Probability and Statistics in Grade XI

Previously we have discussed about probability worksheets and Today we are going to study about Probability and Statistics which comes under maharashtra state education board. Probability as the word means the possibility of an outcome. In probability we can say that it means the random possibility that an event or a condition will get fulfilled. Probability is applied in everyday life as well and can be solved using Probability Calculator. There are many day to day activites in our life which are uncertain and are probable in their occurance to solve them we use probability. We can understand it with the help of an example. Let us suppose we are going to toss a coin, a coin has two sides head and a tail. There are two possible outcomes head or tail. The possibility of head is 1/2 and of tail is 1/2 as well.(want to Learn more about Probability ,click here),
We can also understand statistics related to Grade XI .Statistics is basically the study of collection, organization and representation of data with the help of graphs pictures etc. It includes all the aspects of representations like tables, surveys , experiments as well. Statistics is very important part of mathematics of Grade XI. It talks about the analysis of data as well. If suppose we are given an information about the 12 students of grade V and their respective heights. We can show it or represent it in form of a table and show it in form of a bar graph. Bar graph is an example of a statistical representation. The above mentioned information gives a idea about the Probability and statistics of Grade XI. It is also useful in our day to day life.
This is all about the Probability and Statistics and if anyone want to know about Standard distributions then they can refer to Internet and text books for understanding it more precisely.Read more maths topics of different grades such as Rotations  in the next session here.
 

Learn Cross Sections and Planes

Earlier we have discussed and practiced o probability worksheets  and Today we are going to learn about Planar cross sections; perpendicular lines, planes in XI Grade of gujarat state education board.
Here is about planar cross sections and math questions related to it:
planar cross-sections
We can define planar –cross sections as the figure formed when the plane intersects a solid figure.the result of intersection thus comes out to be a line,line segment or a plane figure such as a circle or a polygon.
There are however certain conditions related to planar cross sections
Like if  plane is parallel to the base of the solid ,the plain figure thus formed will be similar or congruent to the base of solid.(want to Learn more about Cross Sections ,click here),
Now we further move to the next topic of Grade XI called as the perpendicular lines
What are perpendicular lines?
·A perpendicular lines ,is a line that  intersects another line or a plane such that the angle formed by their intersection is exactly 90 degrees . in other words, a right angle. In geometry, such lines are represented by a symbol (⊥) that resembles an upside down T --- that letter being an example of two perpendicular lines joining together. Perpendicular lines and parallel lines share a relationship; if two lines are both perpendicular to a third line, they are parallel to each other . if we extended them indefinitely, the lines will never touch.

Perpendicular Lines Equation

The equation for perpendicular line i.e the linear equation is  "linear equation" is  "y = mx + c" .

 

Perpendicular Lines use in the Real World

·Most perpendicular lines in the real world are man-made; the capital letter "H," for example, consists of two parallel lines with a line connecting them in perpendicular fashion. Like many examples that are perfect mathematically, perpendicularity in the real world is complicated by the lack of any truly flat planes. Eg  A skyscraper.
Now we learn about the planes related to Grade XI
In mathematics a plane is a flat two dimensional suface .A plane is the two dimensional figure of a point,a line and a space.planes can arise as subspaces of some higher dimensional spacesas with walls of rooms.

In a three-dimensional space, another important way of defining a plane is by specifying a point and a normal vector to the plane.
Let r₀ be the position vector of some known point P₀ in the plane, and let n be a nonzero vector normal to the plane. The idea is that a point P with position vector r is in the plane if and only if the vector drawn from P₀ to P is perpendicular to n. Recalling that two vectors are perpendicular if and only if their dot product is zero, it follows that the desired plane can be expressed as the set of all points r such that
n.. (r-r0)= 0
now we have discussed in detail about the three topics related to Grade XI ie.planar cross section,perpendicular lines an planes.
Thus to sum up we can say that all the three topics are related to each other and are the important part of Grade XI mathematics and if you want to know about Correlation coefficient and about grade 10th topic Fundamental theorem of algebra then refer Internet.

Learn Parabolic Functions and Axis of Symmetry.

Previously we have discussed about antiderivative of sinx and In todays lesson we are going to study about Parabolic functions, vertex, axis of symmetry, which comes under Grade XI of maharashtra secondary board.
First we are going to learn about parabolic functions and also learn to solve math problems related to it.
What are parabolic functions?
Parabolic functions is the path traced by a point which moves in a plane I such a way that its distance from a fixed point is always equal to its distance from a fixed line,both lying in the n fixed point does not lie on the given line.
The fixed point is called as the focus of the parabola and the fixed line is called its directrix.
Aline through the focus and perpendicular to the directrix is called the axis of parabola.the poit of intersection of the parabola with its axis is called the vertex of parabola.
There are four  forms of parabola
The first standard form or the rigt handed parabola is given by the formula
Y²=4ax, where a>0
The second form of standard parabola or left handed parabola I sgiven by the formula
Y²=-4ax, where a>0
The third ofrm of parabola is also known as upward parabola an dis given by the equation
X²=4ay where a>0
The last form of parabola is called as the downward parabola
It is given by the formula
X²=-4ay where a>0
Now we move to the second topic of grade XI called as the vertex
What is vertex?
In mathematics ,a vertex most commonly refers to a corner or endpoint where lines meet.it can also refer to the maximum or minimum point on a parabola along its line of symmetry.
Like we can take an example of a triangle which has three vertices.

A parabola's equation can take two different forms. The first, vertex form, is represented as y= a(x-h)²+k, where x and y are the coordinates for individual points on the parabola, a is the scalar that can change the shape and direction of of the parabola, and h and k are the vertex points. The parabola's standard form equation, represented as y = ax² + bx + c, focuses on the parabola's direction and its axis of symmetry, which is the line that bisects the parabola. Converting from the vertex to the standard form can make the equation more easy  for other calculations.
Now we come to other topic for the Grade XI which is called as the axis of symmetry
Every parabola has an axis of symmetry which is the line that runs down its “centre”.this line divided the graph into two perfect halves.
To find the axis of symmetry we can use two different formulas.one fomula works when the parabola’s equation is in vertex form and the other works on the parabola’s equation is in the standard form.(want to Learn more about Axis of Symmetry,click here),
If the equation is in the vertex form then the axis of it is x=h in general vertex form equation y=(x-h)² +k
If the equation is in standard form,then the formula is x=-b/2a from the general standard form equation y=ax² + bx +c
We have now  deep knowledge about the parabolic functions, vertex and axis of symmetry and if anyone want to know about Measures of central tendency
then they can refer to internet and text books for understanding it more precisely. You can also refer Grade XII blog for further reading on Conditional statements.

How to Tackle Right Triangle Problems

Hello friends, today I am going to discuss the topic which is the most important topic for algebra 2 mathematic students of grade XI of maharashtra board. And that topic is “Special Right Triangles, Right Angles”. And this article is from one of the easiest in analytic geometry problems and trigonometric mathematics.
First we discuss about Special Right Triangle:- A special triangle is a sight angle triangle which makes the calculation on triangle easier .Right angle Triangle are those whose one angle is 90^â—¦ and the sum of other two triangle is 180°-90°=90°. side base right  angle triangle  are having length in the ratio of two whole number such as 1:2:3 , 37°-53°-90°.
Angle Base triangle is composed by the relationship of the angles. Angles in right angles are such that larger angle. Is equal to the sum of two other angle examples: - 30°-60°-90°, 45°-45°-90°.Trignometive functions are calculated by the special triangle.(want to Learn more about triangles ,click here),
Right Triangle whose sides are in geometric progression- A kepler triangle is triangle in geometric progression. as this sides are in the ration of
                                         a, ar, ar²
Common ratio r =q where q is golden ratio
Ratio = 1:√L: L
Now friends we discuss on Isosceles Right Triangle: - isosceles right triangles are those in which two angles and two sides are equal.  In isosceles right triangle Ratio of their sides is always 1:1:√2 or x x:x√2.

 Consider an example: - is one of the equal sides of isosceles triangle is -2 other side are
 As we know that ratio of side are       x x:x√2
                  x=2
                         2:2:2√2
It can also be solved by Pythagoras Thermos. in Pythagoras thermos an angle are such sum of the square of two side are equal to the square of larger side.
                          x²+y²=z²
               In isosceles triangle two side are equal
                  2²+2²= z²
                 4+4= z²
                 8=  z²
                    z= √8
           Z= √4*2= 2√2
Then discuss on Different type of triangle is Fibonacci triangle: - Fibonacci series is basically found in Pascal’s triangle. Normally Fibonacci series is the length of hypotenuse of a right triangle with integral side. We can say that  IN Pythagoras triple Fibonacci series is the large number of quantity.
IN Fibonacci series every number below in the triangle is the sum of the two numbers dynamically above it to the left and the right with position outside the triangle counting as zero.  Fibonacci triangle length of the longer leg  is equal to the sum of three side of the preceding triangle in  Fibonacci series of triangle.
Four most important numbers in the Fibonacci series is used to create a right triangle. With the base hypotenuse being determined by the second and third number and the other side being the square root of the product of the first and fourth number. Fibonacci number are (0, 1, 1, 2, 3, 8, 13) etc.
We discuss on properties and Attribute of right Triangle: -
Opposite side of the right angle this will always be the longest side of the right triangle.
When tow side of right triangle are not the hypotenuse. Then they are making up the two side right angle itself.
If the two sides include the right angle in equal length then right triangle can also be isosceles.
In above articles we discuss about the special right triangle and right angle and if anyone want to know about Special right triangles then they can refer to internet and text books for understanding it more precisely. You can also refer Grade XI  blog for further reading on Conditional Statements.

Conditional Statements

Hello friends, today we are going to learn about conditional statement which comes under the syllabus of tamilnadu  education board. Here I am going to tell you the best way of understanding conditional statement and you can get math help online on this topic also.
First we discuss about what is conditional statement: - A conditional statement in math is a statement in the if-then form. Conditional statement, often called conditional for short is used extensively in a form of logic called deductive reasoning. The logical contractor in a conditional statement is denoted by the symbol ----->.  The conditional is defined to be true unless a true hypothesis leads to a false conclusion. A conditional statement is one that can be put in the form if A then B where A is called the hypothesis and B is called the conclusion. We can convert the above statement into this standard form.Get further details on Conditional statement here,
Let we discuss about feature of conditional statement: - The if part of conditional statement is called the hypothesis and then part is called the conclusion. Conditional statement is a part of any programming language .conditional statement executes line of code only if the condition results true. Conditional statement return true or false. Conditional statements are come in pairs.
Then we discuss about the converse of conditional operator: - In traditional logic we use capital letter as place holder for simple statement. Which is simple deceleration of fact?  Any conditional statement we can always derive three other conditional statement for it.
Three basic conditional statement are :- (1)The converse of conditional statement :- To construct the converse of if  A then  B, we simply exchange A with B  , thus obtaining if B than A which here means .If a conditional statement is true that it converse may be true , but it is not necessarily true. The fact that if there is major snowstorm they will cancel school. You can switch both side around, and not negate them like you would in a contra positive this gives
(2)  The inverse of a conditional statement: - in this conditional statement if A than B we simply negate A and B thus obtaining if not A then not B if the figure is not a triangle then it does not have three side. If a conditional statement is true, then its inverse may be true but it not necessarily true. That negating an already negated variable result in the removal of the Not sign this give us A-------->B this is known as the inverse it inverse the truth value of the original statement.
(3)The contra positive of conditional statement: - in contra positive conditional statement if A than B  we can take the inverse of the converse , or the converse of the inverse obtained if not B than not A. if the conditional statement is true than its contra positive is necessarily true as well. A contra positive is formed by turning and if-then statement around and negating both parts. And it is always true given the truth of the original statement.

In above article we discussed about conditional statement. In next lesson we discuss about more feature of conditional statement.To know answers of interesting questions like is pi a rational number wait till next lecture and
You can also refer Grade XII  blog for further reading on Intersection of a plane with 3-d figures.Read more maths topics of different grades such as Permutations and combinations in the next session here.




 

Basic Constructions in Grade XI

Hello friends, today I am going to discuss the important article which is Basic Constructions. This article is important for all Grade XI students.
So friends let’s start with Basic Constructions
In geometry construction means to draw shape, angle, line triangle, square, rectangle (try Formula for Area of a Rectangle here), rhombus, parallelogram, etc. These construction use only straight edge compass and pencil. In this way we can say this is the pure form of geometric construction. The basic construction of the geometric shapes have  some own steps and some own rule  that are to be followed. In this article we would  learn how to construct a line, circle, and triangle.
Let’s start with how to construct a line
Line is easiest basic construction of geometry. These have some steps that is given below
In the first step you have to place the compass at one end of line.
In the second step you have to adjust the compass to slightly longer than the line length
In third step we have draw arc above and below the line.
In fourth step keeping the same compass width, draw arc form the other end of the line.
In fifth step we have to place ruler where the arcs cross, and end of the line.

Let’s know how to construct a circle:
For constructing circle we require either diameter or radius if we know either on of the things we can easily make the circle.
We just need to have a compass and a scale which can measure radius or diameter
Radius is always half of the diameter so when we need to draw the circle we measure the radius by scale and then put that distance into our compass.
We can take any point and draw circle.


Let’s know how to construct a triangle:
You know friends for triangle we need three lines. So first we draw a line with the help of compass and scale.
In second step we make an arc on first point, then make other arc at last point.
In third step where these points are intersecting each other we make point on that then we join both the ends of the line to that intersecting point.
As the name suggest it contains three angles who’s sum is equal to 180⁰ degree. If all the angles are equal then every angle has the value of 60⁰ each as 180/3= 60
If one of the three angles is of 90⁰ then the triangle is called right angle triangle
If two of the side of a triangle is equal then the triangle is called isosceles triangle
If one of the angles is greater than 90⁰ degree then the triangle is called obtuse angle triangle.
In every triangle the sum of any two sides will always be greater than third side.
Note: the distance in which we are making arc should be equal distance from the point.


In the above article we have seen about lines circle and triangle. Hope you have understood all the things that I told to you.

In upcoming posts we will discuss about Conditional Statements and trigonometric ratios. Visit our website for information on CBSE 12th syllabus

Congruent Triangles in Grade XI

Hello Friends, in today's session we all are going to discuss one of Geometry topic. Geometry is one of the most interesting topics of mathematics that is Finding the Area of a Triangle. Today we will discuss triangles more specifically congruence triangle . A grade XI student is aware of the term Geometry. So we don’t need to discuss it. We will quickly move on to our topic.

What is congruence triangle? The term congruence defines the fact that when one object completely coincides with other. That is they are the mirror image of one another. When we put one object on to the other one it completely hides the other one.
In case of triangles, triangles are called congruent triangle when their corresponding sides and interior angles are congruent.
Triangles have three main magnitudes:

1.      Sides

2.      Angles

3.      Area

To prove two triangles are congruent we have to prove:

1.      They have equal corresponding sides.

2.      They have equal corresponding angles.

3.      Their areas are equal.

But we don’t have to prove all of them. We just group some of these. That is certain group of ( Sides, Angles and Area) will do that:

1.      SSS - We have 5 ways to prove two triangles are congruent. SSS that is Side – Side – Side. If all the three sides of a triangle are equal to corresponding 3 sides of other triangle then the triangles are congruent. As we will now prove that two triangles are congruent then their angles are also equal. That is angle P = L, Q = M and R = M 

2.      SAS: Side- Angle- Side. When two corresponding sides and their included angle of one triangle are equal to the other triangle then the two triangles are called congruent triangle. Now if triangles are equal then their remaining sides and angles are also equal. As we will now prove that two triangles are congruent then their third side PQ = LM and two remaining angle, angle P = L and Q = M.

         

3.      ASA:  Angle- Side- Angle. When two angles and their included sides are equal to the other triangle then the two triangles are called congruent. Similarly when triangles are proved congruent then the remaining angles R and N and the two sides are also equal  PR = LN and QR = MN

4.      AAS: Angle- Angle- Side. If two pairs of corresponding angles and a pair of corresponding opposite side of two triangles are equal then both the triangles are congruent. Hence other two side PQ = LM and PR = LN and also angle Q = M.

5.      RHS: Right- Angle- Hypotenuse- Side. When two right-angled triangles have their hypotenuses equal in length and the shorter sides of two right triangles is equal in length, the triangles are called as congruent triangle. Now when triangles are proved congruent then the remaining third side PQ = LM and two angles Q = M and R = N are also equal

           

Now we have discussed congruent triangles and triangle congruence principles. We had also discussed facts which prove two triangles are congruent or not.

In upcoming posts we will discuss about Basic Constructions in Grade XI and reflections. Visit our website for information on CBSE syllabus 12th

Coordinate Geometry in Grade XI

Hello friends, in this article we are going to learn about co-ordinate geometry for the grade XI. We will learn the brief description of the coordinate geometry and some of the basic definitions in them. For more information on coordinate geometry click on it
In the lower grades we have learnt about geometry and its related definitions and facts, but now we are going to learn about some different geometry which is different from that of earlier learnt geometries. Now in the terms of definition of coordinate, when two real numbers are associated in any system with a point then such a system is called as the coordinate of that point and the study of such geometry is called as coordinate geometry. The system in which we perform the operation in coordinate geometry, such geometry is called as system of Cartesian coordinate.
We use the coordinate system to plot any of the line, point or any of geometrical shape in the specific region. The coordinate geometry for the grade XI covers the study of straight lines, so in this part of geometry we will learn about the slopes of a line and the angle between two inclined lines system and also learn how to find the angle and the How to Find Slope of a Line with the system by including some of the older concepts of the geometry. We will learn to find different types of the equation for a line. Like equation for general line, line parallel to axes, equation of intercept form, slope-intercept form of the line, point-slope form and also the normal form of the line. We will also find the distance of any point from the point. We will also learn for the conic sections and will also get introduced with the three dimensional geometry here. The part of conic section includes number of shapes of the geometry like: points, straight lines, pairs of intersecting lines, circle, eclipse, parabolas, hyperbolas and their related issues. In term of parabola, circle and other geometry, we will learn about some of the equations and their properties. In the three dimensional geometry, we will learn here coordinate planes and coordinate axes. We will learn coordinate of a point and distance between two points and other issues of the point.
We are familiar with the lines and points plotting on the coordinates Now we are going to learn about the perpendicular lines and lines on the plane. So taking a perpendicular line in the geometry say two lines XOX’ and YOY’ and these two lines are intersecting at the point O. The line XOX’ is the X-axis or abscissa and YOY’ is called Y-axis or the ordinate of the coordinate geometry.

Let we take any point ‘p’ on the plane and say the coordinate of the point is (x, y) then to find the value of ‘x’ and ‘y’ we have to draw a particular line to the x and y, say M is the point where perpendicular line meets at x axis and N is the point where line meets on the y axis then we can take as:
OM = x and ON = y and thus we can find the coordinate of the point.

In upcoming posts we will discuss about Congruent Triangles in Grade XI and Coordinate geometry. Visit our website for information on CBSE maths syllabus

Trigonometric Ratios in grade XI

Hi friends, today we are going to study about Similarity Properties, Transformations, Trigonometric Ratios, and Pythagorean Triples for Grade XI. Firstly the introduction about trigonometry. Trigonometry deals with triangles, the right-angled triangle, circles, oscillations, and waves. When two right angle triangles are together then the result is supplementary composite triangles.
If one angle of a triangle is 90 degrees and other one is known, the third is fixed, because the sum of three angles of a triangle is 180 degrees. If the angles are known, the ratios of the sides are determined and if the length of one of the sides is known, the other two are determined. Let a, b and c is the length of triangle sides and A is known angle then ratio of trigonometric functions are,
1. Sine function is defined as the ratio of the side opposite the angle to the hypotenuse. (see for more information)
  Sin A= opposite/ hypotenuse=a/c
2. Cosine functions are defined as the ratio of the adjacent leg to the hypotenuse.
  Cos A= adjacent/ hypotenuse=b/c
3. Tangent functions are defined as the ratio of the opposite leg to the adjacent leg.
  Tan A = opposite/ adjacent = a/b = Sin A/Cos A
4. Cosecant (A) is the reciprocal of sin (A). The ratio of the length of the hypotenuse to the length of the opposite side is called Cosecant.
Csc A = 1/sin A = hypotenuse/opposite = c/a
5. Secant (A) is the reciprocal of cos (A) and it is the ratio of the length of the hypotenuse to the length of the adjacent side.
Sec A = 1/cos A = hypotenuse/adjacent = b/c
6. Cotangent (A) is the reciprocal of tan(A) and defined as the ratio of the length of the adjacent side to the length of the opposite side.
Cot A = 1/tan = adjacent/ opposite=b/a=cos/sin.
The values of angles of these trigonometric functions can be easily computed by using the Pythagorean Theorem (Do you know What is Pythagorean Theorem). Similarly, the values of sine, cosine and tangent of an angle of π / 4 radians (45°) can also be found using the Pythagorean Theorem. Some values of angles are given below,
Sin π/4=sin45⁰=cos π/4=cos45⁰=1/√2,
Tan π/4 = tan 45⁰ = 1.
Sin π/4=sin30⁰=cos π/3=cos 60⁰=1/2.
cos π/6=cos30⁰=sin π/3=sin60⁰= Ñ´3/2
tan π/6=tan30⁰=cot π/3=cot60⁰=1/Ñ´3.
 Below table shows the values of different angles of different functions.

Function	00		300	450	600		900
sin	0		1/2	Ñ´2/2	Ñ´3/2		2
Cos	1		Ñ´3/2	Ñ´2/2	1/2		0
Tan	0		Ñ´3/3	1	Ñ´3		infinity
Cot	infinity		Ñ´3	1	Ñ´3/3		0
Sec	1		2Ñ´3/3	Ñ´2	2		infinity
Csc	infinity		2	Ñ´2	2 Ñ´3/3		0

Now I am going to tell you the properties of trigonometry. The trigonometry properties are based on the law of sin, law of cos and law of tan.
Law of sin is a/sin A = b/sin B = c/sin c = 2R. Sine law can be used to calculate the area of a triangle if two angles and one side are known. Area = 1/2( ab sin C).
Law of cos: c²=a²+b²-2ab cos C.  This law used to determine a side of a triangle if two sides and the angle between them are known.
Law of tan: a-b / a+b =tan[1/2(A-B)]/tan[1/2(A+B)].
From the above discussion I hope that it would help you to understand the properties and transformations, trigonometric ratios.

In upcoming posts we will discuss about Coordinate Geometry in Grade XI and Rotations. Visit our website for information on Central Board of Secondary Education

Angles and Polygons in Grade

Hello friends we are back again with another new session. In our today’s session we will discuss angles. Many students find geometry a difficult topic but after today’s session you will find it interesting and easy to deal with.
Today we will discuss Angles of Triangles and Polygons.
A grade XI student is aware that triangle is a type of polygon. As polygon means many sides and triangle is a three side polygon. (Do you want to know What are Adjacent Angles)
Now let’s start with the Angles of Triangles and Polygons. Here we will include triangle in polygons and discuss it inside a polygon.

A triangle has interior and exterior angles. As a grade XI student I think you all know what are interior and exterior angles and what is the difference between them?



The sum of all three interior is 180 degree and sum of interior and exterior angles meets at one point is also 180 degree.

For a polygon the sum of polygon with n sides has interior angle 180 * (n-2). Suppose we have pentagon that is a polygon of 5 sides. Here we have 1, 2, 3, 4, 5 interior angles and 6, 7, 8, 9, 10 exterior angles.



Then the sum of interior angles are 180 * (5-2) = 540. As we know sum of angles meets at one point is 180 degree. From this we can calculate sum of all angles.
Angle 1 + Angle 6 + Angle 2 + Angle 7 + Angle 3 + Angle 8 + Angle 4 + Angle 9 + Angle 5 + Angle 10 = 5*(180) = 900

Now we can calculate sum of exterior angles from the above two calculations.
Exterior angles = sum of all – interior angles
                         =  900 – 540
                         =  360
Angle 1 + Angle 2 + Angle 3 + Angle 4+ Angle 5 (All interior angle) = 540
Angle 6 + Angle 7 + Angle 8 + Angle 8+ Angle 10 (All exterior angle) = 360

There are three types of angles.
What is an Acute Angle? Angle which is less than 90 degree and any triangle which has acute angle is called acute angle triangle.

        
Right Angle: An angle of 90 degree is called right angle and which has acute angle is called right angle triangle.


Obtuse Angle: Angle which is larger than 90 degree and any triangle which has acute angle is called obtuse angle triangle.



Now let’s see some polygons and their angles.

Square or Quadrilaterals:has 4 sides and its interior angle has 360 degree. Because a square has two triangles in it. Didn’t you notice this!



Pentagon:has 5 sides. A pentagon has 3 triangles inside it so the sum of interior angles are 3*180 = 540.


It is not possible to explain all polygons. So here is the list of some polygons with number of sides they have their interior angle.

Polygon	Sides	Sum of interior angles
Triangle	3	180
Quadrilateral	4	360
Pentagon	5	540
Hexagon	6	720
Heptagon	7	900
Octagon	8	1082

 If we want to compute the size of each angle then we can use
(n-2) * 180 / n, where n is the number of sides.
Example: For triangle each angle is
            (3-2) * 180/ 3 = 60 degrees

In upcoming posts we will discuss about Trigonometric Ratios in grade XI and Special right triangles. Visit our website for information on ICSE board syllabus for class 12 math

Types of Events in Grade XI

Hi students, once again I am here to help you and to teach you an interesting math topic called as types of events in probability of Grade XI. You all have gone through the term probability. It describes the attitude of mind towards some proposition of whose truth we are not certain. Event is the outcome or a combination of outcomes of an experiment. in mathematical terms we can define this as a subset of a sample space. Example:

=> a head in the experiment of tossing a coin is an event.

=> a sum equal to 6 in the experiment of throwing a pair of dice is an event.

Now move to the occurrence of event, with help of an example (see for more Conditional Probability Examples), suppose we throw a die. Let ‘E’ be an event of a perfect number. Then E = 1, 4. Let on the upper most face 3 present. In this we say that event has not occurred. ‘E’ will occur only when 1 and 4 appear on the upper side. In a random problems, if E is the event of a sample space” > sample space S and w in the output. In this situation we can easily say that event E has occurred if w € E. Visit this site for more information on Probability.

Now move to different types of events: 

1. Simple Event

2. Compound Event 

3. Null Event

4. Sure Event or Certain Event

5. Compliment of an event

6. Mutually Exclusive Events 

7. Exhaustive Events 

1) Simple Event: it is defined, If an event has one element of the sample space">sample space. 

S= 1,2,3,4,5,6 

If the event is set of elements less than 2, then

E = 1 is a simple event

2) Compound Event: A compound event is an event which has more than one sample points. 

In the above example, of throwing a die, 1, 4 is a compound event.

3) Null Event (f): As null set is a subset of S, it is also an event called the null event or impossible event.

4) The sample space S= 1, 2, 3, 4, 5, 6 in the above experiment is a subset of S. The event represented by S occurs whenever the experiment is performed. Therefore, the event represented by S is called a sure event or certain event.

5) Complement of an Event: The complement of an event E with respect to S is the set of all the elements of S which are not in E. The complement of E is denoted by E' or EC.

In an experiment if E has not occurred then E' has occurred.

We can easily define union, intersection, complement of events and their properties in types of events.  like: 

 A U B, A ∩ A’ are events of the random experiments.

1. A – B is an event, which is same as “A but not B”.
2. A U B = B U A, A ∩ B, A ∩ B = A ∩ B
3. (A U B)’ = A ∩ B’, (A ∩ B)’ = A’ U B’ 

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