of a Sphere Surface Area and Volume of a Hemisphere
If we cut the sphere in two parts then is said to be a hemisphere.
Curved or Lateral surface area of a Sphere 2πr2 Total surface area of a Sphere 3πr2 Volume of a Sphere (2/3) πr3 r = radius Surface Area and Volume of a Hemisphere Example
If we have a metal piece of cone shape with volume 523.33 cm3 and we mould it in a sphere then what will be the surface area of that sphere?
Solution
Given, volume of cone = 523.33 cm3
Volume of cone = Volume of Sphere
Volume of sphere = 100 π cm3
125 = r3
r = 5
Surface area of a sphere = 4πr2
= 314.28 cm2.
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