## Details

"Content
1. Theory of Numbers
Division, G.C.M. Numbers Prime to each other, Prime and Composite Numbers . The Divisors of a Number.Product of n Consecutive Integers (6).Residues of Terms of an
A.P. .Induction .
EXERCISE I
2. Rationals and Irrationals.
Rationals, Fundamental Laws of Order and of Arithmetic, Representation by Points on a Line Absolute Values, Large and Small Numbers, Meaning of â€˜Tends,â€™ Aggregate, Sequence
Approximate Values, Fundamental Inequalities Irrationals, Meaning of Representation of a Number by an Endless Decimal Real Numbers, the Function ax
EXERCISE II
3. Polynomials
Notation Division, Synthetic Division.Remainder Theorem and Applications, Equating Coefficients . Quadratic Functions of x and y
EXERCISE III
Expansion of Products, Binomial Theorem for Positive Integral Index . Expansion of f(x + h), where f(x) = (a0, a1, a2,.....an) (x, 1)n. Multinomial Theorem, Greatest Coefficient in
(a + b + c + ...... + k)n
EXERCISE IV
H.C.F., Prime and Composite Functions .
EXERCISE V
4. Symmetric and Alternating Functions, Substitutions.
Symmetric Functions, Alternating Functions, Cyclic Expressions. Substitutions, Transpositions, Cyclic Substitutions, Inversions .
EXERCISE VI
CHAPTERS
5. Complex Numbers.
Expression of a Complex Number.Definitions of Equality, Addition, etc.Zero Products .Argand Diagram, Modulus, Amplitude . Addition, Subtraction, Products, Quotients, De Moivreâ€™s Theorem .Conjugate Numbers
EXERCISE VII
Roots of Complex Numbers, De Moivreâ€™s Theorem, Factors of xn Â± 1, Imaginary Cube Roots of Unity
EXERCISE VIII
Points representing Products and Quotients.Displacements and Vectors
EXERCISE IX
6. Theory of Equations
Roots of an Equation, Relations connecting the Roots and Coefficients.Transformation of Equations , Cubic and Biquadratic
EXERCISE X
Character and Position of Roots.Descartesâ€™ Rule of Signs
. De Guaâ€™s Role .Limits to the Roots To find the Rational Roots, Newtonâ€™s Method of Divisors
EXERCISE XI
Symmetric Functions of Roots
EXERCISE XII
7. Partial Fractions.
Rational Fractions.Fundamental Theorems on Partial Fractions . Resolution of a Proper Fraction into Partial Fractions
EXERCISE XIII
8. Summation of Series.
Meaning of Summation, Method of Differences .ï“un and ï“1/un, where ï“un and ï“unxn, where un is a polynomial in n. The series
The series Sr = 1r + 2r + 3r + ......... + nr, values of Sr for r = 1, 2, 3,....10,
Bernoulliâ€™s Numbers, Bernoulliâ€™s Theorem on Sr
EXERCISE XIV
CHAPTERS
9. Determinants.
Definitions, Elementary Theorems, Expansion . Minors, Cofactors, Elementary Theorems Examples
EXERCISE XV
Minors, Expansion in Terms of Second Minors Product of Two Determinants .Rectangular Arrays . Reciprocal Determinants, Two Methods of Expansion. Use of Double Suffix, Symmetric and Skew-symmetric Determinants, Pfaffian
EXERCISE XVI
10. Systems of Equations.
Definitions, Equivalent Systems . Linear Equations in Two Unknowns, Line at Infinity . Linear Equations in Three Unknowns, Equation to a Plane, Plane at Infinity
EXERCISE XVII
Systems of Equations of any Degree, Methods of Solution for Special Types
EXERCISE XVIII
11. Reciprocal and Binomial Equations.
Reduction of Reciprocal Equations The Equation
xn â€“ 1 = 0, Special Roots The Equation xn â€“ A = 0 (166). The Equation x17 â€“ 1 = 0, Regular 17-sided Polygon
EXERCISE XIX
12. Cubic and Biquadratic Equations.
The Cubic Equation (roots ï¡ï€¬ï€ ï¢ï€¬ï€ ï§), Equations whose roots are
(ï¢ï€ â€“ï€ ï§)2, etc., Value of ï„, Character of Roots . Cardanâ€™s Solution, Trigonometrical Solution, the Functions ï¡ï€ ï€«ï€ ï·ï¢ï€ ï€«ï€ ï·ï€²ï§ï€¬ï€ ï¡ï€ ï€«ï€ ï·ï€²ï¢ï€ ï€«ï€ ï·ï§,. Cubic as Sum of Two Cubes, the Hessian .Tschirnhausenâ€™s Transformation
EXERCISE XX
The Biquadratic Equation (roots ï¡ï€¬ï€ ï¢ï€¬ï€ ï§ï€¬ï€ ï¤) .The Functions
ï¬ï€ ï€½ï€ ï¢ï§ï€ ï€«ï€ ï¡ï¤, etc., the Functions I, J, ï„, Reducing Cubic, Character of Roots .Ferrariâ€™s Solution and Deductions Descartesâ€™ Solution Conditions for Four Real Roots Transformation into Reciprocal Form .Tschirnhausenâ€™s Transformation
EXERCISE XXI
CHAPTERS
13. Theory of Irrationals.
Sections of the System of Rationals, Dedekindâ€™s Definition
, Equality and Inequality.Use of Sequences in defining a Real Number, Endless Decimals
The Fundamental Operations of Arithmetics, Powers, Roots and Surds . Irrational Indices, Logarithms . Definitions, Interval, Steadily Increasing Functions .Sections of the System of Real Numbers, the Continuum .Ratio and Proportion, Euclidâ€™s Definition
EXERCISE XXII
14. Inequalities.
Weierstrassâ€™ Inequalities .Elementary Methods
(a1b1 + a2b2 + ......)2ï‚£ (a12 + a22 + ......)(b12 + b22 + .......),
For n Numbers a1, a2 .........
Arithmetic and Geometric Means
and Extension Maxima and Minima
EXERCISE XXIII
15. Sequences and Limits.
Definitions, Theorems, Monotone Sequences . Exponential Inequalities and Limits,
and if m>n,
EXERCISE XXIV
General Principle of Convergence . Bounds of a Sequence, Limits of Indetermination
CHAPTERS
Theorems :
(1) Increasing Sequence (un), where un â€“un â€“ 1 0 and un + 1/unï‚®l, then
(3) If then
(4) If and then
Complex Sequences, General Principle of Convergence
EXERCISE XXV
16. Convergence of Series
Definitions, Elementary Theorems, Geometric Series (238).Series of Positive Terms.Introduction and Removal of Brackets, Changing Order of Terms, Comparison Tests, ï“1/np, Dâ€™Alembertâ€™s and Cauchyâ€™s Tests
EXERCISE XXVI
Series with Terms alternately Positive and Negative
Series with Terms Positive or Negative.Absolute Convergence, General Condition for Convergence, Pringsheimâ€™s Theorem, Introduction and Removal of Brackets, Rearrangement of Terms, Approximate Sum, Rapidity of Convergence or Divergence
Series of Complex Terms.Condition of Convergence, Absolute Convergence, Geometric Series, ï“rncosnï±, ï“rn sin nï±.
If un/un + 1 = 1 + an/n, where anï‚®a> 0, then unï‚® 0.
Convergence of Binomial Series
EXERCISE XXVII
17. Continuous Variable.
Meaning of Continuous Variation, Limit, Tending to Â± ï‚¥, Theorems on Limits and Polynomials
Continuous
and Discontinuous Functions.Continuity of Sums, Products, etc., Function of a Function, Rational Functions, xn .Fundamental Theorems . Derivatives, Tangent to a Curve, Notation of the Calculus, Rules of Differentiation Continuity of ï¦{f(x)}, Derivatives of
ï¦{f(x)} and xn . Meaning of Sign of f ï‚¢(x) . Complex Functions, Higher Derivatives (268). Maxima and Minima, Points of Inflexion
EXERCISE XXVIII
CHAPTERS
Inverse Functions, Bounds of a Function, Rolleâ€™s Theorem, Mean Value Theorem.Integration Taylorâ€™s Theorem, Lagrangeâ€™s Form of Remainder Function of a Complex Variable, Continuity
EXERCISE XXIX
18. Theory of Equations, Polynomials , Rational
Fractions
Multiple Roots, Rolleâ€™s Theorem, Position of Real Roots of
f (x) = 0 . Newtonâ€™s Theorem on Sums of Powers of the Roots of f (x) = 0 . Order and Weight of Symmetric Functions . Partial Derivatives, Taylorâ€™s Theorem for Polynomials in x and in x, y, ......... . Eulerâ€™s Theorem for Polynomials,. A Theorem on Partial Fractions
The Equation
EXERCISE XXX
19. Exponential and Logarithmic Functions and Series.
Continuity, Inequalities and Limits The Exponential Theorem, Series for ax Meaning of an Irrational Index, Derivatives of ax, log x and xn Inequalities and Limits, the way in which ex and log x tend to ï‚¥, Eulerâ€™s Constant ï§, Series for log 2 (The Exponential Function E(z), Complex Index Series for sin x, cosx and Exponential Values
Use of Exponential Theorem in Summing Series
EXERCISE XXXI
Logarithmic Series and their Use in Summation of Series, Calculation of Logarithms The Hyperbolic Functions
EXERCISE XXXII
20. Convergence
Series of Positive Terms.Cauchyâ€™s Condensation Test, Test Series .Kummerâ€™s, Raabeâ€™s and Gaussâ€™s Tests . Binomial and Hyper-geometric Series.De Morganâ€™s and Bertrandâ€™s Tests
CHAPTERS
Series with Terms Positive or Negative.Theorem, Abelâ€™s Inequality, Dirichletâ€™s and Abelâ€™s Tests . Power Series, Interval
and Radius of Convergence, Criterion for Identity of Power Series . Binomial Series 1 + nz + .......... when z is complex. Multiplication of Series, Mertenâ€™s and Abelâ€™s Theorems
EXERCISE XXXIII
21. Binomial and Multinomial Theorems.
Statement, Vandermondeâ€™s Theorem.Binomial Theorem.Eulerâ€™s Proof, Second Proof, Particular Instances. Numerically Greatest Term, Approximate Values of
EXERCISE XXXIV .
Use of Binomial Theorem in Summing Series, Multinomial Theorem
EXERCISE XXXV
22. Rational Fractions, Recurring Series and Difference Equations.
Expansion of a Rational Fraction
EXERCISE XXXVI
Expansions of cosnï± and sin nï±/sinï± in Powers of cosï± . Recurring Series, Scale of Relation, Convergence, Generating Function, Sum . Linear Difference Equations with Constant Coefficients EXERCISE XXXVII
Difference Equations, General and Particular Solutions
EXERCISE XXXVIII
23. The Operators ï„, E, D. Interpolation.
The Operators ï„, E, Series for ï„rux, ux+r;u1 + u2 + u3 + ....... in Terms of u1, ï„u1, ï„2u1, ......... Interpolation, Lagrangeâ€™s and Besselâ€™s Formulae The Operator D, Value of
EXERCISE XXXIX
24. Continued Fractions
Definitions, Formation of Convergents, Infinite Continued Fractions Simple and Recurring Continued Fractions
CHAPTERS
EXERCISE XL
Simple Continued Fractions, Properties of the Convergents, an Irrational as a Simple Continued Fraction.Approximations, Miscellaneous Theorems.Symmetric Continued Fractions, Application to Theory of Numbers . Simple Recurring Continued Fractions
EXERCISE XLI
25. Indeterminate Equations of the First Degree.
Solutions of ax Â± by = Â± c . Two Equation in x, y, z; the Equation ax Â± by Â± czÂ± ..... = k
EXERCISE XLII
26. Theory of Numbers
Congruence, Numbers less than and prime to n, Value of ïª(n); ï“ïª(dr) = n, where dr is any Divisor of n Fermatâ€™s Theorem, Eulerâ€™s Extension, Wilsonâ€™s and Lagrangeâ€™s Theorems
EXERCISE XLIII
Roots of a Congruence, the Linear Congruence, Simultaneous Congruences Theorem on Fractions The General Congruence, Division (mod n), Congruences to a Prime Modulus
EXERCISE XLIV
27. Residues of Powers of a Number, Recurring Decimals.
Residues of a, ag, ag2, .... (modn), the Congruence gt 1 (mod n), An Odd Prime Modulus, Primitive Roots Decimal Equivalent of m/n, Number of Figures in the Period. Short Methods of Reckoning, Prime Factors of 10t â€“ 1, (t = 1, 2,..... 10),
EXERCISE XLV
The Congruence Methods of Solution
EXERCISE XLVI
28. Numerical Solution of Equations.
Limits to the Roots, Newtonâ€™s Method
EXERCISE XLVII
Separation of the Roots, Sturmâ€™s Theorem, Fourierâ€™s Theorem
CHAPTERS
EXERCISE XLVIII
Newtonâ€™s Method of Approximating to a Root, Fourierâ€™s Rule, Nearly Equal Roots. Hornerâ€™s Method
EXERCISE XLIX
29. Implicit Functions, Curve Tracing.
Implicit Functions, Rule for Approximations Series for Roots of a Cubic Equation (when all Real)
EXERCISE L
Tangents, Asymptotes, Intersection of a Straight Line and Curve Curve Tracing, Newtonâ€™s Parallelogram
EXERCISE LI
30. Infinite Products.
Convergence, Absolute Convergence, Derangement of Factors, Expansion as a Series Convergence discussed by Use of Logarithms
EXERCISE LII
31. Permutations, Combinations and Distributions.
Combinations with Repetitions, Things not all Different Distributions, Arrangement in Groups, Distribution in Parcels Derangements, General Theorem .Partition of Numbers, Table of pPartitions of n, Eulerâ€™s Use of Series
EXERCISE LIII
EXERCISE LIV
32. Probability.
First Principles, Exclusive Events, Independent and Interdepenent Events Probability estimated by Frequency, Expectation, Successive Events Probability of Causes, Value of Testimony, Application of Geometry
EXERCISE LV
33. Continued Fractions
Expression of a Quadratic Surd as a Simple Continued Fraction, the Form (Â± Â±b1)r1 The form Cycle of Quotients, the b and r Cycles, Solution in Integers of Bx2 â€“ Ay2 = M The Form Integral Solutions of x2 â€“ Ny2 = M The Cycle belonging to ( + b)/r found by the G.C.M. Process
EXERCISE LVI.
MISCELLANEOUS EXERCISES (A), (B)
ANSWERS

## Additional Information

SKU | SK_BKMTG_241 |
---|---|

Delivery Time | 2-3 days (Delhi/NCR), 4-6 days (Rest of India) |

NCERT Book Code | N/A |

Exams | N/A |

Poster Size | No |

ISBN | 9789385966576 |

Author | S.Barnard & J.M.Child |

Class | No |

Publisher | MTG Learning Media (P) Ltd.2 |

Subjects | Mathematics |

Language | English |

Edition | 2016 |

Pages | 584 |