Substance and Its Nature
Introduction
Chemistry is the branch of science which deals with the composition of matter and also the
Physical and Chemical characteristics associated with the different material objects.
A French chemist, Lavoisier (1743 – 1793) is regarded as father of modem chemistry.
Substance and Its Nature
Anything that occupies space, possesses mass and can be felt by any one or more of our senses is called matter.
Anything which has mass and occupies space is called matter.
Matter can be classified as pure substances or mixtures.
A pure substance may either contain constituent particles of only one kind or of different kinds. A pure substance has a fixed composition.
An element is a basic form of matter which cannot be broken down into simpler substances by any physical or chemical means.
Elements can be broadly classified as metals, non-metals and metalloids.
Metals are one category of elements that have lustre. They conduct heat and electricity.
They are sonorous .They are malleable and ductile.
Non metals do not have lustre, are not sonorous and are bad conductors of heat and electricity.
Metalloids are elements having properties intermediate between those of metals and non-metals.
A compound is a pure substance composed of two or more elements chemically combined in a fixed proportion. It can be broken down into simpler substances by chemical or electrochemical methods.
A mixture contains two or more elements or compounds which are mixed together in any proportion. In a mixture no new compound is formed. A mixture shows the properties of the constituent substances.
Mixtures are classified are homogeneous or heterogeneous mixture
Mixtures whose components mix completely with each other to make a uniform composition are called homogeneous mixtures.
A heterogeneous mixture has a non - uniform composition.
The ability of a substance to dissolve in another substance is called solubility.
Homogeneous mixture of two or more substances is called a solution.
Component of a solution present in small quantity is called a solute.
Component of a solution present in large excess is called a solvent.
Solution with high solute concentration is called concentrated solution and those with low concentration is called dilute solution.
A solution that has dissolved maximum amount of solute at any particular temperature is said to be a saturated solution.
The smallest particle of an element is called an atom. An atom can take part in chemical combination and does not occur free in nature. The atom of the hydrogen is the smallest and lightest. Eg - Na, K, Ca, H etc.
A molecule is the smallest particle of an element or compound that can have a stable and independent existence.
\[Eg-{{O}_{2}},{{N}_{2}},C{{I}_{2}}{{P}_{4}},\] etc.
A mole is a collection of \[6.023\times \text{1}{{0}^{3}}\]particles.
The number \[6.023\times {{10}^{23}}\] is called Avogadro's Number,
At the structural level, all living organisms are composed of tiny living units called, cells.
Organisms consisting of only one cell are called unicellular organisms, e.g. Paramedum, Amoeba, etc.
Organisms consisting of more than one (may be millions of cells) cell are known as multicellular organisms/ e.g., plants and animals.
Though, cells are generally microscopic, some cells can also be seen with the naked eye, e.g., ostrich egg measuring about \[170\times 150\] mm.
Cell is the structural and functional unit of all living organisms.
The shape and size of cells are in fact related to the function they perform.
Cells are measured in micrometers (mm) sometimes called microns (m).
The smallest cell is Mycoplasma having 0.1 micron diameter.
The longest cell is nerve cell, measuring about a metre in length.
Cells may be spherical, oval, elliptical, spindle shaped, cuboidal, polygonal, columnar or flat.
E. Purkinje coined the term 'protoplasm' - the life-giving substance present in the cell.
Robert Hooke discovered the basic unit of life 'cell',
Schleiden and Schwann proposed the cell theory in 1839.
Extremely thin, outer boundary of cytoplasm is cell membrane.
Cytoplasm contains different cell organelles like nucleus, mitochondria, endoplasmic reticulum, Golgi body, plastid, lysosome, ribosome, etc.
Nucleus is the most important part of a cell having control over all cellular activities.
Nuclear membrane, nuclear sap, nucleolus, chromatin, etc., are the parts of nucleus.
Animal cells lack cell wall and plastids.
Mitochondria are generally called the "Powerhouse of the cell".
Vacuole, surrounded by a single membrane, is called tonoplast.
Lysosomes containing powerful enzymes are called digestive bags or suicidal bags.
Centrosomes help in cell division.
Vacuoles provide turgidity and rigidity to the cells.
Protoplasm is the physical basis of life.
Diffusion is the movement of molecules of a substance (solid, liquid or gas) from a region of their higher concentration to the region of their lower concentration until they are spread out evenly.
Osmosis is the diffusion of a solvent, usually water, through a semi-permeable membrane from a dilute or weaker solution into a concentrated or stronger solution.
A semi-permeable or partially permeable membrane allows movement of solvent molecules but does not allow the movement of solute molecules.
Endosmosis and exosmosis: In cells, water molecules may diffuse into the cell or out of the cell, depending on whether the cells are kept in a weaker solution or a stronger solution respectively.
Endosmosis (endo = inward) is the inward diffusion of water when the surrounding solution is less concentrated. This brings about swelling of the cell.
Exosmosis (exo = outward) is the outward diffusion of water when the surrounding solution is more concentrated. This brings about shrinkage of the cell.
Hypertonic (hyper = more or higher): The solution outside the cell has more concentration than the cell sap. If a cell is placed in such a solution, water will move out of the cell, i.e., exosmosis will take more...
Multicellular organisms consist of many groups of specialised cells making up their tissues and organs.
Differentiation is the process by which unspecialised structures become modified and specialised for performing specific functions.
Differentiation results in division of labour.
The study of the structure of tissues and organs is known as histology.
Based on ability to divide, plant tissues may be classified as meristematic tissue and permanent tissue.
Meristematic tissues are responsible for growth in plants.
The part of the plant body where meristematic tissues are present is called meristem.
Meristematic cells possess the power of cell division.
Permanent tissues are those which have lost the capacity to divide.
1 Based on function, permanent tissues are classified as protective tissues, supporting tissues, conducting tissues and secretory tissues.
Parenchyma is a widely distributed, simple plant tissue.
Collenchyma is a strong and flexible mechanical tissue.
Like collenchyma, sclerenchyma is also a strengthening and protective tissue.
Xylem and phloem are the conducting tissues or vascular tissues/ also called complex tissues.
Xylem is popularly known as wood.
Xylem is composed of tracheids, vessels, xylem parenchyma and xylem fibres.
In higher plants, xylem and phloem usually occur together forming vascular bundle.
Phloem is composed of sieve tubes; companion cells, phloem parenchyma and phloem fibres.
Protective tissues include epidermis and cork.
In old roots and stem, the epidermal tissue at the periphery is replaced by cork.
Four basic types of animal tissues are - epithelium or epithelial tissue, connective tissue, muscular tissue and nervous tissue.
The epithelial cells lie close together with little or no intercellular substances.
The main function of epithelium is to give protection to the underlying tissues.
Connective tissue serves to 'connect' or 'bind' the cells of other tissues in the body and gives them rigidity and support.
Areolar connective tissue is of two types - white fibres (made of collagen) and yellow fibres (made of elastin).Tendon is made up of white fibres and connects muscles to bones.
Ligaments consist of yellow fibres and connect one bone to another bone.
Cartilage is a non-porous connective tissue.
Bone is very strong, rigid and porous tissue.
Bone is surrounded by a connective tissue known as periosteum,
Bones make up approximately 15% of body mass of an adult.
Blood is a bright, red-coloured fluid connective tissue consisting of plasma and blood cells (erythrocytes, leucocytes and platelets).
Muscular tissue is a contractile tissue which possesses myofibrils, sarcoplasm, sarcolemma, etc.
The main function of muscular tissue is to bring about movement of body parts and locomotion of individual.
Muscular tissue is of three types - striated or voluntary, smooth or involuntary and cardiac muscles.
Nervous tissue is a very specialised tissue for receiving stimuli or sensations and transmitting messages.
Nerve cells or neurons form the most important elements of nervous tissue.
The three main parts of a neuron are cell body or cyton, Dendron and the axon.
Agriculture is the science of growing plants and raising animals useful to man.
Science of growing vegetables, fruits and ornamental plants is called horticulture.
The plants grown and tended or cared for in a field are known as crop plants or crops.
There are two main seasons for cultivating crops in India. These are winter season and summer season crops.
The crops grown in winter season are called Rabi crops and the crops grown in summer season are called Kharif crops.
Wheat, barley, gram, pea, potato and mustard are rabi season crops while rice, maize, groundnut, soyabean, arhar, urad, moong, jowar, and cotton are kharif season crops.
Nutrients which are required in relatively large quantity are called macronutrients while those required in small quantity are called micronutrients.
The major sources of nutrients in the field are manures and fertilisers.
Vermicomposting is composting with the help of earthworms.
Accumulation of wastes and organic matter in the water body leads to excessive growth of phytoplankton and this in turn results into the depletion of free oxygen content in that water body. This phenomenon is called eutrophication.
Biofertilisers are micro-organisms or biologically active products which are used to enrich soil fertility. Generally Rhizobium, Anabaena which are found in symbiotic association with plants like Legumes, Azolla, etc. are used as biofertilisers.
Farming is a process of producing plants and animal products in a farm.
Organic farming is a farming system in which fertilisers, herbicides or pesticides are replaced by manures, recycled farm wastes, and biofertilisers.
Mixed farming is defined as the system of farming in which crop production is combined with the rearing of livestock.
Mixed cropping is the practice of growing two or more crops simultaneously in the same field.
Growing more than two crops in succession in a field during one year is called multiple cropping.
The basic objective of mixed cropping is to achieve insurance against total crop failure under poor rainfall conditions.
Intercropping is the practice of growing two or more crops simultaneously in the same field in rows.
The practice of growing different crops in the same field alternately, in succession, is called crop rotation.
Unwanted plants growing in a field are called weeds, and the process of removing weeds from a field is called weeding.
The method of controlling living organisms by the use of other organisms is called biological control.
Commonly Asked Questions
When green tomatoes turn red then:
(a) New chloroplast are made,
(b) Chromoplasts are changed into chloroplasts
(c) Chloroplasts are disintegrated and get converted into chromoplasts
(d) All of these
(e) None of these
Answer (C)
Explanation: When green tomatoes turn red then chloroplasts are disintegrated and get converted into chromoplasts. Chromoplasts absorb light energy and pass it on to chlorophyll in the chloroplasts.
Which of the following organelles are the cells' garbage disposal system'?
Introduction
Numbers are the basic unit of Mathematics, after all, it with numbers that we perform the various functions which constitute Mathematics. For example: Addition, Subtraction,
Multiplication & Division. The Number system is the backbone of any competitive exam. The correct understanding will help you to solve different and complex problems that appear in these examinations. First and for most, let us have a look at the basic classification of numbers and its various kinds.
Classification of Numbers
Natural Numbers
Natural numbers are all of the whole numbers EXCEPT zero. 1, 1, 3, 4, 5, 6, 7, 8, 9, 10, 11…....
They are also called counting numbers. The lowest natural number is 1.
Whole Numbers
Whole numbers are those numbers which start by 0 or we can say when 0 is Included in the list of natural numbers then we call it whole numbers, for example 0, 1, 2, 3, 7, 4, 5.......
Integers
It is the series of both positive and negative numbers lying on the number line. It is the combination of both positive and negative whole and natural numbers.
Rational Numbers
Rational numbers are those numbers that can be written in the form of a ratio of x and y, where the denominator is not zero.
Real Numbers
The number which lies on the number line is a real number .The number can be positive or negative in nature, for example it may be like as 3, 4, 5, 6, -6, -5, -4, -3, -2.....
Irrational Numbers
Irrational numbers are those which are not rational, that is those numbers that cannot be written in the form of a ratio.
Counting Numbers
Counting numbers are those numbers which are well managed on the number line with the difference of 1. The smallest counting number in the number line is 1.
Complex Numbers
Includes real numbers and imaginary numbers are called complex numbers, eg. a + ib.
Prime numbers
The numbers which don't have any factor other than 1 or itself.
For example: 2, 3, 5, 7, 9, 29, 31, 43 .................. or we can say that the numbers which are not divisible by any number are called prime numbers. There are 24 prime numbers between 1 and 100.
2 is the only even prime number and the least prime number.
Percentage
Percentage
Percentage is a fraction whose denominator is 100. The numerator of the such fraction is called the rate percent.
For example: 15 percent means \[\frac{15}{100}\]and denoted by 15 %.
% of A means ——- and simplifying it. Example: \[45\text{ }%\text{ =}\frac{45}{100}=\frac{9}{20}\]
For conversion of fraction \[\frac{P}{q}\] as percentage, we simply multiply it by 100 and put the
sign of % or mathematically we can write \[\frac{P}{q}=\left( \frac{P}{q}\times 100 \right)%.\]
Application Based Problem on Percentage
The following are the points to remember to solve the problem related to variation in the price of an article.
If the price of an article increases by x % then the reduction in consumption, so that
expenditure remains unaffected, is \[\left( \frac{X}{100+X}\times 100 \right)%\]
If the price of an article decreases by x % then the increase in consumption, so that
expenditure remains unaffected, is \[\left( \frac{X}{100-X}\times 100 \right)%\]
Problem Based on the population of a Locality
Suppose the present population of a locality be ‘A’ and let it increases by x % per annum then
Population after y years \[A{{\left( 1+\frac{x}{100} \right)}^{y}}\]
Population before y years \[=\frac{A}{{{\left( 1+\frac{x}{100} \right)}^{y}}}\]
Ratio and Proportion
In this chapter we will study about the comparison of two or more quantities. When we compare only two quantities of same kind, it is called ratio and more than two quantities is called proportion.
Ratio
A ratio is a relation between two quantities of same kind. Comparison is made between the two quantities by considering what part of one quantity is that of the other quantity. The two quantities are called the terms of ratio.
If x and y are two quantities of same kind then the ratio of x to y is x/y or x : y. It is represented by x : y.
Important Points Related to Ratio
The first term of ratio is called antecedent and the second term is called the consequent.
\[\frac{a}{b}=\frac{c}{d}=\frac{e}{f}=\]………then each ratio is equal to \[\frac{a+c+e........}{b+d+f........}\]
Multiplication and division by the same number (except zero) with antecedent and consequent of the ratios are equal in values, the resultant ratio remains unchanged.
Proportion
It is the equality of two ratios i.e if a : b = c : d, then ad = cd that implies product of extremes
= product of means. Four quantities p, q, r, s are in proportion if ps = qr.
Important Points Related to Proportion
If \[\frac{a}{b}=\frac{c}{d}\] then
\[\frac{a+b}{b}=\frac{c+d}{d}\] (componendo)
\[\frac{a-b}{b}=\frac{c-d}{d}\] (dividendo)
\[\frac{a+b}{a-b}=\frac{c+d}{c-d}\] (componendo and dividendo)
Geometry
Geometry is the visual study of shapes, sizes, patterns, and positions. It occurred in all cultures, through at least one of these five strands of human activities: The following formulas and relationships are important in solving geometry problems.
Angle Relationships
The base angles of an isosceles triangle are equal.
The sum of the measures of the interior angles of any n-sided polygon is 180 (n – 2) degrees.
The sum of the measures of the exterior angles of any n-sided polygon is \[360{}^\circ \].
If two parallel lines are cut by a transversal, the alternate interior angles are equal, and the corresponding angles are equal.
Angle Measurement Theorems
A central angle of a circle is measured by its intercepted arc.
An inscribed angle in a circle is measured by one-half of its intercepted arc.
An angle formed by two chords intersecting within a circle is measured by one-half the sum of the opposite intercepted arcs.
An angle formed by a tangent and a chord is measured by one-half its intercepted arc.
An angle formed by two secants, or by two tangents, or by a tangent and a secant, is measured by one-half the difference of the intercepted arcs.
Proportion Relationships
A line parallel to one side of triangle divides the other two sides proportionally.
In two similar triangles, corresponding sides, medians, altitudes, and angle bisectors are proportional.
If two chords intersect within a circle, the product of the segments of one is equal to the product of the segments of the other.
If a tangent and a secant are drawn to a circle from an outside point, the tangent is the mean proportional between the secant and the external segment.
In similar polygons the perimeters have the same ratio as any pair of corresponding sides.
Right Triangle Relationships
If an altitude is drawn to the hypotenuse of a right triangle it is the mean proportional between the segments of the hypotenuse, and either leg is the mean proportional between the hypotenuse and the segment adjacent to that leg.
In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs. (Remember the Pythagorean triples: 3, 4, 5; 5, 12, 13.)
En a \[30{}^\circ -60{}^\circ \] right triangle, the leg opposite the \[30{}^\circ \]angle is one-half the hypotenuse, and the leg opposite the \[60{}^\circ \] angle is one-half the hypotenuse times\[\sqrt{3}\].
In a right isosceles triangle the hypotenuse is equal to either leg times. \[\sqrt{2}\].
En an equilateral triangle of sides, the altitude equals \[\frac{s}{2}\sqrt{3}\].
Commonly Asked Questions
If the sides of a triangle are produced then the sum of the exterior angles i.e,
\[\angle DAB\,\,+\,\,\angle EBC\,\,+\,\,\angle FCA\] is equal to:
(a) \[180{}^\circ \] (b) \[270{}^\circ \]
(c) more...
Introduction
Mensuration is a science of measurement of the lengths of lines, area of surfaces and volumes of solids.
Some Important Definitions and Formulae:
If any closed figure has three sides then it is called a triangle.
In a triangle the sum of three angles is \[180{}^\circ \].
In a triangle the sum of the lengths of any two sides should be more than the third side.
Similarly the difference between any two sides of a triangle is less than the third side.
The side on which a triangle rests is called the base. The length of the perpendicular drawn on the base from opposite vertex is called the height of the triangle.
If the three sides of a triangle have three different lengths then it is called a scalene triangle.
If exactly two side of a triangle are equal and the third side has different length then it is called an isosceles triangle.
If all the three sides of a triangle are equal then it is called an equilateral triangle.
Area of Plane Geometrical Figures
Triangle
(i) Right Triangle
(ii) Scalene Triangle (Heron's formula)
(iii) Isosceles Triangle
(iv) Equilateral Triangle
Right Triangle: Area of right triangle \[=\frac{1}{2}\times \left( perpendicular \right)\times Base=\frac{1}{2}\times AB\,\times BC\]
Scalene Triangle (Heron's formula): Let, a, b, c be the length of sides of a triangle
then area \[=\sqrt{s(s-a)(s-b)(s-c)}\] sq. unit, where \[s=\frac{1}{2}(a+b+c)\]
Isosceles Triangle: Area ot isosceies trsang Se
\[=\frac{1}{2}\times BC\times AD=\frac{1b}{4}\sqrt{4{{a}^{2}}-{{b}^{2}}}\]
Equilateral triangle:
Area \[=\frac{\sqrt{3}}{4}\,{{(side)}^{2}}=\frac{\sqrt{3}}{4}{{a}^{2}}\]
Circle
A circle is a geometrical figure consisting of all those points in a plane which are at a given distance from a fixed point in the same plane. The fixed point is called the centre and the constant distance is known as the radius.
A circle with centre O and radius r is generally denoted by C (0, r).
Circle Formulas
The circumference C of a circle of radius r is given by the formula \[C=2{}^\circ \,\pi r\].
The area A of a circle of radius r is given by the formula \[A=\pi {{r}^{2}}\].
The areas of two circles are to each other as the squares of their radii.
The length L of an arc of \[n{}^\circ \] in a circle of radius r is given by the formula \[L=\frac{n}{360}\times 2\pi r\]
The area A of a sector of a circle of radius r with central angle of \[n{}^\circ \] is given by \[A=\frac{n}{360}\times \pi {{r}^{2}}\]
Quadrilateral
We know that a geometrical figure bounded by four lines segment more...
Probability
A mathematically measure of uncertainty is known as probability. If there are 'a' elementary events associated with a random experiment and 'b' of them are favourable to event 'E':
Then the probability of occurrence of event E is denoted by P (E).
\[\therefore \,\,\,\,\,\,P(E)=\frac{b}{a}\] \[\Rightarrow \,\,0\le P\left( E \right)\le 1\]
The probability of non-occurrence of event E denoted by P(E) and is defined \[as\,\,\frac{a-b}{a}\].
\[\Rightarrow \,P\left( E \right)+P\left( \overline{E} \right)=1\]
Experiment
An operation which can produce some well- defined outcomes is called an experiment,
Random Experiment: An experiment in which all possible outcomes are known and exact outcome cannot be predicted is called a random experiment.
Example: Rolling an unbiased dice has all six outcomes (1, 2, 3, 4, 5, 6) known but exact outcome can be predicted.
Outcome: The result of a random experiment is called an outcome,
Sample Space: The set of all possible outcomes of a random experiment is known as sample space.
Example: The sample space in throwing of a dice is the set (1, 2, 3, 4, 5, 6).
Trial: The performance of a random experiment is called a trial.
Example: The tossing of a coin is called trial.
Event
An event is a set of experimental outcomes, or in other words it is a subset of sample space.
Example: On tossing of a dice, let A denotes the event of even number appears on top A: {2, 4, 6}.
Mutually Exclusive Events: Two or more events are said to be mutually exclusive if the occurrence of any one excludes the happening of other in the same experiment. E.g. On tossing of a coin is head occur, then it prevents happing of tail, in the same single experiment.
Exhaustive Events: All possible outcomes of an event are known as exhaustive events. Example:
In a throw of single dice the exhaustive events are six {1, 2, 3, 4, 5, 6}.
Equally Likely Event: Two or more events are said to be equally likely if the chances of their happening are equal.
Example: On throwing an unbiased coin, probability of getting Head and Tail are equal.
Playing Cards
Total number of card are 52.
There are 13 cards of each suit named Diamond, Hearts, Clubs and Spades.
Out of which Hearts and diamonds are red cards.
Spades and Clubs are black cards.
There are four face cards each in number four Ace, King, Queen and Jack.
Introduction
The word ANALOGY has been derived from two words-‘ANA’ means ‘Relation’ and ‘LOGUS’ means ‘Study’. The ‘ANALOGY’ literally means:
(i) A similar feature, condition, state etc. shared by two things that are compared.
(ii) A process of reasoning based on similar feature of two things. Thus analogy means “similar feature”, a ‘common feature’ or ‘correspondence’. The analogy test mainly concentrates on relationship, which may be between various elements, things, terms phenomenon etc.
To Solve the Problems on Analogy you may follow the Following Steps:
Step 1: Eliminate choices that represent relations that do not match the relationship between the words.
Step 2: Eliminate choices that have vague relationship. Remember the original relationship will always be clear.
Step 3: Eliminate words pairs that express the same relationship as the given pair, but appear in the opposite word order.
Step 4: If you can’t determine the relationship between two words, try reading them backward.
Example:
(i) Judge : Court :: Peon : Office
Explanation:
A “Judge” works in a ‘Court’ and in the same way a ‘Peon’ works in a ‘Office’, in other words, court (place to give judicial judgment) is a working place for a Judge and in the same way office (place where peon is required for doing petty works) is a working place for a peon.
1st pair - Judge: Court (person and working place relationship).
2nd pair - Peon: Office (person and working place relationship).
Clearly, both pairs show similar relationship in a logical way. Hence, both pairs are analogous and it is said that both pairs exhibit analogy.
(ii) Excess : Shortage :: Straight : Crooked
Explanation:
1st pair - Excess: Shortage (opposite relationship)
2nd pair - Straight: Crooked (opposite relationship)
Clearly, both pairs show similar relationship (opposite relationship). Hence, both pairs exhibit analogy.
Types of Problems
Problems Based on Synonymous Relationship
In such problems, the words given in one pair have same meaning and the same relationship is found in another pair of words.
Example 1
Generous : Kind :: Frank : Candid
Explanation:
1st pair – Generous: Kind (Synonymous relationship).
2nd pair – Frank: Candid (Synonymous relationship).
Example 2
Commence : Begin :: Ordinary : Common
Explanation:
1st pair – Commence: Begin (Synonymous relationship).
2nd pair – Ordinary: Common (Synonymous relationship).
Commonly Asked Questions
Select the pair which is related in the same way as the pair of words given in the question.
Absurd : Silly :: ______ : ______
(a) Fertile: Barren (b) Amend: Improve
(c) Active: Inert (d) Famous: Notorious
(e) None of these
Answer: (b)
Explanation: Option (b) is correct because ‘Absurd’ and ‘Silly’ are synonymous words.
In the same manner ‘Amend’ and ‘Improve’ are synonymous words.
Rest of the options is incorrect more...
Rational Numbers
Rational numbers are those numbers that can be written in the form of a ratio of x and y, where the denominator is not zero.
Real Numbers
The number which lies on the number line is a real number .The number can be positive or negative in nature, for example it may be like as 3, 4, 5, 6, -6, -5, -4, -3, -2.....
Irrational Numbers
Irrational numbers are those which are not rational, that is those numbers that cannot be written in the form of a ratio.
Counting Numbers
Counting numbers are those numbers which are well managed on the number line with the difference of 1. The smallest counting number in the number line is 1.
Complex Numbers
Includes real numbers and imaginary numbers are called complex numbers, eg. a + ib.
Prime numbers
The numbers which don't have any factor other than 1 or itself.
For example: 2, 3, 5, 7, 9, 29, 31, 43 .................. or we can say that the numbers which are not divisible by any number are called prime numbers. There are 24 prime numbers between 1 and 100.
\[\angle DAB\,\,+\,\,\angle EBC\,\,+\,\,\angle FCA\] is equal to:
(a) \[180{}^\circ \] (b) \[270{}^\circ \]
(c) more... 
Scalene Triangle (Heron's formula): Let, a, b, c be the length of sides of a triangle
then area \[=\sqrt{s(s-a)(s-b)(s-c)}\] sq. unit, where \[s=\frac{1}{2}(a+b+c)\]
Isosceles Triangle: Area ot isosceies trsang Se
\[=\frac{1}{2}\times BC\times AD=\frac{1b}{4}\sqrt{4{{a}^{2}}-{{b}^{2}}}\]
Equilateral triangle:
Area \[=\frac{\sqrt{3}}{4}\,{{(side)}^{2}}=\frac{\sqrt{3}}{4}{{a}^{2}}\]
Circle
A circle is a geometrical figure consisting of all those points in a plane which are at a given distance from a fixed point in the same plane. The fixed point is called the centre and the constant distance is known as the radius.
A circle with centre O and radius r is generally denoted by C (0, r).
Circle Formulas
