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Maths Syllabus-1
Engineering Entrance , MANIPAL UGET (Engg)
Mathematics Syllabus (Manipal UGET) Test :
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) log a m+log a n = log a (mn)
(b) log a m - log a n = log a (m/n)
(c) log a mn = n log a m
(d) log b m = logam/log a b (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
(1) In a graph with p vertices and q edges
(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 = (tan?) of a line, where ? 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:
TRIGONOMETRY
Measurement of Angles and Trigonometric Functions
Radian measure - definition, Proofs of:
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.
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