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Manipal UGET

# Mathematics Syllabus Part 2 Manipal University Under Graduate Entrance Test:

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 va

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

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.
(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

MATRICES AND DETERMINANTS:

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 = 4ax ; 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=4ax 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 ellipse x2/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:

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.

Back to Manipal UGET Mathematics syllabus-I .

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