16 ½ to 19 ½ , Unmarried male candidates only

#### PART-1

**Math syllabus****:**

**ALGEBRA:**

- Concept of set, operations on sets, Venn diagrams.
- De Morgan laws, Cartesian product, relation, equivalence relation.
- Representation of real numbers on a line.
- Complex numbers—basic properties, modulus, argument, cube roots of unity.
- Binary system of numbers.
- Conversion of a number in decimal system to binary system and vice-versa.
- Arithmetic, Geometric and Harmonic progressions.
- Quadratic equations with real coefficients.
- Solution of linear in equations of two variables by graphs.
- Permutation and Combination.
- Binomial theorem and its applications.
- Logarithms and their applications.

**MATRICES AND DETERMINANTS:**

- Types of matrices, operations on matrices.
- Determinant of a matrix, basic properties of determinants.
- Adjoint and inverse of a square matrix, Applications-Solution of a system of linear equations in two or three unknowns by Cramer’s rule and by Matrix Method.

**TRIGONOMETRY:**

- Angles and their measures in degrees and in radians.
- Trigonometrical ratios.
- Trigonometric identities Sum and difference formulae.
- Multiple and Sub-multiple angles.
- Inverse trigonometric functions.
- Applications-Height and distance, properties of triangles.

**ANALYTICAL GEOMETRY OF TWO AND THREE DIMENSIONS:**

- Rectangular Cartesian Coordinate system. Distance formula.
- Equation of a line in various forms.
- Angle between two lines.
- Distance of a point from a line.
- Equation of a circle in standard and in general form.
- Standard forms of parabola, ellipse and hyperbola.
- Eccentricity and axis of a conic.
- Point in a three-dimensional space, distance between two points.
- Direction Cosines and direction ratios.
- Equation two points.
- Direction Cosines and direction ratios.
- Equation of a plane and a line in various forms.
- Angle between two lines and angle between two planes.
- Equation of a sphere.

**DIFFERENTIAL CALCULUS:**

- Concept of a real valued function–domain, range and graph of a function.
- Composite functions, one to one, onto and inverse functions.
- Notion of limit, Standard limits—examples.
- Continuity of functions—examples, algebraic operations on continuous functions.
- Derivative of function at a point, geometrical and physical interpretation of a derivative—applications.
- Derivatives of sum, product and quotient of functions, derivative of a function with respect to another function, derivative of a composite function.
- Second order derivatives. Increasing and decreasing functions.
- Application of derivatives in problems of maxima and minima.

**INTEGRAL CALCULUS AND DIFFERENTIAL EQUATIONS:**

- Integration as inverse of differentiation, integration by substitution and by parts, standard integrals involving algebraic expressions, trigonometric, exponential and hyperbolic functions.
- Evaluation of definite integrals—determination of areas of plane regions bounded by curves— applications. Definition of order and degree of a differential equation, formation of a differential equation by examples.
- General and particular solution of a differential equations, solution of first order and first-degree differential equations of various types—examples.
- Application in problems of growth and decay.

**VECTOR ALGEBRA:**

- Vectors in two and three dimensions, magnitude and direction of a vector.
- Unit and null vectors, addition of vectors, scalar multiplication of a vector, scalar product or dot product of two vectors.
- Vector product or cross product of two vectors.
- Applications—work done by a force and moment of a force and in geometrical problems.

**STATISTICS AND PROBABILITY:**

**Statistics:**

- Classification of data, Frequency distribution, cumulative frequency distribution—examples.
- Graphical representation—Histogram, Pie Chart, frequency polygon—examples.
- Measures of Central tendency—Mean, median and mode.
- Variance and standard deviation—determination and comparison.
- Correlation and regression.

**Probability:**

- Random experiment, outcomes and associated sample space, events, mutually exclusive and exhaustive events, impossible and certain events.
- Union and Intersection of events.
- Complementary, elementary and composite events.
- Definition of probability—classical and statistical— examples.
- Elementary theorems on probability—simple problems.
- Conditional probability, Bayes’ theorem—simple problems.
- Random variable as function on a sample space.
- Binomial distribution, examples of random experiments giving rise to Binominal distribution.

##

PART-2

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**General Ability Syllabus**

The general ability section comprises of **two parts** which are mentioned below:

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**(A)**

**English:** The English syllabus covers the topics like vocabulary, Grammar and usage, comprehension and cohesion in extended text to test the candidate’s proficiency in English.

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**(B)**

**General Knowledge:**

The question paper comprises of general knowledge and covers the subjects that include Physics, Chemistry, Social Studies, General Science, Geography and Current Events.

** **

**Section A – Physics**

Motion of objects, Newton’s Laws of Motion, Force and Momentum, Parallelogram of Forces, Stability and Equilibrium of bodies, Gravitation, elementary ideas of work, Velocity and Acceleration, Power and Energy, Sound waves and their properties, Simple musical instruments, Natural and Artificial Magnets, Properties of a Magnet, Earth as a Magnet, Siphon, Levers, Balloon, Pumps, Hydrometer, Pressure Cooker, Thermos Flask, Gramophone, Telegraphs, Telephone, Periscope, Telescope, Microscope, Mariner’s Compass; Lightening Conductors, Safety Fuses, Simple Pendulum. Simple Pulleys, Physical Properties and States of Matter, Mass, Weight, Volume, Density and Specific Gravity, Principle of Archimedes, Pressure Barometer, Effects of Heat, change of State and Latent Heat, Modes of transference of Heat, Rectilinear propagation of Light, Reflection and refraction. Spherical mirrors and Lenses. Human Eye, Static and Current Electricity, Measurement of temperature and heat, conductors and Non-conductors, Ohm’s Law, Simple Electrical Circuits, Heating, Lighting and Magnetic effects of Current, Measurement of Electrical Power, Primary and Secondary Cells, Use of X-Rays.

**Section B – Chemistry**

Preparation and Properties of Hydrogen, Oxygen, Nitrogen and Carbon-Di-oxide, Oxidation and Reduction, Material used in the preparation of substances like soap, Glass, Ink, Paper, Cement, Paints, Safety Matches, and Gun-Powder, Carbon – different forms, Physical and Chemical changes. Symbols, Formulas and simple Chemical Equations, Law of Chemical Combination (excluding problems). Properties of Air and Water, Acids, bases and salts, Elements, Mixtures and Compounds, Fertilizers – Natural and Artificial, Elementary ideas about the Structure of Atom, Atomic, Equivalent and Molecular Weights, Valency.

**Section C – Biology**

Basis of Life – Cells, Protoplasm’s and Tissues, Constituents of food, Balanced Diet, Achievements of Eminent Scientists, Difference between the living and non-living, Elementary knowledge of human Body and its important organs, Food – Source of Energy for man, Growth and Reproduction in Plants and Animals, Common Epidemics, their causes and prevention, The Solar System – Meteors and Comets, Eclipses.

**Section D – History (10-15 questions)**

Freedom Movement in India, Bhoodan, Sarvodaya, National Integration and Welfare State, Basic Teachings of Mahatma Gandhi, A broad survey of Indian History, with emphasis on Culture and Civilisation, Elementary knowledge of Five Year Plans of India, Elementary study of Indian Constitution and Administration, Panchayati Raj, Forces shaping the modern world; Renaissance, Exploration and Discovery; War of American Independence, French Revolution, Industrial Revolution and Russian Revolution, Impact of Science and Technology on Society, Concept of one World, United Nations, Panchsheel, Democracy, Socialism and Communism, Co-operatives and Community Development, Role of India in the present world.

**Section E – Geography (15-20 questions)**

Origin of Earth, Weathering – Mechanical and Chemical, Earthquakes and volcanoes, Atmosphere and its composition, Temperature and Atmospheric Pressure, Rocks and their classification, Planetary Winds, cyclones and Anti-cyclones, Humidity, Condensation and Precipitation, Major Natural regions of the World, Important Sea ports and main sea, land and air routes of India. Main items of Imports and Exports of India, Types of Climate, The Earth, its shape and size, Lattitudes and Longitudes, Concept of time, International Date Line, Movements of Earth and their effects, Ocean Currents and Tides, Regional Geography of India – Climate, Natural vegetation. Mineral and Power resources, location and distribution of agricultural and industrial activities.

**Section F – Indian polity (5-8 questions)**

**Section G – Economy (5-10 questions)**

**Section G– Current Events (10-15 questions)**

Current important world events, Knowledge of Important events that have happened in India in the recent years, prominent personalities – both Indian and International including those connected with cultural activities and sports.

It is not compulsory that the topics mentioned above will surely come in the examination. The question paper does not remain consistent, it changes every year. It may happen that the question paper will consists of the topic not discussed here.

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Written Exam conducted by UPSC SSB Interview conducted by Army/Air Force/Navy

For Army/Navy/Air Force-3 Years at National Defence Academy & 1 Year at IMA/AFA/NA For Naval Academy-4 Years B.Tech Course at Naval Academy