**ASSOCIATE DEGREE IN SCIENCE**

**A-Course of Mathematics**

**Paper-I**

**NOTE:** Each section having four questions, please attempt two questions from each section.

**SECTION-I (4/12: 17,17,17,17)**

Theory of limit and continuity. Solution of Inequalities. Derivatives and its application to business, economics and physics etc. Differentials. Related rates. Higher order derivatives. Leibnitz’s theorem. Limits and continuity of functions of two variables. Partial differentiation and its geometrical meaning for functions of two variables. Euler’s theorem. Increments and differentials. Chain Rule. Extrema by 2^{nd} order derivative test and by Lagrange multiplier method. General theorems and indeterminate forms. L’ Hospital rule and its applications. Increasing and decreasing functions. Intermediate value theorem and its immediate consequence (only statements)

**SECTION-II (4/12: 16,16,16,16)**

Translation and rotation of axes. Second degree equation with reference to conic section. Properties of conics. Tangents and normals (Cartesian Coordinates), Polar equations of conics. Sketching of Curves in polar coordinates, Tangents and normals (Polar Coordinates). Parametric representation of curves. Pedal Equations. Vector spaces and sub spaces. Linearly dependent and independent vectors. Bases and dimension. Linear transformations and matrix of linear transformation. (relevant theorems of bases and linear transformation without proofs) .

** SECTION-III (4/12: 17,17,17,17)**

Sequences. Bounded Sequences. Cauchy sequences. Convergence and divergence of sequences. Cauchy’s theorem. Nth-term test, comparison test, ratio test, root test and integral test for convergence and divergence of infinite series. Convergence and divergence of alternating series. Power series. Complex numbers and their properties. De moivre’s theorem and its applications. Circular, logarithmic and hyperbolic functions. Separation into real and imaginary parts.

**Recommended Books**

- Calculus by H.Anton. John Wiley and Sons New York. (Latest Edition)
- Calculus By C.H Edwards and D.E. Penney. Prentiee Hall. Ine. (Latest Edition)
- Calculus By S.I. Grossman. Academic Press Ine (London) Ltd. (Latest Edition)
- Calculus and Analytic Geometry by S.M. Yousaf. Illmi Kitab Khana. Urdu Bazar Lahore 5
^{th}Edition 1997 - Calculus and analytic geometry by G.B Thomas and R.I. Finney., Adison-Wesley Publishing Company. Lahore. (Latest Edition)
- Elementary Linear Algebra by C.H. Edwards. Jr and Davide penney. Prentic Hall international Ine. (1988)
- Mathematical Techniques by K. H. Dar. Irfan-ul-Haq and M.A. Jajja. The Carvan Book House. Kachehry Road Lahore. 9
^{th}Edition 1997 - Mathematics Methods by S.M. Yousaf. Illmi Kitab Khana. Urdu Bazar Lahore. (2000)
- Set Theory and Logic by Stoll, Robert R.S. Chand & Co. New Delhi
- Number Theory by Dr. Manzoor Hussain. The Carvan Book House. Kachehry Road, Lahore.
- Elementary Linear Algebra (sixth edition) by Howard Anton And Chris Rorres. John Willey & Sons. INC. 10
^{th }Edition 2010.

**ASSOCIATE DEGREE IN SCIENCE**

**A-Course of Mathematics**

**Paper–II**

**NOTE:** Each section having four questions, please attempt two questions from each section.

**SECTION-I (4/12: 17,17,17,17)**

Antiderivatives and indefinite integrals. Methods of integration. Definite integral as limit of sum. Fundamental theorem. Properties. Improper integrals. Reduction formulas. Double and triple integral (simple cases). Area between curves. Length of arc. Intrinsic equations. Asymptotes. Extrema and its application. Singular points. Curvature. Evolute and envelopes. Volume and surfaces of revolution.

**SECTION-II (4/12: 16,16,16,16)**

Definition and examples of metric spaces. Open and closed balls and sets. Neighborhoods. Limit points. Interior, exterior and boundary sets. Closure of a set. Complete metric spaces. Definition and examples of topological spaces. Basic properties. Neighborhoods. Limit points. Interior, exterior and boundary sets. Closure of a set. Divisibility. Euclid theorem. Greatest divisor. Least common multiple. Prime factorization theorem. Introduction to elementary logic. Predicate calculus. Methods of proofs.

**SECTION-III (4/12: 17,17,17,17)**

Definition and examples of a group. Order of an element of a group. Subgroup. Cyclic and permutation groups. Lagrange’s theorm. Rings and fields. Algebra of matrices. Co-factors, minors, adjoint and inverse of a matrix. Elementary row and column operations. Echelon form and rank of matrix. Solution of the system of linear equations(Homogeneous and non-homogeneous) by use of matrices. Net work flow problems. Determinants with properties.

**Recommended Books**

- Calculus by H.Anton. John Wiley and Sons New York. (Latest Edition)
- Calculus By C.H Edwards and D.E. Penney. Prentiee Hall. Ine. (Latest Edition)
- Calculus By S.I. Grossman. Academic Press Ine (London) Ltd. (Latest Edition)
- Calculus and Analytic Geometry by S.M. Yousaf. Illmi Kitab Khana. Urdu Bazar Lahore 1998
- Calculus and analytic geometry by G.B Thomas and R.I. Finney, Adison-Wesley Publishing Company. Lahore. (Latest Edition)
- Elementary Linear Algebra by C.H. Edwards. Jr and Davide penney. Prentic Hall international Ine. (1988)
- Mathematical Techniques by K. H. Dar. Irfan-ul-Haq and M.A. Jajja. The Carvan Book House. Kachehry Road Lahore. 9
^{th}Edition 1997 - Mathematics Methods by S.M. Yousaf. Illmi Kitab Khana. Urdu Bazar Lahore. (2000)
- Set Theory and Logic by Stoll, Robert R.S. Chand & Co. New Delhi
- Number Theory by Dr. Manzoor Hussain. The Carvan Book House. Kachehry Road, Lahore.
- Elementary Linear Algebra (sixth edition) by Howard Anton And Chris Rorres. John Willey & Sons. INC. 10
^{th}Edition 2000

**ASSOCIATE DEGREE IN SCIENCE**

**B-Course of Mathematics**

**Paper –I**

**NOTE:** Each section having four questions, please attempt two questions from each section.

**SECTION-I (4/12: 17,17,17,17)**

Vectors in three-dimensions. Scalar and vector products with applications. Scalar and vector triple products. Differentiation and integration of vector functions. Gradient, divergence and curl. Differential operators. Application to vector analysis. Composition and resolution of co-planar forces. Theorem, Lamy’s Theorem, Varignon’s Theorem, Moments, couples and conditions of equilibrium under the action of co-planar forces.

**SECTION-II (4/12: 16,16,16,16)**

Types of forces, Direction of forces of constraints, Equilibrium of three co-planer forces and related problem. Center of gravity. Symmetry and Center of mass, Center of mass of various bodies. Frictional forces. Laws of friction. Equilibrium of bodies on rough surfaces. Principle of virtual work and related problems.

**SECTION-III (4/12: 17,17,17,17)**

Kinematics of a particle in Cartesian and polar co-ordinates. Laws of mechanics. Linear and angular velocity. Relative velocity. Rectilinear motion with uniform and variable acceleration. Simple harmonic motion. Projectile motion. Motion along horizontal and vertical circles. Orbital motion. Elliptic orbit under a central force, Polar form of the orbit, Apse and Apsidal Distance, Planetary motion and Keplar’s laws.

**Recommended Books.**

- Theory of Differential Equations of Dennis G.Zill. Books Thomson Learning Academic Resource Center. USA. 8
^{th}Edition 2013 - Mathematical Techniques by K.H. Dar, Irfan-ul-Haq and M.A. Jajja. The Carvan Books House. Kachehry Road, Lahore. 9
^{th}Edition 1997 - Mathematics Methods by S.M. Yousaf. Illmi Kitab Khana. Urdu Bazar, Lahore. 1998
- Numerical Analysis by R.L. Burden and J.D. Faires. PES-Kent Publishing Company. Bostan. USA 9
^{th}Edition 2011 - Operations Research by H.A. Taha. Prentice-hall Inc. Englewood. Cliffs USA.4
^{th}Edition 2011 - Mathematical Statistics. By Dr. J.E.Freund. Prentice-hall Inc. Englewood. Cliffs USA. 7
^{th}Edition 2007 - Vector and Tensor Methods, by Chorlton, Ellis Horwood Publishers.1976
- Elementary Vector Analysis. By Dr. Munawar Hussain. S.M. Hafeez. M.A. Saeed and Ch. Bashir Ahmed. The Caravan Book House, Kachhry Road , Lahore.
- A Text Book by Dynamics by Chorlton, Van Nostrand Company Ltd. London. 1967
- Mechanics by O.K. Ghori. West Pakistan Publishing Company, Lahore. 1971

**ASSOCIATE DEGREE IN SCIENCE**

**B-Course of Mathematics**

**Paper –II**

**NOTE:** Each section having four questions, please attempt two questions from each section.

**SECTION-I (4/12: 17,17,17,17)**

Basic concepts of differential equations. Classification and formation of DEs. Various methods of solutions of first order ODE (linear and non-linear). The Bernoulli’s, Ricatti and Clairaut’s equations. Singular solutions. Orthogonal trajectories. Application of first order ODE in problems of decay and growth, population dynamics, logistic equations. Linear DE of higher order (homogeneous and non-homogeneous). Solution by: D-operator and undetermined co-officients Methods. Reduction of order and variation of parameters methods for 2^{nd} order linear DE. Cauchy-Euler equation. Power series solution about an arbitrary point.

**SECTION-II (4/12: 16,16,16,16)**

Laplace Transformation, solution of ODEs. Error analysis. Solution of non-linear (algebraic and transcendental) equation in one variable using bisection method, false position method, Newton – Raphson method and fixed point method. Difference operators. Interpolation (Newton’s and Lagrange’s methods). Numerical differentiation (at a point of the data). Numerical integration (rectangular, trapezoidal and 1/3 Simpson’s rules).

**SECTION-III (4/12: 17,17,17,17)**

Co-ordinates in three dimension. Rectangular, cylindrical and spherical co-ordinates. Equations of plane, straight line, sphere, cylinder, cone, ellipsoid, hyperboloid and paraboloid. Longitude and latitudes. Spherical triangle and direction of Qibla. Inner Product Space, Eigen values and Eigen vectors, Dignalization of matrices.

**Recommended Books.**

- Theory of Differential Equations of Dennis G.Zill. Books Thomson Learning Academic Resource Center. USA.8
^{th}Edition 2013 - Mathematical Techniques by K.H. Dar, Irfan-ul-Haq and M.A. Jajja. The Carvan Books House. Kachehry Road, Lahore.9
^{th}Edition 1997 - Mathematics Methods by S.M. Yousaf. Illmi Kitab Khana. Urdu Bazar, Lahore. 1998
- Numerical Analysis by R.L. Burden and J.D. Faires. PES-Kent Publishing Company. Bostan. USA 9
^{th}Edition 2011 - Operations Research by H.A. Taha. Prentice-hall Inc. Englewood. Cliffs USA.4
^{th}Edition 2011 - Mathematical Statistics. By Dr. J.E.Freund. Prentice-hall Inc. Englewood. Cliffs USA. 7
^{th}Edition 2007 - Vector and Tensor Methods, by Chorlton, Ellis Horwood Publishers. 1976
- Elementary Vector Analysis. By Dr. Munawar Hussain. S.M. Hafeez. M.A. Saeed and Ch. Bashir Ahmed. The Caravan Book House, Kachhry Road , Lahore.
- A Text Book by Dynamics by Chorlton, Van Nostrand Company Ltd. London.1967
- Mechanics by O.K. Ghori. West Pakistan Publishing Company, Lahore. 1971

**ASSOCIATE DEGREE IN ARTS / SCIENCE**

**General Mathematics**

## Paper –I

**NOTE:** Each section having four questions, please attempt two questions from each section.

**SECTION-I (4/12: 17,17,17,17)**

Theory of limit and continuity. Solution of inequalities. Derivatives and its application to business, economics and physics etc. Differentials. Related rates. Newton-Raphson formula. Higher order derivatives. Leibnitz’s theorem. Limits and continuity of functions of two variables. Partial differentiation and its geometrical meaning for functions of two variables. Euler’s theorem. Increments and differentials. Chain Rule. General theorems (without proofs) and indeterminate forms. L’ Hospital rule of functions. Increasing and decreasing functions.

**SECTION-II (4/12: 16,16,16,16)**

Translation and rotation of axes. Second degree equation with reference to conic section. Properties of conics. Polar equations of conics. Tangents and normals. Parametric representation of curves. Pedal Equations. Asymptotes. Extrema and its application. Singular points. Curvature. Evolutes and envelopes.

**SECTION-III (4/12: 17,17,17,17)**

Antiderivatives and indefinite integrals. Methods of integration. Definite integral as limit of sum. Fundamental theorem. Properties. Improper integrals. Reduction formulas. Double and Triple integral (Simple cases). Area between curves. Length of arc. Intrinsic equations. Numerical integration (rectangular, trapezoidal and 1/3 Simpson’s rules). Co-ordinates in three dimension. Rectangular, cylindrical and spherical co-ordinates. Equations of plan, straight line, sphere, cylinder, cone, ellipsoid, hyperboloid and paraboloid. Longitude and latitudes. Spherical triangle and direction of Qibla.

**Note: **A student has to take B-course of Mathematics (both I & II- papers) as an additional course to be eligible to do M.Sc. mathematics as a regular or a private candidate.

**Recommended Books**

- Calculus by H. Anton. John Wiley and Sons New York. (Latest Edition)
- Calculus By C.H Edwards and D.E. Penney. Prentiee Hall. Ine. (Latest Edition)
- Calculus By S.I. Grossman. Academic Press Ine (London) Ltd. (Latest Edition)
- Calculus and Analytic Geometry by S.M. Yousaf. Illmi Kitab Khana. Urdu Bazar Lahore 1998
- Calculus and analytic geometry by G.B Thomas and R.I. Finney., Adison-Wesley Publishing Company. (Latest Edition)
- Elementary Linear Algebra by C.H. Edwards. Jr and Davide penney. Prentic Hall international Ine. 1988
- Mathematical Techniques by K. H. Dar. Irfan-ul-Haq and M.A. Jajja. The Carvan Book House. Kachehry Road Lahore. 9
^{th}Edition 1997 - Mathematics Methods by S.M. Yousaf. Illmi Kitab Khana. Urdu Bazar Lahore. 2000
- Set Theory and Logic by Stoll. Robert R.S chand & Co. New Delhi.
- No Theory by Dr. Manzoor Hussai. The Carvan Book House, Kachachry, Lahore
- Elementary Liner Algebra by Howard Anton and Chris Rorres John Willey & Sons Inc 10
^{th}Edition 2010

**ASSOCIATE DEGREE IN ARTS / SCIENCE**

**General Mathematics**

**Paper–II**

**NOTE:** Each section having four questions, please attempt two questions from each section.

**SECTION-I (4/12: 17,17,17,17)**

Vectors in three-dimensions. Scalar and vector products with applications. Scalar and vector triple products. Differentiation and integration of vector functions. Gradient, divergence and curl. Sequences. Bounded Sequences. Convergence and divergence of sequences. nth-term test, comparison test, ratio test, root test and integral test for convergence and divergence of infinite series. Convergence and divergence of alternating series. Complex numbers and their properties. De moivre’s theorem and its applications. Circular, logarithmic and hyperbolic functions. Separation into real and imaginary parts.

**SECTION-II (4/12: 16,16,16,16)**

Algebra of matrices. Determinants with properties. Co-factors, minors, adjoint and inverse of a matrix. Elementary row and column operations. Echelon form and rank of matrix. Solution of the system of linear equations(Homogeneous and non-homogeneous) by use of matrices. Network flow problems. Vector spaces and sub spaces. Linearly dependent and independent vectors. Bases and dimension. Linear transformation. Matrix of linear transformation.(relevant theorems of bases and linear transformation with out proofs)

**SECTION-III (4/12: 17,17,17,17)**

Basic concepts of differential equations. Classification and formation of DEs. Various methods of solutions of first order ODE (linear and non-linear). The Bernoulli’s, Ricatti and Clairaut’s equations. Singular solutions. Orthogonal trajectories. Application of first order ODE in problems of decay and growth, population dynamics, logistic equations. Linear DE of higher order (homogeneous and non-homogeneous). Solution by: D-operator and undetermined co-officients Methods. Reduction of order and variation of parameters methods for 2^{nd} order linear DE. Cauchy-Euler equation.

Recommended Books

- Calculus by H. Anton. John Wiley and Sons New York. (Latest Edition)
- Calculus By C.H Edwards and D.E. Penney. Prentiee Hall. Ine. (Latest Edition)
- Calculus By S.I. Grossman. Academic Press Ine (London) Ltd. (Latest Edition)
- Calculus and Analytic Geometry by S.M. Yousaf. Illmi Kitab Khana. Urdu Bazar Lahore 1998
- Calculus and analytic geometry by G.B Thomas and R.I. Finney., Adison-Wesley Publishing Company. (Latest Edition)
- Elementary Linear Algebra by C.H. Edwards. Jr and Davide penney. Prentic Hall international Ine. 1988
- Mathematical Techniques by K. H. Dar. Irfan-ul-Haq and M.A. Jajja. The Carvan Book House. Kachehry Road Lahore. 9
^{th}Edition 1997 - Mathematics Methods by S.M. Yousaf. Illmi Kitab Khana. Urdu Bazar Lahore. 2000
- Set Theory and Logic by Stoll. Robert R.S chand & Co. New Delhi.
- Number Theory by Dr. Manzoor Hussai. The Carvan Book House, Kachachry, Lahore
- Elementary Liner Algebra by Howard Anton and Chris Rorres John Willey & Sons Inc 10
^{th}Edition 2010

**ASSOCIATE DEGREE IN ARTS**

** ****MATHEMATICS (OPTIONAL)**

** (Outlines of Course)**** **

**Paper: (Algebra ½ and Calculus ½)** **Marks = 100**

**i) Algebra (½ )**** **

Introduction to Number System, Algebra of Matrices and determinants of order 2×2, 3×3, Addition, subtraction of matrices, scalar multiplication and multiplication of matrices, singular and non-singular matrices, adjoint and inverse of matrices.

Solution of equations by using matrices and Crarner’s Rule.

**Determinants:**

Minor and co-factor of an element of a matrix in its determinants. Determinant of a square matrix of order 2×2 and 3×3 properties of determinants.

** ****Quadratic Equations:**

Some polynomial functions, Remainder and Factor Theorem, Synthetic Division, Relation between Roots and co-efficients of a quadratic equation. System of two equations involving two variables. Arithmetic, Geometric and harmonic sequences. Arithmetic Series, Geometric Series, Infinite Geometric Series.

**Trigonometry:**

Fundamentals of Trigonometry and Trigonometric identities.

**ii) Calculus and Analytic Geometry (½)**** **

Limit, Continuity (Simple Cases), Differentiation and Integration (Simple Cases), Differential equations (Variables separable). Introduction of a vector. Simple cases of dot and cross product of vectors.

**Note:**

Mathematics (Optional) means Mathematics General (Optional), A-Course of Mathematics (Optional) or B-Course of Mathematics (Optional).