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KV. CRPF. AVADI - Study material maths for CLASS-IX IX - Mathematics

Chapter 4: Linear Equations in Two VariablesChapter Notes

Top Definitions

1. An equation of the form ax + by + c = 0, where a, b and c are real numbers, such that a and b are not both zero, is called a linear equation in two variables.

2. A linear equation in two variables is represented geometrically by a straight line the points of which make up the collection of solutions of equation. This is called the graph of the linear equation.

Top Concepts

1. A linear equation in two variables has infinitely many solutions.

2. The graph of every linear equation in two variables is a straight line.

3. x = 0 is the equation of the y – axis and y = 0 is the equation of the x–axis.

4. The graph of x = k is a straight line parallel to the y –axis.

5. The graph of y = k is a straight line parallel to the x – axis.

6. An equation of the type y = mx represents a line passing through the origin, where m is a real number.

7. Every point on the line satisfies the equation of the line and every solution of the equation is a point on the line.

8. The solution of a linear equation is not effected when:(i) The same number is added or subtracted from both the side of

an equation.(ii) Multiplying or dividing both the sides of the equation by the

same non zero number.

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Top Diagrams

1. Graph of a line passing through the origin.

2. Graph of a line parallel to x axis.

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3. Graph of a line parallel to y axis.

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IXMathematics

Chapter 8: QuadrilateralsChapter Notes

Top Definitions1. A quadrilateral is a closed figure obtained by joining four points (with no

three points collinear) in an order.2. A diagonal is a line segment obtained on joining the opposite vertices.3. Two sides of a quadrilateral having no common end point are called its

opposite sides.4. Two angles of a quadrilateral having common arm are called its adjacent

angles.5. Two angles of a quadrilateral not having a common arm are called its

opposite angles.6. A trapezium is quadrilateral in which one pair of opposite sides are parallel.7. In the non – parallel sides of trapezium are equal, it is known as isosceles

trapezium.8. A parallelogram is a quadrilateral in which both the pairs of opposite sides

are parallel.9. A rectangle is a quadrilateral whose each angle is 90°

10. A rhombus is quadrilateral whose all the sides are equal.11. A square is a quadrilateral whose all sides are equal and each angle is 90°.12. A kite is a quadrilateral in which two pairs of adjacent sides are equal.

Top Concepts1. Properties of parallelogram:

i The opposite sides of a parallelogram are parallel.ii A diagonal of a parallelogram divides it in two congruent triangles.iii The opposite sides of a parallelogram are equal.iv The opposite angles of a parallelogram are equal.v The consecutive angles (conjoined angles) of a parallelogram are

supplementary.vi The diagonals of a parallelogram bisect each other.

2. A diagonal of a parallelogram divides the parallelogram into two congruent triangles.

3. In a parallelogram opposite sides are equal.4. If each pair of opposite sides of a quadrilateral is equal, then it is a

parallelogram.5. In a parallelogram opposite angles are equal.6. If in quadrilateral, each pair of opposite angles is equal, then it is a

parallelogram.7. The diagonals of a parallelogram bisect each other.8. If the diagonals of a quadrilateral bisect other, then it is a parallelogram.9. A quadrilateral is a parallelogram, if a pair of opposite sides is equal and

parallel.10. Square, rectangle and rhombus are all parallelograms.11. Kite and trapezium are not parallelogram.12. A square is a rectangle.13. A square is a rhombus.14. A parallelogram is a trapezium.15. Every rectangle is a parallelogram; therefore, it has all the properties of a

parallelogram. Additional properties of a rectangle are:i All the (interior) angles of are rectangle are right angles.ii The diagonals of a rectangle are equal.

16. Every rhombus is a parallelogram; therefore, it has all the properties of a parallelogram. Additional properties of a rhombus are:

i All the sides of rhombus are equal.ii The diagonals of a rhombus intersect at right angles.iii The diagonals bisect the angles of a rhombus.

17. Every square is a parallelogram; therefore, it has all the properties of a parallelogram. Additional properties of a rhombus are:

i All the sides are equalii All the angles are equal to 90° eachiii Diagonals are equal

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iv Diagonal bisect each other at right anglev Diagonals bisects the angles of vertex

18. Sum of all the angles of a quadrilateral is 3600.19. Mid Point Theorem (Basic Proportionality Theorem): The line segment joining

the mid point of any two sides of a triangle is parallel to the third sides and equal to half of it.

20. Converse of mid-point theorem: The line drawn through the mid-point of one side of a triangle parallel to the another side, bisects the third side.

21. If there are three or more parallel lines and the interests made by them on a transversal are equal, then the corresponding intercepts on any other transversal are also equal.

22. A quadrilateral formed by joining the mid-points of the sides of a quadrilateral, in order is a parallelogram.

Top Diagrams1. A quadrilateral ABCD.

2. A trapezium ABCD with sides AB || DC and non parallel sides AD and BC.

3. A parallelogram ABCD in which AB||DC and AD||BC.

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a4. A rectangle ABCD with AD||BC, AB||DC and A = 90° = B = C = D.

5. A rhombus ABCD with AB = BD = CD = DA.

6. A square ABCD in which AB = BC = CD, = DA and A = B = C = D = 90°.

7. A kite ABCD with AB = AD and BC = CD

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8. Diagonal properties of special parallelograms:Properties Parallelogram Rectangle Rhombus SquareDiagonals bisect each other √ √ √ √Diagonals are equal – √ – √Diagonals bisect vertex angles – – √ √Diagonals are perpendicular to each – – √ √otherDiagonals from 4 equal triangle √ √ √ √Diagonals from 4 congruent – – √ √triangle

9. The relations between special parallelograms can be represented by a Veen-diagram.

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IXMathematics

Chapter 9: Area of Parallelograms and Triangles QuadrilateralsChapter Notes

Top Definitions

1. Any side of a parallelogram is called the base.2. The length of perpendicular drawn from any point form the parallel

sides to the base is called the (corresponding) altitude or height.3. The part of the plane enclosed by a simple closed figure is called a

planar region corresponding to that figure.4. The magnitude or measure of that planar region is called its area.5. Two figures are called congruent, if they have the same shape and the

same size.6. Area of a figure is a number (in same unit) associated with the part of

the plane enclosed by the two properties.

Top Concepts

1. If two figures A and B are congruent, they must have equal areas.2. Two figures having equal areas need not be congruent.3. If a planner region formed by a figure T is mad up of two non –

overlapping planner regions formed by figures P and Q, then ar(T) = ar(P) + ar(Q).

4. Two figures are said to be on the same base and between the same parallels, if they have a common base (side) and the vertices (or the vertex) opposite to the common base of each figure lie on a line parallel to the base.

5. Parallelograms on the same base and between the same parallels are equal in area.

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6. Area of a parallelogram is the product of its any side and the corresponding altitude.

7. Parallelograms on the same base or equal bases and between the same parallels are equal in area.

8. Parallelograms on the same base (or equal bases) and having equal areas lie between the same parallels.

9. Two triangles on the same base (or equal base) and between the same parallel are equal in area.

10. Area of triangle is half the product of its base (or any side) and the corresponding altitude (or height).

11. Two triangles with same base (or equal bases) and equal areas will have equal corresponding altitudes.

12. Two triangles having the same base (or equal bases) and equal areas lie between the same parallels.

13. Parallelograms on the same base (or equal bases) and having equal areas lie between the same parallels.

14. A median of a triangle divides it into triangles of equal areas.

Top Diagrams

1. Congruent Figures

2. Parallelograms on the same base and between the same Parallels

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3. Triangles on the same base and between the same parallels

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IXMathematics

Chapter 10: CirclesChapter Notes

Top Definitions

1. A circle is a collection (set) of all those points in a plane, each one of which is at a constant distance from a fixed point in the plane.

2. The fixed point is called the centre and the constant distance is called the radius of the circle.

3. All the points lying inside a circle are called its interior points and all those points which lie outside the circle are called its exterior points.

4. The collection (set) of all interior points of a circle is called the interior of the circle while the collection of all exterior points of a circle is called the exterior of the circle.

5. A line segment joining two points on a circle is called the chord of the circle.

6. A chord passing through the center of the circle is called a diameter of the circle.

7. A line which meets a circle in two points is called a secant of the circle.8. A polygon is a closed figure made up of three or more line segments

(sides) such that each line segment intersects exactly two others at its end – points (vertices) and no two line segments which intersect are collinear.

9. A polygon is called a regular polygon, if it has all its sides equal and has all its angles equal.

10. A (continuous) part of a circle is called an arc of the circle. The arc of a circle is denoted by the symbol ‘ ’.

11. Circumference: The whole arc of a circle is called the circumference of the circle.

12. Semi- circle: One – half of the whole arc of a circle is called a semi – circle of the circle.

13. Minor and Major arcs: An arc less than one - half of the whole arc of a circle is called a minor arc of the circle, and an arc greater than one – half of the whole arc of a circle is called a major arc of the circle.

14. Central Angle: Any angle whose vertex is centre of the circle is called a central angle.

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15. Degree measure of an Arc: The degree measure of a minor arc is the measure of the central angle subtended by the arc.

16. Congruent Circle: Two circles are said to be congruent if and only if either of them can be superposed on the other so as to cover it exactly.

17. Congruent Arc: Two arcs of a circle (or of congruent) circles) are congruent if either of them can be superposed on the other so as to cover it exactly.

18. Sector of a circle: The part of the plane region enclosed by an arc of a circle and its two bounding radii is called a sector of a circle.

19. Segment of a circle: A chord of a circle divides it into two parts. Each part is called a segment.

20. The part containing the minor arc is called the minor segment, and the part containing the major arc is called the major segment.

21. A quadrilateral, all the four vertices of which lie on a circle is called a cyclic quadrilateral. The four vertices A, B, C and D are said to be Concyclic points.

Top Concepts

1. A diameter of circle is its longest chord.2. A line can meet a circle at the most in two points.3. In a circle, perpendicular from the center to a chord bisects the chord.4. In a circle, the line joining the mid – point of a chord to the centre is

perpendicular to the chord.5. Equal chords of a circle are equivalent from the centre of the circle.6. In a circle, chords which subtend equal angles at the centre are equal.7. The two points of intersections determine a chord of the circle.8. In a circle, equal chords subtend equal angles at the centre.9. In a circle, chords which subtend equal angles at the centre are equal.10. Triangle is a polygon with 3 sides.11. Quadrilateral is a polygon with 4 sides.12. The chords corresponding to congruent arcs are equal.

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13. If two arcs of a circle (or of congruent circles) are congruent, then the corresponding chords are equal.

14. If two chords of a circle (or of congruent circles) are equal, then their corresponding arcs (minor, major or semi – circular) are congruent.

15. One and only one circle can be drawn through three non – collinear points.

16. An infinite number of circles can be drawn through a given point P.17. An infinite number of circles can be drawn through the two given

points.18. Perpendicular bisectors of two chords of a circle, intersect each other

at the centre of the circle.19. The angle subtended by an arc at the centre is double the angle

subtended by it at any point on the remaining part of the circle.20. Angles in the same segment of a circle are equal.21. An angle in a semi–circle is a right angle.22. The arc of a circle subtending a right angle at any point of the circle in

its alternate segment is a semi–circle.23. If a line segment joining two points subtends equal angles at two other

points lying on the same side of the line segment, the four points are concyclic, i.e., lie on the same circle.

24. An angle in a semi–circle is a right angle.25. The arc of a circle subtending a right angle at any point of the circle in

its alternate segment is a semi–circle.26. If a line segment joining two points subtends equal angles at two other

points lying on its same side of the line segment, the four points are concyclic i.e., lie on the same circle.

27. If the sum of any pair of opposite angles of a quadrilateral is 180°, then the quadrilateral is cyclic.

28. Any exterior angle of a cyclic quadrilateral is equal to the interior opposite angle.

Top Formulae

1. Diameter = 2 x Radius.

2. If the degree measure of AB is θ°, we write m AB is θ°.

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3. The degree measure of a semi – circles is 180°4. The degree measure of a circle is 360°.5. The degree measure of a major arc is (360° - θ°), where θ° is the

degree measure of the corresponding minor arc.6. For a quad. ABCD, A + C = 180° or B = D = 180°, then ABCD is

cyclic.

7. Area of a circle = r2

Top Diagrams

1. Interior and Exterior of a Circle

2. Concentric circles

3. Secant, Diameter and Chord in a circle.

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4. Arc of a circle

5. Circumference of a circle

6. Semi-Circle

7. Minor and Major arc

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8. Minor and Major Sector

8. Minor and Major Segment

9. Circles passing through a point.

10. Circles passing through two points.

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11. Chord bisectors meet at center.

12. Cyclic Quadrilateral

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IXMathematics

Chapter 11: Geometric ConstructionsChapter Notes

Top Concepts1. To construct an angle equal to a given angle.

Given : Any POQ and a point A.Required : To construct an angle at A equal to POQ.

1. With O as centre and any (suitable) radius, draw an arc to meet OP at R and OQ at S.

2. Through A draw a line AB.3. Taking A as centre and same radius (as in step 1), draw an arc

to meet AB at D.4. Measure the segment RS with compasses.5. With d as centre and radius equal to RS, draw an arc to meet the

previous arc at E.6. Join AE and produce it to C, then BAC is the required angle

equal to POQ

2. To bisect a given angle. Given : Any POQ

1. With O as centre and any (suitable) radius, draw an arc to meet OP at R and OQ at S.

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2. With R as centre and any suitable radius (not necessarily) equal

to radius of step 1 (but > 12 RS), draw an arc. Also, with S as centre and same radius draw another arc to meet the previous arc at T.

3. Join OT and produce it, then OT is the required bisector of POQ.

3. To construct angles of 60°, 30°, 120°, 90°, 45°(i) To construct an angle of 60°

1. Draw any line OP.2. With O as centre and any suitable radius, draw an arc to meet

OP at R.3. With R as centre and same radius (as in step 2), draw an arc to

meet the previous arc at S.4. Join OS and produce it to Q, then POQ = 60°.

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(ii) To construct an angle of 30° Steps of Construction1. Construct POQ = 60° (as above).2. Bisect POQ (as in construction 2). Let OT be the bisector of

POQ, then POT = 30°

(iii) To construct an angle of 120°1. Draw any line OP.2. With O as centre and any suitable radius, draw an arc to meet

OP at R.3. With R as centre and same radius (as in step 2), draw an arc to

meet the previous arc at T. With T as centre and same radius, draw another arc to cut the first arc at S.

4. Join OS and produce it to Q, then POQ = 120°.

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(iv) To construct an angle of 90° Steps of Construction1. Construct POQ = 60°

(as in construction 3(i)).2. Construct POV = 120° (as above).3. Bisect QOV (as in construction 2). Let OU be the bisector of

QOV, then POU = 90°.

(v) To construct an angle of 45° Steps of Construction1. Construct AOP = 90° (as above).2. Bisect AOP (as in construction 2).Let OQ be the bisector of AOP, then AOQ = 45°

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4. To bisect a given line segment.Given : Any line segment AB.Required : To bisect line segment AB.Steps of Construction:1. At A, construct any suitable angle BAC.2. At B, construct ABD = BAC on the other side of the line AB.3. With A as centre and any suitable radius, draw an arc to meet

AC at E.4. From BD, cut off BF = AE.5. Join EF to meet AB at G, then EG is a bisector of the line

segment AB and G is mid – point of AB.

(ii) To divided a given line segment in a number of equal part.

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5. Divided a line segment AB of length 8 cm into 4 equal part.Given : A line segment AB of length 8 cm.Required : To divide line segment 8 cm into 4 equal parts.Steps of Construction:1. Draw lien segment AB = 8 cm.2. At A, construct any suitable angle BAX.3. At B, construct ABY = BAX on the other side of the line AB.4. From AX, cut off 4 equal distances at the points C, D, E and F

such that AC = CD = DE = EF.5. With the same radius, cut off 4 equal distances along BY at the

points H, I, J and K such that BH = HI = IJ = JK.6. Join AK, CJ, DI, EH and FB. Let CJ, DI and EH meet the line

segment AB at the points M, N and O respectively. Then, M, N and O are the points of division of AB such that AM = MN = NO = OB.

6. To draw a perpendicular bisector of a line segment.Given : Any line segment PQ.Required : To draw a perpendicular bisector of lien segment PQ.Steps of Construction:

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1. With P as centre and any line suitable radius draw arcs, one on each side of PQ.

2. With Q as centre and same radius (as in step 1), draw two more arcs, one on each side of PQ cutting the previous arcs at A and B.

3. Join AB to meet PQ at M, then AB bisects PQ at M, and is perpendicular to PQ, Thus, AB is the required perpendicular bisector of PQ.

7. To construct an equilateral triangle when one of its side is given. E.g.: Construct and equilateral triangle whose each side is 5 cm.Given : Each side of an equilateral triangle is 5 cm.Required : To construct the equilateral triangle.

1. Draw any line segment AB = 5 cm.2. With A as centre and radius 5 cm draw an arc.3. With B as centre and radius 5 cm draw an arc to cut the

previous arc at C.4. Join AC and BC. Then ABC is the required triangle.

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8. To construct an equilateral triangle when its altitude is given. E.g.: Construct an equilateral triangle whose altitude is 4 cm. Steps of Construction:1. Draw any line segment PQ.2. Take an point D on PQ and At D, construct perpendicular DR to

PQ. From DR, cut off DA = 4 cm.

3. At A, construct DAS = DAT = 12 60 = 30° on either side of AD. Let AS and AT meet PQ at points B and C respectively. Then, ABC is the required equilateral triangle.

9. Construction of a triangle, given its Base, Sum of the other Two sides and one Base Angle.

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E.g Construct a triangle with base of length 5 cm, the sum of the other two sides 7 cm and one base angle of 60°.Given: In ΔABC, base BC = 5 cm, AB + AC = 7 cm and ABC = 60°Required : To construct the ΔABC.Steps of Construction:1. Draw BC = 5 cm.2. At B, construct CBX = 60°3. From BX, cut off BD = 7 cm.4. Join CD.5. Draw the perpendicular bisector of CD, intersecting BD at a

point A.6. Join AC. Then, ABC is the required triangle.

10. Construction of a triangle, Given its Base, Difference of the Other Two Sides and one Base Angle.Eg: Construct a triangle with base of length 7.5 cm, the difference ofthe other two sides 2.5 cm, and one base angle of 45°

Given : In ΔABC, base BC = 7.5 cm, the difference of the other two sides, AB – AC or AC – AB = 2.5 cm and one base angle is 45°. Required : To construct the ΔABC,CASE (i) AB – AC = 2.5 cm.Steps of Construction:

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1. Draw BC = 7.5 cm.2. At B, construct CBX = 45°.3. From BX, cut off BD = 2.5 cm.4. Join CD.5. Draw the perpendicular bisector RS of CD intersecting BX at a

point A.6. Join AC. Then, ABC is the required triangle.

CASE (ii) AC – AB = 2.5 cmSteps of Construction:1. Draw BC = 7.5 cm.2. At B, construct CBX = 45° and produce XB to form a line XBX’.3. From BX’, cut off BD’ = 2.5 cm.4. Join CD’.5. Draw perpendicular bisector RS of CD’ intersecting BX at a point

A.6. Join AC. Then, ABC is the required triangle.

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e11. Construction of a Triangle of Given Perimeter and Base Angles.

Construct a triangle with perimeter 11.8 cm and base angles 60° and 45°.Given : In ΔABC, AB+BC+CA = 11.8 cm, B = 60° & C = 45°.Required : To construct the ΔABC.

1. Draw DE = 11.8 cm.

2. At D, construct EDP = 12 of 60° = 30° and at E, construct

DEQ = 12 of 45° = 22 12 .

3. Let DP and EQ meet at A.4. Draw perpendicular bisector of AD to meet DE at B.5. Draw perpendicular bisector of AE to meet DE at C.6. Join AB and AC. Then, ABC is the required triangle.

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Class X: MathChapter 13: Surface Areas and Volumes

Chapter Notes

Top Definitions

1. A Cube is a special type of cuboids in which length = breadth = height. Also called an edge of a cube.

2. A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball.

3. A cylinder is a solid or a hollow object that has a circular base and a circular top of the same size.

4. A hemisphere is half of a sphere.5. If a right circular is cut off by a plane parallel to its base, then the

portion of the cone between the plane and the base of the cone is called a frustum of the cone.

Top Concepts

1. The total surface area of the solid formed by the combination of solids is the sum of the curved surface area of each of the individual parts.

2. A solid is melted and converted to another, volume of both the solids remains the same, assuming there is no wastage in the conversions. The surface area of the two solids may or may not be the same.

3. A frustum can be obtained by cutting a cone by a plane, parallel to the base of the cone.

4. The solids having the same curved surface do not necessarily occupy the same volume.

Top Formulae

1. Cuboids:Lateral surface area Or Area of four walls = 2(ℓ + b) h

Total surface area = 2(ℓb + bh + hℓ) Volume = ℓ x b x h

Diagonal of a cuboids = 2 b2 h2

2. Cube

Lateral surface area Or Area of four walls = 4 x (edge)2

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Total surface area = 6 x (edge)²Volume = (edge)3

Diagonal of a cube = 3 x edge.

3. Right circular cylinder:Area of each end or Base area = r²Area of curved surface or lateral surface area

= perimeter of the base x height = 2 r h Total surface area (including both ends)

= 2 rh + 2r² = 2r (h + r)

4. Right circular hollow cylinder: Area of curved surface

= (External surface) + (Internal surface)= (2Rh + 2rh) = 2 (R² - r²)= [2h(R+ r) + 2 (R² - r²)]= [2(R + r) (h + R – r)]

= (External volume) – (Internal volume)= (R²h - r²h) = h (R² - r²)

5. Right circular cone:

Slant height (ℓ) = h2 r2

Area of curved surface = rℓ = r h2 r2

Total surface area = Area of curved surface + Area of base

= rℓ + r² = r (ℓ + r)1 2

Volume = r h36. Sphere:

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Surface area = 4 r²

Volume = 34 r3

7. Spherical shell:Surface area (outer) = 4R² Volume

of material = 34 r³ 34 r³

= 34 R³ r³

8. Hemisphere:Area of curved surface = 2 r²Total surface Area = Area of curved surface + Area of base

= 2 r² + r²= 3r²

Volume =2

r33

9. Frustum of a cone:Total surface area = [R² + r² + ℓ (R + r)]

Volume of the material = 1 h R² r² Rr3

Top Diagrams

1. Cuboid

2. Cube

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3. Right circular cylinder:

4. Right circular hollow cylinder:

5. Right circular cone:

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6. Sphere:

7. Spherical shell:

8. Hemisphere:

9. Frustum of a cone:

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1

Class IX: MathChapter 14: Statistics

Chapter Notes

Top Definitions

1. Facts or figures collected with a definite purpose are called data.2. Statistics deals with collection, presentation, analysis and

interpretation of numerical data.3. Arranging data in a order to study their salient features is called

presentation of data.4. Data arranged in ascending or descending order is called arrayed data

or an array.5. When an investigator with a definite plan or design in mind collects

data first handedly, it is called primary data.6. Data when collected by someone else, say an agency or an

investigator, comes to you, is known as the secondary data.7. Variable is a quantity that assumes different values.8. Range of the data is the difference between the maximum and the

minimum values of the observations.9. The small groups obtained on dividing all the observations are called

classes or class intervals and the size is called the class size or class width.

10. Class mark of a class is the mid value of the two limits of that class.11. A bar graph is the diagram showing a system of connections or

interrelations between two or more things by using bars.12. A histogram is the bar graph such that the area over each class

interval is proportional to the relative frequency of data within this interval.

13. The number of times an observation occurs in the data is called the frequency of the observation.

14. A frequency distribution in which the upper limit of one class differs from the lower limit of the succeeding class is called an Inclusive or discontinuous Frequency Distribution.

15. A frequency distribution in which the upper limit of one class coincides from the lower limit of the succeeding class is called an exclusive or continuous Frequency Distribution.

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16. A bar graph is a pictorial representation of data in which rectangular bars of uniform width are drawn with equal spacing between them on one axis, usually the x axis. The value of the variable is shown on the other axis that is the y axis.

17. A histogram is a set of adjacent rectangles whose areas are proportional to the frequencies of a given continuous frequency distribution.

18. The Cumulative Frequency of a class-interval is the sum of frequencies of that class and the classes which precede (come before) it.

19. The mean value of a variable is defined as the sum of all the values of the variable divided by the number of values.

20. Median is the value of middle most observation(s).21. Mode of a statistical data is the value of that variate which has the

maximum frequency.Top Concepts

1. In case of continuous frequency distribution, the upper limit of a class is not to be included in that class while in discontinuous both the limits are included.

2. The height of rectangles corresponds to the numerical value of the data.

3. Frequency polygons are a graphical device for understanding the shapes of distributions.

4. Bar charts are used for comparing two or more values.5. A histogram differs from a bar chart, as in the former it is the area of

the bar that denotes the value, not the height.6. The height of the rectangle as the ratio of the frequency of the class to

the width or size of the class.7. Last cumulative frequency is always the sum total of all the

frequencies.8. If both a histogram and a frequency polygon are to be drawn on the

same graph, then we should first draw the histogram and then join the mid-points of the tops of the adjacent rectangles in the histogram with line-segments to get the frequency polygon.

9. If classes are not of equal width, then the height of the rectangle is calculated by the ratio of the frequency of that class, to the width of that class.

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10. A measure of central tendency tries to estimate the central value which represents the entire data.

11. The three measures of central tendency for ungrouped data are mean, mode and median.

12. The disadvantage of arithmetic mean is that it is affected by extreme values.

13. The median is to be calculated only after arranging the data in ascending order or descending order.

22. Average height is the modal value.23. Disadvantage of the mode is that it is not uniquely defined in many

cases.24. The data is symmetric about the mean position when the three

averages mean median and mode are all equal.25. The data is asymmetric when the three measures are unequal.14. The variate corresponding to the highest frequency is to be taken as

the mode and not the frequency.Top Formulae

1. Class size =Range

Number of classes

2. Class size = Upper limit – Lower Limit

3. Mean ( x ) 1

n xi

ni1

4. Mean ( x)

f i xi

fi

5. (i) If number of observations (n) is odd, Median = ( n 21

)th observation

n th n ( ) ( 1)th

(ii) If n is even, then median 2 2 observation 2

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Top Diagrams

1. Symmetric Distribution

2. Asymmetrical or skewed distribution

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3. Bar Graph

4. Mean < Mode

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5. Mode < Mean

6. Frequency Polygons

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7. A histogram

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Class IX: MathChapter 15: Probability

Chapter Notes

Top Definitions

1. Probability is a quantitative measure of certainty.2. Any activity associated to certain outcome is called an experiment.

e.g. (i) tossing a coin (ii) throwing a dice (ii) selecting a card.3. A trial is an action which will result in one and several outcomes.4. An event for an experiment is the collection of some outcomes of the

experiment. E.g (i) Getting a head on tossing a coin (ii) getting a face card when a card is drawn from a pack of 52 cards.

Top Concepts

1. Probability of an event lies between 0 and 1.2. Probability can never be negative.3. A pack of playing cards consist of 52 cards which are divided into 4

suits of 13 cards each. Each suit consists of one ace, one king, one queen, one jack and 9 other cards numbered from 2 to 10. Four suits named spades, hearts, diamonds and clubs.

4. King, queen and jack are face cards.5. The two possible outcomes of tossing a coin are head and tail.6. The sum of the probabilities of all elementary events of an experiment

is 1.

Top Formulae

1. The empirical (experimental) probability of an event E denoted as P(E) is given by:

P(E) Number of trialinwhich the event happenendTotalNumber of Outcomes

2

Top Diagrams

1. Suits of Playing Card

Heart Spades Diamond Club

2. Face Cards

A Queen of Heart A Jack of Club A King of Diamond