#### Topics

##### Trigonometric Functions

- Concept of Angle
- Introduction of Trigonometric Functions
- Signs of Trigonometric Functions
- Domain and Range of Trigonometric Functions
- Trigonometric Functions of Sum and Difference of Two Angles
- Trigonometric Equations
- Trigonometric Functions
- Truth of the Identity
- Negative Function Or Trigonometric Functions of Negative Angles
- 90 Degree Plusminus X Function
- Conversion from One Measure to Another
- 180 Degree Plusminus X Function
- 2X Function
- 3X Function
- Expressing Sin (X±Y) and Cos (X±Y) in Terms of Sinx, Siny, Cosx and Cosy and Their Simple Applications
- Graphs of Trigonometric Functions
- Transformation Formulae
- Values of Trigonometric Functions at Multiples and Submultiples of an Angle
- Sine and Cosine Formulae and Their Applications

##### Binomial Theorem

- Introduction of Binomial Theorem
- Binomial Theorem for Positive Integral Indices
- General and Middle Terms
- Proof of Binomial Therom by Pattern
- Proof of Binomial Therom by Combination
- Rth Term from End
- Simple Applications of Binomial Theorem

##### Statistics

- Measures of Dispersion
- Concept of Range
- Mean Deviation
- Introduction of Variance and Standard Deviation
- Standard Deviation
- Standard Deviation of a Discrete Frequency Distribution
- Standard Deviation of a Continuous Frequency Distribution
- Shortcut Method to Find Variance and Standard Deviation
- Introduction of Analysis of Frequency Distributions
- Comparison of Two Frequency Distributions with Same Mean
- Statistics Concept
- Central Tendency - Mean
- Central Tendency - Median
- Concept of Mode
- Measures of Dispersion - Quartile Deviation
- Standard Deviation - by Short Cut Method

##### Sets and Functions

##### Limits and Derivatives

- Intuitive Idea of Derivatives
- Introduction of Limits
- Introduction to Calculus
- Algebra of Limits
- Limits of Polynomials and Rational Functions
- Limits of Trigonometric Functions
- Introduction of Derivatives
- Algebra of Derivative of Functions
- Derivative of Polynomials and Trigonometric Functions
- Derivative Introduced as Rate of Change Both as that of Distance Function and Geometrically
- Limits of Logarithmic Functions
- Limits of Exponential Functions
- Derivative of Slope of Tangent of the Curve
- Theorem for Any Positive Integer n
- Graphical Interpretation of Derivative
- Derive Derivation of x^n

##### Mathematical Reasoning

- Mathematically Acceptable Statements
- New Statements from Old
- Special Words Or Phrases
- Contrapositive and Converse
- Introduction of Validating Statements
- Validation by Contradiction
- Difference Between Contradiction, Converse and Contrapositive
- Consolidating the Understanding

##### Straight Lines

- Slope of a Line
- Various Forms of the Equation of a Line
- General Equation of a Line
- Distance of a Point from a Line
- Brief Recall of Two Dimensional Geometry from Earlier Classes
- Shifting of Origin
- Equation of Family of Lines Passing Through the Point of Intersection of Two Lines

##### Introduction to Three-dimensional Geometry

##### Probability

- Random Experiments
- Introduction of Event
- Occurrence of an Event
- Types of Events
- Algebra of Events
- Exhaustive Events
- Mutually Exclusive Events
- Axiomatic Approach to Probability
- Probability of 'Not', 'And' and 'Or' Events

##### Algebra

##### Relations and Functions

- Cartesian Product of Sets
- Concept of Relation
- Concept of Functions
- Some Functions and Their Graphs
- Algebra of Real Functions
- Ordered Pairs
- Equality of Ordered Pairs
- Pictorial Diagrams
- Graph of Function
- Pictorial Representation of a Function
- Exponential Function
- Logarithmic Functions
- Brief Review of Cartesian System of Rectanglar Co-ordinates

##### Sequence and Series

##### Linear Inequalities

##### Coordinate Geometry

##### Sets

- Sets and Their Representations
- Empty Set (Null or Void Set)
- Finite and Infinite Sets
- Equal Sets
- Subsets
- Power Set
- Universal Set
- Venn Diagrams
- Intrdouction of Operations on Sets
- Union of Sets
- Intersection of Sets
- Difference of Sets
- Complement of a Set
- Practical Problems on Union and Intersection of Two Sets
- Proper and Improper Subset
- Open and Close Intervals
- Disjoint Sets
- Element Count Set

##### Conic Sections

- Sections of a Cone
- Concept of Circle
- Introduction of Parabola
- Standard Equations of Parabola
- Latus Rectum
- Introduction of Ellipse
- Relationship Between Semi-major Axis, Semi-minor Axis and the Distance of the Focus from the Centre of the Ellipse
- Special Cases of an Ellipse
- Eccentricity
- Standard Equations of an Ellipse
- Latus Rectum
- Introduction of Hyperbola
- Eccentricity
- Standard Equation of Hyperbola
- Latus Rectum
- Standard Equation of a Circle

##### Calculus

##### Complex Numbers and Quadratic Equations

- Concept of Complex Numbers
- Algebraic Operations of Complex Numbers
- The Modulus and the Conjugate of a Complex Number
- Argand Plane and Polar Representation
- Quadratic Equations
- Algebra of Complex Numbers - Equality
- Algebraic Properties of Complex Numbers
- Need for Complex Numbers
- Square Root of a Complex Number

##### Permutations and Combinations

- Fundamental Principles of Counting
- Permutations
- Combination
- Introduction of Permutations and Combinations
- Permutation Formula to Rescue and Type of Permutation
- Smaller Set from Bigger Set
- Derivation of Formulae and Their Connections
- Simple Applications of Permutations and Combinations
- Factorial N (N!) Permutations and Combinations

##### Mathematical Reasoning

##### Statistics and Probability

##### Principle of Mathematical Induction

- Mean deviation for grouped data
- Mean deviation for ungrouped data

## Notes

An absolute measure of dispersion is the mean of these deviations. A measure of central tendency lies between the maximum and the minimum values of the set of observations. Therefore, some of the deviations will be negative and some positive. Thus, the sum of deviations may vanish. Moreover, the sum of the deviations from mean ( x ) is zero.

M.D(a) = `("Sum of absolute values of deviations from a")/ ("Number of observations")`**Remark :** Mean deviation may be obtained from any measure of central tendency. However, mean deviation from mean and median are commonly used in statistical studies.

**1) Mean deviation for ungrouped data :**

Let n observations be `x_1, x_2, x_3, ...., x_n`. The following steps are involved in the calculation of mean deviation about mean or median:

**Step 1** - Calculate the measure of central tendency about which we are to find the mean deviation. Let it be ‘a’.

**Step 2**- Find the deviation of each xi from a, i.e., `x_1 – a, x_2 – a, x_3 – a,. . . , x_n– a`

**Step 3**- Find the absolute values of the deviations, i.e., drop the minus sign (–), if it is

there, i.e., `|x_1 - a| , |x_2 - a| , | x_3 - a |, ..., |x_n-a|`

**Step 4**- Find the mean of the absolute values of the deviations. This mean is the mean deviation about a, i.e.,

M.D.(a) = \[\displaystyle\sum_{i=1}^{n} |x_i - a|\]

Thus M.D.`(bar x)` = `1/n` \[\displaystyle\sum_{i=1}^{n} |x_i - \bar{x} | \] , where `bar x` = Mean

and M.D.(M) = `1/n` \[\displaystyle\sum_{i=1}^{n} |x_i - M|\], where M = Median

For example : The mean deviation about the mean for the following data :

12 , 3 , 18 , 17, 4,9,17 ,19,20,15,8,17,2,3,16,11,3,1,0,5

find the mean ( x ) of the given data

`bar x = 1/20 sum _(i=1)^20 x_i = 200 / 20 = 10`

The absolute values of the deviations from mean, i.e., `|x_i - bar x|` are

2,7,8,7,6,1,7,9,10,5,2,7,8,7,6,1,7,9,10,5

Therefore `sum_(i=1)^20 |x_i - bar x| = 124`

and M.D.`(bar x) = 124/20` = 6.2

**2) Mean deviation for grouped data :**Data can be grouped into two ways :

(a) Discrete frequency distribution,

(b) Continuous frequency distribution.

Let us discuss the method of finding mean deviation for both types of the data.

**Discrete frequency distribution:** Let the given data consist of n distinct values `x_1, x_2, ..., x_n` occurring with frequencies `f_1, f_2 , ..., f_n` respectively. This data can be represented in the tabular form as given below, and is called discrete frequency distribution:

`x : x_1 x_2 x_3 ... x_n`

`f : f_1 f_2 f_3 ... f_n`

**Mean deviation about mean:**

First of all we find the mean

`bar x` of the given data by using the formula\[\bar{x} =\frac{\displaystyle\sum_{i=1}^{n} x_i f_i }{ \displaystyle\sum_{i=1}^{n} f_i\ }= \frac {1}{N} \displaystyle\sum_{i=1}^{n}

x_i f_i \] ,

where summation `x_i f_i `denotes the sum of the products of observations xi with their respective frequencies`f_i `and N =\[\displaystyle\sum_{i=1}^{n} f_i\] is the sum of the frequencies.The mean of the absolute values of the deviations, which is the required mean deviation about the mean.

M.D.\[\bar{x} =\frac{\displaystyle\sum_{i=1}^{n} f_i |x_i - \bar{x}| }{ \displaystyle\sum_{i=1}^{n} f_i\ }= \frac {1}{N} \displaystyle\sum_{i=1}^{n}

f_i |x_i - \bar{x}| \] ,**(ii) Mean deviation about median:**

To find mean deviation about median, we find the median of the given discrete frequency distribution. For this the observations are arranged in ascending order. After this the cumulative frequencies are obtained. Then, we identify the observation whose cumulative frequency is equal to or just greater than `N/2`, where N is the sum of frequencies. This value of the observation lies in the middle of the data, therefore, it is the required median. After finding median, we obtain the mean of the absolute values of the deviations from median.Thus

M.D.(M) =\[\frac {1}{N} \displaystyle\sum_{i=1}^{n}

f_i |x_i -M| \]

**(b) Continuous frequency distribution:**

A continuous frequency distribution is a series in which the data are classified into different class-intervals without gaps alongwith their respective frequencies. **(i) Mean deviation about mean:**

The assumption that the frequency in each class is centred at its mid-point.

The mid-point of each given class and proceed further as for a discrete frequency distribution to find the mean deviation.

**(ii) Mean deviation about median:**

The process of finding the mean deviation about median for a continuous frequency distribution is similar as we did for mean deviation about the mean. The only difference lies in the replacement of the mean by median while taking deviations.

The data is first arranged in ascending order. Then, the median of continuous frequency distribution is obtained by first identifying the class in which median lies (median class) and then applying the formula

Median = l + `(N/2 - C)/f xx h`

where median class is the class interval whose cumulative frequency is just greater than or equal to `N/2` , N is the sum of frequencies , l , f , h and C are , respectively the lower limit , the frequency, the width of the median class and C the cumulative frequency of the class just preceding the median class. After finding the median, the absolute values of the deviations of mid-point xi of each class from the median i.e. `|x_i - M|`are obtained.

M.D.(M) = \[\frac {1}{N} \displaystyle\sum_{i=1}^{n}

f_i |x_i -M| \]

**3) Limitations of mean deviation: **

In a series, where the degree of variability is very high, the median is not a representative central tendency. The sum of the deviations from the mean (minus signs ignored) is more than the sum of the deviations from median.

Therefore, the mean deviation about the mean is not very scientific.Thus, in many cases, mean deviation may give unsatisfactory results. Also mean deviation is calculated on the basis of absolute values of the deviations and therefore, cannot be subjected to further algebraic treatment. This implies that we must have some other measure of dispersion. Standard deviation is such a measure of dispersion.