Table Of ContentSSSMathematicsforClass12 (i)
SENIOR
SECONDARY SCHOOL
MATHEMATICS
FORCLASS12
[InaccordancewiththelatestCBSEsyllabus]
RSAggarwal,MSc,PhD
SSS Mathematics for Class 12 (iii)
Preface
It gives me great pleasure in presenting the new edition of this book. In this
edition, the modifications have been dictated by the changes in the CBSE
syllabus.Thestructureandthemethodsusedinthepreviouseditions,which
have been appreciated by teachers using the book in classroom conditions,
remain unchanged.
The main consideration in writing the book was to present the
considerable requirements of the syllabus in as simple a manner as possible.
Special attention has been paid to the gradation of problems. This will help
students gain confidence in problem-solving.
One problem faced by students is the lack of a comprehensive and
carefullyselectedsetofsolvedproblemsintextbooksofthiskind.Ihavegiven
due weightage to this aspect. Each set of solved examples is followed by a
comprehensive exercise section in which students will get enough questions
for practice. Hints have been given to the more difficult questions. Students
should take their help as a last resort.
Ihavereceivedmanysuggestionsandlettersofappreciationfromteachers
alloverthecountry.Ithankthemallforcontributingintheimprovementofthe
book and for their encouragement. I hope they will like this edition as well.
Andasalways,Iwouldliketoheartheirviewsonthebook.
R S Aggarwal
(iii)
SSS Mathematics for Class 12 (v)
Mathematics Syllabus
ForClass12
UNIT I. Relations and Functions
1. Relations and Functions 15 Periods
Types of relations: reflexive, symmetric, transitive and equivalence
relations.Onetooneandontofunctions,compositefunctions,inverse
of a function. Binary operations.
2. Inverse Trigonometric Functions 15 Periods
Definition, range, domain, principal value branches. Graphs of
inverse trigonometric functions. Elementary properties of inverse
trigonometricfunctions.
UNIT II. Algebra
1. Matrices 25 Periods
Concept, notation, order, equality, types of matrices, zero matrix,
transpose of a matrix, symmetric and skew-symmetric matrices.
Addition, multiplication and scalar multiplication of matrices, simple
properties of addition, multiplication and scalar multiplication.
Non-commutativity of multiplication of matrices and existence of
nonzero matrices whose product is the zero matrix (restrict to square
matrices of order 2). Concept of elementary row and column
operations.Invertiblematricesandproofoftheuniquenessofinverse,
if it exists; (Here all matrices will have real entries).
2. Determinants 25 Periods
Determinant of a square matrix (up to 3 (cid:2)3 matrices), properties of
determinants, minors, cofactors and applications of determinants in
finding the area of a triangle. Adjoint and inverse of a square matrix.
Consistency,inconsistencyandnumberofsolutionsofsystemoflinear
equations by examples, solving system of linear equations in two or
three variables (having unique solution) using inverse of a matrix.
UNIT III. Calculus
1. Continuity and Differentiability 20 Periods
Continuity and differentiability, derivative of composite functions,
chainrule,derivativesofinversetrigonometricfunctions,derivativeof
implicit functions. Concept of exponential and logarithmic functions.
Derivatives of logarithmic and exponential functions. Logarithmic
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SSS Mathematics for Class 12 (vi)
differentiation, derivative of functions expressed in parametric forms.
Second-order derivatives. Rolle’s and Lagrange’s Mean Value
Theorems (without proof) and their geometric interpretation.
2. Applications of Derivatives 10 Periods
Applications of derivatives: rate of change of bodies, increasing/
decreasing functions, tangents and normals, use of derivatives in
approximation, maxima and minima (first derivative test motivated
geometricallyandsecondderivativetestgivenasaprovabletool).Simple
problems(thatillustratebasicprinciplesandunderstandingofthesubject
aswellasreal-lifesituations).
3. Integrals 20 Periods
Integrationasinverseprocessofdifferentiation.Integrationofavariety
of functions by substitution, by partial fractions and by parts, simple
integrals of the following type to be evaluated:
(cid:6) dx (cid:6) dx (cid:6) dx (cid:6) dx (cid:6) dx
, , , , ,
x2 (cid:3) a2 x2 (cid:3) a2 a2 (cid:4)x2 ax2 (cid:5) bx(cid:5) c ax2 (cid:5) bx(cid:5) c
(cid:6) (px(cid:5) q) dx,(cid:6) (px(cid:5) q) dx,(cid:6) a2 (cid:3) x2dx, (cid:6) x2(cid:4)a2dx,
ax2 (cid:5) bx(cid:5) c ax2 (cid:5) bx(cid:5) c
(cid:6) ax2 (cid:5) bx (cid:5) c dx,(cid:6)(px(cid:5) q) ax2 (cid:5) bx(cid:5) c dx.
Definite integrals as a limit of a sum, Fundamental Theorem of
Calculus (without proof). Basic properties of definite integrals and
evaluation of definite integrals.
4. Applications of the Integrals 15 Periods
Applications in finding the area under simple curves, especially lines,
areasofcircles/parabolas/ellipses(instandardformonly).Areabetween
thetwoabove-saidcurves(theregionshouldbeclearlyidentifiable).
5. Differential Equations 15 Periods
Definition, order and degree, general and particular solutions of a
differential equation. Formation of a differential equation whose
generalsolutionisgiven.Solutionofdifferentialequationsbymethod
of separation of variables, homogeneous differential equations of first
order and first degree. Solutions of linear differential equation of
the type:
dy
(cid:5) py(cid:7)q,wherepandqare functions ofxor constants.
dx
dx
(cid:5) px(cid:7)q, wherepandqare functions ofyor constants.
dy
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SSS Mathematics for Class 12 (vii)
UNIT IV. Vectors and Three-dimensional Geometry
1. Vectors 15 Periods
Vectors and scalars, magnitude and direction of a vector. Direction
cosines and direction ratios of a vector. Types of vectors (equal, unit,
zero,parallelandcollinearvectors),positionvectorofapoint,negative
ofavector,componentsofavector,additionofvectors,multiplication
ofavectorbyascalar,positionvectorofapointdividingalinesegment
inagivenratio.Scalar(dot)productofvectors,projectionofavectoron
aline.Vector(cross)productofvectors.Scalartripleproductofvectors.
2. Three-dimensional Geometry 15 Periods
Direction cosines and direction ratios of a line joining two points.
Cartesian and vector equation of a line, coplanar and skew lines,
shortestdistancebetweentwolines.Cartesianandvectorequationofa
plane. Angle between (i) two lines, (ii) two planes, (iii) a line and a
plane. Distance of a point from a plane.
UNIT V. Linear Programming
1. Linear Programming 20 Periods
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, feasible and infeasible solutions, optimal feasible
solutions (up to three non-trivial constraints).
UNIT VI. Probability
1. Probability 30 Periods
Conditional probability, multiplication theorem on probability,
independent events, total probability, Bayes’s theorem, Random
variableanditsprobabilitydistribution,meanandvarianceofrandom
variable. Repeated independent (Bernoulli) trials and Binomial
distribution.
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SSS Mathematics for Class 12 (viii)
Weightage
Topic Marks
1. RelationsandFunctions 10
2. Algebra 13
3. Calculus 44
4. Vectorsand3-DimensionalGeometry 17
5. LinearProgramming 06
6. Probability 10
Total 100
Type No.ofQuestions MarksofEach TotalMarks
Very-Short-Answer 4 1 4
Short-Answer 8 2 16
Long-AnswerI 11 4 44
Long-AnswerII 6 6 36
Total 29 100
MarksofEach
Sl. DivisionofTypes
VSA SA LA–I LA–II
No. ofQuestions
(1Mark) (2Marks) (4Marks) (6Marks)
1. Remembering 2 2 2 1
2. Understanding 1 3 4 2
3. Application 1 – 3 2
4. HOTS – 3 1 –
5. Evaluation – – 1* 1
Total 4 8 11 6
*VBQ
(viii)
SSS Mathematics for Class 12 (ix)
Contents
RelationsandFunctions
1. Relations 1
2. Functions 25
3. Binary Operations 67
4. Inverse Trigonometric Functions 82
Algebra
5. Matrices 152
6. Determinants 215
7. Adjointand Inverse of a Matrix 276
8. System of Linear Equations 298
DifferentialCalculus
9. Continuity andDifferentiability 323
10. Differentiation 365
11. Applications of Derivatives 452
IntegralCalculus
12. Indefinite Integral 588
13. Methods of Integration 609
14. Some Special Integrals 698
15. Integration Using Partial Fractions 747
16. Definite Integrals 793
17. Area of Bounded Regions 860
DifferentialEquations
18. Differential Equations and Their Formation 893
19. Differential Equations with Variable Separable 914
20. Homogeneous Differential Equations 941
21. Linear Differential Equations 966
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