Mathematics Karnataka SSLC CBSE Topics and Syllabus

• Introduction of Real Numbers 
• Real Numbers Examples and Solutions 
• Euclid’s Division Lemma 
• Fundamental Theorem of Arithmetic 
  • Significance of the Fundamental Theorem of Arithmetic
• Fundamental Theorem of Arithmetic Motivating Through Examples 
• Proofs of Irrationality 

  of Squareroot of 2

• Concept of Irrational Numbers 
• Revisiting Rational Numbers and Their Decimal Expansions 
  • Decimal Representation of Rational Numbers in Terms of Terminating Or Non-terminating Recurring Decimals

• Euclid’s division lemma, Fundamental Theorem of Arithmetic - statements after reviewing work done earlier and after illustrating and motivating through examples, Proofs of irrationality of squareroot of 2, squareroot of 3, squareroot of 5.
• Decimal representation of rational numbers in terms of terminating/non-terminating recurring decimals.

• Linear Equations in Two Variables 
• Graphical Method of Solution of a Pair of Linear Equations 
  • Graphical Method of Solving a System of Linear Equations
• Algebraic Methods of Solving a Pair of Linear Equations 
  • Substitution Method 
  • Elimination Method 
  • Cross - Multiplication Method 
• Equations Reducible to a Pair of Linear Equations in Two Variables 
  • Equations Reducible to Linear Equations
• Consistency of Pair of Linear Equations 
• Inconsistency of Pair of Linear Equations 
• Algebraic Conditions for Number of Solutions 
• Simple Situational Problems 
• Pair of Linear Equations in Two Variables 

  Pair of Linear Equations in Two Variables Examples and Solutions

• Relation Between Co-efficient 

• Pair of linear equations in two variables and graphical method of their solution, consistency/inconsistency.
• Algebraic conditions for number of solutions.
• Solution of a pair of linear equations in two variables algebraically - by substitution, by elimination and by cross multiplication method.
• Simple situational problems.
• Simple problems on equations reducible to linear equations.

• Arithmetic Progression 
  • Terms and Common Difference of an A.P.
• General Term of an Arithmetic Progression 
• nth Term of an AP 
  • general term of the AP
• Sum of First n Terms of an AP 
  • Sum of the First 'N' Terms of an Arithmetic Progression
• Derivation of the n th Term 
• Application in Solving Daily Life Problems 
• Arithmetic Progressions Examples and Solutions 

• Motivation for studying Arithmetic Progression
• Derivation of the nth term and
• sum of the first n terms of A.P. and
• their application in solving daily life problems.

• Quadratic Equations 
  •  Introduction and Standard Form of a Quadratic Equation - ax² + bx + c = 0, (a ≠ 0)
• Solutions of Quadratic Equations by Factorization 
• Solutions of Quadratic Equations by Completing the Square 
  • method of completing the square
• Nature of Roots 
  • Nature of Roots Based on Discriminant
  • two distinct real roots, two equal real roots, no real roots
  • Solutions of Quadratic Equations by Using Quadratic Formula and Nature of Roots

  • Two distinct real roots if b² – 4ac > 0

  • Two equal real roots if b² – 4ac = 0

  • No real roots if b² – 4ac < 0

• Relationship Between Discriminant and Nature of Roots 
• Situational Problems Based on Quadratic Equations Related to Day to Day Activities to Be Incorporated 
• Quadratic Equations Examples and Solutions 

  Application of quadratic equation

• Standard form of a quadratic equation ax² + bx + c = 0, (a ≠ 0).
• Solutions of quadratic equations (only real roots) by factorization, by completing the square and by using quadratic formula.
• Relationship between discriminant and nature of roots.
• Situational problems based on quadratic equations related to day to day activities to be incorporated.

• Geometrical Meaning of the Zeroes of a Polynomial 
• Relationship Between Zeroes and Coefficients of a Polynomial 
• Division Algorithm for Polynomials 
• Concept of 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.

• Tangent to a Circle 

  Theorem - The tangent at any point of a circle is perpendicular to the radius through the point of contact.

• Number of Tangents from a Point on a Circle 

  Theorem - The Length of Two Tangent Segments Drawn from a Point Outside the Circle Are Equal

• Concept of Circle - Centre, Radius, Diameter, Arc, Sector, Chord, Segment, Semicircle, Circumference, Interior and Exterior, Concentric Circles 

Tangent to a circle at, point of contact

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 a circle are equal.

• Similar Figures 
• Similarity of Triangles 
• Basic Proportionality Theorem Or Thales Theorem 
• Criteria for Similarity of Triangles 
• Areas of Similar Triangles 
• Right-angled Triangles and Pythagoras Property 
• Similarity Triangle Theorem 

  If in a two triangles corresponding angles are equal then their corresponding sides are in same ratio hence two triangle are similar

• Application of Pythagoras Theorem in Acute Angle and Obtuse Angle 
• Triangles Examples and Solutions 
• Angle Bisector 
• Similarity 
  • Similar triangles
  • Criteria of Similarity
    - AA Criterion of similarity
    - SAS Criterion of similarity
    - SSS Criterion of similarity
  • Construction of similar triangles
• Ratio of Sides of Triangle 

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 of 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 angle.

• Division of a Line Segment 
  • Division of Line Segment in a Given Ratio
  • Construction of a Triangle Similar to a Given Triangle
  • To divide a line segment in a given ratio
  • To construct a triangle similar to a given triangle as per given scale factor
• Construction of Tangents to a Circle 
  • Construction of Tangent to the Circle from the Point Out Side the Circle
  • To construct the tangents to a circle from a point outside it
• Constructions Examples and Solutions 

1. Division of a line segment in a given ratio (internally).
2. Tangents to a circle from a point outside it.
3. Construction of a triangle similar to a given triangle.

• Heights and Distances 
  • Problems involving Angle of Elevation
  • Problems involving Angle of Depression
  • Problems involving Angle of Elevation and Depression

Angle of elevation, Angle of Depression

• Simple problems on heights and distances.
• Problems should not involve more than two right triangles.
• Angles of elevation / depression should be only 30°, 45°, 60°.

• Trigonometric Ratios of Complementary Angles 
• Trigonometric Identities 

• Proof and applications of the identity sin²A + cos²A = 1. Only simple identities to be given.
• Trigonometric ratios of complementary angles.

• Trigonometry 
• Trigonometric Ratios 
• Trigonometric Ratios and Its Reciprocal 
• Trigonometric Ratios of Some Special Angles 
• Trigonometric Ratios of Complementary Angles 
• Trigonometric Identities 
• Proof of Existence 

  Well Defined

• Relationships Between the Ratios 

• 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° and 90°.
• Values (with proofs) of the trigonometric ratios of 30°, 45° and 60°.
• Relationships between the ratios.

• Probability - A Theoretical Approach 
  • Classical Definition of Probability
  • Type of Event - Impossible and Sure Or Certain
  • assume that all the experiments have equally likely outcomes, impossible event, sure event or a certain event, complementary events,
• Sample Space 
• Basic Ideas of Probability 
• Concept Or Properties of Probability 
• Simple Problems on Single Events 

  Not using set notation

• Classical definition of probability.
• Simple problems on single events (not using set notation).

• Mean of Grouped Data 
• Mode of Grouped Data 
• Median of Grouped Data 
• Graphical Representation of Cumulative Frequency Distribution 
• Concepts of Statistics 
• Ogives (Cumulative Frequency Graphs) 

• Mean, median and mode of grouped data (bimodal situation to be avoided).
• Cumulative frequency graph.

• Coordinate Geometry 
• Distance Formula 
• Section Formula 
• Area of a Triangle 
• Graphs of Linear Equations 
• Basic Geometric Constructions 

• Concepts of coordinate geometry, graphs of linear equations.
• Distance formula. Section formula (internal division).
• Area of a triangle.

• Areas of Sector and Segment of a Circle 
  • Area of the Sector and Circular Segment
  • Length of an Arc
• Areas of Combinations of Plane Figures 
• Circumference of a Circle 
• Area of Circle 

• 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° and 120° only. Plane figures involving triangles, simple quadrilaterals and circle should be taken.)

• Surface Area of a Combination of Solids 
• Volume of a Combination of Solids 
• Conversion of Solid from One Shape to Another 
• Frustum of a Cone 
• Concept of Surface Area, Volume, and Capacity 

1. 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.
2. 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.)