Thursday, 12 September 2013
Wednesday, 11 September 2013
CBSE Board Syllabus for Class 10 Math
CBSE Board Syllabus for Class 10 Math
CLASS X Mathematics Syllabus-2013
UNITS
UNITS
I.
NUMBER SYSTEMS
II. ALGEBRA
III. GEOMETRY
IV TRIGONOMETRY
V STATISTICS
UNIT I : NUMBER SYSTEMS
1. REAL NUMBERS
Euclid's division lemma, Fundamental Theorem of Arithmetic - statements after reviewing work done earlier
and after il ustrating and motivating through examples, Proofs of results - irrationality of 2, 3, 5, decimal
expansions of rational numbers in terms of terminating/non-terminating recurring decimals.
UNIT II : ALGEBRA
1. POLYNOMIALS
Zeros of a polynomial. Relationship between zeros and coefficients of quadratic polynomials. Statement and
simple problems on division algorithm for polynomials with real coefficients.
PAIR OF LINEAR EQUATIONS IN TWO VARIABLES
Pair of linear equations in two variables and their graphical solution. Geometric representation of different
possibilities of solutions/inconsistency.
Algebraic conditions for number of solutions. Solution of pair of linear equations in two variables algebraical y - by substitution, by elimination and by cross multiplication. Simple situational problems must be included.
UNIT III : GEOMETRY
1. TRIANGLES
Definitions, examples, counter examples of similar triangles.
1. (Prove) If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio.
2. (Motivate) If a line divides two sides of a triangle in the same ratio, the line is paral el to the third side.
3. (Motivate) If in two triangles, the corresponding angles are equal, their corresponding sides are proportional and the triangles are similar.
4. (Motivate) If the corresponding sides of two triangles are proportional, their corresponding angles are equal and the two triangles are similar.
5. (Motivate) If one angle of a triangle is equal to one angle of another triangle and the sides including these angles are proportional, the two triangles are similar.
6. (Motivate) If a perpendicular is drawn from the vertex of the right angle of a right triangle to the hypotenuse, the triangles on each side of the perpendicular are similar to the whole triangle and to each other.
7. (Prove) The ratio of the areas of two similar triangles is equal to the ratio of the squares on their corresponding sides.
8. (Prove) In a right triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides.
9. (Prove) In a triangle, if the square on one side is equal to sum of the squares on the other two sides, the angles opposite to the first side is a right traingle.
UNIT IV : TRIGONOMETRY
1. INTRODUCTION TO TRIGONOMETRY
2. Trigonometric ratios of an acute angle of a right-angled triangle. Proof of their existence (well defined);motivate the ratios, whichever are defined at zero degree & ninty degree. Values (with proofs) of the trigonometric ratios of thirty degree, fourtyfive degree & sixty degree. Relationships between the ratios.
TRIGONOMETRIC IDENTITIES
Proof and applications of the identity sin2 A + cos2 A = 1. Only simple identities to be given. Trigonometric ratios of complementary angles.
UNIT VII : STATISTICS AND PROBABILITY
1. STATISTICS
Mean, median and mode of grouped data (bimodal situation to be avoided). Cumulative frequency graph.
UNITS
II. ALGEBRA (Contd.)II. ALGEBRA
III. GEOMETRY
IV TRIGONOMETRY
V STATISTICS
UNIT I : NUMBER SYSTEMS
1. REAL NUMBERS
Euclid's division lemma, Fundamental Theorem of Arithmetic - statements after reviewing work done earlier
and after il ustrating and motivating through examples, Proofs of results - irrationality of 2, 3, 5, decimal
expansions of rational numbers in terms of terminating/non-terminating recurring decimals.
UNIT II : ALGEBRA
1. POLYNOMIALS
Zeros of a polynomial. Relationship between zeros and coefficients of quadratic polynomials. Statement and
simple problems on division algorithm for polynomials with real coefficients.
PAIR OF LINEAR EQUATIONS IN TWO VARIABLES
Pair of linear equations in two variables and their graphical solution. Geometric representation of different
possibilities of solutions/inconsistency.
Algebraic conditions for number of solutions. Solution of pair of linear equations in two variables algebraical y - by substitution, by elimination and by cross multiplication. Simple situational problems must be included.
UNIT III : GEOMETRY
1. TRIANGLES
Definitions, examples, counter examples of similar triangles.
1. (Prove) If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio.
2. (Motivate) If a line divides two sides of a triangle in the same ratio, the line is paral el to the third side.
3. (Motivate) If in two triangles, the corresponding angles are equal, their corresponding sides are proportional and the triangles are similar.
4. (Motivate) If the corresponding sides of two triangles are proportional, their corresponding angles are equal and the two triangles are similar.
5. (Motivate) If one angle of a triangle is equal to one angle of another triangle and the sides including these angles are proportional, the two triangles are similar.
6. (Motivate) If a perpendicular is drawn from the vertex of the right angle of a right triangle to the hypotenuse, the triangles on each side of the perpendicular are similar to the whole triangle and to each other.
7. (Prove) The ratio of the areas of two similar triangles is equal to the ratio of the squares on their corresponding sides.
8. (Prove) In a right triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides.
9. (Prove) In a triangle, if the square on one side is equal to sum of the squares on the other two sides, the angles opposite to the first side is a right traingle.
UNIT IV : TRIGONOMETRY
1. INTRODUCTION TO TRIGONOMETRY
2. Trigonometric ratios of an acute angle of a right-angled triangle. Proof of their existence (well defined);motivate the ratios, whichever are defined at zero degree & ninty degree. Values (with proofs) of the trigonometric ratios of thirty degree, fourtyfive degree & sixty degree. Relationships between the ratios.
TRIGONOMETRIC IDENTITIES
Proof and applications of the identity sin2 A + cos2 A = 1. Only simple identities to be given. Trigonometric ratios of complementary angles.
UNIT VII : STATISTICS AND PROBABILITY
1. STATISTICS
Mean, median and mode of grouped data (bimodal situation to be avoided). Cumulative frequency graph.
UNITS
III. GEOMETRY (Contd.)
IV. TRIGONOMETRY (Contd.)
V. PROBABILITY
VI. COORDINATE GEOMETRY
VII. MENSU RATION
UNIT II : ALGEBRA (Contd.)
3. QUADRATIC EQUATIONS
4. Standard form of a quadratic equation ax2 + bx + c = 0, (a = 0). Solution of the quadratic equations(only real roots) by factorization, by completing the square and by using quadratic formula. Relationship between discriminant and nature of roots.
ARITHMETIC PROGRESSIONS
Motivation for studying AP. Derivation of standard results of finding the nth term and sum of first n terms.
UNIT III : GEOMETRY (Contd.)
2. CIRCLES
Tangents to a circle motivated by chords drawn from points coming closer and closer to the point.
1. (Prove) The tangent at any point of a circle is perpendicular to the radius through the point of contact.
2. (Prove) The lengths of tangents drawn from an external point to circle are equal.
3. CONSTRUCTIONS
1. Division of a line segment in a given ratio (internal y)
2. Tangent to a circle from a point outside it.
3. Construction of a triangle similar to a given triangle.
UNIT IV : TRIGONOMETRY
3. HEIGHTS AND DISTANCES
Simple and believable problems on heights and distances. Problems should not involve more than two right triangles. Angles of elevation
UNIT V : STATISTICS AND PROBABILITY
2. PROBABILITY
Classical definition of probability. Connection with probability as given in Class IX. Simple problems on single events, not using set notation.
UNIT VI : COORDINATE GEOMETRY
1. LINES (In two-dimensions)
Review the concepts of coordinate geometry done earlier including graphs of linear equations. Awareness of geometrical representation of quadratic polynomials. Distance between two points and section formula
(internal). Area of a triangle.
UNIT VII : MENSURATION
1. AREAS RELATED TO CIRCLES
2. Motivate the area of a circle; area of sectors and segments of a circle. Problems based on areas and perimeter / circumference of the above said plane figures.
SURFACE AREAS AND VOLUMES
(i) Problems on finding surface areas and volumes of combinations of any two of the fol owing: cubes,
cuboids, spheres, hemispheres and right circular cylinders/cones. Frustum of a cone.
(i) Problems involving converting one type of metal ic solid into another and other mixed problems.
West Bengal Board Syllabus for Class 10 Math
West
Bengal Board Syllabus for Class 10 Math
Class 10 Math Syllabus
ALGEBRA
1.) Linear Equation of two variables.
* Problem sums in two variable equations (Answers are integers only. Problems to see that used in daily life to be taken).
2.) Polynomials
* H.C.F & L.C.M.
* Rational Expressions
* Meaning, Simplification using factorization and Basic properties of rational expression.
(Commutative, associative and distributive properties are not need. To see that factor theorem is used.)
3.) Quadratic Equation
* Standard form of quadratic equation.
* To solve quadratic equation using, factorization.
* Problem sums from different areas using quadratic equation. (Application).
(it is necessary that roots of quadratic equation should be real.)
4.) Arithmetic progressions.
* Introduction of Arithmetic progression as a progression of numbers.
* Formula of additions.
* Simple problem sums.
(Proof of formula of addition should not be given. In problem sums common difference
should not be irrational numbers.)
ARITHMETIC
5.) Installments
* Installments - Buying (Purchase) scheme. (Installments not more than 12).
(in payment of installments amount of installments should be equal.)
6.) Income Tax
* Sums based on Income Tax. (income tax on the salary of salaried persons).
GEOMETRY
7.) Similar Triangles
* Introduction
* Similar Triangles
* Congruent Triangles & Similar Triangles.
* Results of proportionality.
* Fundamental theorem of proportionality (Without proof).
* Theorem:-
A line drawn parallel to a side of a triangle to intersect the other two sides in two distinct points, cuts two. line segment from each of these two sides. Then the line segments lying in the same closed half plan of that line are proportional to the corresponding sides of the triangle (with proof).
* Corollary AD the bisector of ∆ of ∆BC meets BC in D. Then AB ÷ BD = AC ÷ DC (without proof).
* Theorem:- A Line intersecting two sides of a triangle in two distinct points in such a way that the line segments cut by it on the two sides lying in the same enclosed half plane are proportional to the corresponding sides then the line is parallel to the third side (without proof).
* Numerical based on similarity and geometrical problems.
8.) Conditions & Similarity
* Introduction
* Theorem on similar triangles.
* Theorem (AAA thm). If for any correspondence between two triangles the corresponding angles are congruent, then the correspondence is a similarity (without proof).
* Corollary: (AA) If for a correspondence between two triangles, two pairs of corresponding angles Are congruent the correspondence is a similarity.
* Theorem (SAS) If for a correspondence between two triangles, two pair of corresponding sides are proportional and the included angles are congruent, then the correspondence is a similarity.
* Theorem (SAS) For a given correspondence of two triangles if the corresponding sides are in proportion then also the correspondence is a similarity and Similarity and area. (Without proof.)
9.) Similarity and Pythagoras theorem
* Introductions
* Right angled triangle and similarity.
* Theorem : If an altitude is drawn on the hypotenuse of a right angled triangle, then the two triangles so formed are similar to each other and each such triangle is similar to the original triangle. (with proof).
* Adjacent line-segment (Definition).
* Thoerem: If an altitude is drawn on the hypotenuse of a right angled triangle then (1) the length of the altitude is the geometric mean of the lengths of the two line segments made by the altitude on the hypotenuse. (2) the length of each side is the geometric mean of the length of line segment of the hypotenuse adjacent to that side. (without proof).
Pythagoras Theorem (with proof) In a right angled triangle the square of the length of the hypotenuse is equal to the sum of the squares of the length of the remaining sides.
Converse of Pythagoras Theorem (with proof). If in a ABC; AC2 = AB2 + BC2, then B is a right angle.
* Explanation of Apollonius Theorem.
* Sum based on conditions of similarity and Pythagoras theorem.
10.) Circle and Chord
* Some definitions:- Circle, Radius, Chord, Diameter, Congruent circles, Concentric Circles, secant etc.
* Separation of the plane of circle by the circle.
* Interical and Extrical of the circle.
* A few theorems on circles.
* Theorem : - A perpendicular drawn through the centre of a circle on a chord bisects the chord (without proof).
* Thorem: - In a circle the line segment joining the mid point of a chord (which is not a diameter) to the centre of the circle is perpendicular to the chord (without proof).
* Result 1 (without proof) Prove that the perpendicular bisector of a chord of a circle lying in the plane of the circle passes through the centre of the circle.
Result 2 (without proof) Prove that three distinct collinear points cannot be the pints on the same circle.
* Theorem:- One and only one circle passes through three non-collinear points.
* Theorems on chords.
* Theorem (without proof):- in the same circle (or in congruent circles) congruent chords are equidistant from the centre of the circle.
* Theorem (without proof): - In the same circle (or congruent circles) congruent chords are equidistant from the centre are congruent.
11.) ARC OF A CIRCLE
* Arc of circle and its length.
* Definition:- Arc, Minor Arc, Major arc, semi circular arc.
* Angle subtended by the minor arc at the centre.
* Congruent arcs.
* Theorems on congruent arcs.
* Theorems:- (without proof) The angles subtended by two congruent minor arcs at the centre are congruent.
* (Without proof) Minor arcs of the same circle, subtending congruent angles at the centre are congruent.
* If two arcs of the same circle are congruent then the chords of the circle corresponding to them are also congruent.
* If two chords of a circle are congruent then minor arcs of the semi-circles corresponding to them are also congruent.
* Angle subtended by an arc of a circle at a point of a circle.
* The measure of the angle subtended by an arc of a circle a the centre is twice the measure of the angle subtended by that arc at any point on the remaining part of the circle.Prove that angle inscribed in a semi circle is a right angle.
* If an angle inscribed in any arc of a circle is a right angle then that arc is a semi circle.
Definitions:- Segment of a circle, minor segment, major segment, semi circular segment.
* Explanation of angle in a segment of a circle.
* Theorems (without proof) If an line segment joining the points, subtend congruent angles at two distinct points lying in the same half plane of the line containing this line segment then all these form arc on the same segment of circle.
* Sums based on arc of a circle and geometric problem sums.
12.) CIRCLE AND ITS TANGENT
* Tangent of a circle.
* Definition of a tangent.
* Theorem (without proof) - A tangent of a circle is perpendicular to the radius drawn through the point of contact.
* Theorem (Without proof) - If two tangents of a circle drawn from P a point in the exterior of the circle touch the circle at the points a and B then PA = PB
* Angles made by a chord with a tangent and in the alternate segment.
* The measure of an angle made by a chord of a circle with the tangent touching the circle at one of the end points of the chord is equal to the measure of an angle made by the chord in the alternate segment.
* Theorem : (with proof) A line passing through one of the end points of a chord of a circle is so drawn in the plane of the circle that the measure of the angle made by the chord in the alternate segment then the line is a tangent of the circle.
* Theorem: (with proof) If the tangent PT at a point T of a circle and a secant AB of the circle passing through points A and B of the circle, intersect each other at a point P in the exterior of the circle then AP.PB = PT2
* Two circles touching each other,.
Theorem(with proof) The common point of contact of two circles touching each other is on the line joining the centres of circles.
* Distance between centres of the circle touching each other. Cyclic quadrilateral and its theorems
* Definition of cyclic quadrilateral
* Theorem(with proof) The opposite angles of a cyclic quadrilateral are supplementary.
* Theorem (without proof) A quadrilateral whose opposite angles are supplementary is a cyclic quadrilateral.
* Sums and geometric problem sums.
ALGEBRA
1.) Linear Equation of two variables.
* Problem sums in two variable equations (Answers are integers only. Problems to see that used in daily life to be taken).
2.) Polynomials
* H.C.F & L.C.M.
* Rational Expressions
* Meaning, Simplification using factorization and Basic properties of rational expression.
(Commutative, associative and distributive properties are not need. To see that factor theorem is used.)
3.) Quadratic Equation
* Standard form of quadratic equation.
* To solve quadratic equation using, factorization.
* Problem sums from different areas using quadratic equation. (Application).
(it is necessary that roots of quadratic equation should be real.)
4.) Arithmetic progressions.
* Introduction of Arithmetic progression as a progression of numbers.
* Formula of additions.
* Simple problem sums.
(Proof of formula of addition should not be given. In problem sums common difference
should not be irrational numbers.)
ARITHMETIC
5.) Installments
* Installments - Buying (Purchase) scheme. (Installments not more than 12).
(in payment of installments amount of installments should be equal.)
6.) Income Tax
* Sums based on Income Tax. (income tax on the salary of salaried persons).
GEOMETRY
7.) Similar Triangles
* Introduction
* Similar Triangles
* Congruent Triangles & Similar Triangles.
* Results of proportionality.
* Fundamental theorem of proportionality (Without proof).
* Theorem:-
A line drawn parallel to a side of a triangle to intersect the other two sides in two distinct points, cuts two. line segment from each of these two sides. Then the line segments lying in the same closed half plan of that line are proportional to the corresponding sides of the triangle (with proof).
* Corollary AD the bisector of ∆ of ∆BC meets BC in D. Then AB ÷ BD = AC ÷ DC (without proof).
* Theorem:- A Line intersecting two sides of a triangle in two distinct points in such a way that the line segments cut by it on the two sides lying in the same enclosed half plane are proportional to the corresponding sides then the line is parallel to the third side (without proof).
* Numerical based on similarity and geometrical problems.
8.) Conditions & Similarity
* Introduction
* Theorem on similar triangles.
* Theorem (AAA thm). If for any correspondence between two triangles the corresponding angles are congruent, then the correspondence is a similarity (without proof).
* Corollary: (AA) If for a correspondence between two triangles, two pairs of corresponding angles Are congruent the correspondence is a similarity.
* Theorem (SAS) If for a correspondence between two triangles, two pair of corresponding sides are proportional and the included angles are congruent, then the correspondence is a similarity.
* Theorem (SAS) For a given correspondence of two triangles if the corresponding sides are in proportion then also the correspondence is a similarity and Similarity and area. (Without proof.)
9.) Similarity and Pythagoras theorem
* Introductions
* Right angled triangle and similarity.
* Theorem : If an altitude is drawn on the hypotenuse of a right angled triangle, then the two triangles so formed are similar to each other and each such triangle is similar to the original triangle. (with proof).
* Adjacent line-segment (Definition).
* Thoerem: If an altitude is drawn on the hypotenuse of a right angled triangle then (1) the length of the altitude is the geometric mean of the lengths of the two line segments made by the altitude on the hypotenuse. (2) the length of each side is the geometric mean of the length of line segment of the hypotenuse adjacent to that side. (without proof).
Pythagoras Theorem (with proof) In a right angled triangle the square of the length of the hypotenuse is equal to the sum of the squares of the length of the remaining sides.
Converse of Pythagoras Theorem (with proof). If in a ABC; AC2 = AB2 + BC2, then B is a right angle.
* Explanation of Apollonius Theorem.
* Sum based on conditions of similarity and Pythagoras theorem.
10.) Circle and Chord
* Some definitions:- Circle, Radius, Chord, Diameter, Congruent circles, Concentric Circles, secant etc.
* Separation of the plane of circle by the circle.
* Interical and Extrical of the circle.
* A few theorems on circles.
* Theorem : - A perpendicular drawn through the centre of a circle on a chord bisects the chord (without proof).
* Thorem: - In a circle the line segment joining the mid point of a chord (which is not a diameter) to the centre of the circle is perpendicular to the chord (without proof).
* Result 1 (without proof) Prove that the perpendicular bisector of a chord of a circle lying in the plane of the circle passes through the centre of the circle.
Result 2 (without proof) Prove that three distinct collinear points cannot be the pints on the same circle.
* Theorem:- One and only one circle passes through three non-collinear points.
* Theorems on chords.
* Theorem (without proof):- in the same circle (or in congruent circles) congruent chords are equidistant from the centre of the circle.
* Theorem (without proof): - In the same circle (or congruent circles) congruent chords are equidistant from the centre are congruent.
11.) ARC OF A CIRCLE
* Arc of circle and its length.
* Definition:- Arc, Minor Arc, Major arc, semi circular arc.
* Angle subtended by the minor arc at the centre.
* Congruent arcs.
* Theorems on congruent arcs.
* Theorems:- (without proof) The angles subtended by two congruent minor arcs at the centre are congruent.
* (Without proof) Minor arcs of the same circle, subtending congruent angles at the centre are congruent.
* If two arcs of the same circle are congruent then the chords of the circle corresponding to them are also congruent.
* If two chords of a circle are congruent then minor arcs of the semi-circles corresponding to them are also congruent.
* Angle subtended by an arc of a circle at a point of a circle.
* The measure of the angle subtended by an arc of a circle a the centre is twice the measure of the angle subtended by that arc at any point on the remaining part of the circle.Prove that angle inscribed in a semi circle is a right angle.
* If an angle inscribed in any arc of a circle is a right angle then that arc is a semi circle.
Definitions:- Segment of a circle, minor segment, major segment, semi circular segment.
* Explanation of angle in a segment of a circle.
* Theorems (without proof) If an line segment joining the points, subtend congruent angles at two distinct points lying in the same half plane of the line containing this line segment then all these form arc on the same segment of circle.
* Sums based on arc of a circle and geometric problem sums.
12.) CIRCLE AND ITS TANGENT
* Tangent of a circle.
* Definition of a tangent.
* Theorem (without proof) - A tangent of a circle is perpendicular to the radius drawn through the point of contact.
* Theorem (Without proof) - If two tangents of a circle drawn from P a point in the exterior of the circle touch the circle at the points a and B then PA = PB
* Angles made by a chord with a tangent and in the alternate segment.
* The measure of an angle made by a chord of a circle with the tangent touching the circle at one of the end points of the chord is equal to the measure of an angle made by the chord in the alternate segment.
* Theorem : (with proof) A line passing through one of the end points of a chord of a circle is so drawn in the plane of the circle that the measure of the angle made by the chord in the alternate segment then the line is a tangent of the circle.
* Theorem: (with proof) If the tangent PT at a point T of a circle and a secant AB of the circle passing through points A and B of the circle, intersect each other at a point P in the exterior of the circle then AP.PB = PT2
* Two circles touching each other,.
Theorem(with proof) The common point of contact of two circles touching each other is on the line joining the centres of circles.
* Distance between centres of the circle touching each other. Cyclic quadrilateral and its theorems
* Definition of cyclic quadrilateral
* Theorem(with proof) The opposite angles of a cyclic quadrilateral are supplementary.
* Theorem (without proof) A quadrilateral whose opposite angles are supplementary is a cyclic quadrilateral.
* Sums and geometric problem sums.
Gujarat Board Syllabus for Class 10 Math
Gujarat
Board Syllabus for Class 10 Math
CLASS X
UNIT I : NUMBER SYSTEMS
1. REAL NUMBERS (15) Periods
Euclid's division lemma, Fundamental Theorem of Arithmetic - statements after reviewing work done earlier and after illustrating and motivating through examples, Proofs of results - irrationality of √2, √3, √5, decimal expansions of rational numbers in terms of terminating/non-terminating recurring decimals.
UNIT II : ALGEBRA
1. POLYNOMIALS (6) Periods
Zeros of a polynomial. Relationship between zeros and coefficients of quadratic polynomials. Statement and simple problems on division algorithm for polynomials with real coefficients.
2. PAIR OF LINEAR EQUATIONS IN TWO VARIABLES (15) Periods
Pair of linear equations in two variables and their graphical solution. Geometric representation of different possibilities of solutions/inconsistency.
Algebraic conditions for number of solutions. Solution of pair of linear equations in two variables algebraically - by substitution, by elimination and by cross multiplication. Simple situational problems must be included. Simple problems on equations reducible to linear equations may be included.
3. QUADRATIC EQUATIONS (15) Periods
Standard form of a quadratic equation ax2 + bx + c = 0, (a ≠ 0). Solution of the quadratic equations (only real roots) by factorization, by completing the square and by using quadratic formula. Relationship between discriminate and nature of roots.
Problems related to day to day activities to be incorporated.
4. ARITHMETIC PROGRESSIONS (8) Periods
Motivation for studying AP. Derivation of standard results of finding the nth term and sum of first n terms.
UNIT III : TRIGONOMETRY
1. INTRODUCTION TO TRIGONOMETRY (12) Periods
Trigonometric ratios of an acute angle of a right-angled triangle. Proof of their existence (well defined); motivate the ratios, whichever are defined at 0o & 90o. Values (with proofs) of the trigonometric ratios of 30o, 45o & 60o. Relationships between the ratios.
2. TRIGONOMETRIC IDENTITIES (16) Periods
Proof and applications of the identity sin2 A + cos2A = 1. Only simple identities to be given. Trigonometric ratios of complementary angles.
3. HEIGHTS AND DISTANCES (8) Periods
Simple and believable problems on heights and distances. Problems should not involve more than two right triangles. Angles of elevation / depression should be only 30o, 45o, 60o.
UNIT IV : COORDINATE GEOMETRY
1. LINES (In two-dimensions) (15) Periods
Review the concepts of coordinate geometry done earlier including graphs of linear equations. Awareness of geometrical representation of quadratic polynomials. Distance between two points and section formula (internal). Area of a triangle.
UNIT V : GEOMETRY
1. TRIANGLES (15) Periods
Definitions, examples, counter examples of similar triangles.
1. (Prove) If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio.
2. (Motivate) If a line divides two sides of a triangle in the same ratio, the line is parallel to the third side.
3. (Motivate) If in two triangles, the corresponding angles are equal, their corresponding sides are proportional and the triangles are similar.
4. (Motivate) If the corresponding sides of two triangles are proportional, their corresponding angles are equal and the two triangles are similar.
5. (Motivate) If one angle of a triangle is equal to one angle of another triangle and the sides including these angles are proportional, the two triangles are similar.
6. (Motivate) If a perpendicular is drawn from the vertex of the right angle of a right triangle to the hypotenuse, the triangles on each side of the perpendicular are similar to the whole triangle and to each other.
7. (Prove) The ratio of the areas of two similar triangles is equal to the ratio of the squares on their corresponding sides.
8. (Prove) In a right triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides.
9. (Prove) In a triangle, if the square on one side is equal to sum of the squares on the other two sides, the angles opposite to the first side is a right traingle.
2. CIRCLES (8) Periods
Tangents to a circle motivated by chords drawn from points coming closer and closer to the point.
1. (Prove) The tangent at any point of a circle is perpendicular to the radius through the point of contact.
2. (Prove) The lengths of tangents drawn from an external point to circle are equal.
3. CONSTRUCTIONS (8) Periods
1. Division of a line segment in a given ratio (internally)
2. Tangent to a circle from a point outside it.
3. Construction of a triangle similar to a given triangle.
UNIT VI : MENSURATION
1. AREAS RELATED TO CIRCLES (12) Periods
Motivate the area of a circle; area of sectors and segments of a circle. Problems based on areas and perimeter / circumference of the above said plane figures. (In calculating area of segment of a circle, problems should be restricted to central angle of 60o, 90o & 120o only. Plane figures involving triangles, simple quadrilaterals and circle should be taken.)
2. SURFACE AREAS AND VOLUMES (12) Periods
(i) Problems on finding surface areas and volumes of combinations of any two of the following: cubes, cuboids, spheres, hemispheres and right circular cylinders/cones. Frustum of a cone.
(ii) Problems involving converting one type of metallic solid into another and other mixed problems. (Problems with combination of not more than two different solids be taken.)
UNIT VII : STATISTICS AND PROBABILITY
1. STATISTICS (15) Periods
Mean, median and mode of grouped data (bimodal situation to be avoided). Cumulative frequency graph.
2. PROBABILITY (10) Periods
Classical definition of probability. Connection with probability as given in Class IX. Simple problems on single events, not using set notation.
INTERNAL ASSESSMENT 20 Marks
Evaluation of activities 10 Marks
Project Work 05 Marks
Continuous Evaluation 05 Marks
UNIT I : NUMBER SYSTEMS
1. REAL NUMBERS (15) Periods
Euclid's division lemma, Fundamental Theorem of Arithmetic - statements after reviewing work done earlier and after illustrating and motivating through examples, Proofs of results - irrationality of √2, √3, √5, decimal expansions of rational numbers in terms of terminating/non-terminating recurring decimals.
UNIT II : ALGEBRA
1. POLYNOMIALS (6) Periods
Zeros of a polynomial. Relationship between zeros and coefficients of quadratic polynomials. Statement and simple problems on division algorithm for polynomials with real coefficients.
2. PAIR OF LINEAR EQUATIONS IN TWO VARIABLES (15) Periods
Pair of linear equations in two variables and their graphical solution. Geometric representation of different possibilities of solutions/inconsistency.
Algebraic conditions for number of solutions. Solution of pair of linear equations in two variables algebraically - by substitution, by elimination and by cross multiplication. Simple situational problems must be included. Simple problems on equations reducible to linear equations may be included.
3. QUADRATIC EQUATIONS (15) Periods
Standard form of a quadratic equation ax2 + bx + c = 0, (a ≠ 0). Solution of the quadratic equations (only real roots) by factorization, by completing the square and by using quadratic formula. Relationship between discriminate and nature of roots.
Problems related to day to day activities to be incorporated.
4. ARITHMETIC PROGRESSIONS (8) Periods
Motivation for studying AP. Derivation of standard results of finding the nth term and sum of first n terms.
UNIT III : TRIGONOMETRY
1. INTRODUCTION TO TRIGONOMETRY (12) Periods
Trigonometric ratios of an acute angle of a right-angled triangle. Proof of their existence (well defined); motivate the ratios, whichever are defined at 0o & 90o. Values (with proofs) of the trigonometric ratios of 30o, 45o & 60o. Relationships between the ratios.
2. TRIGONOMETRIC IDENTITIES (16) Periods
Proof and applications of the identity sin2 A + cos2A = 1. Only simple identities to be given. Trigonometric ratios of complementary angles.
3. HEIGHTS AND DISTANCES (8) Periods
Simple and believable problems on heights and distances. Problems should not involve more than two right triangles. Angles of elevation / depression should be only 30o, 45o, 60o.
UNIT IV : COORDINATE GEOMETRY
1. LINES (In two-dimensions) (15) Periods
Review the concepts of coordinate geometry done earlier including graphs of linear equations. Awareness of geometrical representation of quadratic polynomials. Distance between two points and section formula (internal). Area of a triangle.
UNIT V : GEOMETRY
1. TRIANGLES (15) Periods
Definitions, examples, counter examples of similar triangles.
1. (Prove) If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio.
2. (Motivate) If a line divides two sides of a triangle in the same ratio, the line is parallel to the third side.
3. (Motivate) If in two triangles, the corresponding angles are equal, their corresponding sides are proportional and the triangles are similar.
4. (Motivate) If the corresponding sides of two triangles are proportional, their corresponding angles are equal and the two triangles are similar.
5. (Motivate) If one angle of a triangle is equal to one angle of another triangle and the sides including these angles are proportional, the two triangles are similar.
6. (Motivate) If a perpendicular is drawn from the vertex of the right angle of a right triangle to the hypotenuse, the triangles on each side of the perpendicular are similar to the whole triangle and to each other.
7. (Prove) The ratio of the areas of two similar triangles is equal to the ratio of the squares on their corresponding sides.
8. (Prove) In a right triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides.
9. (Prove) In a triangle, if the square on one side is equal to sum of the squares on the other two sides, the angles opposite to the first side is a right traingle.
2. CIRCLES (8) Periods
Tangents to a circle motivated by chords drawn from points coming closer and closer to the point.
1. (Prove) The tangent at any point of a circle is perpendicular to the radius through the point of contact.
2. (Prove) The lengths of tangents drawn from an external point to circle are equal.
3. CONSTRUCTIONS (8) Periods
1. Division of a line segment in a given ratio (internally)
2. Tangent to a circle from a point outside it.
3. Construction of a triangle similar to a given triangle.
UNIT VI : MENSURATION
1. AREAS RELATED TO CIRCLES (12) Periods
Motivate the area of a circle; area of sectors and segments of a circle. Problems based on areas and perimeter / circumference of the above said plane figures. (In calculating area of segment of a circle, problems should be restricted to central angle of 60o, 90o & 120o only. Plane figures involving triangles, simple quadrilaterals and circle should be taken.)
2. SURFACE AREAS AND VOLUMES (12) Periods
(i) Problems on finding surface areas and volumes of combinations of any two of the following: cubes, cuboids, spheres, hemispheres and right circular cylinders/cones. Frustum of a cone.
(ii) Problems involving converting one type of metallic solid into another and other mixed problems. (Problems with combination of not more than two different solids be taken.)
UNIT VII : STATISTICS AND PROBABILITY
1. STATISTICS (15) Periods
Mean, median and mode of grouped data (bimodal situation to be avoided). Cumulative frequency graph.
2. PROBABILITY (10) Periods
Classical definition of probability. Connection with probability as given in Class IX. Simple problems on single events, not using set notation.
INTERNAL ASSESSMENT 20 Marks
Evaluation of activities 10 Marks
Project Work 05 Marks
Continuous Evaluation 05 Marks
Himachal Pradesh Board Syllabus for Class 10 Math
Himachal Pradesh Board Syllabus for
Class 10 Math
Mathematics
Time: 3 hours
Maximum marks: 85
Unit I : Number Systems
1. Real Numbers (15 Periods)
Euclid's division lemma, Fundamental Theorem of Arithmetic-statements after reviewing work done earlier and after illustrating and motivating through examples, Proofs of results-irrationality of Ö2, Ö3, Ö5, decimal expansions of rational numbers in terms of terminating / non-terminating recurring decimals.
Unit II: Algebra
1. Polynomials (6 Periods)
Zeros of a polynomial. Relationship between zeros and coefficients of a polynomial with particular reference to quadratic polynomials. Statement and simple problems on division algorithm for polynomials with real coefficients.
2. Pair of Linear Equations in Two Variables. (15 Periods)
Pair of linear equations in two variables. Geometric representation of different possibilities of solutions inconsistency.
Algebraic conditions for number of solutions. Solution of pair of linear equations in two variables algebraically by substitution, by elimination and by cross multiplication. Simple situational problems must be included. Simple problems on equations reducible to linear equations may be included.
3. Quadratic Equations
Standard form of a quadratic equation ax2 + bx + c = 0, (a ¹ 0). Solution of the quadratic equations (only real roots) by factorization and by completing the square, i.e by using quadratic formula. Relationship between discriminant and nature of roots.
Problems related to day to day activities to be incorporated.
4. Arithmetic Progression 8 Periods (8 Periods)
Motivation for studying AP. Derivation of standard results of finding the nth term and sum of first n terms.
Unit III : Trigonometry
1. Trigonometric Ratios (15 Periods)
Trigonometric ratios of an acute angle of a right-angled triangle. Proof of their existence (well defined); motivate the ratios, whichever are defined at 0° & 90°. Values (with proofs) of the trigonometric ratios of 30°, 45° & 60°. Relationships between the ratios.
2. Trigonometric Identities (16 Periods)
Prro and applications of the identity sin2 A + cos2 A = 1. Only simple identities to be given. Trigonometric ratios of complementary angles.
3. Heights and Distances (8 Periods)
Simple and believable problems on heights and distances. Problems should not involve more than two right triangles. Angles of elevation / depression should be only 30°, 45°, 60°.
Unit-IV : Coordinate Geometry
1. Lines (In two-dimensions) (15 Periods)
Review the concepts of coordinate geometry done earlier including graphs of linear equations. Awareness of geometrical representation of quadratic polynomials. Distance between two points and section formula (internal). Area of a triangle.
Unit-V : Geometry
1. Triangles (15 Periods)
Definitions, examples, counter examples of similar triangles
1. (Prove) if a line a drawn parallel to one side of a triangle to intersect to other two sides in distinct points, the other two sides are divided in the same ratio.
2. (Motivate) If a line divides two sides of a triangle in the same ratio, the line is parallel to the third side.
3. (Motivate) If in two triangles, the corresponding angles are equal, their corresponding sides are proportional and the triangles are similar.
4. (Motivate) If the corresponding sides of two triangles are proportional, their corresponding angles are equal and the two triangles are similar.
5. (Motivate) If one angles of a triangle is equal to one angle of another triangle and the sides including these angles are proportional, the two triangles are similar.
6. (Motivate) If a perpendicular is drawn from the vertex of the right angle of a right triangle to the hypotenuse, the triangles on each side of the perpendicular are similar to the whole triangle and to each other.
7. (Prove) The ratio of the areas of two similar triangles is equal to the ratio of the squares on their corresponding sides.
8. (Prove) In a right triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides.
9. (Prove) In a triangle, if the square on one side is equal to sum of the squares on the other two sides, the angles opposite to the first side is a right triangle.
2. Circles (8 Periods)
Tangents to a circle motivated by chords drawn from points coming closer and closer and closer to the point.
1. (Prove) The tangent at any point of a circle is perpendicular to the radius through the point of contact.
2. (Prove) The lengths of tangents drawn from an external point to circle are equal.
3. Constructions (8 Periods)
1. Division of a line segment in a given ratio (internally)
2. Tangent to a circle from a point outside it.
3. Construction of a triangle similar to a given triangle.
Unit-VI: Mensuration
1. Areas of Plane Figures (12 Periods)
Motivate the area of a circle ; area of sectors and segments of a circle. Problems based on areas and perimeter/ circumference of the above said plane figures. (In calculating area of segment of a circle, problems should be restricted to central angle of 60°, 90° & 120° only. Plane figures involving triangles, simple quadrilaterals and circle should be taken).
2. Surface Areas And Volumes (12 Periods)
Time: 3 hours
Maximum marks: 85
Unit I : Number Systems
1. Real Numbers (15 Periods)
Euclid's division lemma, Fundamental Theorem of Arithmetic-statements after reviewing work done earlier and after illustrating and motivating through examples, Proofs of results-irrationality of Ö2, Ö3, Ö5, decimal expansions of rational numbers in terms of terminating / non-terminating recurring decimals.
Unit II: Algebra
1. Polynomials (6 Periods)
Zeros of a polynomial. Relationship between zeros and coefficients of a polynomial with particular reference to quadratic polynomials. Statement and simple problems on division algorithm for polynomials with real coefficients.
2. Pair of Linear Equations in Two Variables. (15 Periods)
Pair of linear equations in two variables. Geometric representation of different possibilities of solutions inconsistency.
Algebraic conditions for number of solutions. Solution of pair of linear equations in two variables algebraically by substitution, by elimination and by cross multiplication. Simple situational problems must be included. Simple problems on equations reducible to linear equations may be included.
3. Quadratic Equations
Standard form of a quadratic equation ax2 + bx + c = 0, (a ¹ 0). Solution of the quadratic equations (only real roots) by factorization and by completing the square, i.e by using quadratic formula. Relationship between discriminant and nature of roots.
Problems related to day to day activities to be incorporated.
4. Arithmetic Progression 8 Periods (8 Periods)
Motivation for studying AP. Derivation of standard results of finding the nth term and sum of first n terms.
Unit III : Trigonometry
1. Trigonometric Ratios (15 Periods)
Trigonometric ratios of an acute angle of a right-angled triangle. Proof of their existence (well defined); motivate the ratios, whichever are defined at 0° & 90°. Values (with proofs) of the trigonometric ratios of 30°, 45° & 60°. Relationships between the ratios.
2. Trigonometric Identities (16 Periods)
Prro and applications of the identity sin2 A + cos2 A = 1. Only simple identities to be given. Trigonometric ratios of complementary angles.
3. Heights and Distances (8 Periods)
Simple and believable problems on heights and distances. Problems should not involve more than two right triangles. Angles of elevation / depression should be only 30°, 45°, 60°.
Unit-IV : Coordinate Geometry
1. Lines (In two-dimensions) (15 Periods)
Review the concepts of coordinate geometry done earlier including graphs of linear equations. Awareness of geometrical representation of quadratic polynomials. Distance between two points and section formula (internal). Area of a triangle.
Unit-V : Geometry
1. Triangles (15 Periods)
Definitions, examples, counter examples of similar triangles
1. (Prove) if a line a drawn parallel to one side of a triangle to intersect to other two sides in distinct points, the other two sides are divided in the same ratio.
2. (Motivate) If a line divides two sides of a triangle in the same ratio, the line is parallel to the third side.
3. (Motivate) If in two triangles, the corresponding angles are equal, their corresponding sides are proportional and the triangles are similar.
4. (Motivate) If the corresponding sides of two triangles are proportional, their corresponding angles are equal and the two triangles are similar.
5. (Motivate) If one angles of a triangle is equal to one angle of another triangle and the sides including these angles are proportional, the two triangles are similar.
6. (Motivate) If a perpendicular is drawn from the vertex of the right angle of a right triangle to the hypotenuse, the triangles on each side of the perpendicular are similar to the whole triangle and to each other.
7. (Prove) The ratio of the areas of two similar triangles is equal to the ratio of the squares on their corresponding sides.
8. (Prove) In a right triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides.
9. (Prove) In a triangle, if the square on one side is equal to sum of the squares on the other two sides, the angles opposite to the first side is a right triangle.
2. Circles (8 Periods)
Tangents to a circle motivated by chords drawn from points coming closer and closer and closer to the point.
1. (Prove) The tangent at any point of a circle is perpendicular to the radius through the point of contact.
2. (Prove) The lengths of tangents drawn from an external point to circle are equal.
3. Constructions (8 Periods)
1. Division of a line segment in a given ratio (internally)
2. Tangent to a circle from a point outside it.
3. Construction of a triangle similar to a given triangle.
Unit-VI: Mensuration
1. Areas of Plane Figures (12 Periods)
Motivate the area of a circle ; area of sectors and segments of a circle. Problems based on areas and perimeter/ circumference of the above said plane figures. (In calculating area of segment of a circle, problems should be restricted to central angle of 60°, 90° & 120° only. Plane figures involving triangles, simple quadrilaterals and circle should be taken).
2. Surface Areas And Volumes (12 Periods)
·
Problems on finding surface areas and volumes of combinations of
any two of the following cubes, cuboids, spheres, hemispheres and right
circular cylinders / cones. Frustum of a cone.
·
Problems involving concerting one type of metallic solid into
another and other mixed problems. (Problems with combination of not more than
two different solids be taken)
Unit VII : Statistics And Probability
1. Statistics 15 Periods
Mean, median and mode of grouped data (bimodal situation to be avoided). Cumulative frequency graph.
1. Statistics 15 Periods
Mean, median and mode of grouped data (bimodal situation to be avoided). Cumulative frequency graph.
2. Probability
Classical definition of probability. Connection with probability as given in Class IX. Simple problems on single events, not using set notation.
Prescribed Book:
1. Ganit Published by H.P. Board of School Education.
Tuesday, 10 September 2013
syllabus of 9th Raj Board
UNIT I : NUMBER SYSTEMS
1. REAL NUMBERS
Review of representation of natural numbers, integers, rational numbers on the number
line. Representation of terminating / non-terminating recurring decimals, on the number
line through successive magnification. Rational numbers as recurring /terminating
decimals.
Examples of nonrecurring / non terminating decimals such as Ö2, Ö3, Ö5 etc. Existence
of non-rational numbers (irrational numbers) such as Ö2, Ö3 and their representation
on the number line. Explaining that every real number is represented by unique point
on the number line and conversely, every point on the number line represent a unique
real number.
Existence of Öx for a given positive real number x (visual proof to be emphasized).
Definition of nth root of a real number.
Recall of laws of exponents with integral powers. Rational exponents with positive
real bases (to be done by particular cases, allowing learner to arrive at the general
laws).
Rationalization (with precise meaning) of real numbers of the type (& their combinations)
+ a +b x x y
1 & 1
where x and y are natural number and a, b are integers.
UNIT II : ALGEBRA chtxf.kr
1. POLYNOMIALS cgqiqin
Definition of a polynomial in one variable, its coefficients, with examples and counter
examples, its terms, zero polynomial. Degree of a polynomial. Constant, linear,
quadratic, cubic polynomials; monomials, binomials, trinomials. Factors and multiples.
Zeros/roots of a polynomial / equation. State and motivate the Remainder Theorem
with examples and analogy to integers. Statement and proof of the Factor Theorem.
Factorization of ax2+bx+c, a # 0 where a, b, c are real numbers, and of cubic
polynomials using the Factor Theorem. Recall of algebraic expressions and identities.
Further identities of the type (x+y+z)2 = x2 + y2 + z2 + 2xy + 2 yz + 2zx, (x + y)3 =
x3 + y3 + 3xy (x + y).
x3 + y3 + z3 - 3xyz = (x + y + z) (x2 + y2 + z2 - xy - yz - zx) and their use in
factorization of polynomials. Simple expressions reducible to these polynomials..
2. LINEAR EQUATIONS IN TWO VARIBALES
nks pjksa sa okys jSfSfSf[kd lehdj.k
Recall of linear equations in one variable. Introduction to the equation in two variables.
24 Áflfl⁄UÁáÊ·§Ê ·§ˇÊÊ ~ (×ÊäÿÁ×·§ °fl¢ ¬˝flðÁ‡Ê·§Ê) ¬⁄UˡÊÊ, 2013
Prove that a linear equation in two variables has infinitely many solutions and justify
their being written as ordered pairs of real numbers, plotting them and showing that
they seem to lie on a line. Examples problems from real life, including problems on
Ratio and Proportion and with algebraic and graphical solutions being done
simultaneously.
UNIT III : GEOMETRY T;kfefr
1. INTRODUCTION TO EUCLID’S CEOMETRY
;wfwfDyM dh T;kfefr dk ifjp;
History – Geometry in India and Eulid’s geometry. Euclid’s method of formalizing
observed phenomenon into rigorous mathematics with definitions, common / obvious
notions, axioms/postulates and theorems. The five postulates of Euclid, Equivalent
versions of the fifth postulate. Showing the relationship between axiom and theorem.
1. (Axiom) Given two distinct points, there exists one and only one line through them.
2. (Theorem) (Prove) two distinct lines cannot have more than one point in common.
2. LINES AND ANGLES js[s[kk,a vkSjSjSj dks.s.k
1. (Motivate) if a ray stand on a line, then the sum of the two adjacent angles so
formed is 180o and the converse.
2. (Prove) If two lines intersect, the vertically opposite angles are equal.
3. (Motivate) Results on corresponding angles, alternate angles, interior angles when
a transversal interests two parallel lines.
4. (Motivate) Lines, which are parallel to a given line, are parallel.
5. (Prove) The sum of the angles of a triangle is 180o .
6. (Motivate) If a side of a triangle is produced, the exterior angle so formed is equal
to the sum of the two interiors opposite angles.
3. TRIANGLES f=Hkqtqtqt
1. (Motivate) Two triangles are congruent if any two sides and the included angle of
one triangle is equal to any two sides and the included angle of the other triangle
(SAS Congruence).
2. (Prove) Two triangles are congruent if any two angles and the included side of one
tringle is equal to any two angels and the included side of the other triangle (ASA
Congruence).
3. (Motivate) Two triangles are congruent if the three sides of one triangle are equal
to three sides of the other triangle (SSS Congruence).
4. (Motivate) Two right triangle congruent if the hypotenuse and a side of one triangle
are equal (respectively) to the hypotenuse and a side of the other triangle.
5. (Prove) The angles opposite to equal side of a triangle are equal.
6. (Motivate) The sides opposite to equal angles of a triangle are equal.
7. (Motivate) Triangle inequalities and relation between ‘angle and facing side’
inequalities in triangles.
Áflfl⁄UÁáÊ·§Ê ·§ˇÊÊ ~ (×ÊäÿÁ×·§ °fl¢ ¬˝flðÁ‡Ê·§Ê) ¬⁄UˡÊÊ, 2013 25
4. QUADRILATERALS prqHqHqHkqZtqZqtZt
1. (Prove) The diagonal divides a parallelogram into two congruent triangles.
2. (Motivate) In a parallelogram opposite sides are equal, and conversely.
3. (Motivate) In a parallelogram opposite angels are equal, and conversely.
4. (Motivate) A quadrilateral is a parallelogram if a pair of its opposite sides is parallel
and equal.
5. (Motivate) In a parallelogram, the diagonals bisect each other and conversely.
6. (Motivate) In a triangle, the line segment joining the mid points of any two sides is
parallel to the third side and (motivate) its converse.
5. AREA {ks=s=Qy
Review concept of area, recall area of a rectangle.
1. (Prove) Parallelograms on the same base and between the same parallels have the
same area.
2. (Motivate) Triangles on the same base and between the same parallels are equal in
area and its converse.
6. CIRCLES o`r`r
Through examples, arrive at definitions of circle related concepts, radius, circumference,
diameter, chord, arc, subtended angle.
1. (Prove) Equal chords of a circle subtend equal angles at the center and (motivate)
its converse.
2. (Motivate) The perpendicular from the center of a circle to a chord bisects the
chord and conversely, the line drawn through the center of a circle to bisect a
chord is perpendicular to the chord.
3. (Motivate) There is one and only one circle passing through three given non-collinear
points.
4. (Motivate) Equal chords of a circle (or of congruent circles) are equidistant from
the center(s) and conversely.
5. (Prove) The angle subtended by an arc at the center is double the angle subtended
by it at any point on the remaining part of the circle.
6. (Motivate) Angles in the same segment of a circle are equal.
7. (Motivate) If a line segment joining two points subtends equal angle at two other
points lying on the same side of the line containing the segment, the four points lie
on a circle.
8. (Motivate) The sum of the either pair of the opposite angles of a cyclic quadrilateral
is 180o and its converse.
7. CONSTRUCTIONS jpuk,sasasa
1. Constructions of bisectors of line segments & angels, 60o, 90o, 45o angles etc.,
equilateral triangles.
26 Áflfl⁄UÁáÊ·§Ê ·§ˇÊÊ ~ (×ÊäÿÁ×·§ °fl¢ ¬˝flðÁ‡Ê·§Ê) ¬⁄UˡÊÊ, 2013
2. Construction of a triangle given its base, sum/difference of the other two sides and
one base angle.
3. Construction of a triangle of given perimeter and base angels.
UNIT IV : COORDINATE GEOMETRY funsZs’Zs’Z’kkadadad T;kfefr
1. COORDINATE GEOMETRY funsZ’kkad T;kfefr
The Cartesian plane, coordinates of a point, names and terms associated with the
coordinate plane, notations. Plotting points in the plane, graph of linear equations as
examples; focus on linear equations of the type ax + by + c = 0 by writing it as y =
mx+c .
UNIT V : MENSURATION esUsUsUlwjwjwjs’s’s’ku
1. AREAS {ks=s=Qy
Area of a triangle using Hero’s formula (without proof) and its application in finding the
area of a quadrilateral.
2. SURFACE AREAS AND VOLUMES i`"`"`"Bh; {ks=s=Qy ,oa vk;ru
Surface areas and volumes of cubes, cuboids, spheres (including hemispheres) and
right circular cylinders/cones.
UNIT VI : STATISTICS AND PROBABLILTY
lkaf[;dh ,oa izkf;drk
1. STATISTICS lkafafa [;dh
Introduction to Statistics : Collection of data, presentation of data – tabular form,
ungrouped / grouped, bar graphs, histograms (with varying base length), frequency
polygons, qualitative analysis of data to choose the correct from of presentation for
the collected data. Mean, median, mode of ungrouped data.
2. PROBABILITY izkzkzkf;drk
History, Repeated experiments and observed frequency approach to probability. Focus
is on empirical probability. (A large amount of time to be devoted to group and to
individual activities to motivate the concept; the experiments to be drawn from real –
life situations, and from examples used in the chapter on statistics.
Prescribed Books :
Mathematics -Textbook for class IX NCERT's Book Published under Copyright
Mathematics syllabus of M.P. Board for class X
m.p Board
Mathematics syllabus
for class X_2007-08
Time: 3 hours
Total Marks: 100
Time: 3 hours
Total Marks: 100
|
Unit-1
|
(A) LINEAR EQUATIONS
IN TWO VARIABLES :-
|
1.Linear equation in
two variables system of linear equations
1. Graphically
2. Algebraic Method
(a) Elimination by
Substitution
(b) Elimination by
equating the coefficients
(c) Cross
Multiplication.
(d) Transpose method
of Vedic Mathematics.
3. Application of
Linear equation in two variables in solving simple problems from different
areas
|
12 Marks
|
19 periods
|
|
Unit-2
|
(A) POLYNOMIALS :-
(B) RATIONAL
EXPRESSION
|
Zero of a
Polynomial, Relationship between zero and Coefficients of a polynomial with
particular reference to quadratic polynomials. Statements and Simple problems
on division algorithm for polynomials with real Cofficients
Meaning, addition, subtraction and multiplication, factorization of cyclic order expressions. Introduction of Shridharachrya and his formula Method. |
7 Marks
|
17 Periods
|
|
Unit-3
|
RATIO AND PROPORTION
:-
|
Ratio and
Proportion; Componendo, Dividendo, Alternendo, Invertendo etc, and their
application
|
5 Marks
|
9 Periods
|
|
Unit-4
|
QUADRATIC EQUATIONS
|
(A) Meaning, its
standard ax2 + bx + c = a =0 factorization method and formula method.
Discriminant of
quadratic Equation and nature of roots
(B) Applications of
quadratic equation. Different areas, solutions of equations that are
reducible to quadratic equation. To factorize quadratic polynomial with the
help of formula.
|
10 Marks
|
14 Periods
|
|
Unit-5
|
COMMERCIAL MATHS
|
(A) Compound
Interest :- Rate of growth, depreciation, Conversion period not more than
4Year. (Rate should be 4%, 5% or 10%)
(B) Installments :-
Installment, payments, Installment buying(Numbers of installment should not
more than two in-case of buying) Only equal installment should be taken in
case of payment thought equal installments not more than 3 installments
should be taken.
(C) Income Tax :-
Calculation of Income Tax for salaried class (Salary exclusive of H.R.A.)
(D) LOGARITHM :-
(i) Application in
Mathematics use in compound Interest,increase in Population and depreciation
(ii) Use of
Logarithm mensuration areas of rectangle, square, triangle, rhombus,
trapezium which was taught in earlier classes (Simple Problems)
|
8 Marks
|
9 Periods
|
|
Unit-6
|
SIMILAR TRIANGLES
|
SIMILAR TRIANGLES-
(i) (Motivate) If a
line is drawn Parallel to one side of a triangle, the other two sides are
divided in the same ratio.
(ii) (Prove) If a
line divide any two sides of a triangle in the same ratio, the line is
parallel to the third side.
(iii) (Motivate) If
in two triangles, the corresponding angle are equal, their corresponding
sides are proportionate (axiom).
(iv) (Motivate) If
the corresponding sides of two triangles are proportional, their
corresponding angles are equal (axiom).
(v) (Motivate) If
two triangles are equiangular, the triangles are similar (axiom)
(vi) (Prove) If the
corresponding sides of the two triangles are proportional, the triangle are
similar.
(vii) (Prove) If one
angle of a triangle is equal to one angle of the other and the sides
including these angles are proportional the triangle are similar.
(viii) (Motivate) If
a perpendicular is drawn from the vertex of the right angle of a right
triangle to the hypotenuse, the triangle on each side of the Perpendicular
are similar to the whole triangle and to each other.
(ix) (Prove) The
ratio of the areas of similar triangles is equal to the ratio of the squares
of their corresponding sides.
(x) (a) (Motivate)
In a right triangle, the square on the hypotenuse is equal to the sum of the
square on the othertwo sides.
(b) (Motivate) In a
triangle if the square of greatest side is equal to the sum of the square of
the remaining two, the angle opposite to the greatest side is a right angle.
|
8 Marks
|
13 Periods
|
|
Unit-7
|
CIRCLES
|
CIRCLES :-
Definition of the angle made at the centre, angle= arc/ radius angle = (i) (Motivate) Two circles are Congruent if they have equal radii.
(ii) (Motivate) In a
circle or two congruent circles, if are as are equal the angles subtended by
the areas at the centre are equal and its converse (axiom)
(iii) (Motivate) If
the areas of two congruent circles are equal their corresponding chords are
equal and its converse.
(iv) (Prove)
Perpendicular to a chord from the centre of a circle, bisects the chord and
its converse.
(v) (Motivate) One
and only one circle can be drawn through three non-collinear points.
(vi) (Motivate)
Equal chord are Equidistant from the centre and Conversely if two chord are
Equidistant from the centre, they are equal.
(vii) (Prove) Angle
subtended by an arc at the centre is twice the angle subtended at any other
point on the circle.
(viii) (Prove) Angle
subtended in a semi circle is a right angle and its converse.
(ix) (Prove) Angles
in the same segment of the circle are equal.
(x) If the angles
subtended at two points on the same side of the line segment are equal, then
all the four points are con-cyclic.
(xi) (Motivate)
Equal chords subtend Equal angles at the centre and is converse.
(xii) (Prove) The
sum of the either pair of the opposite angles of a cyclic quadrilateral is
180 °
CONVERSE :- (Prove)
If a pair of
opposite angles of a quadrilateral is supplementary, the quadrilateral is
cyclic.
(xiii) (Prove)
Tangent drawn to a circle at any point is perpendicular to the radius through
the point of contact.
(xiv) (Prove) The
lengths of tangents drawn from an external point to a circle are equal.
(xv) (Motivate) If
two chords of a circle Intersect internally or
externally then the
rectangle formed by the two parts of one chord is equal in area to the
rectangle formed by the two parts of the other.
(xvi) (Prove) If PAB
is a secant to a circle inter sectional it at A and
B and PT is a
tangent, then PA x PB = PT2
(xvii) (Motivate) If
a line to touches a circle and from the point of contact a chord is drawn,
the angles which this chord makes with the given line are equal
respectively to the angles formed in the corresponding alternate
segments and the converse.
(xviii) (Prove) If
two circles touch each other internally or externally, the point of
contact lies on the line Joining their centres. (Concept of common tangents
to two circles should be given.) Information only for the (Motivate) theorem
and proof for (Prove) theorem's are required.
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10 Marks
|
19 Periods
|
|
Unit-8
|
CONSTRUCTIONS
|
(i) Constructions of
Cir-cum circle and in circle of a triangle.
(ii) To construct a
triangle if its base and angle opposite to it is given altitude or median is
given.
(iii) To construct a
Cyclic quadrilateral if its one vertical angle is right angle.
(iv) Construction of
triangles and quadrilaterals Similar to the given figure as per the
given scale factor.
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5 Marks
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11 Periods
|
|
Unit-9
|
TRIGONOMETRY
|
Trigonometrical
functions Trigonometrical identities
Sin2θ + Cos2θ = 1
1 + tan2θ = Sec2θ
1 + Cot2θ = Cosec2θ
Proving simple
identities based on the above trigonometrical ratios of complementary
angle.
Sin(90-θ) = Cosθ
Cos(90-θ ) = sinθ
tan (90-θ )= Cotθ
Cosec(90- θ ) = Secθ
Sec(90-θ ) = Cosecθ
Cot(90-θ ) = tanθ
Simple problem based
on above.
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|
16 Periods
|
|
Unit-10
|
HEIGHTAND DISTANCE
|
Simple problem based
on heights and distances based on angles 30°, 45°, 60° Only
|
5 Marks
|
9 Periods
|
|
Unit-11
|
MENSURATION
|
(i) Area of Circle
:- Area of circle, area of sector of a circle.
(ii) Cube and cuboid
:- Concept of cube cuboid and its four walls, diagonal, surface area and
volume.
(iii) Cylinder Cone
and Sphere - Cylinder, Hollow cylinder, sphere, spherical shell, cone
surface areas, whole surface area and volumes.
(iv) Frustum of a
cone, Problem involving converting one type of metallic solid in to
another and other mixed problems. Problems with combination of not more than
two different solids be taken.
Note :- Use Vedic Mathematics Methods for Calculating Problems of Commercial Maths and mensuration (See Vedic Methods in Mathematics Index) |
10 Marks
|
19 Periods
|
|
Unit-12
|
STATISTICS &
PROBABILITY
|
(A) STATISTICS :-
Mean, Median and
Mode. living index problems related to cost of living index (only).
(B) PROBABILITY :-
Classical definition
of Probability Connection with Probability as given in Class IX Simple
Problems on Single events, not using set notation.
|
10 Marks
|
13 Periods
|
Appendix-
1. Proof in Mathematics
Further discussion on, concept of statement, proof and argument further illustrations, of deductive proof with complete arguments using simple results from arithmetic, algebra and geometry. Simple theorems of the Given ...... and assuming .... prove ..... "Training of using only the given facts (irrespective of their truths) to arrive at the required conclusion. Explanation of converse, negation constructing converses and negations of given results statements.
2. Mathematical Modeling
Reinforcing the concept of mathematical modeling using simple examples of models where some constraints are ignored Estimating probability of occurrence of certain events and estimating averages may be considered. Modeling fair installments payments, using only simple interest and future value (use of AP)
1. Proof in Mathematics
Further discussion on, concept of statement, proof and argument further illustrations, of deductive proof with complete arguments using simple results from arithmetic, algebra and geometry. Simple theorems of the Given ...... and assuming .... prove ..... "Training of using only the given facts (irrespective of their truths) to arrive at the required conclusion. Explanation of converse, negation constructing converses and negations of given results statements.
2. Mathematical Modeling
Reinforcing the concept of mathematical modeling using simple examples of models where some constraints are ignored Estimating probability of occurrence of certain events and estimating averages may be considered. Modeling fair installments payments, using only simple interest and future value (use of AP)
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