# Syllabus for Mathematics [Class -XII]

## Class XII: Mathematics Syllabus

CLASS - XII (Full Marks 100)

ALGEBRA [20 Marks]

Probability :

Concept of Random experiments and their outcomes. Events, certain and uncertain events. Equally likely outcomes. Classical definition of probability, addition rule, multiplication rule. (Venn diagram may be used).

Principle of Mathematical Induction:

Statement of the principle. Proof for the sum of squares, sum of cubes, divisibility properties like 22n- 1 divisible by 3,  $$n \ge 1$$ , 7 divides 32n+1 + 2n+2, n > 1.

Binomial theorem for positive integral index :

Statement of the theorem, proof by method of induction, general term, number of terms, middle term, equidistant terms.

Infinite Series :

Binomial theorem for negative integral and fractional index ; exponential, logarithmic series, infinite G.P. series with ranges of validity (only statement). Simple application of the series. (Expansion of the series should be insisted upon.)

Matrices and Determinants :

Concept of $$m \times n\left( {m \le 3,n \le 3} \right)$$  real matrix. Types of matrices. Operations ofaddition, Scalar multiplication and multiplication of matrices, Inverse of $$2 \times 2$$ matrices.

Determinant of an  $$n \times n\left( {n \le 3}\right)$$ matrix. Statement of the properties of  determinant only.

Minors and Co-factors. Application of determinant in (i) finding area of a triangle (ii) Solving a system of linear equation (not more than three variables) by Cramer's rule.

CO-ORDINATE GEOMETRY OF TWO DIMENSIONS [11 Marks ]

Conies :

Definition of conies (Parabola, ellipse, hyperbola) given focus and directrix :  eccentricity. Classification of standard conies (parabola, ellipse, hyperbola) in terms of eccentricity.

Parabola :

Standard equation, Reduction of the form  x = ay2 + by + c or y = ax2 + bx + c to the standard form y2 = 4ax or x2 = 4ay respectively; elementary properties and parametric equation of parabola.

Ellipse and hyperbola :

Standard equations only. Conjugate Hyperbola. Elementary properties, parametric equations.

DIFFERENTIAL CALCULUS [10 Marks]

Differentiation of a function of a function. Implicit functions (statement only)—their derivations. Differentiation of functions in parametric form. Logarithmic differentiation. Second order derivative of a function.

INTEGRAL CALCULUS [25 Marks]

Indefinite integral:

Integration as the inverse of differentiation. Primitive, integrals of $${x^m}(m \ne 1),\sin mx,\cos mx,\sec mx,\cos ecmx,\tan mx,(m \ne - 0){e^{mx}},1/x,$$ , (Assuming the functions and primitives are defined). Integral of the sum of two functions. Integration by simple substitutions; standard integrals of the form $$\int {{{dx} \over {{x^2} \pm {a^2}}}} \int {{{dx} \over {{x^2} \pm {a^2}}}} \int {{{dx} \over {\sqrt {{a^2} - {x^2}} }}} \int {{{dx} \over {a{x^2} + bx + c}}} ,\int {{{(px + q)dx} \over {a{x^2} + bx + c}}}$$  direct application.

Integration by parts :

Rule of integration by parts. Application in simple cases.

Standard integrals of the form :

$$\int {\sqrt {{x^2} \pm {a^2}} dx,\int {\sqrt {{a^2} - {x^2}} dx,\int {{e^{ax}}\sin bxdx,\int {{e^{ax}}coxbxdx,}}}}$$

$$\int {\sqrt {(a{x^2} + bx + c)} dx,\int {(px + q)\sqrt {a{x^2} + bx + cd} x,\int {{{dx} \over {a + b\cos x}},\int {{{dx} \over {a + b\sin x}}}}}}$$

Integration of rational algebraic functions by partial fractions of the form

$$\int {{{dx} \over {{{(x - a)}^m}{{(x - b)}^n}}}}$$  where m , n are positive integers and $$m \le 2,n \le 2$$

Definite Integral: Definite Integral as the limit of a sum, Definite integrals of x, x2 and of a constant, from above definition. Fundamental theorem of Integral Calculus (statement only). Applications in simple cases. Properties of definite integral

$$\int_a^b f (x)dx = \int_a^b {f(z)dz;\int_a^b f } (x)dx = - \int_b^a f (x)dx;\int_a^b f (x)dx = \int_a^c {f(x)dx + \int_c^b {f(x)dx(a < c < b)} }$$  where c is a point between a and b.

$$\int_0^a {f(x)dx = \int_0^a {f(a - x)dx}}$$  Applications of odd and even functions,

DIFFERENTIAL EQUATIONS [10 Marks]

Periodsl Formation, order and degree of  differential equations.   Solution of first order and first degree differential equation of the form

$${{dy} \over {dx}} = f(x) \times g(y),{{dy} \over {dx}} = {{ax + by} \over {cx + dy}}$$  and of the form  $${{{d^2}y} \over {d{x^2}}} = f(x)$$ . use of initial conditions.

APPLICATION OF CALCULUS [24 Marks]

Tangent and Normal :

Geometric interpretation of differential coefficients. Slope of a tangent. Equations of tangent and normal to curves of the form y = f(x) at the point (x1, y1) and application to circle, parabola, ellipse, hyperbola. Condition that the st.line y = mx + c may be a tangent / normal to a circle or to a curve. Differential coefficient as rate measurer.

Maxima and Minima :

Idea of Maxima and Minima of y = f(x) at a point where  $${{{d^2}y} \over {d{x^2}}} \ne 0$$   ( statement only) Application to algebraic functions, sin x, cos x.

Determination of areas in simple cases :

Interpretation of a definite integral as an area. Calculation of areas bounded by circle, parabola and ellipse, ordinate and abscissa as the case may be. (sketch of the area is needed).

Expression for velocity and acceleration :

Expression for velocity and acceleration of a particle in terms of derivatives,

velocity = $${{ds} \over {dt}}$$ , acceleration = $${{dv} \over {dt}},{{{d^2}s} \over {d{t^2}}},v{{dv} \over {ds}}$$  ; where s represents the displacement.

With the above expression for velocity and acceleration to establish the formula s = vt (v constant velocity ) , v = u + ft, s = ut + 1/2 ft2, v2 = u2 + 2 fs. Simple applications.

Vertical motion under gravity.

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## Class XII: Nutrition Syllabus  (Full Marks—100)

THEORY  ( Full Marks—80 )

PRACTICAL Total Marks—20

A.  Nutritive Process :

## Class XII: History Syllabus

CLASS-XII (Full Marks 100)

Group-A (Marks-50)

Unit—I

1. Impact of the First World War on India :

## Class XII: Geography Syllabus

CLASS - XII (Full Marks 100)

THEORY (Total Marks - 80)

Group - I : BIOSPHER AND PHYSICAL ENVIRONMENT

## Class XII: Computer Science Syllabus  (Full Marks-100)

THEORETICAL   ( Full Marks-70 )

PRACTICAL        ( FullMarks-30 )

## Class XII: Biological Sciences Syllabus

CLASS - XII (Full Marks 100)

Theory-80 Botany (Group - A)

Marks - 25