## Units and Topics

# | Unit/Topic | Marks |
---|---|---|

100 | Sets and Functions | - |

200 | Algebra | - |

300 | Coordinate Geometry | - |

400 | Calculus | - |

500 | Statistics and Probability | - |

600 | Conic Section | - |

700 | Introduction to Three-dimensional Geometry | - |

800 | Mathematical Reasoning | - |

900 | Statistics - 2 | - |

1000 | Correlation Analysis | - |

1100 | Index Numbers and Moving Averages | - |

Total | - |

## Syllabus

- Sets and Their Representations
1) Roster or Tabular method or List method

2) Set-Builder or Rule Method

3) Venn Diagram - The Empty Set
- Empty Set
- null set or the void set

- Finite and Infinite Sets
- Equal Sets
- Subsets
- Subsets of a Set of Real Numbers Especially Intervals - With Notation
- Singleton Set
- Super Set
- Subsets of set of real numbers
- Intervals as subsets of R

- Power Set
- Universal Set
- Venn Diagrams
- Operations on Sets
- Union Set
- Some Properties of the Operation of Union

- Intersection of Sets
- Some Properties of Operation of Intersection

- Difference of Sets
- “ A minus B” Or "A – B"

- Union Set
- Complement of a Set
- Properties of Complement Sets

- Properties of Complement of Sets
- Problems on Union and Intersection of Two and Three Sets.

- Sets and their representations. Empty set. Finite and Infinite sets. Equal sets. Subsets. Subsets of a set of real numbers especially intervals (with notations). Power set. Universal set. Venn diagrams. Union and Intersection of sets.
- Practical problems on union and intersection of two and three sets. Difference of sets. Complement of a set. Properties of Complement of Sets

- Sets
- Cartesian Product of Sets
- Number of Elements in the Cartesian Product of Two Finite Sets
- Cartesian Product of set of the Reals with Itself

- Relation
- Definition of Relation
- Domain
- Co-domain and Range of a Relation

- Concept of Relations
- Concept of Functions
- Function, Domain, Co-domain, Range
- Types of function

1. One-one or One to one or Injective function

2. Onto or Surjective function - Representation of Function
- Graph of a function
- Value of funcation
- Some Basic Functions - Constant Function, Identity function, Power Functions, Polynomial Function, Radical Function, Rational Function, Exponential Function, Logarithmic Function, Trigonometric function

- Exponential Function
Domain and range of this function

- Function as a Type of Mapping
- Types of Functions
- one-one (or injective)
- many-one
- onto (or surjective)
- one-one and onto (or bijective)

- Many to One Function
Type of Function

- Into Function
Type of Function

- Some Functions and Their Graphs
- Identity function - Domain and range of this function
- Constant function - Domain and range of this function
- Polynomial function -Domain and range of this function
- Rational functions - Domain and range of this function
- The Modulus function - Domain and range of this function
- Signum function - Domain and range of this function
- Greatest integer function

Ordered pairs, Cartesian product of sets. Number of elements in the cartesian product of two finite sets. Cartesian product of the set of reals with itself (upto R x R x R).

Definition of relation, pictorial diagrams, domain, co-domain and range of a relation. Function as a special type of relation. Function as a type of mapping, types of functions (one to one, many to one, onto, into) domain, co-domain and range of a function. Real valued functions, domain and range of these functions, constant, identity, polynomial, rational, modulus, signum, exponential, logarithmic and greatest integer functions, with their graphs. Sum, difference, product and quotient of functions

Sets: Self-explanatory

Basic concepts of Relations and Functions

- Ordered pairs, sets of ordered pairs.
- Cartesian Product (Cross) of two sets, cardinal number of a cross product.
- Relations as:
- an association between two sets.
- a subset of a Cross Product.
- Domain, Range and Co-domain of a Relation.
- Functions:
- As special relations, concept of writing “y is a function of x” as y = f(x).
- Introduction of Types: one to one, many to one, into, onto.
- Domain and range of a function.
- Sketches of graphs of exponential function, logarithmic function, modulus function, step function and rational function.

- Magnitude of an Angle
Measure of Angle

Circular measure

- Concept of Angle
- Part of Angle - Initial Side,Terminal Side,Vertex
- Types of Angle - Positive and Negative Angles
- Measuring Angles in Radian
- Measuring Angles in Degrees
- initial side, terminal side, vertex,positive angle, negative angle
- Degree measure
- Radian measure
- Relation between radian and real numbers
- Relation between degree and radian

- Conversion from One Measure to Another
- Introduction of Trigonometric Functions
- Trigonometric Functions with the Help of Unit Circle

- Trigonometric Functions
- Truth of the Identity
sin

^{2}x+cos^{2}x=1, for All X.

- Truth of the Identity
- Signs of Trigonometric Functions
- Domain and Range of Trigonometric Functions
- Domain and Range of Trignometric Functions and Their Graphs

- Trigonometric Functions of Sum and Difference of Two Angles
- Identities Related to Sin 2x, Cos2x, Tan 2x, Sin3x, Cos3x and Tan3x.
- Deducing the Identities
- Deducing the identities like the following:-

`tan(x+-y)=(tanx+-tany)/(1+-tanxtany)", "cot(x+-y)=(cotxcoty+-1)/(coty+-cotx)`

`sinalpha+-sinbeta=2"sin"1/2(alpha+-beta)"cos"1/2(alpha+-beta)`

`cosalpha+cosbeta=2"cos"1/2(alpha+beta)"cos"1/2(alpha-beta)`

`cosalpha-cosbeta=-2"sin"1/2(alpha+beta)"sin"1/2(alpha-beta)`

- Trigonometric Equations
- Solution of Trigonometric Equations (Solution in the Specified Range)
- Graphs of Trigonometric Functions
- The graph of sine function
- The graph of cosine function
- The graph of tangent function

- Trigonometric Functions of Compound Angles
- Trigonometric Functions of Multiple Angles
upto double and triple angles only

- Trigonometric Functions of Multiple Angles
- Convention of Sign of Angles
- The Relation S = rθ Where θ is in Radians
- Relationship Between Trigonometric Functions
- Periods of Trigonometric Functions
- Compound and Multiple Angles- Addition and Subtraction Formula
sin(A B); cos(A B); tan(A B); tan(A + B + C) etc., Double angle, triple angle, half angle and one third angle formula as special cases.

- Trigonometric Functions of All Angles
- Sum and Differences as Products
Sum and differences as products

`sinC + sinD = 2sin((C+D)/2)cos((C-D)/2)", etc."`

- Product to Sum Or Difference
i.e.

2sinAcosB = sin(A + B) + sin(A – B) etc.

- Trigonometric Equations
- Properties of Δ
- Sine formula: `a/sinA=b/sinB=c/sinC`
- Cosine formula:`cosA=(b^2+c^2-a^2)/(2bc)`, etc
- Area of triangle:Δ = `1/2`bc A etc
- Simple applications of the above

Positive and negative angles. Measuring angles in radians and in degrees and conversion from one measure to another.

Definition of trigonometric functions with the help of unit circle. Truth of the identity sin^{2} x+cos^{2} x=1, for all x. Signs of trigonometric functions. Domain and range of trignometric functions and their graphs. Expressing sin (x±y) and cos (x±y) in terms of sinx, siny, cosx & cosy and their simple applications. Deducing the identities like the following:

`tan(x+-y)=(tanx+-tany)/(1+-tanxtany)`

`cot(x+-y)=(cotxcoty+-1)/(coty+-cotx)`

`sinalpha+-sinbeta=2sin `

`cosalpha+cosbeta=2cos`

`cosalpha-cosbeta=-2sin`

Identities related to sin 2x, cos2x, tan 2x, sin3x, cos3x and tan3x. General solution of trigonometric equations of the type siny = sina, cosy = cosa and tany = tana.

Properties of triangles (proof and simple applications of sine rule cosine rule and area of triangle)

**Angles and Arc lengths**

- Angles: Convention of sign of angles.
- Magnitude of an angle: Measures of Angles; Circular measure.
- The relation S = rθ where θ is in radians. Relation between radians and degree.
- Definition of trigonometric functions with the help of unit circle.
- Truth of the identity sin
^{2}x+cos^{2}x=1

** Trigonometric Functions**

- Relationship between trigonometric functions.
- Proving simple identities.
- Signs of trigonometric functions.
- Domain and range of the trigonometric functions.
- Trigonometric functions of all angles.
- Periods of trigonometric functions.
- Graphs of simple trigonometric functions (only sketches)

**Compound and multiple angles**

- Addition and subtraction formula: sin(A ± B); cos(A± B); tan(A ± B); tan(A + B + C) etc., Double angle, triple angle, half angle and one third angle formula as special cases.
- Sum and differences as products
- sinC + sinD = `2sin((C+D)/2)cos((C-D)/2)` , etc
- Product to sum or difference i.e. 2sinAcosB = sin(A + B) + sin(A – B) etc

Trigonometric Equations

- Solution of trigonometric equations (General solution and solution in the specified range).
- Equations expressible in terms of sinθ =0 etc.
- Equations expressible in terms i.e sinθ = sin`alpha` etc
- Equations expressible multiple and sub- multiple angles i.e. sin
^{2}θ = sin^{2}`alpha` etc - Linear equations of the form acosθ +bsinθ = c, where `|c|<=sqrt(a^2+b^2) `
- Properties of Δ
- Sine formula: `a/sinA=b/sinB=c/sinC`
- Cosine formula:`cosA=(b^2+c^2-a^2)/(2bc)`, etc
- Area of triangle:Δ = `1/2`bc A etc
- Simple applications of the above

- Motivation
- Motivating the Application of the Method by Looking at Natural Numbers as the Least Inductive Subset of Real Numbers

- Principle of Mathematical Induction

- Process of the proof by induction, motivating the application of the method by looking at natural numbers as the least inductive subset of real numbers. The principle of mathematical induction and simple applications
- Using induction to prove various summations, divisibility and inequalities of algebraic expressions only

- Concept of Complex Numbers
- Imaginary number
- Complex Number

- The Modulus and the Conjugate of a Complex Number
- Modulus of Complex Number
- Conjugate of Complex Number

- Properties of Conjugate, Modulus and Argument of Complex Numbers
- Argand Plane and Polar Representation
- Representation of Complex Number - Argand Plane Representation
- Representation of Complex Number - Polar Representation of Complex Numbers

- Properties of Cube Roots of Unity
- Algebra of Complex Numbers
- Equality of two Complex Numbers
- Conjugate of a Complex Number
- Properties of `barz`
- Addition of complex numbers - Properties of addition, Scalar Multiplication
- Subtraction of complex numbers - Properties of Subtraction
- Multiplication of complex numbers - Properties of Multiplication
- Powers of i in the complex number
- Division of complex number - Properties of Division

- Locus Questions on Complex Numbers.
- Triangle Inequality

Introduction of complex numbers and their representation, Algebraic properties of complex numbers. Argand plane and polar

representation of complex numbers. Square root of a complexnumber. Cube root of unity

- Conjugate, modulus and argument of complex numbers and their properties.
- Sum, difference, product and quotient of two complex numbers additive and multiplicative inverse of a complex number
- Locus questions on complex numbers.
- Triangle inequality.
- Square root of a complex number.
- Cube roots of unity and their properties

- Quadratic Equations
- Equations Reducible to Quadratic Form
- Nature of Roots
- Product and sum of roots.
- Roots are rational, irrational, equal, reciprocal, one square of the other.
- Complex roots.
- Framing quadratic equations with given roots

- Quadratic Functions
Given `alpha`,`beta` as roots then find the equation whose roots are of the form `alpha^3`, `beta^3` , etc

Case I:a>0 -> 1)Real roots, 2)Complex roots,3)Equal roots

Case II:a<0 -> 1)Real roots, 2)Complex roots,3)Equal roots

Where ‘a’ is the coefficient of x

^{2}in the equations of the form ax^{2}+ bx + c = 0.Understanding the fact that a quadratic expression (when plotted on a graph) is a parabola.

- Sign of Quadratic
Sign when the roots are real and when they are complex

- Quadratic Inequalities
Using method of intervals for solving problems of the type:

A perfect square e.g. `x^2-6x+9>=0`

Inequalities involving rational expression of type

`f(x)/g(x)<=a` etc to be covered

- Algebraic Solutions of Linear Inequalities in One Variable and Their Graphical Representation
- Graphical Solution of Linear Inequalities in Two Variables
Linear Inequalities - Graphical Representation of Linear Inequalities in Two Variables

- Solution of System of Linear Inequalities in Two Variables

Statement of Fundamental Theorem of Algebra, solution of quadratic equations (with real coefficients).

Use of the formula:

`x=(-b+-sqrt(b^2-4ac))/(2a)`

In solving quadratic equations.

Equations reducible to quadratic form.

Nature of roots

- Product and sum of roots.
- Roots are rational, irrational, equal, reciprocal, one square of the other.
- Complex roots.
- Framing quadratic equations with given roots

**Quadratic Functions**

Given `alpha`,`beta` as roots then find the equation whose roots are of the form `alpha^3`, `beta^3` , etc

Case I:a>0 -> 1)Real roots, 2)Complex roots,3)Equal roots

Case II:a<0 -> 1)Real roots, 2)Complex roots,3)Equal roots

Where ‘a’ is the coefficient of x^{2} in the equations of the form ax^{2} + bx + c = 0.

Understanding the fact that a quadratic expression (when plotted on a graph) is a parabola.

**Sign of quadratic**

Sign when the roots are real and when they are complex

**Inequalities**

Linear Inequalities

Algebraic solutions of linear inequalities in one variable and their representation on the number line. Graphical representation of linear inequalities in two variables. Graphical method of finding a solution of system of linear inequalities in two variables.

Quadratic Inequalities

Using method of intervals for solving problems of the type:

A perfect square e.g. `x^2-6x+9>=0`

Inequalities involving rational expression of type

`f(x)/g(x)<=a` etc to be covered

- Introduction of Permutations and Combinations
- Fundamental Principle of Counting
- Concept of Permutations
- Derivation of Formulae and Their Connections
Derivation of formulae for

^{n}P_{r }and^{n}C_{r}and their connections - Introduction of Permutations and Combinations
- Restricted Permutation
- Permutation - Certain Things Always Occur Together
- Permutation - Certain Things Never Occur
- Permutation - Formation of Numbers with Digits
- Permutation - Permutation of Alike Things
- Permutation - Permutation of Repeated Things
- Permutation - Word Building
Repeated Letters

No Letters Repeated

- Properties of Combination
- Combination
^{n}C_{r},^{n}C_{n}=1,^{n}C_{0}= 1,^{n}C_{r}=^{n}C_{n–r},^{n}C_{x}=^{n}C_{y}, then x + y = n or x = y,^{n+1}C_{r}=^{n}C_{r-1}+^{n}C_{r}- When all things are different
- When all things are not different.
- Mixed problems on permutation and combinations.

Fundamental principle of counting. Factorial n (n!) Permutations and combinations, derivation of formulae for ^{n}P_{r }and ^{n}C_{r} and their connections, simple application.

- Factorial notation n! , n! =n (n-1)!
- Fundamental principle of counting
- Permutations

^{n}P_{r..}- Restricted permutation.
- Certain things always occur together.
- Certain things never occur.
- Formation of numbers with digits.
- Word building - repeated letters - No letters repeated.
- Permutation of alike things.
- Permutation of Repeated things.
- Circular permutation – clockwise counterclockwise – Distinguishable /not distinguishable

4. Combinations

^{n}C_{r},^{n}C_{n}=1,^{n}C_{0}= 1,^{n}C_{r}=^{n}C_{n–r},^{n}C_{x}=^{n}C_{y}, then x + y = n or x = y,^{n+1}C_{r}=^{n}C_{r-1}+^{n}C_{r}- When all things are different
- When all things are not different.
- Mixed problems on permutation and combinations.

- Introduction of Binomial Theorem
- History of Binomial Theorem

- Binomial Theorem for Positive Integral Indices
- Statement and Proof of the Binomial Theorem for Positive Integral Indices
- Proof of Binomial Therom by Induction
- Special Case in Binomial Therom
- Pascal's Triangle
- Binomial theorem for any positive integer n
- Some special cases-(In the expansion of (a + b)
^{n})

- General and Middle Terms
- Binomial Theorem

History, statement and proof of the binomial theorem for positive integral indices. Pascal's triangle, General and middle term in binomial expansion, simple applications.

- Significance of Pascal’s triangle.
- Binomial theorem (proof using induction) for positive integral powers,

i.e. `(x+y)^n=^nC_0x^n+^nC_1x^(n-1)y+.....+^nC_ny^n`

- Sequence and Series
- Arithmetic Progression (A.P.)
- Three Terms in Arithematic Progression (A.P.)
- Three terms in A.P. :- a - d, a, a + d

- Four Terms in Arithematic Progression (A.P.)
- Four terms in A.P.:- a - 3d, a - d, a+ d, a + 3d

- Inserting Two Or More Arithmetic Means Between Any Two Numbers
- Geometric Progression (G. P.)
- N
^{th}Term of Geometric Progression (G.P.) - General Term of a Geometric Progression (G.P.)
- Sum of First N Terms of a Geometric Progression (G.P.)
- Sum of infinite terms of a G.P.
- Geometric Mean (G.M.)

- N
- Three Terms in Geometric Progression (G.P.)
- Three terms are in G.P. ar, a, a
^{r-1}

- Three terms are in G.P. ar, a, a
- Four Terms Are in Geometric Progression (G.P.)
- Four terms are in GP ar
^{3}, ar, ar^{-1},ar^{-3}

- Four terms are in GP ar
- Inserting Two Or More Geometric Means Between Any Two Numbers.
- Relationship Between A.M. and G.M.
- Relation Between Arithematic Mean (A.M.) and Geometric Mean (G.M.)

- Arithmetico Geometric Series
- n
^{th}term of A.G.P. - Sum of n terms of A.G.P.
- Properties of Summation

- n

Sequence and Series. Arithmetic Progression (A. P.). Arithmetic Mean (A.M.) Geometric Progression (G.P.), general term of a G.P., sum of first n terms of a G.P., infinite G.P. and its sum, geometric mean (G.M.), relation between A.M. and G.M. Formulae for the following special sums `sumn,sumn^2,sumn^3`

Arithmetic Progression (A.P.)

- T
_{n}= a + (n - 1)d - `S_n=n/2{2a+(n-1)d}`
- Arithmetic mean: 2b = a + c
- Inserting two or more arithmetic means between any two numbers.
- Three terms in A.P. : a - d, a, a + d
- Four terms in A.P.: a - 3d, a - d, a+ d, a + 3d

Geometric Progression (G.P.)

- T
_{n}= ar^{n-1}, S_{n = }`(a(r^n-1))/(r-1),` - `S_oo = a/(1-r;)|r|<1 `
- Inserting two or more Geometric Means between any two numbers
- Three terms are in G.P. ar, a, a
^{r-1} - Four terms are in GP ar
^{3}, ar, ar^{-1},ar^{-3}

Arithmetico Geometric Series

- Identifying series as A.G.P. (when we substitute d = 0 in the series, we get a G.P. and when we substitute r =1 the

A.P).

Special sums `sumn,sumn^2,sumn^3`

- Using these summations to sum up other related expression.

- Straight Lines
- Shifting of Origin
- Slope of a Line
- Slope of a Line Or Gradient of a Line.
- Parallelism of Line
- Perpendicularity of Line in Term of Slope
- Collinearity of Points
- Slope of a line when coordinates of any two points on the line are given
- Conditions for parallelism and perpendicularity of lines in terms of their slopes
- Angle between two lines
- Collinearity of three points

- Various Forms of the Equation of a Line
- General Equation of a Line
- Different forms of Ax + By + C = 0 - Slope-intercept form, Intercept form, Normal form

- Distance of a Point from a Line
- Introduction of Distance of a Point from a Line
- Distance between two parallel lines

- Basic Concepts of Points and Their Coordinates
- Locus

Brief recall of two dimensional geometry from earlier classes. Shifting of origin. Slope of a line and angle between two lines. Various forms of equations of a line: parallel to axis, pointslope form, slope- intercept form, two-point form, intercept form and normal form.

General equation of a line. Equation of family of lines passing through the point of intersection of two lines. Distance of a point from a line.

- Basic concepts of Points and their coordinates
- The straight line

- Slope or gradient of a line.
- Angle between two lines.
- Condition of perpendicularity and parallelism.
- Various forms of equation of lines.
- Slope intercept form.
- Two point slope form.
- Intercept form.
- Perpendicular /normal form.
- General equation of a line.
- Distance of a point from a line.
- Distance between parallel lines.
- Equation of lines bisecting the angle between two lines.
- Equation of family of lines
- Definition of a locus.
- Equation of a locus

- Equations of a Circle in Standard Form
- Circle
- Equations of a Circle in General Form
- Equations of a Circle in Parametric Form
- Conics
- Focus-directrix Property
focus-directrix property of parabola, ellipse, hyperbola, parabola

- Focus-directrix Property
- Given the Equation of a Circle, to Find the Centre and the Radius
- Finding the Equation of a Circle
Finding the equation of a circle.

- Given three non collinear points
- Given other sufficient data for example centre is (h, k) and it lies on a line and two points on the circle are given, etc.

- Condition for Tangency
- Equation of a Tangent to a Circle

Equations of a circle in:

- Standard form.
- Diameter form.
- General form.
- Parametric form.

Given the equation of a circle, to find the centre and the radius.

Finding the equation of a circle.

- Given three non collinear points
- Given other sufficient data for example centre is (h, k) and it lies on a line and two points on the circle are given, etc.

Tangents:

- Condition for tangency
- Equation of a tangent to a circle

- Concept of Limits
- Introduction of Limits
- Limits of Trigonometric Functions
- Limits of Algebraic Functions
- Introduction of Derivatives
- Derivative
- Derivative of Algebraic Functions

Derivative introduced as rate of change both as that of distance function and geometrically.

Intuitive idea of limit. Limits of polynomials and rational functions trigonometric, exponential and logarithmic functions.

Definition of derivative relate it to scope of tangent of the curve, Derivative of sum, difference, product and quotient of functions. Derivatives of polynomial and trigonometric functions

**Limits**

- Notion and meaning of limits.
- Fundamental theorems on limits (statement only).
- Limits of algebraic and trigonometric functions.
- Limits involving exponential and logarithmic functions.

**NOTE**: Indeterminate forms are to be introduced while calculating limits

**Differentiation**

- Meaning and geometrical interpretation of derivative.
- Derivatives of simple algebraic and trigonometric functions and their formulae.
- Differentiation using first principles.
- Derivatives of sum/difference.
- Derivatives of product of functions. Derivatives of quotients of functions

- Central Tendency - Mean
- Concept of Range
- Measures of Dispersion - Range

- Statistics
- Mean Deviation
- Variance and Standard Deviation
- Standard Deviation - by Direct Method
- Standard Deviation - by Step Deviation Method"
- Introduction of Variance and Standard Deviation
- Analysis of Frequency Distributions

Measures of dispersion: range, mean deviation, variance and standard deviation of ungrouped/grouped data. Analysis of frequency distributions with equal means but different variances

- Mean deviation about mean and median
- Standard deviation - by direct method, short cut method and step deviation method.

**NOTE**: Mean, Median and Mode of grouped and ungrouped data are required to be covered.

- Random Experiments
- Event
- Types of Events
- Simple or elementary event
- Occurrence and non-occurrence of event
- Sure Event
- Impossible Event
- Complimentary Event

- Exhaustive Events
- Types of Event - Exhaustive Events

- Mutually Exclusive Events
- Types of Event - Mutually Exclusive Events

- Types of Events
- Probability
- Axiomatic Approach to Probability

Random experiments; outcomes, sample spaces (set representation). Events; occurrence of events, 'not', 'and' and 'or'

events, exhaustive events, mutually exclusive events, Axiomatic (set theoretic) probability, connections with other theories studied in earlier classes. Probability of an event, probability of 'not', 'and' and 'or' events

- Random experiments and their outcomes.
- Events: sure events, impossible events, mutually exclusive and exhaustive events.
- Definition of probability of an event
- Laws of probability addition theorem.

- Sections of a Cone
- Conics as a Section of a Cone
- Definition of Foci, Directrix, Latus Rectum
- Parabola
- Ellipse
- Latus Rectum
- Latus Rectum in Ellipse

- Latus Rectum
- Hyperbola
- Transverse and Conjugate Axes
- Coordinates of Vertices
- Foci and Centre
- Equations of the Directrices and the Axes
- General Second Degree Equation
ax

^{2}+ 2hxy + by^{2}+ 2gx + 2 fy + c = 0- Case 1: pair of straight line if abc+2fgh-af
^{2}-bg^{2}-ch^{2}=0 - Case 2: abc+2fgh-af
^{2}-bg^{2}-ch^{2}≠0 then represents a parabola if h^{2}= ab, ellipse if h^{2}< ab, and hyperbola if h^{2}> ab.

- Case 1: pair of straight line if abc+2fgh-af
- General Equation of Tangents
- Point of Contact and Locus Problems

Sections of a cone, ellipse, parabola, hyperbola, a point, a straight line and a pair of intersecting lines as a degenerated case of a conic section. Standard equations and simple properties of parabola, ellipse and hyperbola.

Conics as a section of a cone.

Definition of Foci, Directrix, Latus Rectum.

PS = ePL where P is a point on the conics, S is the focus, PL is the perpendicular distance of the point from the directrix.

(i) Parabola

e =1, y^{2} = ±4ax, x^{2} = 4ay, y^{2} = -4ax

x^{2} = -4ay, (y - `beta`)^{2} =±4a (x - `alpha`),

(x - `alpha`)2 = ±4a (y - `beta`).

- Rough sketch of the above
- The latus rectum; quadrants they lie in; coordinates of focus and vertex; and equations of directrix and the axis.
- Finding equation of Parabola when Foci and directrix are given, etc
- Application questions based on the above.

(ii) Ellipse

- `x^2/a^2+y^2/b^2=1, e<1,b^2=a^2(1-e^2)`
- `(x-alpha)^2/a^2+(y-beta)^2/b^2=1`
- Cases when a > b and a < b.
- Rough sketch of the above.
- Major axis, minor axis; latus rectum; coordinates of vertices, focus and centre; and equations of directrices and the axes.
- Finding equation of ellipse when focus and directrix are given.
- Simple and direct questions based on the above.
- Focal property i.e. SP + SP' = 2a.

(iii)Hyperbola

- `x^2/a^2-y^2/b^2=1,e>1,b^2=a^2(e^2-1)`
- `(x-alpha)^2/a^2-(y-beta)^2/b^2=1`
- Cases when coefficient y
^{2}is negative and coefficient of x^{2}is negative - Rough sketch of the above
- Focal property i.e. SP - S’P = 2a
- Transverse and Conjugate axes; Latus rectum; coordinates of vertices, foci and centre; and equations of the directrices and the axes.

General second degree equation

ax^{2} + 2hxy + by^{2} + 2gx + 2 fy + c = 0

- Case 1: pair of straight line if abc+2fgh-af
^{2}-bg^{2}-ch^{2}=0 - Case 2: abc+2fgh-af
^{2}-bg^{2}-ch^{2}≠0 then represents a parabola if h^{2}= ab, ellipse if h^{2}< ab, and hyperbola if h^{2}> ab.

Condition that y = mx + c is a tangent to the conics, general equation of tangents, point of contact and locus problems.

- Three - Dimensional Geometry
- Coordinate Axes and Coordinate planes
Coordinate Axes and Coordinate Planes in Three Dimensions

- Distance Between Two Points
- Distance Between Two Points in 3-D Space

- Coordinate Axes and Coordinate planes
- As an Extension of 2-D
- Distance Formula
- Midpoint Formula

Coordinate axes and coordinate planes in three dimensions. Coordinates of a point. Distance between two points and section formula

- As an extension of 2-D
- Distance formula.
- Section and midpoint form

- Mathematically Acceptable Statements
- New Statements from Old
- Special Words Or Phrases
- Mathematical Reasoning
- Consolidating the Understanding
"if and only if (necessary and sufficient) condition", "implies", "and/or", "implied by'', "and", "or'', "there exists" and their use through variety of examples related to real life and Mathematics

- Consolidating the Understanding
- Introduction of Validating Statements
- Validating the Statements Involving the Connecting Words
- statement “p and q”, Statements with “Or”, Statements with “If-then”, Statements with “if and only if ”

- Validation by Contradiction
- Implications

- Mathematically acceptable statements. Connecting words/ phrases - consolidating the understanding of "if and only if (necessary and sufficient) condition", "implies", "and/or", "implied by", "and", "or", "there exists" and their use through variety of examples related to the Mathematics and real life.
- Validating the statements involving the connecting words, Difference between contradiction, converse and contrapositive

- Combined mean and standard deviation
- The Median, Quartiles, Deciles, Percentiles and Mode of grouped and ungrouped data

- Definition and Meaning of Covariance
- Statistics
- Rank Correlation by Spearman’s
Correction Included

Definition and meaning of covariance

Coefficient of Correlation by Karl Pearson

- if `x-barx,y-bary` are small non fractional numbers we use
- `r=(sum(x-barx)(y-bary))/(sqrt(sum(x-barx)^2)sqrt(sum(y-bary)^2)`
- If x and y are small numbers, we use
- `r=(sumxy-1/Nsumxsumy)/(sqrt(sumx^2-1/N(sumx)^2)sqrt(sumy^2-1/N(sumy)^2))`
- Otherwise, we use assumed means
- A and B, where u = x-A, v = y-B
- `r=(sumuv-1/N(sumu)(sumv))/(sqrt(sumu^2-1/N(sumu)^2)sqrt(sumv^2-1/N(sumv)^2))`

Rank correlation by Spearman’s (Correction included)

- Price Index Or Price Relative
- Construction of Index Numbers
- Weighted Aggregate Method
- Laspeyre's Price Index Number
- Paasche’s Price Index Number
- Dorbish-Bowley’s Price Index Number
- Fisher’s Ideal Price Index Number
- Marshall-Edgeworth’s Price Index Number
- Walsh’s Price Index Number

- Weighted Aggregate Method
- Simple Average of Price Relatives
- Weighted Average of Price Relatives
(cost of living index, consumer price index)

- Price index or price relative.
- Simple aggregate method.
- Weighted aggregate method.
- Simple average of price relatives.
- Weighted average of price relatives (cost of living index, consumer price index)

- Meaning and Purpose of the Moving Averages
- Calculation of Moving Averages with the Given Periodicity and Plotting Them on a Graph
- If the Period is Even, Then the Centered Moving Average is to Be Found Out and Plotted

- Meaning and purpose of the moving averages.
- Calculation of moving averages with the given periodicity and plotting them on a graph.
- If the period is even, then the centered moving average is to be found out and plotted.