(TGT/PGT/LT Grade/GIC/GDC/DIET/DSSSB/
KVS/NVS/Bihar Tre/EMRS/JSSC PGT/DSE Odisha
TGT/OAVS TGT/OPSC PGT RPSC//UPHESC
Assistant Professor/Polytechnic Lecturer/UPPSC
Ashram Paddhati/UKPSC/UKSSSC/HTET/HSSC
and other competitive Teaching Examinations)
MATHEMATICS
Chapterwise
Solved Papers
Chief Editor
A. K. Mahajan
Written & Complied by
Subject Expert Team
Computer Graphics by
Balkrishna Tripathi & Charan Singh
Editorial Office
12, Church Lane Prayagraj-211002
9415650134
Email :
website : www.yctbooks.com / www.yctfastbook.com / www.yctbooksprime.com
All Rights Reserved with Publisher
Publisher Declaration
Edited and Published by A.K. Mahajan for YCT Publications Pvt. Ltd.
and E:Book by APP YCT BOOKS In order to Publish the book,
full care has been taken by the Editor and the Publisher,
still your suggestions and queries are welcomed.
In the event of any dispute, the judicial area will be Prayagraj.
.248
, INDEX
PART-1 : ALGEBRA
ALGEBRA ...................................................................................................................................... 9-341
◘ Theory of Equations and Inequations .............................................................................................. 9
◘ Sequences and Series (A.P, G.P, H.P) ........................................................................................... 67
◘ Permutation and Combination ....................................................................................................... 98
◘ Binomial Theorem ....................................................................................................................... 121
◘ Exponential and Logarithmic Series ............................................................................................ 134
◘ Sets, Relations and Aunctions...................................................................................................... 142
◘ Modern Algebra ........................................................................................................................... 204
◘ Matrices and Determinants .......................................................................................................... 242
◘ Linear Algebra ............................................................................................................................. 324
PART-2 : REAL ANALYSIS
REAL ANALYSIS ..................................................................................................................... 342-361
◘ Convergence and Divergence of Sequences and Series of real numbers and functions ............. 342
◘ Functional Analysis and Measure Theory .................................................................................. 352
PART-3 : VECTOR ANALYSIS
VECTOR ANALYSIS ............................................................................................................... 362-428
◘ Operations on Vectors ................................................................................................................ 362
◘ Vector, Gradient and Curl ........................................................................................................... 409
PART-4 : COMPLEX ANALYSIS
COMPLEX ANALYSIS ............................................................................................................ 429-486
◘ Complex number, De-Moiver's theorem, nth root of unity .......................................................... 429
◘ Hyperbolic and logarithmic function of a complex variable ...................................................... 462
◘ Analytic Functions and Power Series .......................................................................................... 468
◘ Complex Integration, Cauchy's Theorem and Calculus of Residue............................................. 472
PART-5 : CALCULUS
CALCULUS................................................................................................................................ 487-805
◘ Limit, continuity and differentiability of function of one variable ............................................. 487
◘ Rolle's Theorem and Lagrange's Mean Value Theorem ............................................................. 564
2
.248
, ◘ Tangent, Normal, Increasing, Decreasing and Maxima/Minima of a Function of one variable . 571
◘ Limit, Continuity and Differentiability of function of two variable and Partial Differentiation . 607
◘ Integration, applications of Integration and area bounded by curve ............................................ 620
◘ Length of curve, surface area and volume of solids revolution .................................................. 693
◘ Ordinary and Partial differential equations, Integral Equations and Calculus of Variations ...... 704
PART-6 : GEOMETRY
GEOMETRY .............................................................................................................................. 806-957
◘ Two Dimension ........................................................................................................................... 806
● The Line and General equation of second degree and its classification................................ 806
● Conic section ......................................................................................................................... 848
◘ Three Dimensions....................................................................................................................... 911
● The Plane and The Line (Cartesian and vector) .................................................................... 911
● Sphere, Cone and Cylinder.................................................................................................... 942
PART-7 : STATISTICS AND TOPOLOGY
STATISTICS AND TOPOLOGY .......................................................................................... 958-1014
◘ Statistics and Probability.............................................................................................................. 958
◘ Topology .................................................................................................................................... 1007
PART-8 : TRIGONOMETRY
TRIGONOMETRY ............................................................................................................... 1015-1073
◘ Trigonometrical Identities and Circular function...................................................................... 1015
◘ Height and Distance .................................................................................................................. 1048
◘ Properties of Triangular and Circular inverse function............................................................. 1054
PART-9 : STATICS AND DYNAMICS
STATICS AND DYNAMICS ................................................................................................ 1074-1151
PART-10 : ARITHMETIC
ARITHMETIC ....................................................................................................................... 1152-1228
PART-11 : NUMERICAL ANALYSIS & OPERATIONS RESEARCH
NUMERICAL ANALYSIS & OPERATIONS RESEARCH ....................................... 1229-1248
◘ Numerical Analysis.................................................................................................................... 1229
◘ Linear Programming Problems .................................................................................................. 1238
3
.248
, TGT Maths : Syllabus
Commerce/Mathematics: Work time and velocity time, Hyperbolic function: Separation into real and imaginary
compound interest, banking, taxation, flow of elementary parts.
rules illustrated. Geometry: Baudhaayan Pythagoras principle and its
Statistics: Frequency distribution, graphical representation explanation, circle and circle segmet, arc and chord of a
of statistical data, measures of central tendency, measures of circle, tangent to the circle, alternate circle segments and
dispersion, birth/death statistics indices. their angles, segments of chords and rectangles formed
Algebra: Surds, polynomials and their factors, from them, symmetry of linear plane figures.
logarithms, linear equations of two unknown quantities, Coordinate geometry: Cartesian plane, line, straight
highest common factor and least common multiples of line pair represented by general exponential equation of
polynomials, simultaneous linear equations of three the second degree. Angle between them and equation of
unknowns, factors of quadratic polynomial quadratic pair of bisectors, standard equations and parametric
equations, ratios and proportions, number system, set and equations of conic (circle, parabola, ellipse and
operations, mapping. hyperbola) in rectangular cartesian, coordinates,
Determinant:Difinition, minors and cofactors, restrictions for representing line pair, circle, parabola,
expansion of determinant upto size 3×3, general ellipse and hyperbola by quadratic general equations,
properties of determinant, solution of system of n-linear obtaining equations of circle, parabola ellipse and
equations by cramer’s rule (for n = 3), Types of matrices, hyperbola with the help of transefer of origin and axes,
addition and multiplication of matrices upto size 3×3. intersection of tangent and normal secant line at any
Transpose of matrix, inverse of symmetric and skew- point of the conic, condition of it being tangent in the
symmetric matrices, solution of simultaneous equation of limit case, parametric equations of tangent, the tangent
three unknown quantities with the help of matrix. Theory pair to the conic from the external point. Equation of
of equation, symmetric functions of roots. Arithmetic, normal at any point of a conic condition of touching or
geometric and harmonic series sum of series formed by being normal, standard equation of a conic in polar
terms of squares and cube of natural numbers, coordinates (binary), three dimensional geometry of
permutation and combination , bionimial theorem, sum sphere, cone and cylinder.
of exponential and logarithmic series. Probability theory Differential calculus: Difinition of differention,
of addition and multiplication. Differention of algebraic, trigonometric, exponential and
Set theory: Algebra of set theory, equivalence, relation logarithmic function, tangent and normal line, tracing
mapping, composition of mappings, Inverse mapping, maximum and minimum of simple curve of a function of
use of Piano’s axiom and induction axiom, group and one variable integration-integration by parts and
group isomorphism, subgroup generated by subsets, substitution, integral with help of partial fractions,
cyclic groups, order of an element, subgroup of cyclic difinite integral its uses in determining the arc,
groups, coset decomposition, Lagrange’s theorem. differential and surface of cylinder, conical sphere under
Real Analysis: Axioms of real numbers, computation of cornes, order and degree of differential equation in
sets, metric spaces, limit, open sets, closed sets, derived example of rectilinear motion under gravity, solving the
set, dense set, perfect set, other general theorems equations in the following from.
including Balzano-Weierstrass theorem, sequence of real 3
dy dy φ( y)
numbers-limit of sequence, convergent sequence, 1. = f ( x ) 2. = f ( x ) 3. = f (x)
divergent sequence bounded sequence, monotonic dx dx dx 2
sequence, operation of convergent sequences, Cauchy Vector analysis: Vectors in the form of consecutive
sequence, Cauchy theorem on limit and cauchy theorem pairs and consecutive triples, displacement vectors, free
of convergence of real numbers, limit and limit of vectors, unit vectors, modulus and cosine, equal vectors,
continuous functions of real variables, left hand limit and combination of sum of vectors (force, velocity,
right hand limit, continuity of functions, characteristics acceleration). Inter-relative velocity of two vectors,
of continuous functions discontinuity and its types. scalar and vectors multiplication of two vectors. Their
Trignometry: Circular measurements and trigonometric use in calculation of work and torque. Triplication of
ratios of specific angles, trigonometric ratios of the sum vectors.
and difference of two angles, trigonometric identities, Positional Science: Equilibrium of bodies applied by
trigonometric equations, solution of triganle, radii and three forces Lami’s theorem, law of triangles,
properties of circuitmscribed and external circles, general trigonometric theorem and planning in two right angled
properties of invere circular functions. forces. General constraints of equilibrium center of
Complex Numbers: Their sum and product, DeMoivre’s gravity.
theorem and its applications, height and distance. Kinematics: Calculation of speed, work, energy and
Exponential functions, circular functions and hyperbolic MKS system of a projectile moving in a vertical plane
of complex quantities. under gravity.
4
.248
KVS/NVS/Bihar Tre/EMRS/JSSC PGT/DSE Odisha
TGT/OAVS TGT/OPSC PGT RPSC//UPHESC
Assistant Professor/Polytechnic Lecturer/UPPSC
Ashram Paddhati/UKPSC/UKSSSC/HTET/HSSC
and other competitive Teaching Examinations)
MATHEMATICS
Chapterwise
Solved Papers
Chief Editor
A. K. Mahajan
Written & Complied by
Subject Expert Team
Computer Graphics by
Balkrishna Tripathi & Charan Singh
Editorial Office
12, Church Lane Prayagraj-211002
9415650134
Email :
website : www.yctbooks.com / www.yctfastbook.com / www.yctbooksprime.com
All Rights Reserved with Publisher
Publisher Declaration
Edited and Published by A.K. Mahajan for YCT Publications Pvt. Ltd.
and E:Book by APP YCT BOOKS In order to Publish the book,
full care has been taken by the Editor and the Publisher,
still your suggestions and queries are welcomed.
In the event of any dispute, the judicial area will be Prayagraj.
.248
, INDEX
PART-1 : ALGEBRA
ALGEBRA ...................................................................................................................................... 9-341
◘ Theory of Equations and Inequations .............................................................................................. 9
◘ Sequences and Series (A.P, G.P, H.P) ........................................................................................... 67
◘ Permutation and Combination ....................................................................................................... 98
◘ Binomial Theorem ....................................................................................................................... 121
◘ Exponential and Logarithmic Series ............................................................................................ 134
◘ Sets, Relations and Aunctions...................................................................................................... 142
◘ Modern Algebra ........................................................................................................................... 204
◘ Matrices and Determinants .......................................................................................................... 242
◘ Linear Algebra ............................................................................................................................. 324
PART-2 : REAL ANALYSIS
REAL ANALYSIS ..................................................................................................................... 342-361
◘ Convergence and Divergence of Sequences and Series of real numbers and functions ............. 342
◘ Functional Analysis and Measure Theory .................................................................................. 352
PART-3 : VECTOR ANALYSIS
VECTOR ANALYSIS ............................................................................................................... 362-428
◘ Operations on Vectors ................................................................................................................ 362
◘ Vector, Gradient and Curl ........................................................................................................... 409
PART-4 : COMPLEX ANALYSIS
COMPLEX ANALYSIS ............................................................................................................ 429-486
◘ Complex number, De-Moiver's theorem, nth root of unity .......................................................... 429
◘ Hyperbolic and logarithmic function of a complex variable ...................................................... 462
◘ Analytic Functions and Power Series .......................................................................................... 468
◘ Complex Integration, Cauchy's Theorem and Calculus of Residue............................................. 472
PART-5 : CALCULUS
CALCULUS................................................................................................................................ 487-805
◘ Limit, continuity and differentiability of function of one variable ............................................. 487
◘ Rolle's Theorem and Lagrange's Mean Value Theorem ............................................................. 564
2
.248
, ◘ Tangent, Normal, Increasing, Decreasing and Maxima/Minima of a Function of one variable . 571
◘ Limit, Continuity and Differentiability of function of two variable and Partial Differentiation . 607
◘ Integration, applications of Integration and area bounded by curve ............................................ 620
◘ Length of curve, surface area and volume of solids revolution .................................................. 693
◘ Ordinary and Partial differential equations, Integral Equations and Calculus of Variations ...... 704
PART-6 : GEOMETRY
GEOMETRY .............................................................................................................................. 806-957
◘ Two Dimension ........................................................................................................................... 806
● The Line and General equation of second degree and its classification................................ 806
● Conic section ......................................................................................................................... 848
◘ Three Dimensions....................................................................................................................... 911
● The Plane and The Line (Cartesian and vector) .................................................................... 911
● Sphere, Cone and Cylinder.................................................................................................... 942
PART-7 : STATISTICS AND TOPOLOGY
STATISTICS AND TOPOLOGY .......................................................................................... 958-1014
◘ Statistics and Probability.............................................................................................................. 958
◘ Topology .................................................................................................................................... 1007
PART-8 : TRIGONOMETRY
TRIGONOMETRY ............................................................................................................... 1015-1073
◘ Trigonometrical Identities and Circular function...................................................................... 1015
◘ Height and Distance .................................................................................................................. 1048
◘ Properties of Triangular and Circular inverse function............................................................. 1054
PART-9 : STATICS AND DYNAMICS
STATICS AND DYNAMICS ................................................................................................ 1074-1151
PART-10 : ARITHMETIC
ARITHMETIC ....................................................................................................................... 1152-1228
PART-11 : NUMERICAL ANALYSIS & OPERATIONS RESEARCH
NUMERICAL ANALYSIS & OPERATIONS RESEARCH ....................................... 1229-1248
◘ Numerical Analysis.................................................................................................................... 1229
◘ Linear Programming Problems .................................................................................................. 1238
3
.248
, TGT Maths : Syllabus
Commerce/Mathematics: Work time and velocity time, Hyperbolic function: Separation into real and imaginary
compound interest, banking, taxation, flow of elementary parts.
rules illustrated. Geometry: Baudhaayan Pythagoras principle and its
Statistics: Frequency distribution, graphical representation explanation, circle and circle segmet, arc and chord of a
of statistical data, measures of central tendency, measures of circle, tangent to the circle, alternate circle segments and
dispersion, birth/death statistics indices. their angles, segments of chords and rectangles formed
Algebra: Surds, polynomials and their factors, from them, symmetry of linear plane figures.
logarithms, linear equations of two unknown quantities, Coordinate geometry: Cartesian plane, line, straight
highest common factor and least common multiples of line pair represented by general exponential equation of
polynomials, simultaneous linear equations of three the second degree. Angle between them and equation of
unknowns, factors of quadratic polynomial quadratic pair of bisectors, standard equations and parametric
equations, ratios and proportions, number system, set and equations of conic (circle, parabola, ellipse and
operations, mapping. hyperbola) in rectangular cartesian, coordinates,
Determinant:Difinition, minors and cofactors, restrictions for representing line pair, circle, parabola,
expansion of determinant upto size 3×3, general ellipse and hyperbola by quadratic general equations,
properties of determinant, solution of system of n-linear obtaining equations of circle, parabola ellipse and
equations by cramer’s rule (for n = 3), Types of matrices, hyperbola with the help of transefer of origin and axes,
addition and multiplication of matrices upto size 3×3. intersection of tangent and normal secant line at any
Transpose of matrix, inverse of symmetric and skew- point of the conic, condition of it being tangent in the
symmetric matrices, solution of simultaneous equation of limit case, parametric equations of tangent, the tangent
three unknown quantities with the help of matrix. Theory pair to the conic from the external point. Equation of
of equation, symmetric functions of roots. Arithmetic, normal at any point of a conic condition of touching or
geometric and harmonic series sum of series formed by being normal, standard equation of a conic in polar
terms of squares and cube of natural numbers, coordinates (binary), three dimensional geometry of
permutation and combination , bionimial theorem, sum sphere, cone and cylinder.
of exponential and logarithmic series. Probability theory Differential calculus: Difinition of differention,
of addition and multiplication. Differention of algebraic, trigonometric, exponential and
Set theory: Algebra of set theory, equivalence, relation logarithmic function, tangent and normal line, tracing
mapping, composition of mappings, Inverse mapping, maximum and minimum of simple curve of a function of
use of Piano’s axiom and induction axiom, group and one variable integration-integration by parts and
group isomorphism, subgroup generated by subsets, substitution, integral with help of partial fractions,
cyclic groups, order of an element, subgroup of cyclic difinite integral its uses in determining the arc,
groups, coset decomposition, Lagrange’s theorem. differential and surface of cylinder, conical sphere under
Real Analysis: Axioms of real numbers, computation of cornes, order and degree of differential equation in
sets, metric spaces, limit, open sets, closed sets, derived example of rectilinear motion under gravity, solving the
set, dense set, perfect set, other general theorems equations in the following from.
including Balzano-Weierstrass theorem, sequence of real 3
dy dy φ( y)
numbers-limit of sequence, convergent sequence, 1. = f ( x ) 2. = f ( x ) 3. = f (x)
divergent sequence bounded sequence, monotonic dx dx dx 2
sequence, operation of convergent sequences, Cauchy Vector analysis: Vectors in the form of consecutive
sequence, Cauchy theorem on limit and cauchy theorem pairs and consecutive triples, displacement vectors, free
of convergence of real numbers, limit and limit of vectors, unit vectors, modulus and cosine, equal vectors,
continuous functions of real variables, left hand limit and combination of sum of vectors (force, velocity,
right hand limit, continuity of functions, characteristics acceleration). Inter-relative velocity of two vectors,
of continuous functions discontinuity and its types. scalar and vectors multiplication of two vectors. Their
Trignometry: Circular measurements and trigonometric use in calculation of work and torque. Triplication of
ratios of specific angles, trigonometric ratios of the sum vectors.
and difference of two angles, trigonometric identities, Positional Science: Equilibrium of bodies applied by
trigonometric equations, solution of triganle, radii and three forces Lami’s theorem, law of triangles,
properties of circuitmscribed and external circles, general trigonometric theorem and planning in two right angled
properties of invere circular functions. forces. General constraints of equilibrium center of
Complex Numbers: Their sum and product, DeMoivre’s gravity.
theorem and its applications, height and distance. Kinematics: Calculation of speed, work, energy and
Exponential functions, circular functions and hyperbolic MKS system of a projectile moving in a vertical plane
of complex quantities. under gravity.
4
.248
