UPKAR PRAKASHAN, AGRA 2

Transcription

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2 Objective Type Questions (According to the Revised Syllabus) Editorial Board Competition Science Vision 016 UPKAR PRAKASHAN, AGRA

3 Other Useful Books Upkar s N.D.A. Exam. 499/- (By : Dr. H.P. Sharma & Yash Srivastava) Upkar s N.D.A. Exam. (By : Jain & Gupta) 375/- Upkar s Practice Work-Book NDA Exam. 85/- Upkar s Upkar s N.D.A. Mathematics NDA Solved Papers 185/- 10/- Upkar s New Pattern Test of Objective English 175/- Upkar s Correct English How to Write It 40/- Upkar s Ever Latest General Knowledge 170/- Upkar s General Knowledge Overview with Current Affairs 60/- Publishers Publishers UPKAR PRAKASHAN (An ISO 9001 : 000 Company) /11A, Swadeshi Bima Nagar, AGRA 8 00 Phone : , , Fax : (056) , care@upkar.in, Website : Branch Offices : 4845, Ansari Road, Daryaganj, New Delhi Phone : /66 8, Chowdhury Lane, Shyam Bazar, Near Metro Station, Gate No. 4 Kolkata (W.B.) Phone : Pirmohani Chowk, Kadamkuan, Patna Phone : B-33, Blunt Square, Kanpur Taxi Stand Lane, Mawaiya, Lucknow (U.P.) Phone : /B, R.R. Complex (Near Sundaraiah Park, Adjacent to Manasa Enclave Gate), Bagh Lingampally, Hyderabad (A.P.) Phone : The publishers have taken all possible precautions in publishing this book, yet if any mistake has crept in, the publishers shall not be responsible for the same. This book or any part thereof may not be reproduced in any form by Photographic, Mechanical, or any other method, for any use, without written permission from the Publishers. All disputes shall be subject to the jurisdiction of courts at Agra. ISBN : Price : (Rs. Five Hundred Seventy Only) Code No. 319 Printed at : Upkar Prakashan (Printing Unit) Bye-pass, AGRA

4 Contents Previous Years Papers Fully Solved MATHEMATICS Algebra Sets 3. Relation 8 3. Complex Numbers Arithmetic Progression Geometric Progression 5 6. Harmonic Progression Miscellaneous Series Permutations and Combinations Quadratic Equations Binomial Theorem Binary Number System Representation of Real Numbers on a line Linear Inequations in two variables 64 Matrices and Determinants 7 85 Trigonometry Identities and Trigonometric Ratios 86. Simple Identities Properties of Triangles Inverse Trigonometrical Functions Height and Distance 109 Coordinate Geometry Rectangular cartesian coordinates and straight lines 116. The Circle The Parabola The Ellipse The Hyperbola Geometry of Three Dimensions The Plane The Sphere 15 Differential Calculus Function 156. Limit and Continuity Differentiation Increasing and Decreasing, Maxima and Minima 194 Integral Calculus and Differential Equations Indefinite Integrals 10. Definite Integrals 3 3. Differential Equations Problems on applications of differential Equation growth and Decay 46 Vector Algebra Statistics and Probability Frequency Distribution, mean, median, mode and standard deviation 64 Graphical Representation Histogram, Frequency Polygon and Pie chart 74. Correlation and Regression Probability 84 GENERAL ENGLISH 1. Common Error 34 Articles, Nouns, Pronouns Adjectives, Adverbs, Adverbial order 8 Verb, Infinitive, Verbal noun, Gerund, Participle 13 Conjunctions, Prepositions 19 Miscellaneous Sentences 4. Antonyms Synonyms Sentence Completion One Word Substitution Comprehension Passage Completion Completion of Paragraphs and Sentences 81 88

5 ( iv ) GENERAL KNOWLEDGE History and Culture 3 17 Indian Polity and Constitution Indian National Movement Geography Geography of India 55 World Geography 63 Indian Economy International Organisation Books and Authors 8 85 Awards Sports Final Population Results Census of India Physics Measurement and Dimensional Analysis 3 7. Rectilinear Motion Motion in Two and Three Dimensions Laws of Motion Work, Energy and Power Rotatory Motion of Rigid Body Gravitation Heat and Thermodynamics Oscillations Wave Motion Electrostatics Current Electricity Thermal and Chemical Effects of Current Magnetic Effect of Current Magnetism Electromagnetic Induction and Alternating Current Electromagnetic Waves Ray Optics and Optical Instruments Electrons and Photons Atoms, Molecule and Nuclei Solids and Semiconductor Devices Primary and Secondary Cells X-rays General Physics Chemistry 1 60 General Chemistry 3 Physical Chemistry 16 Inorganic Chemistry 8 Organic Chemistry 4 General Science 1 16

6 IMPORTANT INSTRUCTIONS AND SYLLABUS A. SCHEME OF THE EXAMINATION 1. The subjects of the written examination, the time allowed and the maximum marks allotted to each subject will be as follows : Subject Subject Code Duration Max. Marks Mathematics 01 1 hours 300 General Ability Test 0 1 hours 600 Total 900 SSB Test/Interview 900. THE PAPERS IN ALL THE SUBJECTS WILL CONSIST OF OBJECTIVE TYPE QUESTIONS ONLY. THE QUESTION PAPERS (TEST BOOK- LETS) OF MATHEMATICS 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 and Measures only will be set. 4. Candidates must write the papers in their own hand. In no circumstances will they be allowed the help of a scribe to write the answers for them. 5. The Commission have discretion to fix qualifying marks in any or all the subjects at the examination. 6. The candidates are not permitted to use calculators or Mathematical or logarithmic table for answering objective type papers (Test Booklets). They should not, therefore, bring the same inside Examination Hall. B. SYLLABUS OF THE EXAMINATION Paper 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, 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 inequations 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. 3. 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. 4. 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 of a plane and a line in various forms. Angle between two lines 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

7 ( vi ) 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. 6. 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 equation, solution of first order and first degree 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 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. 8. Statistics and Probability : Statistics : Classification of data. Frequency distribution, cumulative frequency distributionexamples. Graphical representation Histogram, Pie Chart, frequency polygon examples. Measure 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 statisticalexamples. 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 Binomial distribution. Paper II (Code No. 0) GENERAL ABILITY TEST (Maximum Marks 600) PART A ENGLISH (Maximum Marks 00) The question paper in English will be designed to test the candidate s understanding of English and work-man-like use of words. The syllabus covers various aspects like : Grammar and usage, vocabulary, comprehension and cohesion in extended texts to test the candidate s proficiency in English. PART B GENERAL KNOWLEDGE (Maximum Marks 400) The question paper on General Knowledge will broadly cover the subjects : Physics, Chemistry, General Science, Social Studies, Geography and Current Events. The syllabus given below is designed to indicate the scope of these subjects included in this paper. The topics mentioned are not to be regarded as exhaustive and questions on topics of similar nature not specifically mentioned in the syllabus may also be asked. Candidate s answers are expected to show their knowledge and intelligent understanding of the subject. Section A (Physics) Physical Properties and States of Matter, Mass, Weight, Volume, Density and Specific Gravity, Principle of Archimedes, Pressure Barometer. Motion of objects, Velocity and Acceleration, Newton s Laws of Motion, Force and Momentum, Parallelogram of Forces, Stability and Equilibrium of bodies, Gravitation, elementary ideas of Work, Power and Energy.

8 ( vii ) Effects of Heat. Measurement of Temperature and Heat. Change of State and Latent Heat. Modes of transference of Heat. Sound waves and their properties, Simple musical instruments. Rectilinear propagation of Light. Reflection and refraction, Spherical mirrors and Lenses, Human Eye. Natural and 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 and Magnetic effects of Current. Measurement of Electrical Power, Primary and Secondary Cells. Use of X-rays. General Principles in the working of the following : Simple Pendulum, Simple Pulleys, Siphon, Levers, Balloon, Pumps, Hydrometer, Pressure Cooker, Thermos Flask, Gramophone, Telegraphs, Telephone, Periscope, Telescope, Microscope, Mariner s Compass, Lightning Conductors. Safety Fuses. Section B (Chemistry) Physical and Chemical changes. Elements, Mixtures and Compounds, Symbols, Formulae and Simple Chemical 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 like Soap, 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 Science) Difference between the living and non-living. 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 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. 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 and Volcanoes. Ocean Current and Tides. Atmosphere and its composition; Temperature and Atmospheric Pressure, Planetary winds, Cyclones and Anti-cyclones; Humidity; Condensation and Precipitation; Types of Climate. Major Natural regions of the World.

9 ( viii ) Regional Geography of India Climate, Natural Vegetation. Mineral and Power resources; location and distribution of agricultural and industrial activities. Important Sea Ports and main sea, land and air routes of India. Main items of Imports and Exports of India. Section F (Current Events) Knowledge of Important events that have happened in India in the recent years. Current important world events. Prominent personalities both Indian and International including those connected with cultural activities 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 5%, 15%, 10%, 0%, 0% and 10% weightages respectively. INTELLIGENCE AND PERSONALITY TEST The SSB procedure consists of two stage Selection process-stage-i and stage-ii. Only those candidates who clear the stage-i are permitted to appear for stage II. The details are : (a) Stage-I comprises of Officer Intelligence Rating (OIR) tests are Picture Perception *Description Test (PP & DT). The candidates will be shortlisted based on combination of performance in QIR Test and PP and DT. (b) Stage-II comprises of Interview, Group Testing Officer Tasks, Psychology Tests and the Conference. These tests are conducted over 4 days. The details of these tests are given on the website nic.in. The personality of a candidate is assessed by three different assessors viz. the Interviewing Officer (IO), Group Testing Officer (GTO) and the Psychologist. There are no separate weightage for each test. The mks are allotted by assessors only after taking into consideration the performance of the candidate holistically in all the test. In addition, marks for Conference are also allotted based on the initial performance of the Candidate in the three techniques and decision of the Board. All these have equal weightage. The various tests of IO, GTO and Psych are designed to bring out the presence/absence of Officer Like Qualities and their trainability in a candidate. candidates are Recommended or Not Recommended at the SSB

10 National Defence Academy and Naval Academy Exam. Solved Paper

11 014 (Held on 8 September, 014) Mathematics Directions (Q. 1 and ) Let S n denote the sum of first n terms of an AP and 3S n = S n. 1. What is S 3n : S n equal to? (A) 4 : 1 (B) 6 : 1 (C) 8 : 1 (D) 10 : 1. What is S 3n : S n equal to? (A) : 1 (B) 3 : 1 (C) 4 : 1 (D) 5 : 1 3. What is the length of the latus rectum of the ellipse 5x + 16y = 400? (A) 5/ (B) 5/4 (C) 16/5 (D) 3/5 Directions (Q. 4 and 5) Consider the circles x + y + ax + c = 0 and x + y + by + c = What is the distance between the centres of the two circles? (A) a + b (B) a + b (C) a + b (D) (a + b) 5. The two circles touch each other if (A) c = a + b (B) 1 c = 1 a + 1 b (C) c = 1 a b (D) c = a + b 6. A(3, 4) and B(5, ) are two points and P is a point such that PA = PB. If the area of triangle PAB is 10 square unit, what are the coordinates of P? (A) (1,0) only (B) (7, ) only (C) (1, 0) or (7, ) (D) Neither (1, 0) nor (7, ) 7. What is the product of the perpendiculars drawn from the points ( ± a ) b 0 upon the line bx cos α + ay sin α = ab? (A) a (B) b (C) a + b (D) a + b 8. Which one of the following is correct in respect of the equations x 1 = y and 3 x + 3y = 5? (A) They represent two lines which are parallel (B) They represent two lines which are perpendicular (C) They represent two lines which are neither parallel nor perpendicular (D) The first equation does not represent a line Directions (Q. 9 11) Consider a sphere passing through the origin and the points (, 1, 1), (1, 5, 4), (, 4, 6). 9. What is the radius of the sphere? (A) 1 (B) 14 (C) 1 (D) What is the centre of the sphere? (A) ( 1,, 3) (B) (1,, 3) (C) (1,, 3) (D) ( 1,, 3) 11. Consider the following statements : 1. The sphere passes through the point (0, 4, 0).. The point (1, 1, 1) is at a distance of 5 unit from the centre of the sphere. Which of the above statements is/are correct? (A) 1 only (B) only (C) Both 1 and (D) Neither 1 nor Directions (Q. 1 and 13) The line joining the points (, 1, 3) and (4,, 5) cuts the plane x + y z = Where does the line cut the plane? (A) (0, 4, 1) (B) (0, 4, 1) (C) (1, 4, 0) (D) (0, 4, 1)

12 4 NDA & NA 014 (II) 13. What is the ratio in which the plane divides the line? (A) 1 : 1 (B) : 3 (C) 3 : 4 (D) None of these Directions (Q. 14 and 15) Consider the plane passing through the points A (,, 1), B (3, 4, ) and C (7, 0, 6). 14. Which one of the following points lies on the plane? (A) (1, 0, 0) (B) (1, 0, 1) (C) (0, 0, 1) (D) None of these 15. What are the direction ratios of the normal to the plane? (A) < 1, 0, 1 > (B) < 0, 1, 0 > (C) < 1, 0, 1 > (D) None of these 16. What is 1 + sin θ equal to? (A) cos θ sin θ (B) cos θ + sin θ (C) cos θ + sin θ (D) cos θ + sin θ 17. A lamp post stands on a horizontal plane. From a point situated at a distance 150 m from its foot, the angle of elevation of the top is 30. What is the height of the lamp post? (A) 50 m (C) 50 3 m (B) 50 3 m (D) 100 m 18. If cot A = and cot B = 3, then what is the value of A + B? (A) π/6 (B) π (C) π/ (D) π/4 19. What is sin 66 1 sin 3 1 equal to? (A) sin 47 (B) cos 47 (C) sin 47 (D) cos What is sin sin 1 equal to? 5 (A) π/ (B) π/3 (C) π/4 (D) π/6 cos 7x cos 3x 1. What is equal to? sin 7x sin 5x + sin 3x (A) tan x (B) cot x (C) tan x (D) cot x. In a triangle ABC, c =, A = 45, a =, then what is C equal to? (A) 30 (B) 15 (C) 45 (D) None of these 3. In a triangle ABC, sin A cos B = cos C, then what is B equal to? (A) π (B) π/3 (C) π/ (D) π/4 sin (x + y) 4. If sin (x y) = a + b a b equal to?, then what is tan x tan y (A) b a (B) a b (C) ab (D) 1 5. If sin A sin (60 A) sin (60 + A) = k sin 3A, then what is k equal to? (A) 1/4 (B) 1/ (C) 1 (D) 4 6. The line y = 3 meets the graph y = tan x, where x ( 0 π ), in k points. What is k equal to? (A) One (C) Three (B) Two (D) Infinity 7. Which one of the following is one of the solutions of the equation tan θ. tan θ = 1? (A) π/1 (B) π/6 (C) π/4 (D) π/3 Directions (Q. 8 30) Given that 16 sin 5 x = p sin 5x + q sin 3x + r sin x. 8. What is the value of p? (A) 1 (B) (C) 1 (D) 9. What is the vlaue of q? (A) 3 (B) 5 (C) 10 (D) What is the value of r? (A) 5 (B) 8 (C) 10 (D) Every quadratic equation ax + bx + c = 0 where a, b, c R, a 0 has (A) exactly one real root (B) at least one real root

13 NDA & NA 014 (II) 5 (C) at least two real roots (D) at most two real roots 3. The relation S is defined on the set of integers Z as xsy if integer x divides integer y. Then (A) S is an equivalence relation (B) S is only reflexive and symmetric (C) S is only reflexive and transitive (D) S is only symmetric and transitive 33. If a b c are all positive, then the value of the determinant (A) non-negative (C) negative a b c b c a c a b is (B) non-positive (D) positive 34. Let A and B be two matrices such that AB = A and BA = B. Which of the following statements are correct? 1. A = A. B = B 3. (AB) = AB Select the correct answer using the code given below (A) 1 and only (B) and 3 only (C) 1 and 3 only (D) 1, and What is (1001) equal to? (A) (5) 10 (B) (9) 10 (C) (17) 10 (D) (11) What is i 3 i equal to, where i = 1? (A) 1 (B) 1/6 (C) 6 (D) 37. Let z be a complex number such that z = 4 and arg z = 5π. What is z equal to? 6 (A) 3 + i (B) 3 i (C) 3 + i (D) 3 + i where i = If 6i 3i 1 4 3i i then what is x equal to? (A) 3 (B) (C) 1 (D) 0 = x + iy, where i = 1, 39. If α, β are the roots of ax + bx + c = 0 and α + h, β + h are the roots of px + qx + r = 0, then what is h equal to? (A) 1 b ( a ) q p 1 b (C) ( ) p + q a (B) 1 ( b a ) + q p (D) 1 ( b p ) + q a 40. If the matrix A is such that (A) (C), then what is A equal to? (B) (D) A = 41. Consider the following statements : 1. Determinant is a square matrix.. Determinant is a number associated with a square matrix. Which of the above statements is/are correct? (A) 1 only (B) only (C) Both 1 and (D) Neither 1 nor 4. If A is an invertible matrix, then what is det (A 1 ) equal to? 1 (A) det A (B) det A (C) 1 (D) None of these 43. From the matrix equation AB = AC, where A, B, C are the square matrices of same order, we can conclude B = C provided (A) A is non-singular (B) A is singular (C) A is symmetric (D) A is skew symmetric

14 6 NDA & NA 014 (II) 44. If A = x 3 x + 1 is symmetric, then what is x equal to? (A) (B) 3 (C) 1 (D) 5 a b 0 0 a b 45. If = 0, then which one of the b 0 a following is correct? (A) a is one of the cube roots of unity b a (B) is one of the cube roots of 1 b (C) a is one of the cube roots of unity (D) b is one of the cube roots of unit 46. If a =, b = 5 and a b = 8, then what is a. b equal to? (A) 6 (B) 7 (C) 8 (D) If a + b = a b, then which one of the following is correct? (A) a = b (B) a is parallel to b. (C) a, is perpendicular to b (D) a is a unit vector 48. What is the area of the triangle OAB where O is the origin, OA = 3 ^i ^j + ^k and OB = ^i + ^j 3 ^k? (A) 5 6 square unit (B) 5 6 square unit (C) 6 square unit (D) 30 square unit 49. Which one of the following is the unit vector perpendicular to both a = ^i + ^j + ^k and b = ^i ^j + ^k? ^i + ^j (A) (B) ^k ^j + ^k ^i ^j (C) (D) 50. What is the interior acute angle of the parallelogram whose sides are represented by the vectors ^i + ^j + ^k and ^i ^j + ^k? (A) 60 (B) 45 (C) 30 (D) For what value of λ are the vectors λ^i + (1 + λ) ^j + (1 + λ) ^k and (1 λ) ^i+ λ^j + ^k perpendicular? (A) 1/3 (B) 1/3 (C) /3 (D) 1 Directions (Q. 5 55) a + b + c = 0 such that a = 3 b = 5 and c = What is the angle between a and b? (A) π/6 (B) π/4 (C) π/3 (D) π/ 53. What is a. b + b. c + c. a equal to? (A) 83 (B) 83/ (C) 75 (D) 75/ 54. What is cosine of the angle between b and c? (A) 11/1 (B) 13/14 (C) 11/1 (D) 13/ What is a + b equal to? (A) 7 (B) 8 (C) 10 (D) What is π/ dx 0 a cos x + b sin x (A) ab (B) πab (C) π ab (D) π ab equal to? Directions (Q. 57 and 58) A cylinder is inscribed in a sphere of radius r. 57. What is the height of the cylinder of maximum volume? (A) r 3 (C) r (B) r 3 (D) 3 r

15 NDA & NA 014 (II) What is the radius of the cylinder of maximum volume? (A) r 3 (B) r 3 (C) r (D) 3 r Directions (Q. 59 and 60) Consider the function f " (x) = sec 4 x + 4 with f(0) = 0 and f (0) = What is f (x) equal to? (A) tan x tan3 x + 4x 3 (B) tan x + tan3 x + 4x 3 (C) tan x + sec3 x + 4x 3 (D) tan x tan3 x + 4x What is f(x) equal to? ln sec x (A) + tan x + x ln sec x (B) + cot x + x 6 (C) 4 ln sec x 3 + sec x + x 6 (D) ln sec x + tan4 x 1 + x Directions (Q ) Consider I = π x dx sin x. 61. What is I equal to? (A) π (B) 0 (C) π (D) π 6. What is π (π x)dx equal to? sin x (A) π (B) π/ (C) 0 (D) π 63. What is π dx equal to? sin x (A) 1 (B) (C) 4 (D) Directions (Q. 64 and 65) x tan 1 x dx = A (x + 1) tan 1 x + Bx + C, where C is the constant of integration. 64. What is the value of A? (A) 1 (B) 1/ (C) 1/ (D) 1/4 65. What is the value of B? (A) 1 (B) 1/ (C) 1/ (D) 1/4 Directions (Q. 66 and 67) Consider the integeral I = π ln (sin x) dx What is π/ ln (sin x) dx equal to? 0 (A) 4 I (B) I (C) I (D) I/ 67. What is π/ ln (cos x) dx equal to? 0 (A) I/ (B) I (C) I (D) 4 I Directions (Q. 68 and 69) A rectangular box is to be made from a sheet of 4 inch length and 9 inch width cutting out identical squares of side length x from the four corners and turning up the sides. 68. What is the value of x for which the volume is maximum? (A) 1 inch (B) 1.5 inch (C) inch (D).5 inch 69. What is the maximum volume of the box? (A) 00 cubic inch (B) 400 cubic inch (C) 100 cubic inch (D) None of these 70. What is the degree of the differential equation d ( 3 3/ y d dx ) = ( y)? 3 dx (A) 1 (B) (C) 3 (D) 4 dy 71. What is the solution of the equation ln ( dx) + x = 0? (A) y + e x = c (B) y e x = c (C) y + e x = c (D) y e x = c where c is an arbitrary constant. 7. Eliminating the arbitrary constants B and C in the expression y = 3C (Cx 1)3/ + B, we get

16 8 NDA & NA 014 (II) (A) x[ 1 + ( dy dx) ] = d y dx dy d (B) x ( dx) y dx () = 1 + dy dx dy d (C) () y dx dx = 1 dy d (D) () y + 1 = dx dx Directions (Q ) Let f (x) = ax + bx + c such that f(1) = f( 1) and a, b, c are in Arithmetic Progression. 73. What is the value of b? (A) 1 (B) 0 (C) 1 (D) Cannot be determined due to insufficient data 74. f (a), f (b), f (c) are in (A) A.P. (B) G.P. (C) H.P. (D) Arithemetico-geometric progression 75. f " (a), f " (b), f " (c) are (A) in A.P. only (B) in G.P. only (C) in both A.P. and G.P. (D) neither in A.P. nor in G.P. 76. Suppose A and B are two events. Event B has occurred and it is known that P(B) < 1. What is P (A B c ) equal to? P(A) P(B) P(A) P(AB) (A) (B) 1 P(B) 1 P(B) P(A) + P(B (C) c ) (D) None of these 1 P(B) Directions (Q ) Consider events A, B, C, D, E of the sample space S = {n : n is an integer such that 10 n 0} given by A is the set of all even numbers. B is the set of all prime numbers. C = {15} D is the set of all integers 16. E is the set of all double digit number expressible as a power of. 77. A, B and D are (A) Mutually exclusive events but not exhaustive events (B) Exhaustive events but not mutually exclusive events (C) Mutually exclusive and exhaustive events (D) Elementary events 78. A, B and C are (A) Mutually exclusive events but not exchaustive events (B) Exchaustive events but not mutually exclusive events (C) Mutually exclusive and exhaustive events (D) Elementary events 79. B and C are (A) Mutually exclusive events but not exhaustive events (B) Compound events (C) Mutually exclusive and exhaustive events (D) Elementary events 80. C and E are (A) Mutually exclusive events but not elementary events (B) Exhaustive events but not mutually exclusive events (C) Mutually exclusive and exhaustive events (D) Elementary and mutually exclusive events 81. Consider the following statements in respect of histogram : 1. The histogram is a suitable representation of a frequency distribution of a continuous variable.. The area included under the whole histogram is the total frequency. Which of the above statements is/are correct? (A) 1 only (B) only (C) Both 1 and (D) Neither 1 nor 8. The regression lines will be perpendicular to each other if the coefficient of correlation r is equal to (A) 1 only (B) 1 or 1 (C) 1 only (D) 0

17 NDA & NA 014 (II) For any two events A and B, which one of the following holds? (A) P(A B) P(A) P(A B) P(A) + P(B) (B) P(A B) P(A) P(A B) P(A) + P(B) (C) P(A B) P(B) P(A B) P(A) + P(B) (D) P(A B) P(B) P(A) + P(B) P(A B) 84. The probability that in a random arrangement of the letters of the word UNIVERSITY, the two I s do not come together is (A) 4/5 (B) 1/5 (C) 1/10 (D) 9/ There are 4 white and 3 black balls in a box. In another box, there are 3 white and 4 black balls. An unbiased dice is rolled. If it shows a number less than or equal to 3, then a ball is drawn from the second box, otherwise from the first box. If the ball drawn is black, then the probability that the ball was drawn from the first box is (A) 1/ (B) 6/7 (C) 4/7 (D) 3/7 86. If x and y are the means of two distributions such that x < y and z is the mean of the combined distribution, then which one of the following statements is correct? (A) x < y < z (B) x > y > z x + y (C) z = (D) x < z < y 87. What is the mean deviation about the mean for the data 4, 7, 8, 9, 10, 1, 13, 17? (A).5 (B) 3 (C) 3.5 (D) The variance of 0 observations is 5. If each observation is multiplied by, then what is the new variance of the resulting observations? (A) 5 (B) 10 (C) 0 (D) Two students X and Y appeared in an examination. The probability that X will qualify the examination is 0.05 and Y will qualify the examination is The probability that both will qualify the examination is 0.0. What is the probability that only one of them will qualify the examination? (A) 0.15 (B) 0.14 (C) 0.1 (D) A fair coin is tossed four times. What is the probability that at most three tails occur? (A) 7/8 (B) 15/16 (C) 13/16 (D) 3/4 Directions (Q ) Consider the function f(x) = x 5 x 3 x + 13 x > What is lim f(x) equal to? x 3 (A) (B) 4 (C) 5 (D) Consider the following statements : 1. The function is discontinuous at x = 3.. The function is not differentiable at x = 0. Which of the above statements is/are correct? (A) 1 only (B) only (C) Both 1 and (D) Neither 1 nor 93. What is the differential coefficient of f(x) at x = 1? (A) 5/ (B) 5 (C) 1/5 (D) 1/10 Directions (Q ) The line y = 3x + 1 cuts the parabola 4y = 3x. 94. Where does the line cut the parabola? (A) At (, 3) only (B) At (4, 1) only (C) At both (, 3) and (4, 1) (D) Neither at (, 3) nor at (4, 1) 95. What is the area enclosed by the parabola and the line? (A) 7 square unit (B) 36 square unit (C) 48 square unit (D) 54 square unit 96. What is the area enclosed by the parabola, the line and the y-axis in the first quadrant? (A) 7 square unit (B) 14 square unit (C) 0 square unit (D) 1 square unit

18 10 NDA & NA 014 (II) 97. Consider the function tan kx f(x) = x < 0 x 3x + k x 0 What is the non-zero value of k for which the function is continuous at x = 0? (A) 1/4 (B) 1/ (C) 1 (D) 98. Consider the following statements : 1. The function f(x) = [x], where [.] is the greatest integer function defined on R, is continuous at all points except at x = 0.. The function f(x) = sin x is continuous for all x R. Which of the above statements is/are correct? (A) 1 only (B) only (C) Both 1 and (D) Neither 1 nor Directions (Q. 99 and 100) Consider the curve x = a (cos θ + θ sin θ) and y = a (sin θ θ cos θ). 99. What is dy equal to? dx (A) tan θ (B) cot θ (C) sin θ (D) cos θ 100. What is d y dx equal to? (A) sec θ (B) cosec θ (C) sec 3 θ aθ (D) None of these 101. What is the area of the parabola y = 4bx bounded by its latus rectum? (A) b /3 square unit (B) 4b /3 square unit (C) b square unit (D) 8b /3 square unit 10. If y = x ln x + xe x, then what is the value of dy dx at x = 1? (A) 1 + e (B) 1 e (C) 1 + e (D) None of these log 103. What is lim 5 (1 + x) equal to? x 0 x (A) 1 (B) log 5 e (C) log e 5 (D) What is lim x 1 equal to? x 0 x (A) log e 5 (B) log 5 e (C) 5 (D) n 105. What is lim n n equal to? (A) 5 (B) (C) 1 (D) The function f : N N, N being the set of natural numbers, defined by f(x) = x + 3 is (A) injective and surjective (B) injective but not surjective (C) not injective but surjective (D) neither injective nor surjective (1 + i)4n What is (1 i) 4n + 3 equal to, where n is a natural number and i = 1? (A) (B) i (C) i (D) i 108. What is the number of ways in which one can post 5 letters in 7 letter boxes? (A) 7 5 (B) 3 5 (C) 5 7 (D) What is the number of ways that a cricket team of 11 players can be made out of 15 players? (A) 364 (B) 1001 (C) 1365 (D) A and B are two sets having 3 elements in common. If n(a) = 5, n(b) = 4, then what is n(a B) equal to? (A) 0 (B) 9 (C) 15 (D) If f(x) = ax + b and g(x) = cx + d such that f(g(x)) = g(f(x)), then which one of the following is correct? (A) f(c) = g(a) (B) f(a) = g(c) (C) f(c) = g(d) (D) f(d) = g(b) 11. If A and B are square matrices of second order such that A = 1, B = 3, then what is 3AB equal to? (A) 3 (B) 9 (C) 7 (D) None of these

19 NDA & NA 014 (II) 11 Directions (Q ) Consider the function f(x) = x 1 x What is f(x) x equal to? f(x) 1 (A) 0 (B) 1 (C) x (D) 4x 114. What is f(x) equal to? (A) f(x) + 1 f(x) + 1 (B) f(x) + 3 3f(x) + 1 (C) 3f(x) + 1 f(x) + 3 (D) 115. What is f( f(x)) equal to? (A) x (B) x f(x) + 3 3f(x) + 1 (C) 1 (D) None of these x Directions (Q ) Consider the expansion ( ) x + 1 x What is the independent term in the given expansion? (A) 103 (B) 3003 (C) 4503 (D) None of these 117. What is the ratio of coefficient of x 15 to the term independent of x in the given expansion? (A) 1 (B) 1/ (C) /3 (D) 3/ Consider the following statements : 1. There are 15 terms in the given expansion.. The coefficient of x 1 is equal to that of x 3. Which of the above statements is/are correct? (A) 1 only (B) only (C) Both 1 and (D) Neither 1 nor 119. Consider the following statements : 1. The term containing x does not exist in the given expansion.. The sum of the coefficients of all the terms in the given expansion is 15. Which of the above statements is/are correct? (A) 1 only (B) only (C) Both 1 and (D) Neither 1 nor 10. What is the sum of the coefficient of the middle terms in the given expansion? (A) C(15, 9) (B) C(16, 9) (C) C(16, 8) (D) None of these Answers with Explanations For the questions 1 and S n Given that 3S n = S n = n [a + (n 1)d] 3 n [a + (n 1)d] = n [a + (n 1)d] 6a + 3(n 1)d = 4a + (4n )d a = (n + 1)d...(i) 1. (B) S 3n S n = = 3n [a + (3n 1)d] n [a + (n 1)d] 3[(n + 1)d + (3n 1)d] (n + 1)d + (n 1)d [From (i)] 1 nd = nd = 6 S 3n : S n = 6 : 1 3n. (A) S [a + (3n 1)d] 3n = S n n [a + (n 1) d] 3[(n + 1)d + (3n 1)d] = [(n + 1)d + (n 1)d] [From(i)] = 1 nd 6 nd = S 3n : S n = : 1 3. (D) Given ellipse is 5x + 16y = 400 x 16 + y 5 = 1 a = 4, b = 5 Length of Latus Rectum = a b = (4) 5 = 3 5 ( a < b)

20 1 NDA & NA 014 (II) For the questions 4 and 5 Given two circles are x + y + ax + c = 0...(i) Centre C 1 ( a, 0), radius (r 1 ) = a c and x + y + by + c = 0...(ii) Centre C (0, b), radius (r )= b c 4. (A) Distance between centres of two circles : C 1 C = ( a 0) + (0 + b) = a + b 5. (B) Two circles touch each other, if distance between two circles = sum of radii C 1 C = r 1 + r a + b = a c + b c Squaring both sides, we get a + b = (a c) + (b c) + a c b c a c b c = c Squaring again, we get (a c) (b c) = c a b c(a + b ) + c = c c(a + b )= a b a + b a b = 1 c 1 a + 1 b = 1 c 6. (C) Let co-ordinates of P be (x, y). Given that, PA = PB (x 3) + (y 4) = (x 5) + (y + ) (x 3) + (y 4) = (x 5) + (y + ) 4x 1y = 4 x 3y = 1...(i) Also given, Area of Δ PAB = 10 1 [x (4 + ) + 3 ( y) + 5 (y 4)] = ± 10 Taking +ve sign 6x 6 3y + 5y 0 = 0 6x + y = 46 3x + y = 3...(ii) Solving equations (i) and (ii), we get x = 7, y = Taking ve sign 6x + y = 6 3x + y = 3 (iii) Solving equations (i) and (iii), we get x = 1, y = 0 Co-ordinates of are (7, ) or (1, 0). 7. (B) The length of perpendicular from (x 1, y 1 ) to the line ax + by + c = 0 is given by p = ax 1 + by 1 + c a + b Product of lengths of prependiculars from ( a b, 0) and ( a b, 0) to the line bx cos α + ay sin α ab = 0 p 1 p = b cos α a b + a sin α 0 ab b cos α + a sin α b cos α a b + a sin α 0 ab b cos α + a sin α b ( a b cos α a) b ( a = b cos α + a) b cos α + a sin α = b [a (a b ) cos α] b cos α + a sin α = b [a (1 cos α) + b cos α] b cos α + a sin α = b 8. (B) The given lines are : x 1 = y 3 3x y + 1 = 0...(i) and x + 3y 5 = 0...(ii) Slopes of lines (i) and (ii) are m 1 = 3 and m = 3 m 1 m = 3 () 3 = 1 The given lines are perpendicular to each other. For the questions 9 to 11 Let the equation of the sphere be x + y + z + ux + vy + wz + d = 0 Which passes through (0, 0, 0), (, 1, 1), (1, 5, 4) and (, 4, 6).

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