Study material for Manipal entrance exam

Will you please provide me the Study material I want the syllabus for Manipal entrance exam as I have to give an entrance exam for that?

[Quick Sam]

Manipal University is known as Sikkim Manipal University.

It is located in Manipal, Karnataka, India.

It was established in the year of 1995.

It is affiliated to UGC, ACU, AIU.

It is type of Deemed University.

It is recognized by the University Grants Commission and approved by the Govern- ment of India.

It offers quality education to the students from North and North Eastern parts of India.

It is the first government-private initiative in the region.

The university has branch campuses in Bangalore, Mangalore, Sikkim, Jaipur, Dubai, Malaysia and Antigua.

So, here I am providing you the Syllabus for the entrance exam and I thought that it ca be very help full for you.

Syllabus of Chemistry:

Stoichiometry:

Equivalent mass of elements – definition, principles involved in the determination of equivalent masses of elements by hydrogen displacement method, oxide method, chloride method and inter conversion method (experimental determination not needed). Numerical problems.

Atomic Structure:

Introduction- constituents of atoms, their charge and mass.Atomic number and atomic mass.
Wave nature of light, Electromagnetic spectrum-emission spectrum of hydrogen-Lyman series, Balmer series, Paschen series, Brackett series and Pfund series. Rydberg’s equation.

Periodic Properties

Periodic table with 18 groups to be used.Atomic radii (Van der Waal and covalent) and ionic radii: Comparison of size of cation and anion with the parent atom, size of isoelectronic ions. Ionization energy, electron affinity, electronegativity- Definition with illustrations.

Physics Syllabus



Chemisty Syllabus
MANIPAL ENAT 2014 SYLLABUS FOR CHEMISTRY STOICHIOMETRY Equivalent mass of elements - definition, principles involved in the determination of equivalent masses of elements by hydrogen displacement method, oxide method, chloride method and inter conversion method (experimental determination not needed). Numerical problems. Equivalent masses of acids, bases and salts. Atomic mass, Moleqular mass, vapour density-definitions. Relationship between molecular mass and vapour density. Concept of STP conditions. Gram molar volume. Experimental determination of molecular mass of a volatile substance by Victor Meyer’s method. Numerical problems. Mole concept and Avogadro number, numerical problems involving calculation of: Number of moles when the mass of substance is given, the mass of a substance when number of moles are given and number of particles from the mass of the substance. Numerical problems involving mass-mass, mass-volume relationship in chemical reactions. Expression of concentration of solutions-ppm, normality, molarity and mole fraction. Principles of volumetric analysis- standard solution, titrations and indicators-acid-base (phenolphthalein and methyl orange) and redox (Diphenylamine). Numerical problems. ATOMIC STRUCTURE Introduction- constituents of atoms, their charge and mass. Atomic number and atomic mass. Wave nature of light, Electromagnetic spectrum-emission spectrum of hydrogen-Lyman series, Balmer series, Paschen series, Brackett series and Pfund series. Rydberg’s equation. Numerical problems involving calculation of wavelength and wave numbers of lines in the hydrogen spectrum. Atomic model- Bhor’s theory, (derivation of equation for energy and radius not required). Explanation of origin of lines in hydrogen spectrum. Limitations of Bhor’s theory. Dual nature of electron- distinction between a particle and a wave. de Broglie’s theory. Matter-wave equation (to be derived). Heisenberg’s uncertainty principle (Qualitative). Quantum numbers - n, l, m and s and their significance and inter relationship. Concept of orbital- shapes of s, p and d orbitals. Pauli’s exclusion principle and aufbau principle. Energy level diagram and (n+1) rule. Electronic configuration of elements with atomic numbers from 1 to 54. Hund’s rule of maximum multiplicity. General electronic configurations of s, p and d block elements. PERIODIC PROPERTIES Periodic table with 18 groups to be used. Atomic radii (Van der Waal and covalent) and ionic radii: Comparison of size of cation and anion with the parent atom, size of isoelectronic ions. Ionization energy, electron affinity, electronegativity- Definition with illustrations. Variation patterns in atomic radius, ionization energy, electron affinity, electronegativity down the group and along the period and their interpretation. OXIDATION NUMBER Oxidation and reduction-Electronic interpretation. Oxidation number: definition, rules for computing oxidation number. Calculation of the oxidation number of an atom in a compound/ion. Balancing redox equations using oxidation number method, calculation of equivalent masses of oxidising and reducing agents. GASEOUS STATE GAS LAWS: Boyle’s Law, Charle’s Law, Avogadro’s hypothesis, Dalton’s law of partial pressures, Graham’s law of diffusion and Gay Lussac’s law of combining volumes. Combined gas equation. Kinetic
molecular theory of gases-postulates, root mean square velocity, derivation of an equation for the pressure exerted by a gas. Expressions for r.m.s velocity and kinetic energy from the kinetic gas equation. Numerical problems. Ideal and real gases, Ideal gas equation, value of R (SI units). Deviation of real gases from the ideal behaviour. PV-P curves. Causes for the deviation of real gases from ideal behavior. Derivation of Van der Waal’s equation and interpretation of PV-P curves CHEMICAL KINETICS Introduction. Commercial importance of rate studies. Order of a reaction. Factors deciding the order of a reaction-relative concentrations of the reactants and mechanism of the reaction. Derivation of equation for the rate constant of a first order reaction. Unit for the rate constant of a first order reaction. Half-life period. Relation between half-life period and order of a reaction. Numerical problems. Determination of the order of a reaction by the graphical and the Ostwald’s isolation method. Zero order, fractional order and pseudo first order reactions with illustrations. Effect of temperature on the rate of a reaction-temperature coefficient of a reaction. Arrhenius interpretation of the energy of activation and temperature dependence of the rate of reaction. Arrhenius equation. Influence of catalyst on energy profile. Numerical problems on energy of activation. ORGANIC COMPOUNDS WITH OXYGEN-2, AMINES Phenols: Uses of phenol. Classification: Mono, di and tri-hydric Phenols Isolation from coal tar and manufacture by Cumene process. Methods of preparation of phenol from - Sodium benzene sulphonate,Diazonium salts Chemical properties: Acidity of Phenols-explanation using resonance-Effect of substituents on Acidity(methyl group and nitro group as substituents), Ring substitution reactions-Bromination, Nitration, Friedel-craft’s methylation, Kolbe’s reaction, Reimer-Tiemann reaction. Aldehydes and Ketones: Uses of methanal,benzaldehyde and acetophenone Nomenclature General methods of preparation of aliphatic and aromatic aldehydes and ketones from Alcohols and Calcium salts of carboxylic acids Common Properties of aldehydes and ketones a) Addition reactions with - Hydrogen cyanide, sodium bisulphate b) Condensation reactions with-Hydroxylamine, Hydrazine, Phenyl hydrazine, Semicarbazide c) Oxidation. Special reactions of aldehydes:Cannizzaro’s reaction-mechanism to be discussed, Aldol condensation, Perkin’s reaction, Reducing properties-with Tollen’s and Fehling’s reagents. Special reaction of ketones-Clemmensen’s reduction Monocarboxylic Acids: Uses of methanoic acid and ethanoic acid. Nomenclature and general methods of preparation of aliphatic acids From Alcohols, Cyanoalkanes and Grignard reagent General properties of aliphatic acids: Reactions with - Sodium bicarbonate, alcohols, Ammonia, Phosphorus pentachloride and soda lime Strength of acids-explanation using resonance. Effect of substituents (alkyl group and halogen as substituents) Amines: Uses of Aniline Nomenclature Classification-Primary, Secondary, Tertiary-aliphatic and aromatic. General methods of preparation of primary amines from - Nitro hydrocarbons, Nitriles(cyano hydrocarbons), Amides(Hoffmann’s degradation)
General Properties - Alkylation,Nitrous acid, Carbyl amine reaction, Acylation Tests to distinguish between-Primary, secondary, Tertiary amines-Methylation method. Interpretaion of Relative Basicity of-Methylamine, Ammonia and Aniline using inductive effect. HYDROCARBONS-2 Stability of Cycloalkanes-Baeyer’s Strain theory-interpretation of the properties of Cycloalkanes, strain less ring. Elucidation of the structure of Benzene - Valence Bond Theory and Molecular Orbital Theory. Mechanism of electrophilic substitution reactions of Benzene-halogenations, nitration, sulphonation and Friedel Craft’s reaction. HALOALKANES Monohalogen derivaties: Nomenclature and General methods of preparation from-Alcohols and alkenes. General properties of monohalogen derivatives: Reduction, with alcoholic KOH, Nucleophilic substitution reactions with alcoholic KCN, AgCN and aqueous KOH, with Magnesium, Wurtz reaction, Wurtz-Fittig’s reaction, Friedal-Craft’s reaction Mechanism of Nucleophilic Substitution reactions- SN1 mechanism of Hydrolysis of teritiary butyl bromide and SN2 mechanism of Hydrolysis of methyl bromide. COORDINATION COMPOUNDS Co-ordination compound: Definition, complex ion, ligands, types of ligands-mono, bi, tri and polydentate ligands. Co-ordination number, isomerism (ionization linkage, hydrate), Werner’s theory, Sidgwick’s theory, and E A N rule, Nomenclature of coordination, compounds.Valance Bond Theory: sp3, dsp2 and d2sp3 hybridisation taking [Ni(Co)4], [Cu(NH3)4]SO4, K4[Fe(CN)6] respectively as examples. CHEMICAL BONDING – 2 Covalent bonding-molecular orbital theory :linear combination of atomic orbitals (Qualitative approach), energy level diagram, rules for filling molecular orbitals, bonding and anti bonding orbitals, bond order, electronic configuration of H2, Li2 and O2 Non existence of He2 and paramagnetism of O2. Metallic bond: Electron gas theory (Electron Sea model), definition of metallic bond, correlation of metallic properties with nature of metallic bond using electron gas theory. CHEMICAL THERMODYNAMICS-2 Spontaneous and nonSpontaneous process. Criteria for spontaneity-tendency to attain a state of minimum energy and maximum randomness. Entropy-Entropy as a measure of randomness, change in entropy, unit of entropy. Entropy and spontaneity. Second law of thermodynamics. Gibbs’ free as a driving force of a reaction Gibbs’ equation. Prediction of feasibility of a process in terms of • G using Gibbs’ equation. Standard free energy change and its relation to Kp(equation to be assumed). Numerical problems. SOLID STATE Crystalline and amorphous solids, differences. Types of crystalline solids - covalent, ionic, molecular and metallic solids with suitable examples. Space lattice, lattice points, unit cell and Co- ordination number. Types of cubic lattice-simple cubic, body centered cubic, face centered cubic and their coordination numbers. Calculation of number of particles in cubic unit cells. Ionic crystals-ionic radius, radius ratio and its relation to co-ordination number and shape. Structures of NaCl and CsCl crystals. ELECTROCHEMISTRY Electrolytes and non electrolytes. Electrolysis-Faraday’s laws of electrolysis. Numerical problems. Arrhenius theory of electrolytic dissociation, Merits and limitations. Specific conductivities and molar conductivity-definitions and units. Strong and weak electrolytes-examples. Factors affecting conductivity. Acids and Bases: Arrhenius’ concept, limitations. Bronsted and Lowry’s concept, merits and limitations.
Lewis concept, Strengths of Acids and Bases - dissociation constants of weak acids and weak bases. Ostwald’s dilution law for a weak electrolytes-(equation to be derived) - expression for hydrogen ion concentration of weak acid and hydroxyl ion concentration of weak base - numerical problems. Ionic product of water. pH concept and pH scale. pKa and pkb values-numerical problems. Buffers, Buffer action, mechanism of buffer action in case of acetate buffer and ammonia buffer. Henderson’s equation for pH of a buffer (to be derived). Principle involved in the preparation of buffer of required pH-numerical problems. Ionic equilibrium: common ion effect, solubility.2B and AB2product, expression for Ksp of sparingly soluble salts of types AB, A B2Relationship between solubility and solubility product of salts of types AB, A. Applications of common ion effect and solubility product in inorganic2and AB qualitative analysis. Numerical problems. Electrode potential: Definition, factors affecting single electrode potential. Standard electrode potential. Nernst’s equation for calculating single electrode potential (to be assumed). Construction of electro-chemical cells-illustration using Daniel cell. Cell free energy change [•Go =-nFEo (to be assumed)]. Reference electrode: Standard Hydrogen Electrode-construction, use of SHE for determination of SRP of other single electrodes. Limitations of SHE. Electrochemical series and its applications. Corrosion as an electrochemical phenomenon, methods of prevention of corrosion. ORGANIC CHEMISTRY Inductive effect, Mesomeric effect and Electromeric effect with illustrations, Conversion of methane to ethane and vice versa and Methanol to ethanol and vice versa ISOMERISM-2 Stereo isomerism:geometrical and optical isomerism Geometrical isomerism-Illustration using 2-butene, maleic acid and fumaric acid as example, Optical Isomerism-Chirality, optical activity-Dextro and Laevo rotation(D and L notations). CARBOHYDRATES Biological importance of carbohydrates, Classification into mono, oligo and poly saccharides. Elucidation of the open chain structure of Glucose. Haworth’s structures of Glucose, Fructose, Maltose and Sucrose(elucidation not required). OILS AND FATS Biological importance of oils and fats, Fatty acids-saturated, unsaturated, formation of triglycerides. Generic formula of triglycerides. Chemical nature of oils and fats-saponification, acid hydrolysis, rancidity refining of oils, hydrogenation of oils, drying oils, iodine value. AMINO ACIDS AND PROTEINS AminoacidsaBiological importance of proteins, - General formula Formulae and unique feature of glycine, alanine, serine, cysteine, aspartic acid, lysine, tyrosine and proline. Zwitter ion, amphiprotic nature, isoelectric point, peptide bond, polypeptides and proteins. Denaturation of proteins Structural features of Insulin - a natural polypeptide. METALLURGY – 2 Physico-chemical concepts involved in the following metallurgical operations - Desilverisation of lead by Parke’s process-Distribution law. Reduction of metal oxides - Ellingham diagrams - Relative tendency to undergo oxidation in case of elements Fe Ag, Hg, Al, C. Cr, and Mg. Blast furnace - metallurgy of iron - Reactions involved and their role, Maintenance of the temperature gradient, Role of each ingredient and Energetics
INDUSTRIALLY IMPORTANT COMPOUNDS: Manufacture of Caustic soda by Nelson’s cell Method, Ammonia by Haber’s process, Sulphuric acid by Contact process and Potassium dichromate from chromite. Uses of the above compounds. Chemical properties of Sulphuric acid: Action with metals, Dehydrating nature, Oxidation reactions and Reaction with PCI Chemical properties of potassium dichromate: With KOH, Oxidation reactions, formation of chromyl chloride. GROUP 18, NOBEL GASES Applications of noble gases. Isolation of rare gases from Ramsay and Raleigh’s method and separation of individual gases from noble gas mixture (Dewar’s charcoal adsorption method).Preparation of Pt XeF6 by Neil Bartlett. d - BLOCK ELEMENTS (TRANSITION ELEMENTS) Definition. 3d series: electronic configurations, size, variable oxidation states, colour, magnetic properties, catalytic behaviour, complex formation and their interpretations. THEORY OF DILUTE SOLUTIONS Vant Hoffs theory of dilute Solutions. colligative property. Examples of colligative properties-lowering of vapour pressure, elevation in boiling points, depression in freezing point and osmotic pressure. Lowering of vapour pressure-Raoult’s law (mathematical form to be assumed). Ideal and non ideal solutions (elementary idea) - measurement of relative lowering of vapour pressure-ostwald and Walker’s dymnamic method. Determination of molecular mass by lowering of vapour pressure). Numerical problems. COLLOIDS Introduction. Colloidal system and particle size. Types of colloidal systems. Lyophilic and lyiphobic sols, examples and differences. Preparation of sols by Bredig’s arc method and peptisation. Purification of sols-dialysis and electro dialysis. Properties of sols-Tyndall effect, Brownian movement electrophoresis, origin of charge, coagulation, Hardy and Schulze rule, Protective action of sols. Gold number. Gold number of gelatin and starch. Applications of colloids. Electrical precipitation of smoke, clarification of drinking water and formation of delta.

Mathematics Syllabus
MANIPAL ENAT 2014 SYLLABUS FOR MATHEMATICS - I ALGEBRA PARTIAL FRACTIONS Rational functions, proper and improper fractions, reduction of improper fractions as a sum of a polynomial and a proper fraction. Rules of resolving a rational function into partial fractions in which denominator contains (i) Linear distinct factors, (ii) Linear repeated factors, (iii) Non repeated non factorizable quadratic factors [problems limited to evaluation of three constants]. LOGARITHMS (i) Definition Of logarithm (ii) Indices leading to logarithms and vice versa (iii) Laws with proofs: (a) logam+logan = loga(mn) (b) logam - logan = loga(m/n) (c) logamn = nlogam (d) log b m = logam/logab (change of base rule) (iv) Common Logarithm: Characteristic and mantissa; use of logarithmic tables,problems theorem MATHEMATICAL INDUCTION (i) Recapitulation of the nth terms of an AP and a GP which are required to find the general term of the series (ii) Principle of mathematical Induction proofs of a. ∑n =n(n+1)/2 b.∑n2 =n(n+1)(2n+1)/6 c. ∑n3 = n2 (n+1)2/4 By mathematical induction Sample problems on mathematical induction SUMMATION OF FINITE SERIES (i) Summation of series using ∑n, ∑n2, ∑n3 (ii) Arithmetico-Geometric series (iii) Method of differences (when differences of successive terms are in AP) (iv) By partial fractions THEORY OF EQUATIONS (i) FUNDAMENTAL THEOREM OF ALGEBRA: An nth degree equation has n roots(without proof) (ii) Solution of the equation x2 +1=0.Introducing square roots, cube roots and fourth roots of unity (iii) Cubic and biquadratic equations, relations between the roots and the co-efficients. Solutions of cubic and biquadratic equations given certain conditions (iv) Concept of synthetic division (without proof) and problems. Solution of equations by finding an integral root between - 3 and +3 by inspection and then using synthetic division. Irrational and complex roots occur in conjugate pairs (without proof). Problems based on this result in solving cubic and biquadratic equations. BINOMIAL THEOREM
Permutation and Combinations: Recapitulation of nPr and nCr and proofs of (i) general formulae for nPr and nCr (ii) nCr = nCn-r (iii) nCr-1 + n C r = n+1 C r (1) Statement and proof of the Binomial theorem for a positive integral index by induction. Problems to find the middle term(s), terms independent of x and term containing a definite power of x. (2) Binomial co-efficient - Proofs of (a) C 0 + C 1 + C 2 + …………………..+ C n = 2 n (b) C 0 + C 2 + C 4 + …………………..= C 1+ C 3 + C 5 + ………2 n – 1 MATHEMATICAL LOGIC Proposition and truth values, connectives, their truth tables, inverse, converse, contrapositive of a proposition, Tautology and contradiction, Logical Equivalence - standard theorems, Examples from switching circuits, Truth tables, problems. GRAPH THEORY Recapitulation of polyhedra and networks (i) Definition of a graph and related terms like vertices, degree of a vertex, odd vertex, even vertex, edges, loop, multiple edges, u-v walk, trivial walk, closed walk, trail, path, closed path, cycle, even and odd cycles, cut vertex and bridges. (ii) Types of graphs: Finite graph, multiple graph, simple graph, (p,q) graph, null graph, complete graph, bipartite graph, complete graph, regular graph, complete graph, self complementary graph, subgraph, supergraph, connected graph, Eulerian graph and trees. (iii) The following theorems: p p (1) In a graph with p vertices and q edges ∑deg n i = 2 q i=1 (2) In any graph the number of vertices of odd degree is even. (iv) Definition of connected graph, Eulerian graphs and trees - simple problems. ANALYTICAL GEOMETRY 1. Co-ordinate system (i) Rectangular co-ordinate system in a plane (Cartesian) (ii) Distance formula, section formula and mid-point formula, centroild of a triangle, area of a triangle - derivations and problems. (iii) Locus of a point. Problems. 2 .Straight line (i)Straight line: Slope m = (tanθ) of a line, where θ is the angle made by the line with the positive x-axis, slope of the line joining any two points, general equation of a line - derivation and problems. (ii) Conditions for two lines to be (i) parallel, (ii) perpendicular. Problems. (iii) Different forms of the equation of a straight line: (a) slope - point form (b) slope intercept form (c) two points form(d) intercept form and (e) normal form - derivation; Problems. (iv) Angle between two lines point of intersection of two lines condition for concurrency of three lines. Length of the perpendicular from the origin and from any point to a line. Equations of the internal and external bisectors of the angle between two lines- Derivations and Problems. 3. Pair of straight lines (i) Pair of lines, homogenous equations of second degree. General equation of second degree. Derivation of (1) condition for pair of lines (2) conditions for pair of parallel lines, perpendicular lines and distance between the pair of parallel lines.(3) Condition for pair of co-incidence lines and (4) Angle and point of
intersection of a pair of lines. LIMITS AND CONTINUTY (1) Limit of a function - definition and algebra of limits. (2) Standarad limits (with proofs) (i) Lim x n - a n/x - a= na n-1 (n rational) x→a (ii) Lim sin θ / θ = 1 (θ in radian) and Lim tan θ / θ = 1 (θ in radian) θ→0 θ →0 (3) Statement of limits (without proofs): (i) Lim (1 + 1/n) n = e (ii) Lim (1 + x/n) n = ex n→ ∞ n→∞ (iii) Lim (1 + x)1/x = e (iv) Lim log(1+x)/x = 1 x→0 x→0 v) Lim (e x - 1)/x= 1 vi) Lim (a x - 1)/x = logea x→0 x→0 Problems on limits (4) Evaluation of limits which tale the form Lim f(x)/g(x)[0/0 form]’ Lim f(n)/g(n) x→0 x→∞ [∞ /∞ form] where degree of f(n) ≤ degree of g(n). Problems. (5) Continuity: Definitions of left- hand and right-hand limits and continuity. Problems. TRIGONOMETRY Measurement of Angles and Trigonometric Functions Radian measure - definition, Proofs of: (i) radian is constant (ii) p radians = 1800 (iii) s = rθ where θ is in radians (iv) Area of the sector of a circle is given by A = ½ r2θ where θ is in radians. Problems Trigonometric functions - definition, trigonometric ratios of an acute angle, Trigonometric identities (with proofs) - Problems.Trigonometric functions of standard angles. Problems. Heights and distances - angle of elevation, angle of depression, Problems. Trigonometric functions of allied angles, compound angles, multiple angles, submultiple angles and Transformation formulae (with proofs). Problems. Graphs of sinx, cosx and tanx. Relations between sides and angles of a triangle Sine rule, Cosine rule, Tangent rule, Half-angle formulae, Area of a triangle, projection rule (with proofs). Problems. Solution of triangles given (i) three sides, (ii) two sides and the included angle, (iii) two angles and a side, (iv) two sides and the angle opposite to one of these sides. Problems. MATHEMATICS - II ALGEBRA ELEMENTS OF NUMBER THEORY (i) Divisibility - Definition and properties of divisibility; statement of division algorithm. (ii) Greatest common divisor (GCD) of any two integers using Eucli’s algorithm to find the GCD of any two integers. To express the GCD of two integers a and b as ax + by for integers x and y. Problems. (iii) Relatively prime numbers, prime numbers and composite numbers, the number of positive divisors of a number and sum of all positive division of a number - statements of the formulae without proofs. Problems. (iv) Proofs of the following properties: (1) the smallest divisor (>1) of an integer (>1) is a prime number (2) there are infinitely many primes
(3) if c and a are relatively prime and c| ab then c|b (4) if p is prime and p|ab then p|a or p|b (5) if there exist integers x and y such that ax+by=1 then (a,b)=1 (6) if (a,b)=1, (a,c)=1 then (a,bc)=1 (7) if p is prime and a is any ineger then either (p,a)=1 or p|a (8) the smallest positive divisor of a composite number a does not exceed √a Congruence modulo m - definition, proofs of the following properties: (1) ≡mod m" is an equivalence relation (2) a ≡ b (mod m) => a ± x ≡ b ± x (mod m) and ax ≡ bx (mod m) (3) If c is relatively prime to m and ca ≡ cb (mod m) then a ≡ b (mod m) - cancellation law (4) If a ≡ b (mod m) - and n is a positive divisor of m then a ≡ b (mod n) (5) a ≡ b (mod m) => a and b leave the same remainder when divided by m Conditions for the existence of the solution of linear congruence ax ≡ b (mod m) (statement only), Problems on finding the solution of ax ≡ b (mod m) GROUP THEORY Groups - (i) Binary operation, Algebraic structures. Definition of semigroup, group, Abelian group - examples from real and complex numbers, Finite and infinite groups, order of a group, composition tables, Modular systems, modular groups, group of matrices - problems. (ii) Square roots, cube roots and fourth roots of unity from abelian groups w.r.t. multiplication (with proof). (iii) Proofs of the following properties: (i) Identity of a group is unique (ii)The inverse of an element of a group is unique (iii) (a-1)-1 = a, " a Є G where G is a group (iv)(a*b)-1 = b-1*a-1 in a group (v)Left and right cancellation laws (vi)Solutions of a* x = b and y* a = b exist and are unique in a group (vii)Subgroups, proofs of necessary and sufficient conditions for a subgroup. (a) A non-empty subset H of a group G is a subgroup of G iff (i) " a, b Є H, a*b Є H and (ii) For each a Є H,a-1Є H (b) A non-empty subset H of a group G is a subgroup of G iff a, b Є H, a * b-1 Є H. Problems. VECTORS (i) Definition of vector as a directed line segment, magnitude and direction of a vector, equal vectors, unit vector, position vector of point, problems. (ii) Two-and three-dimensional vectors as ordered pairs and ordered triplets respectively of real numbers, components of a vector, addition, substraction, multiplication of a vector by a scalar, problems. (iii) Position vector of the point dividing a given line segment in a given ratio. (iv) Scalar (dot) product and vector (cross) product of two vectors. (v) Section formula, Mid-point formula and centroid. (vi) Direction cosines, direction ratios, proof of cos2 α + cos2β +cos2γ = 1 and problems. (vii) Application of dot and cross products to the area of a parallelogram, area of a triangle, orthogonal vectors and projection of one vector on another vector, problems. (viii) Scalar triple product, vector triple product, volume of a parallelepiped; conditions for the coplanarity of 3 vectors and coplanarity of 4 points. (ix) Proofs of the following results by the vector method: (a) diagonals of parallelogram bisect each other (b) angle in a semicircle is a right angle (c) medians of a triangle are concurrent; problems (d) sine, cosine and projection rules (e) proofs of 1. sin(A±B) = sinAcosB±cosAsinB 2. cos(A±B) = cosAcosB μ sinAsinB
MATRICES AND DETERMINANTS (i) Recapitulation of types of matrices; problems (ii) Determinant of square matrix, defined as mappings ∆: M (2,R) → R and ∆ :M(3,R)→ R. Properties of determinants including ∆(AB)=∆(A) ∆(B), Problems. (iii) Minor and cofactor of an element of a square matrix, adjoint, singular and non-singular matrices, inverse of a matrix,. Proof of A(Adj A) = (Adj A)A = |A| I and hence the formula for A-1. Problems. (iv) Solution of a system of linear equations in two and three variables by (1) Matrix method, (2) Cramer’s rule. Problelms. (v) Characteristic equation and characteristic roots of a square matrix. Cayley-Hamilton therorem |statement only|. Verification of Cayley-Hamilton theorem for square matrices of order 2 only. Finding A-1 by Cayley-Hamilton theorem. Problems. ANALYTICAL GEOMETRY CIRCLES (i) Definition, equation of a circle with centre (0,0) and radius r and with centre (h,k) and radius r. Equation of a circle with (x1 ,y1) and (x2,y2) as the ends of a diameter, general equation of a circle, its centre and radius - derivations of all these, problems. (ii) Equation of the tangent to a circle - derivation; problems. Condition for a line y=mx+c to be the tangent to the circle x2+y2 = r2 - derivation, point of contact and problems. (iii) Length of the tangent from an external point to a circle - derivation, problems (iv) Power of a point, radical axis of two circles, Condition for a point to be inside or outside or on a circle - derivation and problems. Poof of the result “the radical axis of two circles is straight line perpendicular to the line joining their centres”. Problems. (v) Radical centre of a system of three circles - derivation, Problems. (vi) Orthogonal circles - derivation of the condition. Problems CONIC SECTIONS (ANANLYTICAL GEOMETRY) Definition of a conic 1. Parabola Equation of parabola using the focus directrix property (standard equation of parabola) in the form y2 = 4 ax ; other forms of parabola (without derivation), equation of parabola in the parametric form; the latus rectum, ends and length of latus rectum. Equation of the tangent and normal to the parabola y2 = 4 ax at a point (both in the Cartesian form and the parametric form) (1) derivation of the condition for the line y=mx+c to be a tangent to the parabola, y2 = 4 ax and the point of contact. (2) The tangents drawn at the ends of a focal chord of a parabola intersect at right angles on the directix - derivation, problems. 2. Ellipse Equation of ellipse using focus, directrix and eccentricity - standard equation of ellipse in the form x2/a2 +y2/b2 = 1(a>b) and other forms of ellipse (without derivations). Equation of ellips in the parametric form and auxillary circle. Latus rectum: ends and the length of latus rectum. Equation of the tangent and the normal to the ellipse at a point (both in the cartesian form and the parametric form) Derivations of the following: (1) Condition for the line y=mx+c to be a tangrent to the ellipsex2/a2 +y2/b2 = 1 at (x1,y1) and finding the point of contact (2) Sum of the focal distances of any point on the ellipse is equal to the major axis (3) The locus of the point of intersection of perpendicular tangents to an ellipse is a circle (director circle)
3 Hyperbola Equation of hyperbola using focus, directrix and eccentricity - standard equation hyperbola in the form x2/a2 -y2/b2 = 1 Conjugate hyperbola x2/a2 -y2/b2 = -1 and other forms of hyperbola (without derivations). Equation of hyperbola in the parametric form and auxiliary circle. The latus rectum; ends and the length of latus rectum. Equations of the tangent and the normal to the hyperbola x2/a2 -y2/b2 = 1 at a point (both in the Cartesian from and the parametric form). Derivations of the following results: (1) Condition for the line y=mx+c to be tangent to the hyperbola x2/a2 -y2/b2 = 1 and the point of contact. (2) Differnce of the focal distances of any point on a hyperbola is equal to its transverse axis. (3) The locus of the point of intersection of perpendicular tangents to a hyperbola is a circle (director circle) (4) Asymptotes of the hyperbola x2/a2 -y2/b2 = 1 (5) Rectangular hyperbola (6) If e1 and e2 are eccentricities of a hyperbola and its conjugate then 1/e12+1/e22=1 TRIGONOMETRY COMPLEX NUMBERS (i) Definition of a complex number as an ordered pair, real and imaginary parts, modulus and amplitude of a complex number, equality of complex numbers, algebra of complex numbers, polar form of a complex number. Argand diagram, Exponential form of a complex number. Problems. (ii) De Moivre’s theorem - statement and proof, method of finding square roots, cube roots and fourth roots of a complex number and their representation in the Argand diagram. Problems. DIFFERENTIATION (i) Differentiability, derivative of function from first principles, Derivatives of sum and difference of functions, product of a constant and a function, constant, product of two functions, quotient of two functions from first principles. Derivatives of Xn , e x, a x, sinx, cosx, tanx, cosecx, secx, cotx, logx from first principles, problems. (ii) Derivatives of inverse trigonometric functions, hyperbolic and inverse hyperbolic functions. (iii) Differentiation of composite functions - chain rule, problems. (iv) Differentiation of inverse trigonometric functions by substitution, problems. (v) Differentiation of implicit functions, parametric functions, a function w.r.t another function, logarithmic differentiation, problems. (vi) Successive differentiation - problems upto second derivatives. Applications Of Derivatives (i) Geometrical meaning of dy\dx, equations of tangent and normal, angle between two curves. Problems. (ii) Subtangent and subnormal. Problems. (iii) Derivative as the rate measurer. Problems. (iv) Maxima and minima of a function of a single variable - second derivative test. Problems. Inverse Trigonometric Functions (i) Definition of inverse trigonometric functions, their domain and range. Derivations of standard formulae. Problems. (ii) Solutions of inverse trigonometric equations. Problems. General Solutions Of Trigonometric Equations General solutions of sinx = k, cosx=k, (-1≤ k ≤1), tanx = k, acosx+bsinx= c - derivations. Problems.
Integration (i) Statement of the fundamental theorem of integral calculus (without proof). Integration as the reverse process of differentiation. Standarad formulae. Methods of integration, (1) substitution, (2) partial fractions, (3) integration by parts. Problems. (4) Problems on integrals of: 1/(a+bcosx); 1/(a+bsinx); 1/(acosx+bsinx+c); 1/asin2x+bcos2x+c; [f(x)]n f ' (x); f'(x)/ f(x); 1/√(a2 - x2 ) ; 1/√( x2 - a2); 1/√( a2 + x2); 1/x √( x2± a2 ) ; 1/ (x2 - a2); √( a2 ± x2); √( x2- a2 ); px+q/(ax2+bx+c; px+q/√(ax2+bx+c); pcosx+qsinx/(acosx+bsinx); ex[f(x) +f1 (x)] DEFINITE INTEGRALS (i) Evaluation of definite integrals, properties of definite integrals, problems. (ii) Application of definite integrals - Area under a curve, area enclosed between two curves using definite integrals, standard areas like those of circle, ellipse. Problems. DIFFERENTIAL EQUATIONS Definitions of order and degree of a differential equation, Formation of a first order differential equation, Problems. Solution of first order differential equations by the method of separation of variables, equations reducible to the variable separable form. General solution and particular solution. Problems.