# ICSE CLASS 12 MATHS SYLLABUS

MATHEMATICS

ISC  (2021-22)

CLASS XII

There will be two papers in the subject:

Paper I: Theory (3 hours) ……80 marks

Paper II: Project Work ……20 marks

PAPER I (THEORY) – 80 Marks

The syllabus is divided into three sections A, B and C.

Section A is compulsory for all candidates. Candidates will have a choice of attempting questions from EITHER Section B OR Section C.

There will be one paper of three hours duration of 80 marks.

Section A (65 Marks): Candidates will be required to attempt all questions. Internal choice will be provided in two questions of two marks, two questions of four marks and two questions of six marks each.

Section B/ Section C (15 Marks): Candidates will be required to attempt all questions EITHER from Section B or Section C. Internal choice will be provided in one question of two marks and one question of four marks.

DISTRIBUTION OF MARKS FOR THE THEORY PAPER

 S.No. UNIT TOTAL WEIGHTAGE SECTION A: 65 MARKS 1. Relations and Functions 10 Marks 2. Algebra 10 Marks 3. Calculus 32 Marks 4. Probability 13 Marks SECTION B: 15 MARKS 5. Vectors 5 Marks 6. Three – Dimensional Geometry 6 Marks 7. Applications of Integrals 4 Marks OR SECTION C: 15 MARKS 8. Application of Calculus 5 Marks 9. Linear Regression 6 Marks 10. Linear Programming 4 Marks TOTAL 80 Marks

SECTION A

1. Relations and Functions

(i) Types of relations: reflexive, symmetric, transitive and equivalence relations. One to one and onto functions, composite functions.

Relations as:

– Relation on a set A

– Identity relation, empty relation, universal relation.

– Types of Relations: reflexive, symmetric, transitive and equivalence relation.

Functions:

– As special relations, concept of writing “y is a function of x” as y = f(x).

– Types: one to one, many to one, into, onto.

– Real Valued function.

– Composite functions (algebraic functions only).

(ii) Inverse Trigonometric Functions

Definition, domain, range, principal value branch. Elementary properties of inverse trigonometric functions.

– Principal values. $\sin ^{-1} x, \cos ^{-1} x, \tan ^{-1} x$ etc.

– $\sin ^{-1} x=\cos ^{-1} \sqrt{1-x^{2}}=\tan ^{-1} \frac{x}{\sqrt{1-x^{2}}} .$

– $\sin ^{-1} x=\operatorname{cosec}^{-1} \frac{1}{x} ; \sin ^{-1} x+\cos ^{-1} x=\frac{\pi}{2}$ and similar relations for $\cot ^{-1} x, \tan ^{-1} x$, etc.

$\sin ^{-1} x \pm \sin ^{-1} y=\sin ^{-1}\left(x \sqrt{1-y^{2}} \pm y \sqrt{1-x^{2}}\right)$

$\cos ^{-1} x \pm \cos ^{-1} y=\cos ^{-1}\left(x y m \sqrt{1-y^{2}} \sqrt{1-x^{2}}\right)$

similarly $\tan ^{-1} x+\tan ^{-1} y=\tan ^{-1} \frac{x+y}{1-x y}, x y<1$

$\tan ^{-1} x-\tan ^{-1} y=\tan ^{-1} \frac{x-y}{1+x y}, x y>-1$

– Formulae for $2 \sin ^{-1} x, 2 \cos ^{-1} x, 2 \tan ^{-1} x$, $3 \tan ^{-1} x$ etc. and application of these formulae.

2. Algebra

Matrices and Determinants

(i) Matrices

Concept, notation, order, equality, types of matrices, zero and identity matrix, transpose of a matrix, symmetric and skew symmetric matrices. Operation on matrices: Addition and multiplication and multiplication with a scalar. Simple properties of addition, multiplication and scalar multiplication. Non- commutativity of multiplication of matrices. Invertible matrices, if it exists (here all matrices will have real entries).

(ii) Determinants

Determinant of a square matrix (up to 3 x 3 matrices), properties of  determinants, minors, co-factors. Adjoint and inverse of a square matrix. Solving system of linear equations in two or three variables (having unique solution) using inverse of a matrix.

– Types of matrices $(m \times n ; m, n \leq 3)$, order; Identity matrix, Diagonal matrix. Symmetric, – – Skew symmetric.

– Operation – addition, subtraction, multiplication of a matrix with scalar, multiplication of two matrices (the compatibility).

E.g. $\left[\begin{array}{ll}1 & 1 \\ 0 & 2 \\ 1 & 1\end{array}\right]\left[\begin{array}{cc}1 & 2 \\ 2 & 2\end{array}\right]=A B($ say $)$ but $B A$ is not possible.

– Singular and non-singular matrices.

– Existence of two non-zero matrices whose product is a zero matrix.

– Inverse $(2 \times 2,3 \times 3) A^{-1}=\frac{A d j A}{|A|}$

Martin’s Rule (i.e. using matrices)

\begin{aligned}&a_{1} x+b_{1} y+c_{1} z=d_{1} \\&a_{2} x+b_{2} y+c_{2} z=d_{2} \\&a_{3} x+b_{3 y}+c_{3} z=d_{3} \\&\quad A=\left[\begin{array}{lll}\mathrm{a}_{1} & \mathrm{~b}_{1} & \mathrm{c}_{1} \\\mathrm{a}_{2} & b_{2} & c_{2} \\\mathrm{a}_{3} & b_{3} & c_{3}\end{array}\right] \quad B=\left[\begin{array}{l}d_{1} \\d_{2} \\d_{3}\end{array}\right] \quad X=\left[\begin{array}{l}x \\y \\ z\end{array}\right] \\&A X=B \Rightarrow X=A^{-1} B\end{aligned}

Problems based on above.

Determinants

– Order.

– Minors.

– Cofactors.

– Expansion.

3. Calculus

(i) Continuity, Differentiability and Differentiation. $\quad$ Continuity $\quad$ and differentiability, derivative of composite functions, chain rule, derivatives of inverse trigonometric functions, derivative of implicit functions. Concept of exponential and logarithmic functions.

Derivatives of logarithmic and exponential functions. Logarithmic differentiation, derivative of functions expressed in parametric forms. Second order derivatives.

Continuity

– Continuity of a function at a point $x=a$.

– Continuity of a function in an interval.

– Algebra of continues function.

– Removable discontinuity.

Differentiation

– Concept of continuity and differentiability of $|x|,[x]$, etc.

– Derivatives of $\quad$ trigonometric functions.

– Derivatives of exponential functions.

– Derivatives of logarithmic functions.

– Derivatives of inverse trigonometric functions – differentiation by means of substitution.

– Derivatives of implicit functions and chain rule.

– e for composite functions.

– Derivatives of Parametric functions.

– Differentiation of a function with respect to another function e.g. differentiation of $\sin x^{3}$ with respect to $x^{3}$.

– Logarithmic $\quad$ Differentiation Finding $d y / d x$ when $y=x^{x^{x}}$.

– Successive differentiation up to $2^{\text {nd }}$ order.

NOTE : Derivatives of composite functions using chain rule.

– $L^{\prime}$ Hospital’s theorem. $\frac{0}{0}, \frac{\infty}{\infty}$ forms only.

(ii) Applications of Derivatives

Applications of derivatives: increasing/decreasing functions,  tangents and normals, maxima and minima (first derivative test motivated geometrically and second derivative test given as a provable tool). Simple problems (that illustrate basic principles and understanding of the subject as well as real-lifesituations).

Equation of Tangent and Normal

Increasing and decreasing functions.

Maxima and minima.

– Stationary/turning points.

– Absolute maxima/minima

– local maxima/minima

– First derivatives test and second derivatives test

– Application problems based on maxima and minima.

(iii) Integrals

Integration as inverse process of differentiation. Integration of a variety of functions by substitution, by partial fractions and by parts, Evaluation of simple integrals of the following types and problems based on them.

Fundamental Theorem of Calculus (without proof). Basic properties of definite integrals and evaluation of definite integrals.

Indefinite integral

– Integration as the inverse of differentiation.

– Anti-derivatives of polynomials and functions $(a x+b)^{n}, \sin x, \cos x, \sec ^{2} x$,

$\operatorname{cosec}^{2} x$ etc

– Integrals of the type $\sin ^{2} x, \sin ^{3} x$, $\sin ^{4} x, \cos ^{2} x, \cos ^{3} x, \cos ^{4} x$.

– Integration of $I / x, e^{x}$.

– Integration by substitution.

– Integrals of the type $f^{\prime}(x)[f(x)]^{n}$, $\frac{f^{\prime}(x)}{f(x)}$

– Integration of tanx, cotx, secx, $\operatorname{cosecx} .$

– Integration by parts.

– Integration using partial fractions. Expressions of the form $\frac{f(x)}{g(x)}$ when degree of $f(x)<$ degree of $g(x)$

E.g. $\frac{x+2}{(x-3)(x+1)}=\frac{A}{x-3}+\frac{B}{x+1}$

$\frac{x+2}{(x-2)(x-1)^{2}}=\frac{A}{x-1}+\frac{B}{(x-1)^{2}}+\frac{C}{x-2}$

When degree of $f(x) \geq$ degree of $g(x)$,

e.g.

$$\frac{x^{2}+1}{x^{2}+3 x+2}=1-\left(\frac{3 x+1}{x^{2}+3 x+2}\right)$$

Integrals of the type:

$\int \frac{d x}{x^{2} \pm a^{2}} \int \frac{p x+q}{a x^{2}+b x+c} d x$

Definite Integral

– Fundamental theorem of calculus (without proof)

– Properties of definite integrals.

– Problems based on the following properties of definite integrals are to be covered.

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

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

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

where $a<c<b$ $\int_{a}^{b} f(x) d x=\int_{a}^{b} f(a+b-x) d x$

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

$\int_{0}^{2 a} f(x) d x=\left\{\begin{array}{l}2 \int_{0}^{a} f(x) d x, i f \quad f(2 a-x)=f(x) \\ 0, \quad f(2 a-x)=-f(x)\end{array}\right.$

$\int_{-a}^{a} f(x) d x=\left\{\begin{array}{c}2 \int_{0}^{a} f(x) d x, \text { if } f \text { is an even function } \\ 0, \text { if } f \text { is an odd function }\end{array}\right.$

(iv) Differential Equations

Definition, order and degree, general and particular solutions of a differential equation. Solution of differential equations by method of separation of variables solutions of homogeneous differential equations of first order and first degree.

Solutions of linear differential equation of the type: $\frac{d y}{d x}+\mathrm{py}=\mathrm{q}$, where $\mathrm{p}$ and $\mathrm{q}$ are functions of $x$ or constants. $\frac{d x}{d y}+p x=q$, where $\mathrm{p}$ and $\mathrm{q}$ are functions of $\mathrm{y}$ or constants.

– Differential equations, order and degree.

– Solution of differential equations.

– Variable separable.

– Homogeneous equations.

– Linear form $\frac{d y}{d x}+P y=Q$ where $P$ and $Q$ are functions of $x$ only. Similarly, for $d x / d y$

NOTE : The second order differential equations are excluded.

4. Probability

Conditional probability, multiplication theorem on probability, independent events, total probability, Bayes’ theorem.

– Independent and dependent events conditional events.

– Laws of Probability, addition theorem, multiplication theorem, conditional probability.

– Theorem of Total Probability.

– Baye’s theorem.

SECTION B

5. Vectors

Vectors and scalars, magnitude and direction of a vector. Direction cosines and direction ratios of a vector. Types of vectors (equal, unit, zero, parallel and collinear vectors), position vector of a point, negative of a vector, components of a vector, addition of vectors, multiplication of a vector by a scalar. Definition, Geometrical Interpretation, properties and application of scalar (dot) product of vectors, vector (cross) product of vectors.

– As directed line segments.

– Magnitude and direction of a vector. Types: equal vectors, unit vectors, zero vector

Position vector.

– Components of a vector.

– Vectors in two and three dimensions.

– $\hat{i}, \hat{j}, \hat{k}$ as unit vectors along the $x, y$ and the $z$ axes, expressing a vector in terms of the unit vectors.

– Scalar (dot) product of vectors and its geometrical significance.

– Cross product – its properties – area of a triangle, area of parallelogram, collinear vectors.

NOTE: Proofs of geometrical theorems by using Vector algebra are excluded.

6. Three – dimensional Geometry

Direction cosines and direction ratios of a line joining two points. Cartesian equation and vector equation of a line, coplanar and skew lines. Cartesian and vector equation of a plane. Angle between two lines. Distance of a point from a plane.

– Equation of $x$-axis, $y$-axis, $z$ axis and lines parallel to them.

– Equation of $x y$ – plane, yz – plane, $z x=$ plane.

– Direction cosines, direction ratios.

– Angle between two lines in terms of direction cosines /direction ratios.

– Condition for lines to be perpendicular/ parallel.

Lines

– Cartesian and vector equations of a line through one and two points.

– Coplanar and skew lines.

– Conditions for intersection of two lines. Distance of a point from a line.

Planes

– Cartesian and vector equation of $a$ plane.

– Direction ratios of the normal to the plane.

– One point form.

– Normal form.

– Intercept form.

– Distance of a point from a plane. Intersection of the line and plane.

7. Application of Integrals

Application in finding the area bounded by simple curves and coordinate axes. Area enclosed between two curves.

– Application of definite integrals – area bounded by curves, lines and coordinate axes is required to be covered.
– Simple curves: lines, parabolas, polynomial functions.

SECTION C

8. Application of Calculus
Application of Calculus in Commerce and Economics in the following:
– Cost function,
– average cost,
– marginal cost and its interpretation
– demand function, revenue function,
– marginal revenue function and its interpretation, Profit function and breakeven point.
– AR, MR,R,C,AC,MC and their mathematical interpretation using the concept of increasing- decreasing functions.
Self-explanatory
NOTE: Application involving differentiation, increasing and decreasing function to be covered.

9. Linear Regression
– Lines of regression of x on y and y on x. Lines of best fit. Regression coefficient of x on y and y on x

– $b_{x y} \times b_{y x}=r^{2}, 0 \leq b_{x y} \times b_{y x} \leq 1$
– Identification of regression equations
– Estimation of the value of one variable using the value of other variable from appropriate line of regression. Self-explanatory

10. Linear Programming

Introduction, related terminology such as constraints, objective function, optimization, different types of linear programming (L.P.) problems, mathematical formulation of L.P. problems, graphical method of solution for problems in two variables, feasible and infeasible regions (bounded and unbounded), feasible and infeasible solutions, optimal feasible solutions (up to three non-trivial constraints).
Introduction, definition of related terminology such as constraints, objective function, optimization, advantages of linear programming; limitations of linear programming: application areas of linear programming; different types of linear programming (L.P.) problems, mathematical formulation of L.P problems, graphical method of solution for problems in two variables, feasible and infeasible regions, feasible and infeasible solutions, optimum feasible solution.
Note: Transportation problem is excluded.

PAPER II – PROJECT WORK – 20 Marks

Candidates will be expected to have completed two projects, one from Section A and one from either Section B or Section C.

The project work will be assessed by the subject teacher and a Visiting Examiner appointed locally and approved by the Council.
Mark allocation for each Project [10 marks]:

 Overall format 1 mark Content 4 marks Findings 2 marks Viva-voce based on the Project 3 marks Total 10 marks

List of suggested assignments for Project Work:

Section – A

1. Using a graph, demonstrate a function which is one-one but not onto.

2. Using a graph demonstrate a function which is invertible.

3. Draw the graph of $y=\sin ^{-1} x$ (or any other inverse trigonometric function), using the graph of $y=\sin x$ (or any other relevant trigonometric function). Demonstrate the concept of mirror line (about $y=x)$ and find its domain and range.

4. Explore the principal value of the function $\sin ^{-1} x$ (or any other inverse trigonometric function) using a unit circle.

5. Explain the concepts of increasing and decreasing $\quad$ functions, using geometrical significance of $d y / d x$. Illustrate with proper examples.

6. Explain and illustrate (with suitable examples) the concept of local maxima and local minima using graph.

7. Demonstrate application of differential equations to solve a given problem (example, population increase or decrease, bacteria count in a culture, etc.).

8. Explain the conditional probability, the theorem of total probability and the concept of Bayes’ theorem with suitable examples.

Section – B

9. Using vector algebra, find the area of a parallelogram/triangle. Also, derive the area analytically and verify the same.

10. Find the image of a line with respect to a given plane.

11. Find the area bounded by a parabola and an oblique line/parabola.

(Any other pair of curves which are specified in the syllabus may also be taken.)

Section – C

12. Draw a rough sketch of Cost (C), Average Cost (AC) and Marginal Cost (MC)

Or

Revenue (R), Average Revenue (AR) and Marginal Revenue (MR).

Give their mathematical interpretation using the concept of increasing – decreasing functions.

13. For a given data, find regression equations by the method of least squares. Also find angles between regression lines.

14. Using any suitable data, find the Optimum cost/ profit by formulating a linear programming problem (LPP).

NOTE: No question paper for Project Work will be set by the Council.

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