Dr. H. P. Sharma&

Yash Srivastava

By

Upkar Prakashan, AGRA-2

(According to the Syllabus)

Printed at : UPKAR PRAKASHAN (Printing Unit) Bye-pass, AGRA

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CONTENTS

● Previous Year’s Solved Paper

● Objective General Knowledge 1–64— History and Culture of India 3–11— Indian National Movement 12–13— Indian Polity 14–26— Indian Geography 27–31— World Geography 32–41— Indian Economy 42–53— Sports 54–56— Current Affairs 57–64

● Physics 1–152Unit 1 : Mechanics 3–30Unit 2 : General Properties of Matter 31–39Unit 3 : Heat 40–52Unit 4 : Optics 53–69Unit 5 : Wave Motion 70–80Unit 6 : Electricity 81–99Unit 7 : Electromagnetism 100–124Unit 8 : Modern Physics 125–142Unit 9 : Radioactivity and Nuclear

Physics 143–148Unit 10 : General Principles in the

Working of Some AppliancesInstruments etc. 149–152

● Chemistry 1–87Unit 1 : Physical and Chemical Changes 3–4Unit 2 : Elements, Mixtures and

Compounds 5–11Unit 3 : Laws of Chemical Combination 12–15Unit 4 : Atomic Structure 16–23Unit 5 : Oxidation and Reduction 24–31Unit 6 : Acids, Bases and Salts 32–36Unit 7 : Chemistry of Non-Metals

(Hydrogen, Oxygen & Nitrogen) 37–45Unit 8 : Some Important Chemical

Compounds 46–52Unit 9 : Carbon and its Different Forms 53–56Unit 10 : Fertilizers 57–60Unit 11 : Air and Water 61–64Unit 12 : Carbon Dioxide 65–67Unit 13 : Concept of Atomic, Molecular

and Equivalent Masses 68–73Unit 14 : Valency 74–78Unit 15 : Periodic Classification of

Elements 79–87

● Biology 1–56

● Objective English Language 1–56— Comprehension 3–17— English Usage and Grammar 18–21— Spotting Errors 22–30— Sentence Completion 31–37— Sequence of Words in a Sentence 38–42— Ordering of Sentences 43–44— Inappropriate Words/Expressions 45–47— Identification of Error 48–49— Idioms and Proverbs 50–53— Vocabulary (Synonyms & Antonyms) 54–56

● Mathematics 1–192Unit 1 : Sets Relations and Functions 3–10Unit 2 : Complex Numbers 11–19Unit 3 : Binary Numbers 20–22Unit 4 : Sequence and Series 23–31Unit 5 : Quadratic Equations and

inequations 32–37Unit 6 : Permutations and Combinations 38–42Unit 7 : Binomial Theorem 43–48Unit 8 : Logarithms 49–51Unit 9 : Matrices 52–60Unit 10 : Determinants 61–67Unit 11 : Angles and Trigonometric Ratios 68–74Unit 12 : Inverse Circular Functions 75–79Unit 13 : Heights and Distances 80–83Unit 14 : Properties of Triangles 84–89Unit 15 : Co-ordinate System and

Straight Lines 90–97Unit 16 : Circles 98–104Unit 17 : Conic Section (Parabola,

Ellipse, Hyperbola) 105–119

Unit 18 : Three Dimensional Geometry 120–124

Unit 19 : Limit’s Continuity andDifferentiability 125–131

Unit 20 : Differentiation 132–136

Unit 21 : Functions and Maxima-Minima 137–147

Unit 22 : Indefinite Integral 148–155

Unit 23 : Definite Integral and Area

under Curves 156–162

Unit 24 : Differential Equations 163–168

Unit 25 : Vector Algebra 169–176

Unit 26 : Probability 177–181

Unit 27 : Statistics 182–192

SYLLABUS

A. SCHEME OF THE EXAMINATION1. The subjects of the written examination, the time

allowed and the maximum marks allotted to each subjectwill be as follows :

SubjectSubject

CodeDuration

Max.

Marks

Mathematics 01 212 hours 300

General Ability Test 02 212 hours 600

Total 900

SSB Test/Interview 900

2. THE PAPERS IN ALL THE SUBJECTS WILL

CONSIST OF OBJECTIVE TYPE QUESTIONS ONLY. THE

QUESTION PAPERS (TEST BOOKLETS) OF MATHE-

MATICS AND PART ‘B’ OF GENERAL ABILITY TEST

WILL BE SET BILINGUALLY IN HINDI AS WELL AS

ENGLISH.

3. In the question papers, wherever necessary,questions involving the Metric System of Weights andMeasures only will be set.

4. Candidates must write the papers in their ownhand. In no circumstances will they be allowed the helpof a scribe to write the answers for them.

5. The Commission have discretion to fix qualifyingmarks in any or all the subjects at the examination.

6. The candidates are not permitted to use calculatorsor Mathematical or logarithmic table for answeringobjective type papers (Test Booklets). They should not,therefore, bring the same inside Examination Hall.

B. SYLLABUS OF THE EXAMINATIONPaper–I (Code No. 01)

MATHEMATICS(Maximum Marks–300)

1. Algebra :Concept of a set, operations on sets, Venn diagrams.

De Morgan laws. Cartesian product, relation, equivalencerelation.

Representation of real numbers on a line Complexnumbers—basic properties, modulus, argument, cuberoots of unity. Binary system of numbers. Conversion ofa number in decimal system to binary system and vice-versa. Arithmetic, Geometric and Harmonic progressions.

Quadratic equations with real coefficients. Solution oflinear inequations of two variables by graphs. Permutationand Combination. Binomial theorem and its applications.Logarithms and their applications.

2. 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 threeunknowns by Cramer’s rule and by Matrix Method.

3. Trigonometry :Angles and their measures in degrees and in radians.

Trigonometrical ratios. Trigonometric identities Sum anddifference formulae. Multiple and Sub-multiple angles.Inverse trigonometric functions. Applications—Heightand distance, properties of triangles.

4. Analytical Geometry of Two and ThreeDimensions :

Rectangular Cartesian Coordinate system. Distanceformula. Equation of a line in various forms. Anglebetween 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 threedimensional space, distance between two points.Direction Cosines and direction ratios. Equation of aplane and a line in various forms. Angle between twolines and angle between two planes. Equation of a sphere.

5. 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, Standardlimits—examples. Continuity of functions—examples,algebraic operations on continuous functions. Derivativeof function at a point, geometrical and physicalinterpretation of a derivative-applications. Derivatives ofsum, product and quotient of functions, derivative of afunction with respect to another function, derivative of acomposite function. Second order derivatives. Increasingand decreasing functions. Application of derivatives inproblems of maxima and minima.

6. Integral Calculus and Differential Equations :Integration as inverse of differentiation, integration

by substitution and by parts, standard integrals involving

( vii )

algebraic expressions, trigonometric, exponential andhyperbolic functions. Evaluation of definite integrals—determination of areas of plane regions bounded bycurves—applications. Definition of order and degree of adifferential equation, formation of a differential equationby examples. General and particular solution of adifferential equation, solution of first order and firstdegree differential equations of various types—examples.Application in problems of growth and decay.

7. Vector Algebra :Vectors in two and three dimensions, magnitude and

direction of a vector. Unit and null vectors, addition ofvectors, scalar multiplication of a vector, scalar productor dot product of two vectors. Vector product or crossproduct of two vectors. Applications-work done by aforce and moment of a force, and in geometricalproblems.

8. Statistics and Probability :Statistics : Classification of data. Frequency

distribution, cumulative frequency distribution-examples.Graphical representation—Histogram, Pie Chart,frequency polygon—examples. Measure of Centraltendency—Mean, median and mode. Variance andstandard deviation—determination and comparison.Correlation and regression.

Probability : Random experiment, outcomes andassociated sample space, events, mutually exclusive andexhaustive events, impossible and certain events. Unionand intersection of events. Complementary, elementaryand composite events. Definition of probability—classicaland statistical-examples. Elementary theorems onprobability— simple problems. Conditional probability,Bayes’ theorem-simple problems. Random variable asfunction on a sample space. Binomial distribution,examples of random experiments giving rise to Binomialdistribution.

Paper–II (Code No. 02)

GENERAL ABILITY TEST

(Maximum Marks–600)

PART ‘A’

ENGLISH

(Maximum Marks–200)

The question paper in English will be designed totest the candidate’s understanding of English and work-man-like use of words. The syllabus covers variousaspects like : Grammar and usage, vocabulary,comprehension and cohesion in extended texts to test thecandidate’s proficiency in English.

PART ‘B’

GENERAL KNOWLEDGE

(Maximum Marks–400)The question paper on General Knowledge will

broadly cover the subjects : Physics, Chemistry, GeneralScience, Social Studies, Geography and Current Events.

The syllabus given below is designed to indicate thescope of these subjects included in this paper. The topicsmentioned are not to be regarded as exhaustive andquestions on topics of similar nature not specificallymentioned in the syllabus may also be asked. Candidate’sanswers are expected to show their knowledge andintelligent understanding of the subject.

Section ‘A’ (Physics)Physical Properties and States of Matter, Mass,

Weight, Volume, Density and Specific Gravity, Principleof Archimedes, Pressure Barometer.

Motion of objects, Velocity and Acceleration,Newton’s Laws of Motion, Force and Momentum,Parallelogram of Forces, Stability and Equilibrium ofbodies, Gravitation, elementary ideas of Work, Powerand Energy.

Effects of Heat. Measurement of Temperature andHeat. Change of State and Latent Heat. Modes oftransference of Heat.

Sound waves and their properties, Simple musicalinstruments.

Rectilinear propagation of Light. Reflection andrefraction, Spherical mirrors and Lenses, Human Eye.

Natural and Artificial Magnets. Properties of aMagnet, Earth as a Magnet.

Static and Current Electricity. Conductors and Non-conductors, Ohm’s Law. Simple Electrical Circuits.Heating, Lighting and Magnetic effects of Current.Measurement of Electrical Power, Primary andSecondary Cells. Use of X-rays.

General Principles in the working of the following :

Simple Pendulum, Simple Pulleys, Siphon, Levers,Balloon, Pumps, Hydrometer, Pressure Cooker, ThermosFlask, Gramophone, Telegraphs, Telephone, Periscope,Telescope, Microscope, Mariner’s Compass, LightningConductors. Safety Fuses.

Section ‘B’ (Chemistry)

Physical and Chemical changes. Elements, Mixturesand Compounds, Symbols, Formulae and SimpleChemical Equations. Law of Chemical Combination(excluding problems). Properties of Air and Water.

Preparation and Properties of Hydrogen, Oxygen,Nitrogen and Carbondioxide, Oxidation and Reduction.

Acids, Bases and Salts.

Carbon–Different forms.

Fertilizers–Natural and Artificial.

Materials used in the preparations of substances likeSoap, Glass, Ink, Paper, Cement, Paints, Safety Matchesand Gunpowder.

Elementary ideas about the Structure of Atom,Atomic Equivalent and Molecular Weights. Valency.

Section ‘C’ (General Science)Difference between the living and non-living.

( viii )

Basis of Life—Cells Protoplasms and Tissues.Growth and Reproduction in Plants and Animals.Elementary knowledge of human Body and its

important organs.Common Epidemics, their causes and prevention.Food—Source of Energy for Man, Constituent of

food, Balanced Diet.The Solar System Meteors and Comets, Eclipses.Achievements of Eminent Scientists.

Section ‘D’ (History, Freedom Movement etc.)A broad survey of Indian History, with emphasis on

Culture and Civilisation.Freedom Movement in India.Elementary study of Indian Constitution and

Administration.Elementary knowledge of Five Year Plans of India.Panchayati Raj, Co-operatives and Community

Development.Bhoodan, Sarvodaya, National Integration and

Welfare State, Basic teachings of Mahatma Gandhi.Forces shaping the modern World; Renaissance

Exploration and Discovery. War of AmericanIndependence, French Revolution, Industrial Revolutionand Russian Revolution, Impact of Science andTechnology on Society. Concept of One World, UnitedNations Panchsheel, Democracy, Socialism andCommunism. Role of India in the Present World.

Section ‘E’ (Geography)The Earth, its shape and size, Latitudes and

Longitudes. Concept of Time, International Date line,Movements of Earth and their effects.

Origin of Earth, Rocks and their classification;Weathering—Mechanical and Chemical, Earthquakes andVolcanoes.

Ocean Current and Tides.Atmosphere and its composition; Temperature and

Atmospheric Pressure, Planetary winds, Cyclones andAnticyclones; Humidity; Condensation and Precipitation;Types of Climate. Major Natural regions of the World.

Regional Geography of India—Climate, NaturalVegetation. Mineral and Power resources; location anddistribution of agricultural and industrial activities.

Important Sea Ports and main sea, land and air routesof India. Main items of Imports and Exports of India.

Section ‘F’ (Current Events)

Knowledge of Important events that have happenedin India in the recent years.

Current important world events.

Prominent personalities—both Indian andInternational including those connected with culturalactivities and sports.

Note—Out of maximum marks assigned to Part ‘B’of this paper questions on Sections ‘A’, ‘B’, ‘C’, ‘D’, ‘E’and ‘F’ will carry approximately 25%, 15%, 10%, 20%,20% and 10% weightages respectively.

INTELLIGENCE AND PERSONALITY TEST

The SSB procedure consists of two stage Selectionprocess-stage-I and stage-II. Only those candidates whoclear the stage-I are permitted to appear for stage II. Thedetails are :

(a) Stage-I comprises of Officer Intelligence Rating(OIR) tests are Picture Perception *Description Test(PP & DT). The candidates will be shortlisted basedon combination of performance in QIR Test and PPand DT.

(b) Stage-II comprises of Interview, Group TestingOfficer Tasks, Psychology Tests and the Conference.These tests are conducted over 4 days. The details ofthese tests are given on the website www.joinin-dianarmy.nic.in.

The personality of a candidate is assessed by threedifferent assessors viz. the Interviewing Officer (IO),Group Testing Officer (GTO) and the Psychologist.There are no separate weightage for each test. The mksare allotted by assessors only after taking into considera-tion the performance of the candidate holistically in allthe test. In addition, marks for Conference are alsoallotted based on the initial performance of the Candidatein the three techniques and decision of the Board. Allthese have equal weightage.

The various tests of IO, GTO and Psych are designedto bring out the presence/absence of Officer LikeQualities and their trainability in a candidate.Accordingly candidates are Recommended or NotRecommended at the SSB.

National Defence AcademyExamination

Solved Paper

2013(Held on 11 August, 2013)

Mathematics

1. What is the seventh term of the sequence 0, 3, 8, 15,24, …… ?

(A) 63 (B) 48

(C) 35 (D) 33

2. What is tan 15° equal to ?

(A) 2 – √⎯ 3 (B) 2 + √⎯ 3

(C) 1 – √⎯ 3 (D) 1 + √⎯ 3

3. Consider the following—

1. tan ( )π6

2. tan ( )3π4

3. tan ( )5π4

4. tan ( )2π3

What is the correct order ?

(A) 1 < 4 < 2 < 3 (B) 4 < 2 < 1 < 3

(C) 4 < 2 < 3 < 1 (D) 1 < 4 < 3 < 2

4. If cosx = 13 , then what is

sin x · cot x · cosec x · tan x

equal to ?

(A)23

(B)32

(C) 2 (D) 1

5. What is log81 243 equal to ?

(A) 0·75 (B) 1·25

(C) 1·5 (D) 3

6. If the three vertices of the parallelogram ABCD areA(1, a), B(3, a), C(2, b), then D is equal to—

(A) (3, b) (B) (6, b)

(C) (4, b) (D) (5, b)

7. The value of k for which the lines 2x + 3y + a = 0and 5x + ky + a = 0, represent family of parallel linesis—

(A) 3 (B) 4·5

(C) 7·5 (D) 15

8. What is the equation of the line which passes through(4, –5) and is perpendicular to 3x + 4y + 5 = 0 ?

(A) 4x – 3y – 31 = 0

(B) 3x – 4y – 41 = 0

(C) 4x + 3y – 1 = 0

(D) 3x + 4y + 8 = 0

9. The foci of the hyperbola

4x2 – 9y2 – 1 = 0 are—

(A) ( )± 13 ‚ 0 (B) ⎝⎜⎛

⎠⎟⎞

± 136

‚ 0

(C) ⎝⎜⎛

⎠⎟⎞

0‚ ± 136

(D) None of these

10. For what value of k are the two straight lines 3x + 4y= 1 and 4x + 3y + 2k = 0 equidistant from the point(1, 1) ?

(A)12

(B) 2

(C) – 2 (D) – 12

11. A point P moves such that its distances from (1, 2)and (– 2, 3) are equal. Then the locus of P is—

(A) Straight Line (B) Parabola

(C) Ellipse (D) Hyperbola

12. The equation of the locus of a point which isequidistant from the axes is—

(A) y = 2x (B) x = 2y

(C) y = ±x (D) 2y + x = 0

13. What angle does the line segment joining (5, 2) and(6, – 15) subtend at (0, 0) ?

(A)π6

(B)π4

(C)π2

(D)3π4

14. The length of latus rectum of the ellipse 4x2 + 9y2 =36 is—

(A)43

(B)83

(C) 6 (D) 12

15. What is the equation to the straight line passingthrough (5, – 2) and (– 4, 7) ?

(A) 5x – 2y = 4 (B) – 4x + 7y = 9

(C) x + y = 3 (D) x – y = – 1

16. What is sin2 20° + sin2 70° equal to ?

(A) 1 (B) 0

(C) – 1 (D)12

NDA (II) 2013 | 3

17. What is the value of the determinant

⎪⎪⎪⎪⎪⎪

⎪⎪⎪⎪⎪⎪1 bc a(b + c)

1 ca b(c + a)1 ab c(a + b)

?

(A) 0 (B) abc(C) ab + bc + ca (D) abc (a + b + c)

18. Consider the following—1. A ∪ (B ∩ C) = (A ∩ B) ∪ (A ∩ C)2. A ∩ (B ∪ C) = (A ∪ B) ∩ (A ∪ C)Which of the above is/are correct ?(A) 1 only (B) 2 only(C) Both 1 and 2 (D) Neither 1 nor 2

19. Let n be a positive integer and(1 + x)n = a0 + a1x + a2x2 + …… + anxn

What is a0 + a1 + a2 + …… + an equal to ?

(A) 1 (B) 2n

(C) 2n – 1 (D) 2n + 1

20. If α, β are the roots of the equation ax 2 + bx + b = 0,

then what is the value of

αβ

+ βα

+ ba ?

(A) – 1 (B) 0

(C) 1 (D) 2

21. A number in binary system is 110001. It is equal towhich one of the following numbers in decimalsystem ?

(A) 45 (B) 46

(C) 48 (D) 49

22. The roots of the equation x2 – 8x + 16 = 0

(A) are imaginary (B) are distinct and real

(C) are equal and real (D) cannot be ascertained

23. What is tan–1 ( )12

+ tan–1 ( )13

equal to ?

(A)π2

(B)π3

(C)π4

(D)π6

24. What is (1 – sin2 θ) (1 + tan2 θ) equal to ?

(A) sin2 θ (B) cos2 θ(C) tan2 θ (D) 1

25. How many terms are there in the expansion of (1 +2x + x2)10 ?

(A) 11 (B) 20

(C) 21 (D) 30

26. The sum of the first five terms and the sum of thefirst ten terms of an AP are same. Which one of thefollowing is the correct statement ?

(A) The first term must be negative

(B) The common difference must be negative

(C) Either the first term or the common difference isnegative but not both

(D) Both the first term and the common differenceare negative

27. If A = {1, 3, 5, 7}, then what is the cardinality of thepower set P(A) ?

(A) 8 (B) 15

(C) 16 (D) 17

28. What is the difference in the roots of the equationx2 – 10x + 9 = 0 ?

(A) 2 (B) 3

(C) 5 (D) 8

29. If 8x – 9y = 20 and 7x – 10y = 9, then what is 2x – yequal to ?

(A) 10 (B) 11

(C) 12 (D) 13

30. The quadratic equation x2 + bx + 4 = 0 will have realroots if—

(A) b ≤ – 4 only (B) b ≥ 4 only

(C) – 4 < b < 4 (D) b ≤ – 4, b ≥ 4

31. What is the positive square root of 7 + 4√⎯ 3 ?

(A) √⎯ 3 – 1 (B) √⎯ 3 + 1

(C) √⎯ 3 – 2 (D) √⎯ 3 + 2

32. If A = {1, 2}, B = {2, 3} and C = {3, 4}, then whatis the cardinality of (A × B) ∩ (A × C) ?

(A) 8 (B) 6

(C) 2 (D) 1

33. If α, β are the roots of the equation x2 + x + 2 = 0,

then what is α10 + β10

α–10 + β–10 equal to ?

(A) 4096 (B) 2048

(C) 1024 (D) 512

34. If a and b are rational and b is not perfect square,then the quadratic equation with rational coefficients

whose one root is 3a + √⎯ b is—

(A) x2 – 6ax + 9a2 – b = 0

(B) 3ax2 + x – √⎯ b = 0

(C) x2 + 3ax + √⎯ b = 0

(D) √⎯ bx2 + x – 3a = 0

35. If A is a finite set having n elements, then thenumber of relations which can be defined in A is—

(A) 2n (B) n2

(C) 2n2 (D) nn

36. If the positive integers a, b, c and d are in AP, thenthe numbers abc, abd, acd, bcd are in—

(A) HP (B) AP

(C) GP (D) None of these

National Defence Academy Examination

Publisher : Upkar Prakashan ISBN : 9788174825087 Author : Dr.H.P.Sharma &Yash Srivastava

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