There are two parts of the SLET written test, and candidates need to attempt both Paper 1 and Paper 2 to crack the exam. Paper 1 covers general subjects, while Paper 2 comprises the core subject chosen by the candidates. For the students who wish to select Mathematics and Science, we have covered all the topics and issues under Mathematics and Science in the list below.

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**SLET Syllabus****: Mathematical Science**

**Basic Concepts of Real and Complex Analysis**

- Sequences and Series
- Continuity
- Uniform Continuity
- Differentiability
- Residues
- Contour Integration
- Mean Value Theorem
- Sequences and Series of Functions
- Uniform Convergence
- Riemann Integral-definition and Simple Properties
- Algebra of Complex Numbers
- Analytic Functions
- Cauchy’s Theorem and Integral Formula
- Power Series
- Taylor’s and Laurent’s Series

**2. Basic Concepts of Linear Algebra**

- Space of n-vectors
- Quadratic Forms
- Characteristic Roots and Vectors
- Linear Dependence
- Basic, Linear Transformation
- Algebra of Matrices
- The Rank of a Matrix
- Determinants
- Linear Equations

**3. Basic Concepts of Probability**

- Sample Space
- Discrete and Continuous Random Variables
- Binomial, Poisson and Normal Distributions
- Expectation and Moments
- Independence of Random Variables
- Chebyshev’s Inequality
- Discrete Probability
- Simple Theorem on Probability
- Independence of Events
- Bayes Theorem

**4. Linear Programming Basic Concepts**

- Convex Sets
- Extreme Point and Graphical Method
- Linear Programming Problem
- Hyperplane, Open and Closed Half-spaces
- Feasible, Essential Feasible, and Optimal Solutions

**5. Real Analysis**

- Finite, Countable, and Uncountable Sets
- Bounded and Unbounded Sets
- Functions of Bounded Variation
- Elements of Metric Spaces
- Archimedean Property
- Ordered Field
- Completeness of R
- Extended Real Number System
- Lump and Limit of a Sequence
- The Epsilon-delta Definition of Continuity and Convergence
- The Algebra of Continuous Functions, Monotonic Functions
- Types of Discontinuities
- Infinite Limits and Limits at Infinity

**6. Complex Analysis**

- Riemann Sphere and Stereographic Projection
- Zero- sets of Analytic Functions
- Exponential
- Sine and Cosine Functions
- Power Series Representation
- Classification of Singularities
- Conformal Mapping
- Lines, Circle Cross Ratio
- Mobius Transformations
- Analytic Functions
- Cauchy-Riemann Equations
- Line Integrals
- Cauchy’s Theorem
- Morera’s Theorem
- Liouville’s Theorem
- Integral Formula

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**7. Algebra**

- Group, Subgroups, Normal Subgroups, Quotient Groups, Homomorphisms, Cyclic Groups, Permutation Groups
- Cayley’s Theorem
- Rings, Ideals, Integral Domains, Fields, Polynomial Rings

**8. Linear-Algebra**

- Vector Spaces, Subspaces, Quotient Spaces
- Linear Independence
- Bases, Dimension
- Invariant Subspaces
- Canonical Forms; Diagonal Form, Triangular Form, Jordan Form, Inner Product Spaces
- The Algebra of Linear Transformations, Kernel, Range, Isomorphism
- Matrix Representation of a Linear Transformation, Change of Bases
- Linear Functionals, Dual Space, Projection, Determinant Function
- Eigenvalues and Eigenvectors
- Cayley-Hamilton Theorem

**9. Differential Equations**

- First-order ODE
- Singular Solutions
- PDE’s of Higher Order with Constant Coefficients
- Initial Value Problems of First Order ODE
- The General Theory of Homogenous and Non-homogeneous Linear ODE
- Variation of Parameters
- Lagrange’s and Charpit’s Methods of Solving First-order Partial Differential Equations

**10. Data Analysis Basic Concepts**

- Graphical Representation
- Measures of Central Tendency and Dispersion
- Bivariate Data, Correlation, and Regression
- Least Squares-polynomial Regression
- Application of Normal Distribution

The SLET Syllabus for mathematical science is a vast subject that covers many topics, and the ones mentioned above are some of its subtopics. It also includes Probability, Probability Distribution, Theory of Statistics, Statistical Methods and Data Analysis, Operational Research Modelling, Linear Programming, Finite Population, Design of Experiments, and much more.

**SLET Syllabus****: Science**

The SLET syllabus for science subjects is further divided into four parts mentioned below.

**Chemical Science**

- Structure and Bonding
- Acids and Bases
- Redox Reactions
- Introductory Energetics and Dynamics of Chemical Reactions
- Aspects of S, P, D, F Block Elements
- IUPAC Nomenclature of Simple Organic and Inorganic Compounds
- Concept of Chirality
- Common Organic Reactions and Mechanisms
- Elementary Principles and Applications of Electronic, Vibrational, NMR, EPR, Mossbauer, and Mass Spectral Techniques to Simple Structural Problems
- Data Analysis

**2. Life Science**

- Cell Biology
- Biochemistry
- Physiology
- Genetics
- Evolutionary Biology
- Environmental Biology
- Biodiversity and Taxonomy

**3. Physical Science**

- Basic Mathematical Methods
- Classical Dynamics
- Electromagnetics
- Quantum Physics and Applications
- Thermodynamics and Statistical Physics
- Experimental Techniques, Measurement of Fundamental Physical Constants, Temperature, Pressure Humidity Sensors, Photon and Particle Detectors

**4. Environmental Science**

- Fundamentals of Environmental Sciences
- Environmental Chemistry
- Environmental Biology
- Environmental Assessment, Management, and Legislation
- Statistical Approaches and Modelling in Environmental Sciences
- Contemporary Environmental Issues
- Environmental Geosciences
- Energy and Environment
- Environmental Pollution and Control
- Solid and Hazardous Waste Management

Candidates must know about the subjects they have chosen as core subjects for Paper 2, and they can learn about this with the help of the SLET syllabus. The list mentioned above was a rundown of all the topics under the SLET mathematics and science syllabus. For a detailed syllabus, candidates can download the entire SLET syllabus for mathematics and science in PDF from the official website.

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