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(ii) Gynoecium - part of gynoecium, concept of carpel, Types of gynoecium - apocarpous and syncarpous gynoecium. Types of gynoecium based on number of carpels - monocarpellary, bicarpellary, tricarpellary and multicarpellary conditions.Nature of ovary of gynoecium with reference to locule - unilocular, bilocular, trilocular and multilocular conditions. Placentation - definition, types - marginal, axile, basal and parietal. International structure of essential parts: a) T.S of mature anther and structure of the pollen grain (Microsporogenesis not needed) b) Structure of a mature anatropous ovule (Megasporogenesis not needed). Pollination in Angiosperms: Definition, self and cross pollination, types (Autogamy, Allogamy, Geitonogamy, Xenogamy, Cleistogamy, Homogamy). Agents (Anemophily, Zoophily - Entomophily - Ornithophily and Hydrophily) with examples. (Pollination mechanisms not needed). Fertilization in Angiosperms: Definition, a brief account of double fertiltzation and its significance (Embroyogeny not required). The Angiosperm fruit: Definition, types of fruits - Simple fruits - fleshy fruits (drupe and berry), Dry fruits (capsule, cypsela and cremocarp) and Pome (apple). Aggregate fruits - etaerio of follicles. Multi fruits - Scrosis. The Angiosperm seed: Concept of seed. A typical dicotyledonous seed (Example: Bean seed). A typical monocotyledonous seed (Example: Maize grain). Taxonomy and Economic Botany: Taxonomy: An outline of classification system of Engler and Prantl. Distinguishing characters and plants of economic interest of the following families of angiosperms: Malvaceae - (Hibiscus, Cotton, Lady’s finger). Apocynaceae - ( Catheranthus roseus, Rauwolfia serpentiana, Plumeria alba and Nerium indicum) Musaceae - (Musa paradisiaca and Ravenala madagascariensis). Economic Botany: Introduction. Oil yielding plants - Groundnut and Sunflower. Cereals and millets - Rice and Jowar. Pulses - Pigeon pea and Bengal gram. Medicinal plants - Adathoda vasica, Ephedra gerardiana, Dryopteris, Santalum album, Gymnema sylvestre, Ocimum sanctum, Phyllanthus emblica. Spices - Pepper, cloves and cardamom. Beverages - Coffee, cocoa and tea. (Mentioning scientific names, family, parts used and uses only). Elements of plant pathhology: Symptoms, etiology, type and nature of pathogens, and methods of control with reference to the following diseases: (i) Banana bunchy top (ii)Tikka disease of groundnut (iii)Crown gall (of any common dicot plant). GENERAL BIOLOGY TOPICS Introduction to Biology: Definition of Biology and its main branches - Botany and Zoology. Scope of Biology. Branches of Biology(definition only). Classical branches - morphology, cytology, histology, anatomy, physiology, developmental biology, biosystamatics, genetics, ecology, organic evolution and palaeontology. Interdisciplinary branches - biophysics, biochemistry and biostatistics. Applied branches and career prospects - agriculture, entomology, silviculture, pathology, apiculture, microbiology and bioinformatics. Role of biology in dispelling myths and disbeliefs. Biomolecules: Carbohydrates: Definition. Classification - monosaccharides (ribose, deoxyribose, glucose, fructose and galactose), oligosaccharides (maltose, sucrose and lactose) and polysaccharides (starch, glycogen, cellulose, pectin, chitin and agar agar). Biological significance. Proteins: Definition. Classification - simple proteins (albumins, globulins, histones, actin, myosin and keratin), conjugate proteins - Chromoproteins (haemoglobin), glycoproteins (mucin of saliva), phospoproteins (casein of milk) and lipoproteins (lipovitelline of egg yolk). Biological significance of amino acid and proteins. Lipids: Definition. Classification - Simple lipids - oils (vegetable oil and oil of animal origin), fats (butter) and waxes (beeswax), Compound lipids - phospholipids (lecithin and cephalin) and sphingolipids (cerebrosides),Related compounds - steroids (estrogen, progesterone and testosterone), sterols (cholestoral) and prostaglandins. Biological significance. Enzymes: Definition, properties, classification based on functions. Mode of action - induced fit theory of Koshland. Nucleic acid: Occurrence, basic chemical composition (nucleoside and nucleotide), mention of type (DNA and RNA) and functions (structural details are not required). [*Note: Details of chemical structure of biomolecules are not required]. Origin of life and organic evolution: Origin of life: Introduction. Concept of abiogenesis and biogenesis (experimental evidences not required).A.I.Oparin’s Theory of chemical evolution of life (Views of Haldane and Sidney Fox to be mentioned).Stanley Miller’s experiment in support of chemical evolution. Organic evolution: Introduction. Darwin’s theory (DDT resistance in mosquitoes and industrial melanism in Peppered moth, to illustrate natural selection to be quoted as examples).Brief account of Mutation theory. NeoDarwininism - Introduction, Darwinian concept vs NeoDarwinian concept (gene pool and gene frequency), Hardy - Weinberg law and sources of variations as evolutionary force - sexual reproduction, genetic drift, gene flow, mutation and isolation (reproductive and geographic). ZOOLOGY TOPICS Diversity of animal life: Introduction. Outline classification of kingdom Animalia (only the major phyla to be considered). Major animal phyla: Outline classification as treated in ‘A Manual of Zoology’ Vol. I and Vol. II (1971) by Ekambarantha Ayyar. Non-chordata (animals without backbone) - General characters and classification up to classes [salient features of classes of Invertebrate phyla not to be given] with suitable examples of the following phyla: Protozoa, Porifera, Coelenterata, Platyhelminthes, Nematoda, Annelida, Arthropoda, Mollusca and Echinodermata. Chordata (Animals with backbone) - Fundamental characters and classification of chordata up to subphyla - Hemichordata, Urochordata, Cephalochordata and Vertebrata with suitable examples. Subphylum Vertebrata - Salient features with examples of (i) Subphylum Pisces: Class Chondreichthyes and Class Osteichthyes); (ii) Superclass Tetrapoda: Amphibia, Reptilia, Aves and Mammalia. Differences between non-chordates and chordates. Study of Morphology: Cockroach - Periplaneta sp. Morphology (Structure of head capsule and compound eye not required).Digestive and nervous systems. Animal resources: Sericulture; Definition. Main aspects - moriculture, rearing of silkworms and reeling. Brief account of moriculture: definition, methods (row and pit systems) and its importance. Types of silk - mulberry and non-mulberry (Tasar, Eri and Muga). Diseases of mulberry silkworm - Pebrine, Muscardine or Calcino, Flacherie and Grasserie[Listing of diseases and causative organisms only]. Aquaculture: Definition. Areas - fin fisheries and shell fisheries. Pisciculture: definition, capture fisheries and culture fisheries. Inland fisheries - procedure. Monoculture, monosex culture and polyculture (composite fish farming) - meaning with examples. Dairy: Definition. Types of indigenous cattle with examples based on utility - draught, milching and dual purpose (Cow breeds - Sindhi, Sahiwal, Amrithmahal, Hallikar, Ongole and Haryana; Buffalo breeds - Murrah, Surti, Mehsana and Nagpuri). Examples of high yielding exotic breeds (Holstein, Red Dane, Jersey and Brown Swiss). Nutritive value of milk. Utility of cattle - biogas, leather, gelatin and organic manure. Poultry: Definition. Types of indigenous fowls with examples based on utility - layers, broilers and dual purpose (Aseel, Chittagong, Ghagus, Basra and Kadaknath). Examples of exotic breeds (White Leghorn, Cornish, Rhode Island Red Plymouth Rock and Newhampshire). Giriraj - origin and salient features. Nutritive value of egg. Diseases ( Respiratory mycoplasmosis, Fowl pox candidiasis, Raniketh and Fowl cholera) - Mentioning of diseases and causative organisms only. Vermiculture: Definition and procedure. Vermicompost - degradation of organic wastes and role of Earthworm in soil fertility. BIOLOGY - II GENERAL BIOLOGY TOPICS Molecular Biology: Nucleic acids: DNA - Occurrence, DNA as the genetic material (with the experiment of Avery as evidence), chemical composition, structure (Watson - Crick model), Semiconservative method of replication. RNA - Occurrence, chemical composition, brief account of structure and functions of genetic RNA, rRNA, mRNA and tRNA (clover - leaf model). Gene: The gene, the genetic code and genetic control of protein synthesis - Concept of gene (prokaryotic and eukaryotic), genetic code and its characteristics, genetic control of protein synthesis (transcription and translation) and Lac operon concept. Biotechnology: Introduction: Scope of biotechnology. Genetic Engineering: Introduction; Tools used in genetic engineering - Vectors (plasmid - pUC18), Enzymes (REN and Ligase), Host cell (E.coli) and Bioreactors. Recombinant DNA technology and its applications: Insulin synthesis to be used as an example. A brief account of: DNA fingerprinting, Gene therapy, Human genome project, Monoclonal antibodies. Improvement of crop plants: Breeding techniques; Tissue culture technique - organ culture example: stem; transgenic plants example: Golden rice. Improvement of animals: Breeding techniques and stem cell culture, transgenic animals example: Cattle. Hazards and safeguards of genetic engineering. BOTANY TOPICS Plant history & anatomy: Introduction: Definition and general classification of plant tissues. Meristems: Definition, structure and classification based on position, origin and function (theories an apical organization not required). Permanent Tissues - Distribution, structure and functions of: Simple tissues: Parenchyma (Chorenchyma and Aerenhyma), Collenchyma (angular, lacunar & lamellar) and Sclerenchyma - Fibres (Intraxylary and Extraxylary), Sclereids (Macrosclereids, Brachysclereids, Astrosclereids and Osteosclereids). Complex tissues: Xylem and Phloem. Definition of the terms: Primary and secondary vascular tissues, exarch xylem, endarch xylem, collateral conjoint open and collateral conjoint closed vascular bundles, radial arrangement of vascular tissues. Secondary growth in dicot stem: intrastelar and extrastelar secondary growth. Plant physiology. Water relations of plants: Fundamental concepts: Importance of water to plants. Significance and definitions of the following: Imbibition, Diffusion, Osmosis, Endosmosis, Exosmosis, Plasmolysis, Deplasmolysis, Turgor pressure, Well pressure, Osmotic pressure. Water potential and its components. Absorption of water: Structure of root hair. Sources of water for plants (available water and nonavailable water). Region of absorption of water in plants. Entry of water from soil into xylem of root. Active and passive absorption of water (active absorption to show osmotic and non osmotic processes). Ascent of sap: Definition and evidences to show the involvement of xylem (the Balsam plant experiment). Composition of xylem sap. Transpiration pull theory - merits and demerits. Loss of water in plants: Transpiration - Definition and types. Structure of a typical stomatal apparatus (dicot example only). Mechanism of stomatal movement - Steward’s Starch hydrolysis theory and K+ pump theory. Factors influencing the rate of transpiration (external). Significance of transpiration. A brief note on antitranspirants. Guttation: A brief account of guttation - occurrence, causes and structure of hydathode. Translocation of solutes: Definition and evidences in support of involvement of phloem in the process (Girdling experiment and Tracer method). Composition of phloem sap. Munch’s mass flow hypothesis with merits and demerits. Vein loading. Bioenergetics: Introduction: Light as the source of energy and ATP as energy currency. Photosynthesis: Definition. Ultrastructure of the chloroplast. Photosynthetic pigments and their role; composition of photsystems I & II. (Molecular structures and formulae not required). Mechanism - light reaction - cyclic and noncyclic photophosprylations; Dark reaction (C3 pathway - Calvin cycle) - (details of regeneration steps not required); C4 pathway and CAM (definition and examples only). Influence of external factors on photosynthesis; Blackman’s law of limiting factors. Significance of photosynthesis. Respiration: Definition and types (aerobic and anaerobic). Ultra structure of mitochondrion. Mechanism of aerobic respiration - Glycolysis, Krebs cycle and Terminal oxidation. Anaerobic respiration - Mechanism of fermentation in the presence of yeast and lactic acid bacteria. Role of external factors, respiratory quotient (RQ) and its significance and Pasteur effect. Growth and growth regulators in plants: Growth: Definition, regions of growth, phases of growth and growth curve. Growth regulators: Definition. Role of the following plant hormones (Details of experiments on discovery of hormones not required): i. Auxins. ii. Gibberellins. iii. Cytokinins. iv. Abscissic acid. v. Ethylene. Synthetic growth regulators and their applications (with reference to IAA, IBA, NAA, 2, 4-D, BAP and Ethephon). GENERAL BIOLOGY TOPICS Genetics: Mendelian genetics: Mendel and his work. Definitions of the following terms: Allele, Phenotype, Genotype, Homozygous and Heterozygous. Principles of inheritance: Unit characters, dominance, law of segregation (purity of gametes) and law of independent assortment. Monohybrid cross, Dilhybrid cross and Test cross. Deviations from Mendelian laws: Incomplete dominance: Example - Flower colour in Mirabilis jalapa. Multiple allelism: Example - ABO blood groups and their inheritance in man: Blood typing; Rh factor with a note on erythroblastosis foetalis. Sex linked inheritance in man: Example - Inheritance of colourblindness and hypertrichosis in man. Genetic disorders in man: Chromosomal disorders - Down’s syndrome, Klinefelter’s syndrome, Turner’s syndrome and Cri-du-Chat syndrome. Gene disorders - Sickle cell anemia, haemophilia. Biodiversity: Definition and Types: Ecosystem or habitat diversity, Species diversity and Genetic diversity. Biodiversity profiles of India and Karnataka: Species diversity, Endemic species, Threatened species and Endangered species. Benefits of biodiversity: Economic - Traditional crop varieties and lesser known plants and animals of food value, medicinal plants harvested from wild habitat. Ecological/Social - For controlling soil - water regimes and hydrology, for efficient organic residue management and soil fertility management. Ethical - Cultural, Spiritual and Religious belief systems centred around the concept of sacred species, sacred groves and sacred landscapes. Biodiversity depletion: Anthropocentric causes - urbanization, expansion of agriculture, deforestation, pollution, acidification of soil and water, mining activities, desertification and loss of soil fertility. Intellectual property rights: Patenting life forms. Concept of ecosystem sustainability: Conservation of natural resources based on traditional ecological knowledge (TEK): Conservation of Water - rainwater harvesting and watershed management. Conservation of soil - Prevention of soil erosion and maintenance of soil fertility: methods of soil conservation. Conservation of forests - Afforestation and maintenance of biosphere reserves. Conservation of wild life - (i) Setting up of national parks, sanctuaries, bioreserves and zoos (ii) Habitat improvement. Global issues: Concept, causes, effects and control measures of the following: Global warming and greenhouse effect, Ozone layer depletion, Acid rain, Nuclear winter. BOTANY TOPICS Man in health and diseases: Concept of Homeostasis - The central Dogma in physiology: Definition. Meaning of internal environment. Factors to be kept constant to achieve homeostasis. An example to illustrate homeostasis - regulation of blood glucose level by liver and pancreas through negative feed back. A note on diabetes mellitus. Body defence and immunity: Introduction. Nonspecific body defences : a) Surface barriers b) Cellular and bio-chemical defences: phagocytosis, natural killer cells, interferons and inflammatory response. Specific body defences (immunity): Antigen and antibody, role of B and T lymphocytes. Types of immunity: Active (infection and vaccination) and Passive (from mother and immune serum Y-globulins). Digestion: Gross anatomy of human digestive system (structure of tooth not required). Components of food (concept of balanced diet). Physiology of digestion of carbohydrates, proteins and fats. Disorders: Causes, symptoms and prevention of hyperacidity and ulcer, jaundice and its types and hepatitis. Circulation: Introduction. Gross anatomy of the human heart. Mechanism of working of heart - cardiac cycle, stroke volume, cardiac out-put, complete double circulation. Origin and conduction of heart beat. Mechanism of blood clotting (Best and Taylor theory). Blood pressure - hypotension and hypertension. Disorders - causes and symptoms of myocardial infarction and cyanosis. Respiration: Gross anatomy of human respiratory system. Mechanism of respiration: (i) Breathing (inspiration and expiration) (ii) External respiration (exchange of oxygen and carbon dioxide between alveoli and blood) (iii) Internal respiration (exchange of oxygen and carbon dioxide between blood and body cells) (iv) Cellular respiration. Disorders: Rhinitis, Asthma and bronchogenic carcinoma. Artificial breathing. Excretion: Introduction. Gross structure of nephron, Physiology of urine formation. Chemical composition of urine. Disorders: a. Renal failure - acute and chronic b. Renal calculi. Kidney replacement therapy: a brief note on dialysis (haemodialysis and continuous ambulatory peritoneal dialysis) and kidney transplantation. Nervous system: Components - CNS, PNS & ANS. Human brain - structure (sagittal section only) and functions (functional areas of cerebrum not required). Human spinal cord - structure and functions. Meaning of reflex arc and reflex action. A brief study of the endocrine functions of the pituitary. Disorders: Meaning, causes and symptoms of epilepsy, Parkinson’s disease, Alzheimer’s disease and Huntington’s chorea. Alcoholism and its effects. Narcotic drugs - meaning, listing of types (stimulants, depressants, analgesics and hallucinogens) and their effects. Drug abuse and addiction, Efforts to counter alcoholism and drug menace Continuity of life: Developmental biology (basics of sexual reproduction) - Gametogenesis: Spermatogenesis - formation of spermatids and spermiogenesis (details of spermiogenesis are not required). Ultrastructure of human sperm. Oogenesis. Generalized structure of ovum. Fertilization - Definition. Types - external and internal. Mechanism. Significance. Early development of frog - Structure of egg. Cleavage. Blastulation. Gastrulation. Derivatives of primary germ layers. Human Reproduction: A brief account of Fertilization, Implantation, Placenta. Role of gonadotropins and sex hormones in males and females (meaning of menstrual cycle to be highlighted). Fertility control - Need for fertility control. Survey of family planning methods: Spacing methods (Barriers, IUDs, Hormonal and Physiological) and Terminal methods (Tubectomy and Vasectomy). Infertility control - Meaning and causes of infertility in males and females. Remedical methods (Assisted conception methods) - IVF,ET,GIFT and ZIET. (details of GIFT AND ZIFT not required). Sexually transmitted diseases - Meaning, causative organisms, mode of infection, symptoms and preventive measures of gonorrhoea, syphilis and AIDS. 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 probles. 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 Euclid'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. GENERAL ENGLISH Broadly, this paper includes questions on general English like spotting of errors, sentence improvement, vocabulary etc. Contact Details: Manipal Institute Of Technology Eshwar Nagar, Manipal, Karnataka 576104 India Map Location:
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