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Here i am giving you syllabus of the mathematics for MBA Entrance IP University as below:- Mathematics - Syllabus for PUC-I/XI Std. or equiv. ALGEBRA 1. THEORY OF INDICES AND LOGARITHMS • Recapitulation of theory of Indices - problems • Laws of logarithms (with proof) - problems 2. PROGRESSIONS • Recapitulation of sequences of real numbers, finite and infinite sequences as mappings. • Definition of infinite series, A.P., G.P., H.P,. nth term of an AP, GP, HP, sum to n terms of an AP, GP (with proof) - problems • Sum to infinity of a G.P. when the common ratio r is such that -1 < r < 1. Recurring decimal numbers - problems. • A.M., G.M., H.M. of two numbers a and b. Proofs of G2 = AH and A P G P H , where A, G H are the A.M., G.M., and H.M. respectively of any two numbers a and b. To insert n arithmetic means, n geometric means and n harmonic means between any two given numbers - problems 3. MATHEMATICAL INDUCTION • Principle of mathematical induction. Problems on induction including Sn, Sn2, Sn3 4. THEORY OF EQUATIONS • Recapitulation of quadratic equations and the formula for the roots of a quadratic equation. • The equation x2 + 1 = 0 and introducing complex numbers, square roots, cube roots and fourth roots of unity. • The relations between the roots and coefficients of a quadratic equation, a cubic equation and a biquadratic - equation. Solutions of quadratic, cubic and biquadratic equations given certain conditions and given that the roots are in A.P., G.P., H.P. - problems. • Symmetric functions of the roots of quadratic, cubic and biquadratic equations - problems. • Proofs of (i) irrational roots of a polynomial equation occur in conjugate pairs, (ii) complex roots of a polynomial equation occur in conjugate pairs - Problems of solving equations given an irrational root and given a complex root - problems. • Solution of a standard cubic equation X3 + 3HX + G = 0 by Cardan's method only - problems. 5. PERMUTATIONS AND COMBINATIONS • Definition of linear permutation, derivation of the formula for nPr from first principles. Formula for the number of permutations when some things are alike of one kind, etc. - problems • Circular permutation - formula - problems. • Definition of combination, derivation of the formula for nCr, from first principles. Proofs of nCr = nCn-r and nCr-1 + nCr = n+1Cr - problems 6. BINOMIAL THEOREM • Statement and proof of Binomial theorem for a positive integral index by induction. To find the middle terms, terms independent of x and term containing a definite power of x - problems. • Binomial coefficients - problems. 7. PARTIAL FRACTIONS • Rational fractions, proper and improper fractions, reduction of an improper fraction into a sum of a polynomial and a proper fraction - problems • Rules for resolving a proper fraction into partial fractions. - problems 8. ELEMENTS OF NUMBER THEORY AND CONGRUENCES • Divisibility - Definition and properties of divisibility, statement of Division Algorithm. • Greatest Common Divisor (G.C.D.) of any two integers, using Euclid,s Algorithm., to find the G.C.D. of any two integers. To express the G.C.D. of two integers a and b as ax + by for integers x and y - problems • Relatively prime numbers, prime numbers and composite numbers, the number of positive divisors of a number and sum of all positive divisors of a number - statements of the formulae without proof - problems. • Proofs of the following properties (1) The smallest divisor > 1 of an integer > 1 is a prime number. (2) There are infinity of 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 integer then either (p,a) = 1 or p | a (8) The smallest positive divisor of a composite number "a" does not axceed a • Congruence modulo m - Definition, Proofs of the following properties (1) "Lmode m" is an equivalence relation (2) a L b (mod m) => a ExLbEx (mod m) and ax L bx (mod m) (3) If c is relatively prime to m and ca Lcb (mod m) then a L b (mod m) - cancellation law (4) If a L b (mod m) and n is a positive divisor of m then a L b (mod n) (5) a L 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 L b (mod m) (statements only). to find the solution of ax L b (mod m) - problems ANALYTICAL GEOMETRY AND CALCULUS 9. COORDINATE GEOMETRY • Coordinate system in a plane (cartesian) • Distance formula, section formula, mid-point formula, centroid of a triangle, area of a triangle - Derivations, problems • Locus of a point, problems • Straight lines, slope of a line m = tanq where q 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 • Conditions for parallelism and perpendicularity of two lines - problems • Various forms of the equation of a straight line : slope - point form, slope - intercept form , two point form, intercept form, Normal form - Derivations - problems • 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 a point to a line, Equation of the inernal and external bisector of the angle between two lines - Derivations, problems • Pair of lines - Homogeneous equation of second degree, general equation of second degree, derivations of (1) condition for pair of lines, (2) condition for a pair of parallel lines, perpendicular lines and distance between the pair of parallel lines, (3) condition for a pair of coincident lines (4) angle and point of intersections of a pair of lines - problems. 10. CALCULUS • Functions of a real variable, types of functions, periodic functions, functional value - problems. • Limit of a function - definition, statements of the algebra of limits - problems • Standard limits (with proofs) (1) lim x -> a, x^n - a^n x - a =n a^(n-1) when n is rational (2) lim q -> 0, sin q = 1 when q is radians q (3) lim q -> 0, (tan q)/q= 1 when q is radians (4) Statements of the limits (i) lim n->infinty (1+ 1/n)^n = e (ii) lim x->0 (1 + x)^ 1/x = e (iii) lim x->0 loge (1 + x) /x = 1 (iv) lim x->0 e^x - 1 / x = 1 (v) lim x->0 a^x - 1 / x = loge a Problems on these limits Evaluation of limits if lim x -> 0 f (x) / g (x) OR 0/0 form lim n -> h f (n) / g (n) OR infinity/infinity form where degree of f(n) O degree g (n) problems TRIGONOMETRY 11. MEASUREMENT OF ANGLES AND TRIGONOMETRIC FUNCTIONS • Radian measure - Definition. Proofs of (i) p radians = 1800 (ii) 1 radian is constant (iii ) s = rq where q is in radians (iv) Area of the sector of a circle given by A=1/2 r2q where q is in radians - problems • Trigonometric functions - definitions. • Trigonometric ratios of an acute angle. • Trigonometric identities (with proofs), problems • Trigonometric functions of standard angles, problems. • Trigonometric functions of allied angles, compound angles, multiple angles, submultiple angles and Transformation formulae (with proofs) - problems. • Heights and distances - Angle of elevation, angle of depression, problems. • Graphs of Trigonometric functions 12. 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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