Time: 3 hours
Maximum Marks: 100

1. Partial Fractions -( 5 Marks) – 7 Periods

Definition of Partial Fractions, Resolving a fraction in to partial fractions when

$Vedic\ Method\ for\ solution\ of\ partial\ fraction. \left\{\begin{matrix} (i). Denominator\ contains\ non repeated\ real\ linear\ fractors.\\ (ii). Denominator\ contains\ some repeated\ real\ linear\ fractors.\\ (iii). Denominator\ has\ real\ non repeated\ quadratic\ fractors. \end{matrix}\right.$

2. Three Dimensional Geometry (co-ordinate System direction cosines etc.)-(15 Marks)- 7 Periods

Coordinate, Coordinate axes, Cartesian system, Coordinate plane, Coordinate of a point in space, Distance between two points, Intercept Formula, Projection of a point on a line and a plane, Projection of a line segment on a given line. Direction cosines of a line, Direction cosines of the axes, Projection of the line segment, joining two, points on a given line.

3. Plane -( 15 Marks) - 7 Periods

Concept of a plane equation of a plane in normal form. Intercept form of the equation of a plane, General equation of a plane as a linear equation in three variables. Angle between two planes, Distance of a point from a plane, Equation to a plane through the intersection of two planes, Plane bisecting angle between two planes.

4. Straight line and Sphere -( 15 Marks) - 7 Periods

Equation of the line joining two points, Angle between a line and a plane. The condition for a line to lie on a plane . Plane containing a line, Cartesian equation of a sphere in the following cases. When the coordinates of centre and the radius of a sphere is given, General equations of a sphere, equations of a sphere when the coordinates of the extremities of a diameter is given.

5. Vectors : ( 15 Marks) - 7 Periods

Scalar and Vector quantities, Vectors as a directed line segment, the magnitude and
directions of a vector. Equal vectors, Unit Vectors, Zero Vectors, Position Vector of a point, Components of a vector, Two dimesional and three dimesional vectors, Resolved parts of a vector in two or three mutually perpendicular directions. Addition of vectors, product of a scalar with a vector. Position vector of the point dividing a line segment in a specified ratio.

6. Product of Vectors ( 15 Marks) - 7 Periods

Scalar Product of vectors, Vector product of vectors scalar and vector triple products and their properties. Application of vectors in geometrical Problems.

Application of vectors

Work = Force . Displacement (W = $\vec{f}$ . $\vec{r}$),

Moment of a force, Moment of a couple , Verifications of cosine law.

Proof of the fact:

Angle subtended by a semi circle is a right angle. Expressing area of a triangle as. $\frac{1}{2}$ a x b │.Verification of sine law. Application of triple product in determining the volume of a cuboid, Coplanarity of vectors.

7. Application of Vectors in three dimesional Geometry: ( 15 Marks) - 7 Periods

Application of vectors in three dimesional Geometry mentioned in unit number 2,,3,4.
Shortest distance between two non coplanar lines (Skew lines) using vectors.

8. Inverse Trigonometric Functions : (5 Marks)-8 Periods

Inverse of a function, Inverse of trigonometrical function, Principal value, branch, Graph of inverse trigonometric functions. Properties and their proof such as : Simple transformation of trigonometric functions e.g.

sin-sin x = x sin-1(-x) = - sin-1sin-1x + cos-1x = $\frac{\pi}{2}$

sin-1$\frac{1}{x}$ = cosec

tan-1x $\pm$ tan-1y = $tan^{-1}\left(\frac{x \pm y}{1 \mp xy}\right)$

2 tan-1x = sin-1$\left(\frac{2x}{1 + x^{2}}\right)$ = cos-1$\left(\frac{1 - x^{2}}{1 + x^{2}}\right)$$tan^{-1}\left(\frac{2x}{1 - x^{2}}\right)$

Simplified transformation of trigonometric functions such as:

tan-1$\frac{x}{\sqrt{a^{2} - x^{2}}}$ sin-1$\frac{x}{a}$

9. Functions, Limit and continuity : (5 Marks)-8 Periods

Concept of real functions Domain and range of a function. Graph of a function.

Composition of functions ;

Meaning of x → a , x →a+, x→a-,

$\lim_{x \to a}$ f(x), $\lim_{x \to a^{+}}$ f(x) , $\lim_{x \to a^{-}}$ f(x)

fundamental theorems on limits (without proof)

Proof of the following

(Without treatment)

$\lim_{x \to a}$ $\frac{x^{n} - a^{n}}{x - a}$ = n a(n -1) (a < 0)

$\lim_{x \to 0}$ sin x = 0    $\lim_{x \to 0}$ $\frac{sin x}{x}$ = 1

$\lim_{x \to 0}$ cos x = 1

$\lim_{x \to 0}$ $\frac{1}{x}$ loge (1 + x) = 1

$\lim_{x \to 0}$ $\frac{e^{x} -1}{x}$ = 1, $\lim_{x \to 0}$ $\left(1 + \frac{1}{x}\right)$ = e

$\lim_{x \to 0}$ (1 + x)1/x = ex$\lim_{x \to 0}$ $\left(\frac{a^{x} - 1}{x}\right)$ = loge a

Continuity of a function at a point, in open and closed intervals Sum, Product, Quotient, of continuous functions, polynomials, trigonometricals, functions, exponential functions. Logarithm functions and inverse trigonometric functions and composite functions.

10. Differentiation:- (10 Marks)- 8 Periods

Differentiation of a function. Its physical and geometric meaning, relation between continuity
and differentiability, Differentiation of xn, sinx, cosx, tanx, ex, log ex. (By first principles ),Theorems involving differentiation of sum difference, product and quotients of functions. Differentiation of trigonometric functions, inverse trigonometrical functions, logarithmic functions,and exponential functions, Differential coefficient function of function (Chain rule). Differentiation by transformation of Inverse trigonometrical functions, differentiation by first principles (ab initio).

11. Harder Differentiation: (10 Marks)- 8 Periods

Differentiation of higher order, successive differentiation, nth derivative of sinx, cosx, xn ,$\frac{1}{ax + b}$
, a x, e x log e(ax+b) differentiation of implicit function, logarithmic differentiation, Parametric differentiation.

12. Applications of Derivatives :( 5 Marks)-8 Periods

Motion is a straight line, motion under gravity, rate of change, increasing and decreasing functions and the sign of the derivatives, maxima and minima (absolute and local). Rolle's theorem, Mean Value theory, solution of quadartic equations by differentiations.

Note: simple level questions may be asked.

13. Integration : (15 Marks)-8 Periods

Indefinite integral as an intensive differentiation or properties of anti derivatives.

ʃ [ f(x)+g(x)] dx = ʃ f(x)dx + ʃg(x)dx

ʃ c f(x)dx= cʃ f(x)dx

Fundamental of integration which includes algabraic trigometrical, exponential functions, integration by standard transformation, integration by substitutions, area of a region by integration.

Integration of some standard forms:

$\int$ $\frac{dx}{a^{x} + x^{2}}$, $\int$ $\frac{dx}{x^{x} - a^{2}}$, $\int$ $\frac{dx}{\sqrt{x^{x} + a^{2}}}$

$\int$ $\frac{dx}{\sqrt{a^{x} - x^{2}}}$, $\int$ $\frac{dx}{\sqrt{x^{x} - a^{2}}}$

$\int$ $\sqrt{x^{2} \pm a^{2}}$ dx, $\int$ $\sqrt{a^{2} - x^{2}}$ dx,

Integration by trigonometrical substitution, integration by parts.

14. Harder Integration : (15 Marks)-8 Periods

Integration by splitting into partial fraction integration of rational functions, Derivation and application of integration in the following form.

$\int$ $\frac{dx}{ax^{2} + bx + c}$, $\int$ $\frac{px + q}{ax^{2} + bx + c}$ dx

$\int$ $\frac{dx}{\sqrt{ax^{2} + bx + c}}$,  $\int$ $\frac{px + q}{\sqrt{ax^{2} + bx + c}}$

$\int$ (px + q) $\sqrt{ax^{2} + bx + c}$ dx, $\int$ $\frac{dx}{\sqrt{ax^{2} + bx + c}}$

$\int$\frac{dx}{a + b cosx}$,$\int $\frac{dx}{a + b sinx}$

$\int$ $\frac{dx}{a cosx + b sinx + c}$

$\int$ $\frac{a cosx + b sinx}{c cosx + d sinx}$ dx

$\int$ $\frac{dx}{a + b cos^{2}x}$,  $\int$ $\frac{dx}{a + b sin^{2}x}$

$\int$ $\frac{dx}{a sin^{2}x + b sin^{2} x}$

$\int$ sinm x cosn x dx (where m and n both are odd)

Note :- Question based on a reduction formula are not included.

15. Definite Integration : - (15 Marks)-8 Periods

Definition of definite integral as a limited a sum, fundamental theorem of integration,
transformation of definite integral by substitution, following properties of definite integrals.

$\int_{a}^{b}$ f(x) dx = - $\int_{b}^{a}$ f(x) dx

$\int_{a}^{b}$ f(x) dx = $\int_{a}^{c}$ f(x) dx + $\int_{c}^{b}$ f(x) dx

$\int_{a}^{b}$ f(x) dx = $\int_{a}^{b}$ f(t) dt

$\int_{0}^{a}$ f(x) dx = $\int_{0}^{a}$ f(a -x) dx

$\int_{-a}^{a}$ f(x) dx = $\left\{\begin{matrix} 2 \int_{0}^{a} f(x)dx\ when\ f(x)\ is\ even\\ when\ f(x)\ is\ odd \end{matrix}\right.$

$\int_{0}^{a}$ f(x) dx = $\left\{\begin{matrix}2 \int_{0}^{a} f(x)dx\ when\ f(2a - x)=f(x)\\ when f(2a -x) = -f(x) \end{matrix}\right.$

Integration by using the above properties . Area Bounded by x axis and two ordinates ': Y axis and two horizontal and the curves such as circle parabola ellipse (in standard forms )

16. Differential equations :- (5 Marks)-8 Periods

Definition of differential equation order and degree formation of differential equations, general and particular solution, variable separable method of solving differential equations, solution of linear differential equations of first order dy/dx + Py = Q , Where P and Q can both be constant or functions of x . simple questions based on them .

Note : Equations solvable for P ,X, Y and clairut's equations are not included.

17. Correlation: (5 Marks)-8 Periods

Meaning of bivariate data, explanation of correlation table for bivariate distribution (in same units). Bivariate frequency distribution, Difference between the values of dependent variable with respect to conditional distribution , correlation analysis as a measure of strength of relation between two numerical variables, Types of correlation on the basis of direction, positive and negative correlation meaning of simple, partial and multiple correlation based on number of variates total correlation and correlation coefficient and their interpretation by scattered diagram.

Use of correlation coefficient.

18. Regression: (5 Marks)-8 Periods

Meaning of regression, coeff of regression. Types of regression coefficients and their calculation. Lines of regression. Calculation of regression coefficients of two straight lines (By least square method).

Note: Regression error and standard errors are not included.

19. Probability: (5 Marks)-8 Periods

Meaning Random experiment and Sample space sample points, Types of sample space, Events and types of events. Events as subsets of sample space, Occurrence of an event sure events, impossible events, mutually exclusive events, simple events, favorable equally likely events. Definition of probability of an events, probability as ratio of favorable equally likely and Exhaustive events.

Addition rule for mutually exclusive events. Operations "OR" and "AND" for favorable equally likely events. Conditional Probability, Independent Events and Probability related to independent Experiments. Random variable as a function of finite sample space. Probability distribution of a random variable; Binomial distribution, Examples of different random experiments.

20. Numerical Methods: (5 Marks)-8 Periods

Revision of solving equations.
(i). Successive division method.
(ii). Method of false position.
(iii). Newton Raphsons Method. Partial Integration, Partial integration method.
(iv). Trapezoidal rule'
(v). Simpson's Rule (Simple questions should be asked)
(vi) To find cube root of numbers which are not perfect cubes by vedic maths method.
(vii) Contribution and life history of Swami Bharti Krishna Tirth. Apendix-Contribution and life

(i) Proof in mathematics (ii) Mathematical modeling.
(i) Proof in mathematics- Through a variety of examples related to mathematics and already familiar to the learner bring out different kinds of proofs: direct, contrapofitike, by contradiction by counter-example.

(ii) Mathematical modeling- Modeling real-life problems where many constraint and may really need to be ignored ( Continuing from class XI). However, now the models concerned would use technique/ results of matrices, calculus and linear programming.

For Revision : 20 Periods

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