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An angle is positive if its rotation is in the anticlockwise and negative if its rotation is in the clockwise direction.
An angle is positive if its rotation is in the anticlockwise and negative if its rotation is in the clockwise direction.
(Scroll down to continue …)
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If one of the trigonometric ratios of an acute angle is known, the remaining trigonometric ration of the angle can be determined.
Two angles are said to be complementary, if their sum is 900 and each one of them is called the complement of the other.
sin (900 – θ) = Cos θ
Cos (900– θ)= Sin θ
tan (900– θ) = Cot θ
Cot(900– θ) = tan θ
sec (900– θ)= cosec θ
cosec (900– θ) = sec θ
An equation with trigonometric ratios of an angle θ, which is true for all values of ‘ θ ‘, for which the given trigonometric ratios are defined, is called an identity.
The three fundamental trigonometric identities are
⇒ sin2 θ =1-cos2 θ
⇒ sin2 θ =(1-cos θ)(1+cos θ)
⇒ (1- cos θ) = (sin2 θ) /(1+ cos θ)
⇒ (1+ cos θ) = (sin2 θ) /(1- cos θ)
⇒ cos2 θ + sin2 θ = 1
cos2 θ =1- sin2 θ
⇒ cos2 θ =(1- sin θ)(1+ sin θ)
⇒ (1+ sin θ) = (cos2 θ) /(1- sin θ)
⇒ (1- sin θ) = cos2 θ /(1+sin θ)
(b) sec2 θ = 1 + tan2 θ
⇒ sec2 θ – tan2 θ =1
⇒ (sec θ – tan θ)(sec θ + tan θ) = 1
⇒ (sec θ – tan θ) = 1/ (sec θ + tan θ)
⇒ (sec θ + tan θ) = 1/ (sec θ – tan θ)
⇒ sec2 θ – 1 = tan2 θ
⇒ (sec θ – 1)( sec θ – 1) = tan2 θ
(c) cosec2 θ = 1+cot2θ
⇒ cosec2 θ – cot2 θ = 1
⇒ (cosec θ – cot θ)(cosec θ + cot θ)=1
(cosec θ+ cot θ) =1cosec θ – cot θ
(cosec θ- cot θ) = 1cosec θ + cot θ
⇒ Cosec2 θ – 1 = cot2 θ
⇒ (Cosec θ – 1)( Cosec θ – 1) = Cot2 θ
(a+b)2=+a2+b2+2ab
(a-b)2 = a2+b2-2ab
(a+b)2+(a-b)2= 2 (a2+b2)
(a+b)2– (a-b)2= 4ab
(a-b)2– (a+b)2= – 4ab
(a+b)2 = (a-b)2+ 4ab
(a-b)2 = (a+b)2– 4ab
(a2-b2)=(a+b)(a-b)
a+b=(a2-b2) /(a-b)
a-b=(a2-b2) /(a-b)
(a+b)2= (a-b)2+ 4ab
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Introduction :
One of the main applications of trigonometry is to find the distance between two or more than two places or to find the height of the object or the angle subtended by any object at a given point without actually measuring the distance or heights or angles.
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Trigonometry is useful to astronomers, navigators, architects and surveyors etc. in solving problems related to heights and distances.
The directions of the objects can be described by measuring :
(i) angle of elevation and (ii) angle of depression
Angles of elevation or angles of depression of the objects are measured by an instrument called The odolite.
The odolite is based on the principles of trigonometry, which is used for measuring angles with a rotating telescope.
In 1856, Sir George Everest first used the giant theodolite, which is now on display in the Museum of the survey of India in Dehradun.
Angle of Elevation:
Let P be the position of the object above the horizontal line OA and O be the eye of the observer, then angle AOP is called angle of elevation. It is called the angle of elevation, because the observer has to elevate (raise) his line of sight from the horizontal OA to see the object P. [ When the eye turns upwards above the horizontal line.]
Angle of Depression:
Let P be the position of the object below the horizontal line OA and O be the eye of the observer, then angle AOP is called angle of depression.
It is called the angle of depression because the observer has to depress (lower) his line of sight from the horizontal OA to see the object P.
[When the eye turns downwards below the horizontal line].
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Circle:
Circle is a round shaped figure has no corners or edges.
A circle is the locus of all points in a plane which are at constant distance
(called radius) from a fixed point (called centre). A circle with centre O and radius r is denoted by C (O, r).
Radius:
A line segement that joins the centre and circumference or boundary of the circles is called the radius of the circles.
A line segement that divides the circle into two halves is called he diameter of the circle.
Diameter = 2x radius
Radius = Diameter/2
Chord:
A line segment joining any two points on a circle. The largest chord of a circle is a diameter.
Position of A point With Respect To a Circle:
In a plane a point P can lie either inside, or on the circle or outside the given circle.
Position of A Line With Respect to A Circle
If a circle C(O, r) and a straight line ‘l’ are in the same plane, then only three possibilities are there. These are :
Outside The Circle:
(i) The line ‘l’ does not intersect the circle at all. The line ‘l’ is called a non-intersecting line with respect to the circle.
Inside of the Circle – Secant To A Circle:
The line ‘ l ‘ intersects the circle in two distinct points say A and B. The line which intersects the circle in two distinct points is called a secant line.
Touching The Circle – Tangent To Circle:
A tangent to a circle is a special case of the secant when the two end points of the corresponding chord are coincide.
That is the line ‘ l ‘ touches the circle in only one point. Such a line which touches the circle only in one point is called a tangent line.
Tangent To Circle :
Etimology of Tangent:
The word ‘tangent’ comes from the Latin word ‘tangere’, which means to touch and was introduced by the Danish mathematician Thomas Fincke in 1583.
Tangent is a line that intersects the circle in exactly one point.
A tangent to a circle is the limiting position of a secant when its two points of intersection with the circle coincide.
The common point of the circle and the tangent is called the point of contact.
In other words the point, at which the tangent touches the circle is called
the point of contact.
Number of Tangents From A Point To Circle:
Number of tangents to a circle from a point (say P) depends upon the position of the point P.
(a)
When point ‘P’ lies outside the circle: There are only two lines, which touch the circle in one point only, all the remaining lines either intersect in two points or do not intersect the, circle. Hence, there are only two tangents from point P to the circle.
(b)
When point ‘ P ‘ lies on the circle : There is only one line which touches the circle in one point, all other lines meet the circle in more than one point. Hence, there is one and only one tangent to the circle through the point P lies on the circle.
(c)
When point ‘ P ‘ lies inside the circle: Every line passing through the point P (lies inside the circle) intersect the circle in two points. Hence, there is no tangent through the point P lies inside the circle
There is only one tangent at a point on the circumference of the circle.
Point of contact is the common point of the tangent and the circle.
The tangent at any point of a circle is perpendicular to the radius through the point of
contact.
Theorems :
(i) Tangent-Radius Theorem
The line perpendicular to the tangent and passing through the point of contact, is known as the normal.
Statement: The tangent at any point of a circle is perpendicular to the radius through the point of contact.
The converse of above theorem is also true.
Theorem :
The tangents at any point of a circle is perpendicular to the radius through the point of contact. Or At the point of contact the angle between radius and tangents to a circle is 90^0 .
Theorem :
The length of tangents drawn from an external point to a circle are equal.
Important Results:
If two circles touch internally or externally, the point of contact lies on the straight line through the two centres.
The tangent at any point of a circle is perpendicular to the radius through the point of contact.
The length of the tangents drawn from an external point to a circle are equal.
PT =PT’ =
The distances between two parallel tangents drawn to a circle is equal to the diameter of the circle.
Facts:
In two concentric circles, the chord of the larger circle, which touches the smaller circle, is bisected at the point of contact.
More Points To Remember !
There is no tangent to a circle passing through a point lying inside the circle.
At any point on the circle there can be one and only one tangent.
The tangent at any point of a circle is perpendicular to the radius through the point of contact.
There are exactly two tangents to a circle through a point outside the circle.
The length of the segment of the tangent from the external point P and the point of contact with the circle is called the length of the tangent.
The lengths of the tangents drawn from an external point to a circle are equal.
The line containing the radius through the point of contact of tangent is called the normal to the circle at the point.
There is no tangent to the circle passing through a point lying inside the cirele.
There are exactly two tangents to a cirele through a point lying outside the circle
The length of the segment of the tangent from the external point and the point of contact
with the circle is called the length of the tangent.
The length of tangents drawn from an external point to a circle are equal.
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Introduction :
Circumference of a circle Or Perimeter of a circle :
The distance around the circle or the length of a circle is called its circumference or perimeter. (Scroll down to continue …)
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AREAS AND PERI METER RELATED TO CIRCLES
Introduction :
Circumference of a circle Or Perimeter of a circle :
The distance around the circle or the length of a circle is called its circumference or perimeter.
Circumference (perimeter) of a circle = d or 2r, where d is a diameter and r is a radius of the circle and = 227 or 3.14
Area of a circle:
Area of a circle = πr2
Area of a semicircle :
Area of semicircle = 12 πr2
Area of a quadrant :
Area of a quadrant (quarter circle) = r24
Perimeter of a semicircle:
Perimeter of a semicircle or protractor =πr+2r
Area of the ring :
Area of the ring or an annulus
= πR2 – πr2
= π(R2 – r2)
= π(R- r)(R+r)
Length of the arc AB
= 2πrθ3600 or πrθ1800
Area of a sector:
Area of sector OACBO =πrθ3600
OR
Area of sector OACBO = 12(r×l)
Perimeter of a sector :
Perimeter of sector OACBO
= Length of arc AB+2r
= πrθ1800+2r
Other important formulae :
(i) Distance moved by a wheel in 1 revolution = Circumference of the wheel.
(ii) Number of revolutions in one minute = Distance moved in 1 minuteCircumference
(iii) Angle described by minute hand in 60 minutes =3600
(iv) Angle described by hour hand in 12 hours = 3600
The midpoint of the hypotenuse of a right triangle is equidistant from the vertices of the triangle.
Angle subtended at the circumference by a diameter is always a right angle.
Area of a segment:
(i) Area of minor segment ACBA
= Area of sector OACBO- Area of ∆OAB
= πrθ3600- 12r2sinθ
(ii) Area of major segment BDAB
= Area of the circle – Area of minor segment ACBA
=πr2 – Area of minor segment ACBA
If a chord subtends a right angle at the center, then
Area of the corresponding segment = 4-12r2
If a chord subtends an angle of 600 at the center, then
Area of the corresponding segment = 6-32r2
If a chord subtends an angle of 1200 at the center, then Area of the corresponding segment = 3-34r2
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What is a cuboid?
The flat surfaces of a cuboid are known as its faces.
A cuboid has six faces, and each face is a rectangle.
These faces are arranged such that three pairs of opposite faces are parallel to each other.
The adjacent faces are perpendicular to each other (i.e., the angle between any two touching faces of a cube is right angle, 90°.
An edge is a line segment where the two surfaces of a cuboid meet.
There are 12 edges in a cuboid, where three edges meet at each vertex.
All edges form right angles with the adjacent edges and faces.
A vertex is a point where the three edges meet. Vertices is the plural of vertex.
Cuboid has eight vertices.
Diagonal of a cuboid is a line segment that joins two opposite vertices.
The cuboid has four space diagonals.
Length of the diagonal of cuboid = √(length2 + breadth2 + height2) units.
Cuboids exhibit high symmetry.
They have rotational symmetry of order 4, meaning that you can rotate them by 90 degrees about their centre and they will look the same.
It is a three-dimensional, Rectangular figure.
It has 6 faces, 12 edges, and 8 vertices.
All 6 faces are rectangles.
Each vertex meets three faces and three edges.
The edges run parallel to those parallel to it.
All angles of a cuboid are right angles.
The total surface area of a cuboid is defined as the area of its surface (Appearing face).
Imagine yourself sitting in a cuboid shaped room. You can then see the four walls around you. This denotes the lateral surface area of that room.
That is, the lateral surface area of a cuboid shaped room is the area of its four walls, excluding the ceiling and the floor.
The lateral surface area of the cuboid is the sum of areas of its square faces, excluding the area of the top and the bottom face.
So the lateral surface area of a cube = sum of areas of 4 faces = (Length ✕ Height) + (Length ✕ Height) + (Length ✕ Height) + (Breadth ✕ Height) + (Breadth✕ Height)
Derivation of Total Surface Area of a Cuboid
Since the total surface area of a cuboid (TSA) is the area of its surface.
Total surface area of a cuboid = Lateral Surface Area + Area Of Bottom Surface + Area Of Top Surface
Total surface area of a cuboid = Area Of Front Surface + Area Of Back Surface + Area Of Left Srface + Area Of Right Surface + Area Of Bottom Surface + Area Of Top Surface
Total surface area of a cuboid = Lateral Surface Area 2[Area Of Bottom Surface]
Since Area Of Top Surface = + Area Of Bottom Surface We get, Total surface area of a cuboid = Lateral Surface Area + 2[Area Of Top Surface]
TSA = (Length ✕ Height) + (Length ✕ Height) + (Length ✕ Height) + (Breadth ✕ Height) + (Breadth✕ Height) + (Length ✕ Breadth) + (Length ✕ Breadth)
TSA = 2(Length ✕ Height) + 2(Breadth ✕ Height) + 2(Length ✕ Breadth)
TSA = 2[(Length ✕ Height) + (Breadth ✕ Height) + (Length ✕ Breadth)]
The volume of a three-dimensional object can be defined as the space required for it.
Similarly, Volume of a cuboid is defined as the space required for the cuboid or the Space occupied by the cuboid.
The volume of a cuboid can be calculated using the formula, V = lbh, where,
l = length, b = breadth or width, h = height
This formula shows that the volume of a cuboid is directly proportional to its length, breadth and height.
The volume is calculated by multiplying the object’s length, breadth, and height.
Hence the volume of the cube = lbh = lenth ✕ breadth ✕ height
Similar Shapes:
Real-world Examples:
Fun Fact:
Remember that these notes provide an overview of cuboids, and there are more advanced topics and applications related to this shape in various fields of study.
A cube is a three-dimensional regular polyhedron characterised by its 6 Identical (Congruent) Squares in which 3 Pairs of them parallel.
The flat surfaces of a cube are known as its faces.
A cube has six faces, and each face is a perfect square. These faces are arranged such that three pairs of faces are parallel to each other.
The adjacent faces are perpendicular to each other (the angle between any two touching faces of a cube is right angle, 90°.
All the edges have the same length.
A cube also has 8 vertices and 12 edges.
An edge is a line segment where the two surfaces of a cube meet.
There are twelve edges in a cube, where three edges meet at each vertex.
All edges have equal length and form right angles with the adjacent edges and faces.
A vertex is a point where the three edges meet. Vertices is the plural of vertex.
Cube has eight vertices.
The cube has four space diagonals that connect opposite vertices, each of which has a length of √3 times the length of an edge.
Cubes exhibit high symmetry.
They have rotational symmetry of order 4, meaning that you can rotate them by 90 degrees about their centre and they will look the same.
It is a three-dimensional, square-shaped figure.
It has 6 faces, 12 edges, and 8 vertices.
All 6 faces are squares with equal area.
All sides have the same length.
Each vertex meets three faces and three edges.
The edges run parallel to those parallel to it.
All angles of a cube are right angles.
The total surface area of a cube is defined as the area of its outer surface.
Derivation of Total Surface Area of a Cube
Since the total surface area of a cube is the area of its outer surface.
total surface area of a cube = 6 ✕ area of one face.
We know that the cube has six square faces and each of the square faces is of the same size, the total surface area of a cube = 6 ✕ area of one face.
Let the length of each edge is “s”.
Area of one square face = length of edge ✕ length of edge
Area of one square face == s ✕ s = s²
Therefore, the total surface area of the cube = 6s²
The total surface area of the cube will be equal to the sum of all six faces of the cube.
Imagine yourself sitting in a cube shaped room. You can then see the four walls around you. This denotes the lateral surface area of that room.
That is, the lateral surface area of a cube shaped room is the area of its four walls, excluding the ceiling and the floor.
The lateral surface area of the cube is the sum of areas of its square faces, excluding the area of the top and the bottom face.
So the lateral surface area of a cube = sum of areas of 4 faces = 4a²
The Volume of a Cube
The volume of a three-dimensional object can be defined as the space required for it.
Similarly, Volume of a cube is defined as the space required for the cube or the Space occupied by the cube.
The volume of a cube can be calculated using the formula V = s3, where “s” represents the length of one side of the cube.
This formula shows that the volume of a cube is directly proportional to the cube of its side length.
The volume is calculated by multiplying the object’s length, breadth, and height. In the case of a cube shape, the length, width, and height are all of the same length. Let us refer to it as “s”.
Hence the volume of the cube is s ✕ s ✕ s = s³
We encounter many cubes in our daily life such as Ice cubes, sugar cubes, dice and the building blocks used in games.
Cubes play a fundamental role in the study of geometry and serve as a basis for understanding three-dimensional space and concepts such as volume and surface area.
Also, Cubes have many applications in mathematics, engineering, architecture and art etc.
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Cylinder is an important topic in Mathematics. It is a three-dimensional solid shape that has two parallel circular bases connected by a curved surface.
In this post, we will explore the properties of a cylinder and how to calculate its volume and surface area.
Let’s start with the basic definition of a right circular cylinder.
A cylinder is a solid shape that has two parallel circular bases of equal size and shape.
The curved surface that connects the two bases is called the lateral surface.
The axis of the cylinder is a line passing through the center of both bases.
(i). Solid Cylinder
(ii). Hollow Cylinder
Total Surface Area of Right Circular Cylinder = Curved Surface Area + Cicular Base Area + Circular Top Surface Area.
Since the Cicular Base Area And The Circular Top Surface Area of a cylinder are equal
Total Surface Area of Right Circular Cylinder = Curved Surface Area + 2(Cicular Top Surface Area)
Or
Total Surface Area of Right Circular Cylinder = Curved Surface Area + 2(Cicular Base Surface Area)
Therefore the formula to calculate the surface area of a cylinder is expressed as the following:
$$SA = 2\pi r h + 2\pi r^2$$
where SA is the surface area, r is the radius of the base, and h is the height of the cylinder.
Here, Curved Surface Area, $$CSA = 2\pi r h$$ and
Cicular Base Area = Top Surface Area = $$\pi r^2$$
The formula to calculate the volume of a cylinder is given by:
$$V = \pi r^2 h$$
where V is the volume, r is the radius of the base, and h is the height of the cylinder.
Now let’s take an example to understand how to use these formulas. Suppose we have a cylinder with a radius of 4 cm and a height of 10 cm. To calculate its volume, we can use the formula:
$$V = \pi (4)^2 (10) = 160\pi$$
Therefore, the volume of the cylinder is 160π cubic cm.
To calculate its surface area, we can use the formula:
$$SA = 2\pi (4) (10) + 2\pi (4)^2 = 120\pi$$
Therefore, the surface area of the cylinder is 120π square cm.
In conclusion, understanding the properties of a cylinder and how to calculate its volume and surface area is important in CBSE Class 10 Mathematics. By using the formulas mentioned above, you can easily solve problems related to cylinders.
A hollow cylinder is a three-dimensional object with a circular base and a cylindrical shape. It is also known as a cylindrical shell. The cylinder has two circular faces and a curved surface. The thickness of the cylinder is uniform and it is hollow from inside.
The volume of a hollow cylinder can be calculated using the formula V = πh(R2-r2), where h is the height of the cylinder, R is the radius of the outer circle, and r is the radius of the inner circle.
The surface area of a hollow cylinder can be calculated using the formula A = 2πh(R+r), where h is the height of the cylinder, R is the radius of the outer circle, and r is the radius of the inner circle.
Hollow cylinders are used in various applications such as pipes, drums, and containers. They are also used in engineering structures such as bridges and towers.
In conclusion, a hollow cylinder is a useful shape in various fields and can be easily calculated using mathematical equations.
A cone is a three-dimensional geometric shape that has a circular base and a single vertex. It can be visualized as a pyramid with a circular base. In this note, we will cover the basic concepts and formulas related to cones.
The surface area of a cone is the sum of the areas of its base and lateral surface. The formula to calculate the surface area of a cone is:
$$A = \pi r (r + l)$$
Where:
The volume of a cone is the amount of space enclosed by it. The formula to calculate the volume of a cone is:
$$V = \frac{1}{3} \pi r^2 h$$
Where:
Here are a few example equations related to cones:
Solid Sphere
Introduction – A solid sphere is a three-dimensional geometric figure in which all points inside the sphere are at the same distance from its center. – It is a type of 3D shape known as a “sphere” with a uniform density throughout.
Characteristics – The solid sphere has a well-defined volume, surface area, and mass. – It is completely filled with matter.
Volume of Solid Sphere The formula to calculate the volume (\(V_s\)) of a solid sphere is given by: \[ V_s = \frac{4}{3} \pi r^3 \] Where: \(V_s\) = Volume of the solid sphere, \(\pi\) (\(\pi\)) ≈ 3.14159, \(r\) = Radius of the sphere.
Surface Area of Solid Sphere The formula to calculate the surface area (\(A_s\)) of a solid sphere is given by: \[ A_s = 4 \pi r^2 \]
Where: \(A_s\) = Surface area of the solid sphere, \(\pi\) (\(\pi\)) ≈ 3.14159, \(r\) = Radius of the sphere
Mass of Solid Sphere The mass (\(m_s\)) of a solid sphere can be calculated using the formula: \[ m_s = \text{Density} \times V_s \] Where: \(m_s\) = Mass of the solid sphere, \(\text{Density}\) = Density of the material making up the sphere (usually in \(kg/m^3\)), \(V_s\) = Volume of the solid sphere (calculated using the previous formula)
Introduction – A hollow sphere is also a three-dimensional geometric figure, but unlike a solid sphere, it has an empty space inside. – It consists of an outer shell or surface with a certain thickness and an inner empty region.
Characteristics – The hollow sphere has a well-defined outer radius (\(R\)), inner radius (\(r\)), volume, surface area, and mass. – It is partially filled with matter, mainly in the form of the outer shell.
Volume of Hollow Sphere The formula to calculate the volume (\(V_h\)) of a hollow sphere is given by: \[ V_h = \frac{4}{3} \pi (R^3 – r^3) \] Where: – \(V_h\) = Volume of the hollow sphere, \(\pi\) (\(\pi\)) ≈ 3.14159, \(R\) = Outer radius of the sphere, \(r\) = Inner radius of the sphere
Surface Area of Hollow Sphere The formula to calculate the surface area (\(A_h\)) of a hollow sphere is given by: \[ A_h = 4 \pi (R^2 – r^2) \] Where: \(A_h\) = Surface area of the hollow sphere, \(\pi\) (\(\pi\)) ≈ 3.14159, \(R\) = Outer radius of the sphere, \(r\) = Inner radius of the sphere
Mass of Hollow Sphere The mass (\(m_h\)) of a hollow sphere can be calculated using the formula: \[ m_h = \text{Density} \times V_h \] Where: \(m_h\) = Mass of the hollow sphere, \(\text{Density}\) = Density of the material making up the sphere (usually in \(kg/m^3\)), \(V_h\) = Volume of the hollow sphere (calculated using the previous formula)
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Probability:
Probability is a measure of uncertainity (an event may happen or may not happen).
Probability is classified into the following two types.
(i) Empirical probability
(ii) theoretical probability.
Theoretical probability Or Classical Probability:
Theoretical probability Or Classical Probability is the probability of events based on the results obtained from theoretical approach.
Theoretical Approach
Here, we try to predict the outcomes without performing an actual experiment.
We assume that the outcomes of an experiment are equally likely.
We find that the experimental probability of an event approaches its theoretical probability if the number of trials of an experiment is very large.
Key Terms
Experiment:
Experiment is An activity that causes some well defined outcomes.
Random Experiment
Random experiment is an experiment that
may not give the same result on repitition
under identical conditions,
Trial: Each repition of the activity is called A Trial.
Outcome
Each possible result of a random experiment or a Trial is called its outcome.
Sample Space
The collection of all possible outcomes in a random experiment is called Sample space.
Or
The total number of possible outcomes of a random experiment.
As the number of trials in an experiment increases we may expect the empirical and theoretical probabilities to be nearly the same.
Event:
Any subset of smaple space is called an Event.
Compound Event An event connected to a random experiment is a compound event if it is obtained by combining two or more elementary events connected to the random experiment.
Occurrence of an event An event corresponding to a random experiment is said to occur if any one of the elementary events corresponding to the event is the outcome.
Impossible Events The event which never occurs is an impossible event.
So the probability ofan impossible eventis always zero.
Sure Event The event which certainly occurs is a sure event.In general, it is truethat for anevent E, ( ̅) ( ) Here theevent E is representing “not E”. This is called the compound ofthe event ‘E’.So ‘E’ and E are complementary events.
Theoretical Probability of An Event:
If there are n events associated with a random experiment and m of them are favorable to an event E, then the probability of the event E is denoted by P(E) and is calculated using the follwoing formula.
i.e.,
Where the event
representing non-occurrence of event E, is called the complement of the event E.
Hence, E and
are complementary events.
Sum of the probabilities of all the elementary events of an experiment is
i.e.,
P (E1) + P (E2) + P (E3) … + P (En) = 1
Playing Cards:
A pack of cards consists of four suits. They are
Each suit consists of 13 cards, nine cards numbered 2, 3, 4 ….10, an Ace (B) a Jack (J), a Queen (Q), and a King (K).
Spades and Clubs are black in colour.
Hearts and Diamonds are red in colour.
So there are 26 black cards and 26 red cards.
King, queen and jack are called face cards.
There are totally 12 (4 x3) face cardsin a packof 52 cards.
I.e. in eachsuit we have3 face cards.
Coins:
A coin has two sides namely head and tail.
In the experiment of tossing a coin for once, there are 2 possible outcomes. They are 1 head, 1 tail.
P (Head) = 1/2 = P (Tail)
Dice:
A die is a well-balanced cubewith six facesnumbered from 1 to 6.
Dice is the pluralform. There are six equally likely outcomes 1, 2,3, 4, 5,6 in a single throw. Geometric Probability:
If the total number of outcomes of a trial in a random experiment is infinite, then the above definition is not sufficient to find the probability of an event.
In such cases, the definition of probability is modified and probability so obtained is called Geometric Probability.
The geometric probability p of an event is given by
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Rational numbers are closed under the operations of addition, subtraction and multiplication, But not in division. (Scroll down to continue …)
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The operations addition and multiplication are
(i) commutative for rational numbers.
(ii) associative for rational numbers.
The rational number 0 is the additive identity for rational numbers.
The additive inverse of the rational number a/b is -a/b and vice- versa.
The reciprocal or multiplicative inverse of the rational number
is if a/b is c/d if (a/b)(c/d) =1
Distributive property of rational numbers:
For all rational numbers a, b and c, a(b + c) = ab + ac
and a(b – c) = ab – ac.
Rational numbers can be represented on a number line.
Between any two given rational numbers there are countless rational numbers.
The idea of mean helps us to find rational numbers between two rational numbers.
Positive Rationals: Numerator and Denominator both are either positive or negative.
Example: 2/3, -4/-5
Positive Rationals: Numerator and Denominator both are either positive or negative.
Example: -2/3, 4/-5
.
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A statement of equality of two algebraic expressions involving one or more variables. Example: x + 2 = 3
Linear Equation in One variable: The expressions which form the equation that contain single variable and the highest power of the variable in the equation is one. (Scroll down to continue …)
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Linear Equations in One Variable
An algebraic equation is an equality involving variables. It says that the value of the expression on one side of the equality sign is equal to the value of the expression on the other side.
The equations we study in Classes VI, VII and VIII are linear equations in one variable. In such equations, the expressions which form the equation contain only one variable. Further, the equations are linear, i.e., the highest power of the variable appearing in the equation is 1.
A linear equation may have for its solution any rational number.
An equation may have linear expressions on both sides. Equations that we studied in Classes VI and VII had just a number on one side of the equation.
Just as numbers, variables can, also, be transposed from one side of the equation to the other.
Occasionally, the expressions forming equations have to be simplified before we can solve them by usual methods. Some equations may not even be linear to begin with, but they can be brought to a linear form by multiplying both sides of the equation by a suitable expression.
The utility of linear equations is in their diverse applications; different problems on numbers, ages, perimeters, combination of currency notes, and so on can be solved
using linear equations.
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A quadrilateral has 10 parts – 4 sides, 4 angles and 2 diagonals. Five measurements can determine a quadrilateral uniquely. (Scroll down to continue …)
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Practical Geometry
Five measurements can determine a quadrilateral uniquely.
A quadrilateral can be constructed uniquely if the lengths of its four sides and a diagonal is given.
A quadrilateral can be constructed uniquely if its two diagonals and three sides are
known.
A quadrilateral can be constructed uniquely if its two adjacent sides and three angles
are known.
A quadrilateral can be constructed uniquely if its three sides and two included angles
are given.
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Data Handling: Deals with the process of collecting data, presenting it and getting result.
Data mostly available to us in an unorganised form is called raw data. (Scroll down to continue …)
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Grouped data can be presented using histogram. Histogram is a type of bar diagram, where the class intervals are shown on the horizontal axis and the heights of the bars show the frequency of the class interval. Also, there is no gap between the bars as there is no gap between the class intervals.
In order to draw meaningful inferences from any data, we need to organise the data systematically.
Frequency gives the number of times that a particular entry occurs.
Raw data can be ‘grouped’ and presented systematically through ‘grouped frequency distribution’.
Statistics: The science which deals with the collection, presentation, analysis and interpretation of numerical data.
Observation: Each entry (number) in raw data.
Range: The difference between the lowest and the highest observation in a given data.
Array: Arranging raw data in ascending or descending order of magnitude. Data can also presented using circle graph or pie chart. A circle graph shows the relationship between a whole and its part.
There are certain experiments whose outcomes have an equal chance of occurring. A random experiment is one whose outcome cannot be predicted exactly in advance. Outcomes of an experiment are equally likely if each has the same chance of occurring.
Frequency: The number of times a particular observation occurs in the given data.
Class Interval: A group in which the raw data is condensed.
(i) Continuous: The upper limit of a class interval coincides with the lower limit of the next class.
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Square: Number obtained when a number is multiplied by itself. It is the number raised to the power 2. 22 = 2 x 2=4(square of 2 is 4).
If a natural number m can be expressed as n2, where n is also a natural number, then m is a square number. (Scroll down to continue …)
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All square numbers end with 0, 1, 4, 5, 6 or 9 at unit’s place. Square numbers can only have even number of zeros at the end. Square root is the inverse operation of square.
There are two integral square roots of a perfect square number.
Positive square root of a number is denoted by the symbol For example, 32=9 gives
Perfect Square or Square number: It is the square of some natural number. If m=n2, then m is a perfect square number where m and n are natural numbers. Example: 1=1 x 1=12, 4=2 x 2=22.
Properties of Square number:
triplet, if m2 + n2 = p2: 32 + 42 = 25 = 52
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Cube number: Number obtained when a number is multiplied by itself three times. 23 = 2 x 2 x 2 = 8, 33 = 3 x 3 x 3=27, etc.
Numbers like 1729, 4104, 13832, are known as Hardy – Ramanujan Numbers. They
can be expressed as sum of two cubes in two different ways.
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Numbers obtained when a number is multiplied by itself three times are known as cube numbers. For example 1, 8, 27, … etc.
If in the prime factorisation of any number each factor appears three times, then the
number is a perfect cube.
The symbol
denotes cube root. For example
Perfect Cube: A natural number is said to be a perfect cube if it is the cube of some natural number. Example: 8 is perfect cube, because there is a natural number 2 such that 8 = 23, but 18 is not a perfect cube, because there is no natural number whose cube is 18.
The cube of a negative number is always negative.
Properties of Cube of Number:
ends in 3.
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Ratio: Comparing by division is called ratio. Quantities written in ratio have the sameunit. Ratio has no unit. (Scroll down to continue …)
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Equality of two ratios is called proportion. Product of extremes = Product of means
Percentage: Percentage means for every hundred. The result of any division in
whichthe divisor is 100 is a percentage. The divisor is denoted by a special
symbol %, read as percent. Profit and Loss:
(i) Cost Price (CP): The amount for which an article is bought. (ii) Selling Price (SP): The amount for which an article is sold. Additional expenses made after buying an article are included in the cost price
and are known as overhead expenses. These may include expenses like amount
spent onrepairs, labour charges, transportation, etc. Discount is a reduction given on marked price. Discount = Marked Price – Sale
Price. Discount can be calculated when discount percentage is given. Discount
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Expressions are formed from variables and constants.
Constant: A symbol having a fixed numerical value.
Example: 2,, 2.1, etc. (Scroll down to continue …)
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Variable: A symbol which takes various numerical values. Example: x, y, z, etc.
Algebric Expression: A combination of constants and variables connected by the sign
+, -, and is called algebraic expression.
Terms are added to form expressions.
Terms themselves are formed as product of factors.
Expressions that contain exactly one, two and three terms are called monomials, binomials and trinomials respectively.
In general, any expression containing one or more terms with non-zero coefficients (and with variables having non- negative exponents) is called a polynomial.
Like terms are formed from the same variables and the powers of these variables are the same, too.
Coefficients of like terms need not be the same.
While adding (or subtracting) polynomials, first look for like terms and add (or subtract) them; then handle the unlike terms.
There are number of situations in which we need to multiply algebraic expressions: for example, in finding area of a rectangle, the sides of which are given as expressions.
Monomial: An expression containing only one term. Example: -3, 4x, 3xy, etc.
Binomial: An expression containing two terms. Example: 2x-3, 4x+3y, xy-4, etc.,
Polynomial: In general, any expression containing one or more terms with non-zero coefficients (and with variables having non-negative exponents).
A polynomial may contain any number of terms, one or more than one.
A monomial multiplied by a monomial always gives a monomial.
Multiplication of a Polynomial and a monomial:
While multiplying a polynomial by a monomial, we multiply every term in the polynomial by the mononomial.
Trinomial: An expression containing three terms.
Example:
3x+2y+5z, etc.
In carrying out the multiplication of a polynomial by a binomial (or trinomial), we multiply term by term, i.e., every term of the polynomial is multiplied by every term in the binomial (or trinomial).
Note that in such multiplication, we may get terms in the product which are like and have to be combined.
An identity is an equality, which is true for all values of the variables in the equality.
On the other hand, an equation is true only for certain values of its variables.
An equation is not an identity.
The following are the standard identities:
(a + b)2 = a2 + 2ab + b2
(a – b)2 = a2 – 2ab +b2
(a + b)(a – b) = a2 – b2
(x + a) (x + b) = x2 + (a + b) x + ab
The above four identities are useful in carrying out squares and products of algebraic expressions.
They also allow easy alternative methods to calculate products of numbers and so on.
Coefficients: In the term of an expression any of the factors with the sign of the term is called the coefficient of the product of the other factors.
Terms: Various parts of an algebraic expression which are separated by + and – signs. Example: The expression 4x + 5 has two terms 4x and 5.
(iii) Unlike term: The terms having different literal factors.
Example:
are unlike terms.
and 3xy
Factors: Each term in an algebraic expression is a product of one or more number (s) and/or literals. These number (s) and/or literal (s) are known as the factor of that term. A constant factor is called numerical factor, while a variable factor is known as
a literal factor. The term 4x is the product of its factors 4 and x.
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Perimeter: Length of boundary of a simple closed figure.
Perimeter of Rectangle = 2(l +b)
Perimeter of Square = 4a
Perimeter of Parallelogram = 2(sum of two adjacent sides)
Area: The measure of region enclosed in a simple closed figure.
Area of a trapezium = half of the sum of the lengths of parallel sides × perpendiculardistance between them.
Area of a rhombus = half the product of its diagonals. (Scroll down to continue …)
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Numbers with exponents obey the following laws of exponents.
Very small numbers can be expressed in standard form using negative exponents. (Scroll down to continue …)
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Use of Exponents to Express Small Number in Standard form:
(iii) A number written as number such that is said to be in standard form if m is a decimal and n is either a positive or a negative integer.
Examples: 150,000,000,000 = 1.5 x 1011.
Exponential notation is a powerful way to express repeated multiplication of the same number.
For any non-zero rational number ‘a’ and a natural number n, the product a x a x a x x a(n times) = an.
It is known as the nth power of ‘a’ and is read as ‘a’ raised to the power n’.
The rational number a is called the base and n is called exponent.
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Variations: If the values of two quantities depend on each other in such a way that a change in one causes corresponding change in the other, then the two quantities are said to be in variation. (Scroll down to continue …)
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Direct Variation or Direct Proportion:
Extra:
Two quantities x and y are said to be in direct proportion if they increase (decrease) together in such a manner that the ratio of their corresponding values remains
constant. That is if
=k [k is a positive number, then x and y are said to vary directly.
In such a case if y1, y2 are the values of y corresponding to the values x1, x of x
respectively then = .
If the number of articles purchased increases, the total cost also increases. More than money deposited in a bank, more is the interest earned.
Quantities increasing or decreasing together need not always be in direct proportion, same in the case of inverse proportion.
When two quantities x and y are in direct proportion (or vary directly), they are
written as
. Symbol
stands for ‘is proportion to’.
Inverse Proportion: Two quantities x and y are said to be in inverse proportion if an increase in x causes a proportional decrease in y (and vice-versa) in such a manner that the product of their corresponding values remains constant. That is, if xy
= k, then x and y are said to vary inversely. In this case if y1, y2 are the values of y
corresponding to the values x1, x2 of x respectively then
x1, Y1 = x2, y2 or
=
When two quantities x and y are in inverse proportion (or vary inversely), they are
written as x
. Example: If the number of workers increases, time taken to finish
the job decreases. Or If the speed will increase the time required to cover a given distance decreases.
| Time (hours) | Distance (km) | Formula |
|---|---|---|
| 0 | 0 | |
| 1 | 60 | |
| 2 | 120 | |
| 3 | 180 |
| Number of Apples | Total Cost ($) | Formula |
|---|---|---|
| 0 | 0 | |
| 1 | 2 | |
| 2 | 4 | |
| 5 | 10 |
| Number of Servings | Amount of Flour (cups) | Formula |
|---|---|---|
| 0 | 0 | |
| 4 | 2 | |
| 6 | 3 | |
| 8 | 4 |
| Distance (km) | Fuel Consumed (liters) | Formula |
|---|---|---|
| 0 | 0 | |
| 15 | 1 | |
| 30 | 2 | |
| 45 | 3 |
| Hours Worked | Total Earnings ($) | Formula |
|---|---|---|
| 0 | 0 | |
| 1 | 15 | |
| 5 | 75 | |
| 10 | 150 |
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Factorisation: Representation of an algebraic expression as the product of two or more expressions is called factorization. Each such expression is called a factor of the given algebraic expression. (Scroll down to continue …)
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When we factorise an expression, we write it as a product of factors. These factors may be numbers, algebraic variables or algebraic expressions.
An irreducible factor is a factor which cannot be expressed further as a product of factors.
A systematic way of factorising an expression is the common factor method. It consists of three steps:
Sometimes, all the terms in a given expression do not have a common factor; but the terms can be grouped in such a way that all the terms in each group have a common factor. When we do this, there emerges a common factor across all the groups leading to the required factorisation of the expression. This is the method of regrouping.
In factorisation by regrouping, we should remember that any regrouping (i.e., rearrangement) of the terms in the given expression may not lead to factorisation. We must observe the expression and come out with the desired regrouping by trial and error.
A number of expressions to be factorised are of the form or can be put into the form: a2 + 2ab + b2, a2 – 2ab + b2, a2 – b2 and x2 + (a + b)x + ab. These expressions can be easily factorised using Identities I, II, III and IV
a2 + 2ab + b2 = (a + b)2
a2 – 2ab + b2 = (a – b)2
a2 – b2 = (a + b) (a – b)
Factorisation
x2 + (a + b)x + ab = (x + a)(x + b)
In expressions which have factors of the type (x + a) (x + b), remember the numerical term gives ab.
Its factors, a and b, should be so chosen that their sum, with signs taken care of, is the coefficient of x.
We know that in the case of numbers, division is the inverse of multiplication. This idea is applicable also to the division of algebraic expressions.
In the case of division of a polynomial by a monomial, we may carry out the division either by dividing each term of the polynomial by the monomial or by the common factor method.
In the case of division of a polynomial by a polynomial, we cannot proceed by dividing each term in the dividend polynomial by the divisor polynomial. Instead, we factorise both the polynomials and cancel their common factors.
In the case of divisions of algebraic expressions that we studied in this chapter, we have Dividend = Divisor × Quotient.
In general, however, the relation is Dividend = Divisor × Quotient + Remainder
Thus, we have considered in the present chapter only those divisions in which the remainder is zero.
There are many errors students commonly make when solving algebra exercises.
You should avoid making such errors.
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Graphical presentation of data is easier to understand.
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Introduction to Graphs
A line graph displays data that changes continuously over periods of time. A line graph which is a whole unbroken line is called a linear graph.
For fixing a point on the graph sheet we need, x-coordinate and y-coordinate.
The relation between dependent variable and
through a graph.
independent variable is shown
A Bar Graph: A pictorial representation of numerical data in the form of bars (rectangles) of uniform width with equal spacing. The length (or height) of each bar
represents the given number.
A Pie Graph: A pie graph is used to compare parts of a whole. The various
observations or components are represented by the sectors of the circle.
A Histogram: Histogram is a type of bar diagram, where the class intervals are shown on the horizontal axis and the heights of the bars (rectangles) show the frequency of the class interval, but there is no gap between the bars as there is no gap between the
class intervals.
Linear Graph: A line graph in which all the line segments form a part of a single line. Coordinates: A point in Cartesian plane is represented by an ordered pair of numbers.
Ordered Pair: A pair of numbers written in specified order.
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Write An Article On Chandrayaan 3 with SEO For www.therightmentor.com
Chandrayaan 3: India’s Next Mission to the Moon
India is all set to launch its third lunar mission, Chandrayaan 3, in the near future. The Indian Space Research Organisation (ISRO) has been working tirelessly on this project since the success of Chandrayaan 2 in September 2019. This mission is expected to be a major milestone in India’s space exploration journey, and it has generated a lot of interest and excitement among space enthusiasts.
What is Chandrayaan 3?
Chandrayaan 3 is India’s third lunar mission, which aims to land a rover on the moon’s surface. The mission is a follow-up to Chandrayaan 2, which was partially successful. The main objective of Chandrayaan 3 is to conduct scientific experiments and collect data on the moon’s surface and atmosphere. The mission will also help India to develop its technological capabilities in space exploration.
What are the key features of Chandrayaan 3?
Chandrayaan 3 will consist of an orbiter, a lander, and a rover. The orbiter will be responsible for mapping the moon’s surface, while the lander will carry the rover to the moon’s surface. The rover will then explore the moon’s surface, collect data, and conduct experiments. Chandrayaan 3 will also carry a range of scientific instruments that will help in studying the moon’s geology, mineralogy, and atmosphere.
What are the challenges of Chandrayaan 3?
Chandrayaan 3 is a complex mission that involves several challenges. One of the biggest challenges is the landing of the rover on the moon’s surface. The lander has to be designed in such a way that it can withstand the harsh lunar environment and land safely on the moon’s surface. Another challenge is the communication between the lander and the orbiter. The lander has to communicate with the orbiter in order to transmit data back to Earth.
When will Chandrayaan 3 be launched?
The launch date for Chandrayaan 3 has not been announced yet. However, ISRO has stated that it is working towards launching the mission as soon as possible. The COVID-19 pandemic has delayed the mission’s progress, but ISRO is confident that it will be able to launch Chandrayaan 3 in the near future.
Conclusion
Chandrayaan 3 is an important mission for India’s space exploration journey. It will help India to develop its technological capabilities in space exploration and contribute to scientific research on the moon’s surface and atmosphere. The success of Chandrayaan 3 will be a major milestone for India and will inspire future generations of space enthusiasts.
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