NDA/NANDA/NATOPIC-WISE SOLVED PAPERS

National DefenceAcademy/National Academy

FULLY SOLVED PAPERS

Typeset by Disha DTP Team

Compiled & Edited by Disha Experts

DISHA PUBLICATIONALL RIGHTS RESERVED

© Publisher

No part of this publication may be reproduced in any form without prior permission of the publisher. The author and the

publisher do not take any legal responsibility for any errors or misrepresentations that might have crept in. We have tried

and made our best efforts to provide accurate up-to-date information in this book.

• Head Office : B-32, Shivalik Main Road, Malviya Nagar, New Delhi-110017

• Sales Office : B-48, Shivalik Main Road, Malviya Nagar, New Delhi-110017

Tel. : 011-26691021 / 26691713

For further information about the books from DISHA

Log on to www.dishapublication.com or email to info@dishapublication.co.in

EB

D_9

059

Chapter 1 Set, Relation, Function

and Number System M-1-M-24

Chapter 2 Polynomial, Quadratic Equation

& Inequalities M-25-M-42

Chapter 3 Sequence and Series M-43-M-58

Chapter 4 Complex Numbers M-59-M-68

Chapter 5 Binomial Theorem,

Mathematical Induction M-69-M-74

Chapter 6 Permutation & Combination M-75-M-80

Chapter 7 Cartesian Co-ordinate System,

Straight Line M-81-M-94

Chapter 8 Pair of Straight Lines M-95-M-96

Chapter 9 Circles M-97-M-102

Chapter 10 Conics - Parabola, Ellipse

& Hyperbola M-103-M-110

Chapter 11 Trigonometry : Ratio & Identity,

Trigonometric Equation M-111-M-136

Paper - I : MATHEMATICSChapter 12 Properties of Triangle, Inverse

Trigonometric Function M-137-M-150

Chapter 13 Height & Distance M-151-M-160

Chapter 14 Functions, Limits, Continuity

and Differentiability M-161-M-178

Chapter 15 Derivative M-179-M-190

Chapter 16 Application of DerivativesM-191-M-202

Chapter 17 Indefinite Integration M-203-M-208

Chapter 18 Definite Integration &

its Applications M-209-M-220

Chapter 19 Differential Equation M-221-M-234

Chapter 20 Matrices & Determinants M-235-M-262

Chapter 21 Probability andProbability Distribution M-263-M-276

Chapter 22 Vectors M-277-M-302

Chapter 23 3D-Geometry M-303-M-318

Chapter 24 Statistics M-319-M-334

F NDA/NA Solved Paper 2016-II 2016-1-2016-34F NDA/NA Solved Paper 2016-I 2016-35-2016-72F NDA/NA Solved Paper 2015-II 1-36F NDA/NA Solved Paper 2015-I 2015-1-2015-40F NDA/NA Solved Paper 2014-I 2014-1-2014-34F NDA/NA Solved Paper 2014-II 2014-35-2014-70F NDA/NA Solved Paper 2013-II 2013-1-2013-30

CONTENT

PHYSICS

Chapter 1 Properties of Matter G-1-G-4

Chapter 2 Mechanics G-5-G-16

Chapter 3 Heat & Thermodynamics G-17-G-20

Chapter 4 Waves G-21-G-24

Chapter 5 Optics G-25-G-32

Chapter 6 Electricity & Magnetism G-33-G-44

Chapter 7 Modern Physics G-45-G-48

CHEMISTRY

Chapter 8 Basic Concepts of Chemistry,

Atomic Structure G-49-G-54

Chapter 9 Carbon and different forms,

Carbon Dioxide G-55-G-58

Chapter 10 Chemistry of Metal and

Non-metals (H2, O2 and N2) G-59-G-64

Chapter 11 Some Important

Chemical Compounds G-65-G-70

Chapter 12 Acids, Bases and Salts,

Oxidation and Reduction G-71-G-74

BIOLOGY

Chapter 13 Botany G-75-G-80

Chapter 14 Zoology G-81-G-90

HISTORY

Chapter 15 Ancient India G-91-G-94

Chapter 16 Medieval India G-95-G-96

Chapter 17 Modern India G-97-G-110

Chapter 18 World History G-111-G-112

GEOGRAPHY

Chapter 19 World & Physical GeographyG-113-G-128

Chapter 20 Geography of India G-129-G-142

POLITICAL SCIENCE

Chapter 21 Indian Polity G-143-G-150

Chapter 22 Indian Economy G-151-G-154

GENERAL KNOWLEDGE

Chapter 23 General Knowledge G-155-G-166

Chapter 24 Current Affairs G-167-G-172

Paper - II : GENERAL ABILITY TESTPART A : ENGLISH

Chapter 1 Synonyms E-1-E-10

Chapter 2 Antonyms E-11-E-20

Chapter 3 Fill in the Blanks E-21-E-22

Chapter 4 Spotting Errors E-23-E-32

Chapter 5 Selecting Words / Phrases E-33-E-34

Chapter 6 Ordering of Sentences E-35-E-42

Chapter 7 Ordering of Words in

a Sentence E-43-E-58

Chapter 8 Comprehension E-59-E-74

PART B : GENERAL KNOWLEDGE

EB

D_9

059

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

8. Statistics: Classification of data, Frequency distribution,cumulative frequency distribution - examples Graphicalrepresentation – Histogram, Pie Chart, Frequency Polygon -examples. Measures of Central tendency – mean, medianand mode. Variance and standard deviation - determinationand comparison. Correlation and regression.

9. Probability : Random experiment, outcomes and associatedsample space, events, mutually exclusive and exhaustiveevents, impossible and certain events. Union andIntersection of events. Complementary, elementary andcomposite events. Definition of probability - classical andstatistical - examples. Elementary theorems on probability- simple problems. Conditional probability, Bayes’ theorem- simple problems. Random variable as function on a samplespace. Binomial distribution, examples of randomexperiments giving rise to Binominal distribution.

PAPER - II : GENERAL ABILITYPart A - ENGLISHThe question paper in English will be designed to test thecandidate’s understanding of English and workman like use ofwords. The syllabus covers various aspects like Grammar andusage, vocabulary, comprehension and cohesion in extendedtext to test the candidate’s proficiency in English.Part B - GENERAL KNOWLEDGEThe question paper on General Knowledge will broadly coverthe subjects: Physics, Chemistry, General Science, Social Studies,Geography and Current Events.The syllabus given below is designed to indicate the scope ofthese subjects included in this paper. The topics mentioned arenot to be regarded as exhaustive and questions on topics ofsimilar nature not specifically mentioned in the syllabus mayalso be asked. Candidate’s answers are expected to show theirknowledge and intelligent understanding of the subject.

Section A - PhysicsPhysical Properties and States of Matter, Mass, Weight, Volume,Density and Specific Gravity, Principle of Archimedes, PressureBarometer. Motion of objects, Velocity and Acceleration, Newton’sLaws of Motion, Force and Momentum, Parallelogram of Forces,Stability and Equilibrium of bodies, Gravitation, elementaryideas of work, Power and Energy. Effects of Heat, Measurementof temperature and heat, change of State and Latent Heat, Modesof transference of Heat. Sound waves and their properties, Simplemusical instruments. Rectilinear propagation of Light, Reflectionand refraction. Spherical mirrors and Lenses. Human Eye. Naturaland Artificial Magnets, Properties of a Magnet, Earth as a Magnet.Static and Current Electricity, conductors and Non-conductors,Ohm’s Law, Simple Electrical Circuits, Heating, Lighting andMagnetic effects of Current, Measurement of Electrical Power,Primary and Secondary Cells, Use of X-Rays. General Principlesin the working of – Simple Pendulum, Simple Pulleys, Siphon,Levers, Balloon, Pumps, Hydrometer, Pressure Cooker, ThermosFlask, Gramophone, Telegraphs, Telephone, Periscope,Telescope, Microscope, Mariner ’s Compass; LighteningConductors, Safety Fuses.

The National Defence Academy (NDA) founded in 1957, is apremier Inter Service training institution where future cadetsare trained. The training involves an exacting schedule of 3 yearsbefore the cadets join their respective Service Academies, viz.Indian Military Academy, Naval Academy and Air Force Academy.The examination of the National Defence Academy is conductedby UPSC twice a year, after which the selected candidates aresent to National Defence Academy for training.

S Y L L A B U SPAPER - I : MATHEMATICS

1. Algebra : Concept of a set, operations on sets, Venndiagrams. De Morgan laws. Cartesian product, relation,equivalence relation. Representation of real numbers ona line. Complex numbers – basic properties, modulus,argument, cube roots of unity. Binary system of numbers.Conversion of a number in decimal system to binary systemand vice-versa. Arithmetic, Geometric and Harmonicprogressions. Quadratic equations with real coefficients.Solution of linear inequations of two variables by graphs.Permutation and Combination. Binomial theorem and itsapplication. Logarithms and their applications.

2. Matrices and Determinants : Types of matrices, operations onmatrices Determinant of a matrix, basic properties ofdeterminant. Adjoint and inverse of a square matrix,Applications – Solution of a system of linear equations intwo or three unknowns by Cramer’s rule and by Matrix Method.

3. Trigonometry : Angles and their measures in degrees andin radians. Trigonometrical ratios. Trigonometric identitiesSum and difference formulae. Multiple and Sub-multipleangles. Inverse trigonometric functions. Applications –Height and distance, properties of triangles.

4. Analytical Geometry of two and three dimensions : RectangularCartesian Coordinate system. Distance formula. Equationof a line in various forms. Angle between two lines. Distanceof a point from a line. Equation of a circle in standard andin general form. Standard forms of parabola, ellipse andhyperbola. Eccentricity and axis of a conic.

5. Differential Calculus : Concept of a real valued function -domain, range and graph of a function. Compositefunctions, one to one, onto and inverse functions. Notionof limit, Standard limits - examples. Continuity of functions- examples, algebraic operations on continuous functions.Derivative of a function at a point, geometrical and physicalinterpreatation of a derivative - applications. Derivativesof sum, product and quotient of functions, derivative of afunction with respect of 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 asinverse of differentiation, integration by substitution andby parts, standard integrals involving algebraic expressions,trigonometric, exponential and hyperbolic functions.Evaluation of definite integrals - determination of areas ofplane regions bounded by curves - applications. Definitionof order and degree of a differential equation, formation ofa differential equation by examples. General and particularsolution of a differential equation, solution of first orderand first degree differential equations of various types -examples. Application in problems of growth and decay.

NATIONAL DEFENCE ACADEMY EXAMINATION

Section B - ChemistryPhysical and Chemical changes. Elements, Mixtures andCompounds, Symbols, Formulae and simple Chemical Equations,Law of Chemical Combination (excluding problems). Propertiesof Air and Water. Preparation and Properties of Hydrogen, Oxygen,Nitrogen and Carbondioxide, Oxidation and Reduction. Acids,bases and salts. Carbon - different forms. Fertilizers - Naturaland Artificial Material used in the preparation of substances likesoap, Glass, Ink, Paper, Cement, Paints, Safety Matches, and Gun-Powder. Elementary ideas about the Structure of Atom, Atomic,Equivalent and Molecular Weights, Valency.

Section C - General ScienceDifference between the living and non- living. Basis of Life –Cells, Protoplasms and Tissues. Growth and Reproduction inPlants and Animals. Elementary knowledge of human Body andits important organs. Common Epidemics, their causes andprevention. Food – Source of Energy for man. Constituents offood, Balanced Diet. The Solar System – Meteors and Comets,Eclipses. Achievements of Eminent Scientists.

Section D - History, Freedom MovementA broad survey of Indian History, with emphasis on Culture andCivilisation. Freedom Movement in India. Elementary study ofIndian Constitution and Administration. Elementary knowledgeof Five Year Plans of India. Panchayati Raj, Co-operatives andCommunity Development. Bhoodan, Sarvodaya, NationalIntegration and Welfare State, Basic Teachings of MahatmaGandhi. 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 oneWorld, United Nations, Panchsheel, Democracy. Socialism andCommunism. Role of India in the present world.

Section E - GeoegraphyThe Earth, its shape and size. Lattitudes and Longitudes, Conceptof time. International Date Line. Movements of Earth and theireffects. Origin of Earth. Rocks and their classification; Weathering– Mechanical and Chemical, Earthquakes and volcanoes. OceanCurrents and Tides Atmosphere and its composition;Temperature and Atmospheric Pressure, Planetary Winds,cyclones and Anti-cyclones; Humidity; Condensation andPrecipitation; Types of Climate. Major Natural regions of theWorld. Regional Geography of India – Climate, Naturalvegetation. Mineral and Power resources; location anddistribution of agricultural and industrial activities. ImportantSea ports and main sea, land and air routes of India. Mainitems of Imports and Exports of India.

Section F - Current AffairsKnowledge of Important events that have happened in India inthe recent years. Current important world events. Prominentpersonalities – both Indian and International including thoseconnected with cultural activities and sports.

v HOW TO APPLY

Candidates are required to apply online by using thewebsite www.upsconline.nic.in. Brief instructions for fillingup the online application form name been given inadvertisement itself.

EXAMINATION PATTERNSTAGE I : Written Examination held by the UPSCThe subjects of the written examination, the time allowed and themaximum marks allotted to each subject will be as follows :

Subject Code Duration Maximum MarksMathematics 01 2-½ Hours 300General Ability Test 02 2-½ Hours 600

Total 900

NOTE : The papers in all the subjects will consist of objective typequestions only. The question papers (test booklets) of Mathematicsand Part-B of General Ability Tests will be set bilingually in Hindi aswell as English.

STAGE II : An interview by the Service Selection Board to assess acandidate’s officer potentialities (Intelligence and personalitytest) – 900 marks

INTELLIGENCE AND PERSONALITY TESTIn addition to the interview the candidates will be put toIntelligence Tests both verbal and non-verbal, designed toassess their basic intelligence. They will also be put to GroupTests such as group discussions, group planning, outdoor grouptasks, and asked to give brief lectures on specified subjects. Allthese tests are intended to judge the mental calibre of acandidate. In broad terms, this is really an assessment of notonly his intellectual qualities but also his social traits andinterests in current affairs.STAGE III : Medical Examination by the Service Medical Board.

TRAININGThe selected candidates for the three services, viz., Army, Navyand Air Force are given preliminary training both academic andphysical for a period of 3 years at the National Defence Academywhich is an Inter-Service Institution. The training during thefirst two and half years is common to the cadets of three wings.The cadets on passing out will be awarded B.Sc./B.A. degreefrom Jawaharlal Nehru University, Delhi. On passing out fromthe National Defence Academy, Army Cadets go to the IndianMilitary Academy, Dehra Dun, Naval Cadets to the CadetsTrainingship and Air Force cadets to AFA, HYDERABAD/BFTS,ALLAHABAD.At the I.M.A. Army Cadets are known as Gentlemen Cadets andare given strenuous military training for a period of one yearaimed at turning officer capable to leading infantry Sub-units.On successful completion of training Gentlemen Cadets aregranted Permanent Commission in the rank of Lt. subject to beingmedically fit in “SHAPE” one. The Naval cadets are selected forthe Executive, Engineering and Electrical Branches of the Navy,on passing out from the National Defence Academy and aregiven sea training.

EXAMINATION CENTRESAgartala, Ahmedabad, Aizawl, Allahabad, Bangalore, Bareilly,Bhopal, Chandigarh, Chennai, Cuttack, Dehradun, Delhi, Dharwar,Dispur, Gangtok, Hyderabad, Imphal, Itanagar, Jaipur, Jammu,Jorhat, Kochi, Kohima, Kolkata, Lucknow, Madurai, Mumbai,Nagpur, Panaji (Goa), Patna, Port Blair, Raipur, Ranchi, Sambalpur,Shillong, Shimla, Sringar, Thiruvananthapuram, Tirupati, Udaipur,Vishakhapatnam.

EB

D_9

059

DETAILED BREAKUP OF QUESTIONS (2007-16)PAPER-I : MATHEMATICS

CHAPTERS NAME I II I II I II I II I II I II I II I II I II I II

Set, Relation, Function and Number System

10 13 11 9 13 18 9 9 16 5 14 7 13 12 13 8 8 8 17 13

Polynomial, Quadratic Equation & Inequalities

11 11 6 9 7 7 7 5 11 11 7 3 8 7 2 2 5 5 6 7

Sequence and Series 3 10 4 9 4 5 6 3 7 7 7 7 2 3 2 5 2 2 5 9Complex Numbers 4 2 6 3 3 5 5 5 3 8 2 3 2 2 4 3 5 6 4 2Binomial Theorem, Mathematical Induction

2 2 4 1 1 2 1 4 1 2 2 0 1 2 4 5 1 3 1 2

Permutation and Combination 1 3 3 3 4 3 3 3 6 2 1 1 2 3 5 0 4 2 5 4

Cartesian Co-ordinate System, Straight Line

6 3 4 3 3 3 3 6 8 6 6 10 8 9 0 4 7 7 4 4

Pair of Straight Lines 0 0 0 1 0 0 0 0 0 0 0 1 0 5 0 1 2 1 0 2Circles 2 2 3 2 1 1 1 2 1 1 0 0 1 2 0 0 0 0 2 3Conics - Parabola, Ellipse & Hyperbola

2 2 1 3 3 2 5 2 2 2 2 2 2 4 5 5 2 2 4 2

Trigonometry : Ratio & Identity, Trigonometric Equation

10 8 10 7 13 14 7 12 4 10 19 8 9 7 11 11 5 6 8 8

Properties of Triangle, Inverse Trigonometric Function

3 6 5 3 5 4 5 2 7 6 5 1 2 1 4 6 9 4 5 4

Height & Distance 1 1 1 1 2 1 1 2 1 4 4 2 3 2 2 1 2 2 1 2Functions, Limits, Continuity and Differentiability

7 3 8 9 5 4 10 7 3 0 5 8 7 6 11 9 7 9 9 9

Derivative 0 3 7 2 5 8 4 4 1 2 3 4 6 4 0 2 2 2 5 5Application of Derivatives 7 4 3 5 7 4 6 5 3 6 5 3 2 3 5 4 8 6 4 3Indefinite Integration 2 1 2 3 3 2 4 3 3 1 2 3 2 3 0 0 4 6 1 2Definite Integration & its applications 3 5 2 3 2 3 3 6 4 4 3 5 6 7 5 6 8 5 3 3

Differential Equation 6 5 2 6 2 3 5 6 5 2 3 7 5 3 13 4 6 6 5 4Matrices and Determinants 8 11 11 7 10 8 9 9 8 7 11 8 7 9 6 10 7 8 11 11Probability and Probability Distribution

6 4 6 7 8 6 8 4 6 5 3 10 11 10 5 13 8 8 2 6

Vectors 9 6 9 10 7 7 8 8 8 6 7 8 6 6 2 10 7 7 12 63D-Geometry 6 7 5 7 6 6 8 2 5 4 5 8 7 1 13 7 4 8 2 8Statistics 7 6 6 7 6 6 4 10 7 19 5 11 8 9 8 4 7 7 4 5

2013 2014 2015 20162007 2008 2009 2010 2011 2012

PAPER-II : GENERAL ABILITY TEST

I II I II I II I II I II I II I II I II I II I IISynonym s 8 9 8 9 8 8 8 6 10 10 10 10 10 10 0 10 4 0 10 5Antonym s 8 9 8 9 8 8 8 6 10 10 10 10 10 10 4 10 4 4 11 5Fill in the Blanks 0 0 0 0 0 0 8 10 0 0 0 0 0 10 16 10 6 4 9 10Spotting Errors 8 5 7 5 11 8 8 10 10 15 10 10 10 10 23 20 30 23 20 23Selecting Words /Phrases 0 0 0 0 0 0 0 0 0 0 0 0 20 0 0 0 0 12 0 0Ordering of Sentences 8 8 9 0 7 0 0 0 0 0 0 0 0 6 0 0 6 0 0 7Ordering of Words in a Sente nce 8 8 8 8 7 6 8 6 10 8 10 12 0 0 0 0 0 0 0 0Com prehens ion 10 11 10 11 9 12 10 3 3 2 3 2 0 4 0 7 0 7 0 0

UNIT ‘I’ : PHYSICS 25 26 24 24 25 21 27 25 23 24 24 23 23 25 26 30 19 22 25 23Proper ties of M atte r 2 3 1 4 4 2 2 7 1 2 1 0 0 0 1 4 1 1 1 0M echanics 5 4 7 3 3 4 6 7 4 7 6 7 2 7 13 9 2 2 3 2Heat & Therm odynam ics 3 3 1 1 3 1 3 1 3 2 2 2 6 3 0 1 0 5 6 4Waves 3 2 1 4 3 3 5 1 2 3 2 2 2 1 0 1 4 3 1 3Optics 3 4 1 3 3 6 2 4 5 2 9 5 6 3 3 3 4 3 5 4Electr icity & M agnetism 7 7 8 7 6 4 6 5 5 7 4 6 6 9 7 7 4 3 3 2M odern Phys ics 2 3 5 2 3 1 3 0 3 1 0 1 1 2 2 5 4 5 6 8

UNIT ‘II’ : CHEM ISTRY 15 15 16 16 15 17 19 16 12 15 16 18 16 14 9 12 18 19 17 16Bas ic concepts of Chem is try, Atom ic Structure

7 4 9 4 4 5 6 3 1 3 1 5 7 3 2 6 9 8 5 3

Carbon and diffe re nt form s, Carbon dioxide

2 2 2 2 0 2 3 4 2 1 3 2 0 1 2 0 1 2 4 4Chem is try of M etals and Non M etals 2 4 0 2 3 3 5 3 6 4 7 5 5 3 2 3 0 1 1 3Som e im portant Che m ical Com pounds 3 3 5 5 7 5 4 3 3 2 4 3 0 3 2 2 4 4 5 3Acids , Bases and Salts , Oxidation and Reduction

1 2 0 3 1 2 1 3 0 5 1 3 4 4 1 1 4 4 2 3

UNIT ‘III’ : BIOLOGY 10 9 10 11 10 7 11 9 10 10 7 10 10 10 10 10 7 9 10 11Botany 4 4 5 4 4 0 5 5 3 2 3 7 4 0 2 5 2 5 7 7Zoology 6 5 5 7 6 7 6 4 7 8 4 4 6 10 8 5 5 4 3 4

UNIT ‘IV ’ : HISTORY 13 10 15 7 8 10 14 13 11 12 13 12 14 18 12 18 13 12 8 7Ancient India 1 2 1 0 2 2 1 4 2 4 2 2 1 0 0 0 4 3 4 2M edieval India 2 1 3 1 2 1 0 0 0 0 1 0 1 4 0 3 5 3 3 2M odern India 7 4 8 6 4 6 10 8 9 6 10 9 11 13 9 3 3 6 8 2World His tory 3 3 3 0 0 1 3 1 0 2 0 1 1 1 3 2 1 0 2 1

UNIT ‘V’ : GEOGRAPHY 19 20 19 20 12 18 20 19 18 19 18 17 12 18 17 16 9 20 19 19World & Phys ical Geography 12 9 7 16 7 13 16 12 13 13 12 7 8 17 16 15 6 11 5 5Geography of India 7 11 12 4 5 5 4 7 5 6 6 10 4 1 1 1 3 9 14 14

UNIT ‘VI’ : POLITICAL SCIENCE 4 8 4 10 12 11 5 6 9 4 4 7 7 0 5 2 3 5 10 10Indian Polity 3 6 2 7 7 7 5 6 3 4 3 5 4 0 5 1 3 5 6 4Indian Econom y 1 2 2 3 5 4 0 0 6 0 1 2 3 0 0 1 0 0 4 6

UNIT ‘VII’ : GENERAL KNOWLEDGE 14 11 12 13 8 10 11 12 11 9 12 13 11 12 22 24 23 13 13 14General Know ledge 5 8 6 9 7 7 11 7 10 2 8 4 5 4 15 15 13 5 8 8Current Affairs 9 3 6 4 1 3 0 5 1 7 4 9 6 8 7 9 10 8 5 6

2013 2014PART ‘A’ - ENGLISH

20162015

PART ‘B’ - GENERAL KNOWLEDGE

2007 2008 2009 2010 2011 2012

EB

D_9

059

8. Out of 15 points in a plane, n points are in the same straightline. 445 triangles can be formed by joining these points.What is the value of n?(a) 3 (b) 4(c) 5 (d) 6

9. If z = 107 107

3 i 3 i+ + – ,2 2 2 2

æ ö æ öç ÷ ç ÷ç ÷ ç ÷è ø è ø

then what is the

imaginary part of z equal to?

(a) 0 (b)12

(c)3

2(d) 1

10. If both the roots of the equation x2 – 2kx + k2 – 4 = 0 liebetween –3 and 5, then which one of the following iscorrect?(a) –2 < k < 2 (b) –5 < k < 3(c) –3 < k < 5 (d) –1 < k < 3

11. What is the number of distinct solutions of the equationz2 + |z| = 0(where z is a complex number)?(a) One (b) Two(c) Three (d) Five

12. How many geometric progressions is/are possible containing27, 8 and 12 as three of its/their terms?(a) One (b) Two(c) Four (d) Infinitely many

13. Let R be a relation from A = {1, 2, 3, 4} to B = {1, 3, 5} such thatR = [( a, b) :a < b, where a ÎA and bÎ B ]. What is RoR–1

equal to?(a) {(1, 3), (1, 5), (2, 3), (2, 5), (3, 5), (4, 5)}(b) {(3, 1), (5, 1), (3, 2), (5, 2), (5, 3), (5, 4)}(c) {(3, 3), (3, 5), (5, 3), (5, 5)}(d) {(3, 3), (3, 4), (4, 5)}

14. A five-digit number divisible by 3 is to be formed using thedigits 0, 1, 2, 3 and 4 without repetition of digits. What is thenumber of ways this can be done?(a) 96 (b) 48(c) 32 (d) No number can be formed

15. What is5

47 51 52– j4 3 3

j=2C + C + Cå equal to?

(a) 52C4 (b) 51C5(c) 53C4 (d) 52C5

MATHEMATICS

1. Let S be a set of all distinct numbers of the form PQ

, where

p, q 1, 2, 3, 4, 5, 6Î { } . What is the cardinality of the set S?(a) 21 (b) 23(c) 32 (d) 36

2. If c > 0 and 4a + c < 2b, then ax2 – bx + c = 0 has a root inwhich one of the following intervals?(a) (0, 2) (b) (2, 3)(c) (3, 4) (d) (–2, 0)

3. If 2A = {x R : x + 6x – 7 < 0}Î and2B = {x R : x +9x +14 > 0}Î , then which of the following

is/are correct?1. A B {X R : –2 < x < 1}Ç = Î2. A \ B = {x R : –7 < x < –2}ÎSelect the correct answer using the code given below:(a) 1 only (b) 2 only(c) Both 1 and 2 (d) Neither 1 nor 2

4. If A is semy square matrix of order 3 and det A= 5, then whatis det[(2A)–1] equal to?(a) 1/10 (b) 2/5(c) 8/5 (d) 1/40

5. What is w100 + w200 + w300 equal to, where w is the cuberoot of unity?(a) 1 (b) 3 w(c) 3 w2 (d) 0

6. If z –1Re = 0,z +1

æ öç ÷è ø

where 2 = x + iy is a complex number, then

which one of the following is correct?(a) z = 1+ i (b) z = 2

(c) z = 1 – i (d) z = 1

7. What is [x y z] a h gh b fg f c

é ùê úê úê úë û

equal to ?

(a) [ax + hy + gz h + b + f g+ f+ c]

(b)a h ghx by fzg f c

é ùê úê úê úë û

(c)ax + hy + gzhx + by + fzgx + fy + cz

é ùê úê úê úë û

(d) [ax+ hy+ gz hx+ by+ fz gx + fy+ cz ]

NDA/NA SOLVED PAPER 2016-II(Held on: 18 September, 2016)

2016-2 NDA/NA Solved Paper 2016-IIConsider the following for the next three (03) items that follow:

Let a, x, y, z, b be in AP, where x + y + z = 15. Let a, p, q, r, b be

in HP, where –1 –1 –1 5p + q + r =3

16. What is the value of ab?(a) 10 (b) 9(c) 8 (d) 6

17. What is the value of xyz?(a) 120 (b) 105(c) 90 (d) Cannot be determined

18. What is the value of pqr?(a) 35/243 (b) 81/35(c) 243/35 (d) Cannot be determined

Consider the following for the next two (02) items that follow:The sixth term of an AP is 2 and its common difference isgreater than 1.

19. What is the common difference of the AP so that the productof the first, fourth and fifth terms is greatest?(a) 8/5 (b) 9/5(c) 2 (d) 11/5

20. What is the first term of the AP so that the product of thefirst, fourth and fifth terms is greatest?(a) – 4 (b) – 6(c) – 8 (d) – 10

Consider the following for the next two (02) items that follow:

Let 3 2x +1 2x 3x

ax + bx + cx + d = 2x +3 x +1 x2 – x 3x + 4 5x –1

then

21. What is the value of c?(a) – 1 (b) 34(c) 35 (d) 50

22. What is the value of a + b + c + d ?(a) 62 (b) 63(c) 65 (d) 68

Consider the following for the next two (02) items that follow:The interior angles of a polygon of n sides are in AP. Thesmallest angle is 120° and the common difference is 5°.

23. How many possible values can n have?(a) One (b) Two(c) Three (d) Infinitely many

24. What is the largest interior angle of the polygon?(a) 160° only (b) 195° only(c) Either 160° or 195° (d) Nither 160 nor 195°

25. If m= 1 00 1

é ùê úë û

and n = 0 1–1 0

é ùê úë û

, then what is the value of the

determinant of m cosq – n sinq ?(a) – 1 (b) 0(c) 1 (d) 2

26. If f{x}=cos x – sinx 0sinx cosx 0 ,0 0 1

é ùê úê úê úë û

then which of the following are

correct ?

1. ( ) ( ) ( )f θ ×f = f θ+f f

2. The value of the determinant of the matrix f(q) ´ f(f) is 1.3. The determinant of f(x) is an even function.Select the correct answer using the code given below:(a) 1 and 2 only (b) 2 and 3 only(c) 1 and 3 only (d) 1, 2 and 3

27. Which of the following are correct in respect of the systemof equations x + y + z =8, x –y +2z = 6 and 3x –y +5z = k?1. They have no solution. if k = 15.2. They have infinitely many solutions, if k = 20.3. They have unique solution, if k = 25Select the correct answer using the code given below:(a) 1 and 2 only (b) 2 and 3 only(c) 1 and 3 only (d) 1, 2 and 3

28. A = 1 – 12 3

é ùê úë û

and B= 2 3

–1 – 2é ùê úë û

, then which of the following

is/are correct?

1. –1 –1AB(A B ) is a unit matrix.

2. ( )–1 –1 –1AB = A B

Select the correct answer using the code given below:(a) 1 only (b) 2 only(c) Both 1 only 2 (d) Neither 1 nor 2

29. If ( )xy nn n XZ y 4 n yz 2

x .y .z = yæ öæ öç ÷ç ÷è ø è ø

lll l for any x >1, y >1 and z >1,

then which one of the following is correct?(a) ln y is the GM of ln x, ln x, ln x and ln z(b) ln y is the AM of ln x, ln x, ln x and ln z(c) ln y is the HM of ln x, ln x and ln z(d) ln y is the AM of ln, In x, ln z and ln z

30. If the number 235 in decimal system is converted into binarysystem, then what is the resulting number ?(a) (11110011)2 (b) (11101011)2(c) (11110101)2 (d) (11011011)2

Consider the following for the next two (O2) items that follow:Let a and b be the roots of the equationx2 – (1 – 2a2) x + (1 – 2a2) = 0

31. Under what condition does the above equation have realroots ?

(a)2 1a <

2 (b) 2 1a >2

(c) 2 1a2

£ (d) 2 1a2

³

EB

D_9

059

NDA/NA Solved Paper 2016-II 2016-3

32. Under what condition is 2 2

1 1 < 1+a b

?

(a) 2 1a <2

(b) 2 1a >2

(c) 2a >1 (d) 2 1 1a ,3 2

æ öÎç ÷è ø

only

33. What is 21

1++ww

equal to, where w is the cube root of unity?

(a) 1 (b) w

(c) w2 (d) iw, where 1i = -34. In an examination, 70% students passed in Physics, 80%

students passed in Chemistry, 75% students passed inMathematics and 85% students passed in Biology, and x%students failed in all the four subjects. What is the minimumvalue of x ?(a) 10 (b) 12(c) 15 (d) None of the above

Consider the following for the next two (02) items that follow:For the system of linear equations 2x + 3y + 5z = 9, 7x + 3y –2z = 8 and 2x + 3y + lz = µ

35. Under what condition does the above system of equationshave infinitely many solutions ?(a) l = 5 and 9m ¹ (b) l =5 and 9m =

(c) l = 9 and 5m = (d) l = 9 and 5m ¹36. Under what condition does the above system of equations

have unique solutions?(a) l =5 and m = 9(b) l ¹ 9 and m = 7 only(c) l ¹ 5 and m has any real value(d) l has any real value and m ¹ 9

37. What is the number of odd integers between 1000 and 9999with no digit repeated?(a) 2100 (b) 2120(c) 2240 (d) 3331

38. What is the greatest value of the positive integer n satisfyingthe condition

n–11 1 1 1 11+ + + +...... + < 2 – ?2 4 8 10002

(a) 8 (b) 9(c) 10 (d) 11

Consider the following for the next two (02) items that follow:2x2 + 3x – a – 0 has roots –2 and b while the equation x2 – 3mx+ 2m2 = 0 has both roots positive, where a > 0 and b > 0.

39. What is the value of a ?(a) 1/2 (b) 1(c) 2 (d) 4

40. If b , 2, 2m are in GP, then what is the value of β m ?(a) 1 (b) 2(c) 4 (d) 6

41. sin A + 2 sin 2A + sin 3A is equal to which of the following?

1. 4 sin 2A 2 Acos2

æ öç ÷è ø

2. 2 sin 2A2A Asin + cos

2 2æ öç ÷è ø

3. 8 sin A cos A 2 Acos2

æ öç ÷è ø

Select the correct answer using the code given below:(a) 1 and 2 only (b) 2 and 3 only(c) 1 and 3 only (d) 1, 2 and 3

42. If 0 0x = sin70 .sin50 and 0 0y = cos60 .cos80 , then what is xyequal to ?(a) 1/16 (b) 1/8(c) 1/4 (d) 1/2

43. If sin q1 + sin q2 + sin q3 + sin q4 = 4, then what is the valueof cos q1 + cos q2 + cos q3 + cos q4 = ?(a) 0 (b) 1(c) 2 (d) 4

44. What is the value of

π 3π 5π 7π1+ cos 1+ cos 1+ cos 1+ cos ?8 8 8 8

æ öæ öæ öæ öç ÷ç ÷ç ÷ç ÷è øè øè øè ø

(a)12

(b)1 1

+2 2 2

(c)1 1

–2 2 2

(d) 18

45. If xcos q + ysin q = z, then what is the value of (x sin q – y cos q)2 ?

(a) 2 2 2x + y – z (b) 2 2 2x – y – z

(c) 2 2 2x – y + z (d) 2 2 2x + y + z46. What is the value of cos(2 cos–110.8) ?

(a) 0.81 (b) 0.56(c) 0.48 (d) 0.28

47. The top of a hill when observed from the top and bottom ofa building of height h is at angles of elevation p and qrespectively. What is the height of the hill?

(a)hcot q

cot q – cot p (b)hcot p

cot p – cot q

(c)2htan p

tan p – tan q (d) 2htan q

tanq – tan p

48. If sin 0 5 – 118 = ,

4then what is the value of sin 81° ?

(a) 3 + 5 + 5 – 54

(b) 3 + 5 + 5 + 54

(c) 3 – 5 + 5 – 54

(d) 3 5 5 54

+ - -

2016-4 NDA/NA Solved Paper 2016-II49. A moving boat is observed from the top of a cliff of 150 m

height. The angle of depression of the boat changes from60° to 45° in 2 minutes. What is the speed of the boat inmetres per hour?

(a)4500

3 (b) 4500( 3 –1)3

(c) 4500 3 (d) 4500( 3 +1)3

50. What is 0 0

0 0

1– tan 2 cot 62tan152 – cot 88 equal to?

(a) 3 (b) – 3

(c) 2 – 1 (d) 1 – 251. An equilateral triangle has one vertex at (0,0) and another at

(3, 3 ). What are the coordinates of the third vertex?

(a) (0, 2 3) only

(b) (3, 3)- only

(c) (0, 2 3) or (3, 3)-

(d) Neither (0, 2 3) nor (3, 3)-52. What is the equation of the right bisector of the line segment

joining (1, 1) and (2, 3)?(a) 2x + 4y –11= 0 (b) 2x – 4y – 5= 0(c) 2x – 4y –11= 0 (d) x– y + 1 = 0

53. What is the radius of the circle passing through the point(2, 4) and having centre at the intersection of the lines x – y= 4 and 2x + 3y + 7 = 0?(a) 3 units (b) 5 units

(c) 3 3 units (d) 5 2 units54. What is the equation of the hyperbola having latus rectum

and eccentrieity 8 and 35 respectively?

(a)2 2x y– = 1

25 20(b)

2 2x y– = 140 20

(c)2 2x y– = 1

40 30(d)

2 2x y– = 130 25

55. If the point (a, a) lies between the lines x + y = 2, thenwhich one of the following is correct?

(a) a < 2 (b) a < 2

(c) a < 1 (d)1a2

<

56. What is the equation of the straight line which passesthrough the point of intersection of the straight lines x + 2y = 5and 3x + 7y = 17 and is perpendicular to the straight line3x + 4y = 10?(a) 4x + 3y +2 = 0 (b) 4x – y+2 = 0(c) 4x – 3y – 2 =0 (d) 4x –3y +2 = 0

57. If (a, b) is at unit distance from the line 8x + 6y + 1 = 0, thenwhich of the following conditions are correct?1. 3a – 4b – 4 = 02. 8a + 6b + 11= 03. 8a + 6b – 9 = 0Select the correct answer using the code given below:(a) 1 and 2 only (b) 2 and 3 only(c) 1 and 3 only (d) 1, 2 and 3

58. If the ellipse 9x2 + 16y2 = 144 intercepts the line 3x + 4y = 12,then what is the length of the chord so formed?(a) 5 units (b) 6 units(c) 8 units (d) 10 units

59. A straight line cuts off an intercept of 2 units on the positivedirection of x-axis and passes through the point (–3, 5).What is the foot of the perpendicular drawn from the point(3, 3) on this line?(a) (1, 3) (b) (2, 0)(c) (0, 2) (d) (1, 1)

60. What is the eccentricity of rectangular hyperbola?(a) 2 (b) 3(c) 5 (d) 6

Consider the following for the next two (02) items that follow:Let Q be the image of the point P ( –2, 1, –5 ) in the plane3x –2y + 2z + 1 = 0.

61. Consider the following:1. The coordinates of Q are (4, –3, –1).2. PQ is of length more than 8 units.3. The point (1, –1, –3) is the mid-point of the line segment

PQ and lies on the given plane.Which of the above statements are correct?(a) 1 and 2 only (b) 2 and 3 only(c) 1 and 3 only (d) 1, 2 and 3

62. Consider the following:1. The direction ratios of the line segment PQ are <3, –2, 2 >.2. The sum of the squares of direction cosines of the line

segment PQ is unity.Which of the above statements is/are correct ?(a) 1 only (b) 2 only(c) Both1 and 2 (d) Neither 1 nor 2

Consider the following for the next two (02) items that follow:A line L passes through the point P(5, – 6, 7) and is parallelto the planes x + y + z = 1 and 2x – y – 2z = 3.

63. What are the direction ratios of the line of intersection of thegiven planes?(a) < 1, 4, 3 > (b) < –1, – 4, 3>(c) <1, – 4, 3> (d) <1, – 4, – 3>

64. What is the equation of the line L ?

(a)x – 5 y + 6 z – 7

= =–1 4 –3

(b)x + 5 y – 6 z + 7= =

–1 4 –3

(c)x – 5 y + 6 z – 7= =

–1 –4 3

(d)x – 5 y + 6 z – 7

= =–1 –4 –3

EB

D_9

059

NDA/NA Solved Paper 2016-II 2016-5Consider the following for the next two (02) items that follow:

Let a = i + j, b = 3i + 4kr r

ˆ ˆ ˆ ˆ and b = c + d,r r r

where cr

is

parallel to ar

and dr

is perpendicular to ar

.

65. What is cr

equal to?

(a)3(i + j)

2ˆ ˆ

(b)ˆ ˆ2(i + j)3

(c)ˆ ˆ(i + j)

2(d)

ˆ ˆ(i + j)3

66. If d = x i + y j + z k,r

ˆ ˆ ˆ then which of the following equationsis/are correct?1. y – x = 42. 2z – 3 = 0Select the correct answer using the code given below:(a) 1 only (b) 2 only(c) Both 1 and 2 (d) Neither 1 nor 2

Consider the following for the next two (02) items that follow:

Let a, br r

and cr

be three vectors such that a + b + c = 0r r r

and a = 10, b = 6r r

and cr

= 14.

67. What is a b + b c. + c a.. . .r r r r r ur

equal to?(a) –332 (b) – 166(c) 0 (d) 166

68. What is the angle between ar

and br

?(a) 30° (b) 45°(c) 60° (d) 75°

69. In a right-angled triangle ABC, if the hypotenuse AB = p,then what is AB AC + BC. BA + CA CB

uuur uuur uuur uuur uuur uuur

. . equal to?(a) p (b) p2

(c) 2p2 (d)2p

270. A force F = 3 i + 2 j 4 k-

rˆ ˆ ˆ is applied at the point (1, –1, 2).

What is the moment of the force about the point (2, –1, 3)?(a) + 4 j + 4 kˆ ˆ ˆi (b) 2 i + j + 2 kˆ ˆ ˆ

(c) 2 i – 7 j – 2 kˆ ˆ ˆ (d) 2 i + 4 j – kˆ ˆ ˆ

71. What is the domain of the function f(x) = 1 ?

x – x

(a) ( )0- ¥, (b) ( )0 ¥,(c) 0 < x < 1 (d) x > 1

72. Consider the following in respect of the function

f(x) = 2 + x, x 02 – x, x < 0

³ìíî

1.1

lim f ( )x

x®

does not exist.

2. f(x) is differentiable at x = 03. f(x) is continuous at x = 0Which of the above statements is/are correct?(a) 1 only (b) 3 only(c) 2 and 3 only (d) 1 and 3 only

73. Let f : A R® , where A = R\ {0} is such that f(x) = x + x

x.

On which one of the following sets is f(x) continuous?

(a) A (b) B = { }x R x 0Î ³:

(c) C = { }x R x 0Î £: (d) D = R74. Which one of the following statements is correct in respect

of the function 3f(x) = x sinx?(a) It has local maximum at x = 0.(b) It has local minimum at x = 0.(c) It has neither maximum nor minimum at x = 0.(d) It has maximum value 1.

75. What is the area bounded by the curves 2y = 1 – x ?

(a)43

square units (b)83

square units

(c) 4 square units (d)163

square units

Consider the following function for the next two (02) items thatfollow:

23 12 –1 1 2(x)37 , 2 3

ì + - £ £ï= í- < £ïî

x x xfx x

76. Which of the following statements is/are correct?1. f(x) is increasing in the interval [–1, 2].2. f(x) is decreasing in the interval (2, 3].Select the correct answer using the code given below:(a) 1 only (b) 2 only(c) Both 1 and 2 (d) Neither 1 nor 2

77. Which of the following statements are correct?1. f(x) is continuous at x = 2.2. f(x) attains greatest value at x = 2.3. f(x) is differentiable at x = 2.Select the correct answer using the code given below:(a) 1 and 2 only (b) 2 and 3 only(c) 1 and 3 only (d) 1, 2 and 3

Consider the following for the next three (03) items that follow:

Let f(x) = 2x – x –1é ùë û78. What is f '(x) equal to when x>1?

(a) 0 (b) 2x – 1(c) 4x – 2 (d) 8x – 4

79. What is f '(x) equal to when 0< x <1?(a) 0 (b) 2x – 1(c) 4x – 2 (d) 8x – 4

80. Which of the following equations is/are correct?1. f(–2) = f (5)2. f "(–2) + f "(0.5) + f "(3) = 4Select the correct answer using the code given below:(a) 1 only (b) 2 only(c) Both 1 and 2 (d) Neither 1 nor 2

2016-6 NDA/NA Solved Paper 2016-IIConsider the following for the next three (03) items that follow:

Let f (x) = [x], where [.] is the greatest integer function andg(x) = sin x be two real valued functions over R.

81. Which of the following statements is correct?(a) Both f(x) and g(x) are continuous at x = 0.(b) f(x) is continuous at x = 0, but g(x) is not continuous at

x = 0.(c) g(x) is continuous at x = 0, but f(x) is not continuous at

x = 0.(d) Both f(x) and g(x) are discontinuous at x = 0.

82. Which one of the following statements is correct?

(a)x 0lim (fog)(x)exists®

(b)x 0lim (gof)(x)exists®

(c)x 0+ x 0 –lim (fog)(x) = lim (gof)(x)® ®

(d)x 0+ x 0+lim (fog)(x)= lim (gof)(x)® ®

83. Which of the following statements are correct?1. (fof) (x) = f(x).2. (gog) (x) = g(x) only when x = 0.3. (go (fog)) (x) can take only three values.Select the correct answer using the code given below:(a) 1 and 2 only (b) 2 and 3 only(c) 1 and 3 only (d) 1, 2 and 3

Consider the following for the next two (02) items that follow:

Let f(x) =

xe –1, x > 0x

0, x = 0

ìïíïî

be a real valued function.

84. Which one of the following statements is correct?(a) f(x) is a strictly decreasing function in (0, x),(b) f(x) is a strictly increasing function in (0, x),(c) f(x) is neither increasing nor decreasing in (0, x)(d) f(x) is not decreasing in (0, x).

85. Which of the following statements is/are correct?1. f(x) is right continuous at x = 0.2. f(x) is discontinuous at x = 1.Select the correct answer using the code given below:(a) 1 only (b) 2 only(c) Both 1 and 2 (d) Neither 1 nor 2

Consider the following for the next two (02) items that follow:

Consider the parabola 2y = x + 7x + 2 and the straight liney = 3x – 3.

86. What are the coordinates of the point on the parabola whichis closest to the straight line?(a) (0, 2) (b) (–2, – 8)(c) (–7, 2) (d) (1, 10)

87. What is the shortest distance from the above point on theparabola to the line?

(a)102

(b)105

(c)110 (d)

54

Consider the following for the next three (03) items that follow:

Let f(x) = –2, –3 x 0x – 2, 0 < x 3

£ £ìí

£î and ( )g(x) = f x + f(x)

88. Which of the following statements is/are correct?1. g(x) is differentiable at x = 0.2. g(x) is differentiable at x = 2.Select the correct answer using the code given below:(a) 1 only (b) 2 only(c) Both 1 and 2 (d) Neither 1 nor 2

89. What is the value of the differential coefficient of g(x) at x = –2?(a) –1 (b) 0(c) 1 (d) 2

90. Which of the following statements are correct?1. g(x) is continuous at x = 0.2. g(x) is continuous at x = 2.3. g(x) is continuous at x = –1.Select the correct answer using the code given below:(a) 1 and 2 only (b) 2 and 3 only(c) 1 and 3 only (d) 1, 2 and 3

91. Let f(x) be a function such that 31f ' + x f '(x)x

æ öç ÷è ø

= 0, What is

1

1f(x)dx

-ò equal to?

(a) 2 f (1) (b) 0(c) 2 f (–1) (d) 4 f (1)

92. What is 4

2 4 2

x –1

x x + x +1ò dx equal to?

(a)4 2x + x +1 + c

x(b) 4

21x + 2 – + c

x

(c) 22

1x + +1 + cx

(d)4 2x – x +1 + c

x93. What are the degree and order respectively of the differential

equation satisfying 2 2y 1–x +x 1–y xe = ce ,

( )where c > 0, x < 1, y < 1 ?

(a) 1, 1 (b) 1, 2(c) 2, 1 (d) 2, 2

94. What is the curve which passes through the point (1,1) and

whose slope is 2y ?x

(a) Circle (b) Parabola(c) Ellipse (d) Hyperbola

95. If 2x dy = ydx + y dy, y > 0and y(1)= 1, then what isy (–3) equal to?(a) 3 only (b) –1 only(c) Both –1 and 3 (d) Neither –1 nor 3

EB

D_9

059

NDA/ NA 11 Years Topic-Wise SolvedPapers (2006 - 2016) 4th Edition

Publisher : Disha Publication ISBN : 9789386323217 Author : Disha Publication

Type the URL : http://www.kopykitab.com/product/11217

Get this eBook

25%OFF