UPKAR PRAKASHAN, AGRA–2

ByDr. H.P. Sharma

&Dr. M.B. Lal

According to the Latest Syllabus

Hindi Editions are Also Available

© Publishers

Publishers

UPKAR PRAKASHAN2/11A, Swadeshi Bima Nagar, AGRA–282 002Phone : 4053333, 2530966, 2531101Fax : (0562) 4053330E-mail : care@upkar.in, Website : www.upkar.in

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● The publishers have taken all possible precautions in publishing this book, yet if any mistakehas crept in, the publishers shall not be responsible for the same.

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ISBN : 978-81-7482-300-7

Code No. 346

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

Contents

● Practice Set 1....................................................................................................................... 3–36

● Practice Set 2..................................................................................................................... 37–69

● Practice Set 3................................................................................................................... 70–106

● Practice Set 4................................................................................................................. 107–143

● Practice Set 5................................................................................................................. 144–177

● Practice Set 6................................................................................................................. 178–213

● Practice Set 7................................................................................................................. 214–242

● Practice Set 8................................................................................................................. 243–274

● Practice Set 9................................................................................................................. 275–302

● Practice Set 10.. ............................................................................................................. 303–332

IMPORTANT INSTRUCTIONS AND SYLLABUS

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

time allowed and the maximum marks allotted to eachsubject will be as follows :

SubjectSubject

CodeDuration

Max.

Marks

Mathematics 01 212 hours 300

General Ability Test 02 212 hours 600

Total 900

2. 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 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 fixqualifying marks in any or all the subjects at theexamination.

6. The candidates are not permitted to usecalculators or Mathematical or logarithmic table foranswering objective type papers (Test Book-lets). Theyshould not, therefore, bring the same insideExamination 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,equivalence relation.

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 progres-

sions. Quadratic equations with real coefficients.Solution of linear inequations of two variables bygraphs. Permutation and Combination. Binomialtheorem and its applications. Logarithms and theirapplications.

2. Matrices and Determinants :Types of matrices, operations on matrices.

Determinant of a matrix basic properties ofdeterminants. Adjoint and inverse of a square matrixApplications—Solution of a system of linear equationsin two or three unknowns by Cramer’s rule and byMatrix Method.

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

radians. Trigonometrical ratios. Trigonometric identitiesSum and difference formulae. Multiple and Sub-multiple angles. Inverse trigonometric functions.Applications—Height and distance, properties oftriangles.

4. Analytical Geometry of Two and ThreeDimensions :Rectangular Cartesian Coordinate system. Distance

formula. 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 asphere.

5. Differential Calculus :Concept of a real valued function-domain, range

and graph of a function. Composite functions, one toone, 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 andphysical 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 orderderivatives. Increasing and decreasing functions.Application of derivatives in problems of maxima andminima.

( vii )

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

by substitution and by parts, standard integralsinvolving algebraic expressions, trigonometric,exponential and hyperbolic functions. Evaluation ofdefinite integrals—determination of areas of planeregions bounded by curves—applications. Definition oforder and degree of a differential equation, formation ofa differential equation by examples. General andparticular solution of a differential equation, solution offirst order and first degree differential equations ofvarious types—examples. Application in problems ofgrowth and decay.

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

and direction of a vector. Unit and null vectors, additionof vectors, scalar multiplication of a vector, scalarproduct or dot product of two vectors. Vector product orcross product of two vectors. Applications-work doneby a force and moment of a force, and in geometricalproblems.

8. Statistics and Probability :

Statistics : Classification of data. Frequencydistribution, cumulative frequency distribution-examples. Graphical representation—Histogram, PieChart, frequency polygon—examples. Measure ofCentral tendency—Mean, median and mode. Varianceand standard 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—classical and statistical-examples. Elementary theoremso n probability— simple problems. Conditionalprobability, Bayes’ theorem-simple problems. Randomvariable as function on a sample space. Binomialdistribution, examples of random experiments givingrise to Binomial distribution.

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 testthe candidate’s proficiency in English.

PART ‘B’

GENERAL KNOWLEDGE

(Maximum Marks–400)

The question paper on General Knowledge willbroadly cover the subjects : Physics, Chemistry, GeneralScience, Social Studies, Geography and Current Events.

The syllabus given below is designed to indicatethe scope of these subjects included in this paper. Thetopics mentioned are not to be regarded as exhaustiveand questions on topics of similar nature not specificallymentioned in the syllabus may also be asked.Candidate’s answers are expected to show theirknowledge 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 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,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 andSimple Chemical Equations. Law of ChemicalCombination (excluding problems). Properties of Airand Water.

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

( viii )

Acids, Bases and Salts.

Carbon–Different forms.

Fertilizers–Natural and Artificial.

Materials used in the preparations of substanceslike Soap, Glass, Ink, Paper, Cement, Paints, SafetyMatches 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 Movementetc.)

A broad survey of Indian History, with emphasis onCulture and Civilisation.

Freedom Movement in India.

Elementary study of Indian Constitution andAdministration.

Elementary knowledge of Five Year Plans of India.

Panchayati Raj, Co-operatives and CommunityDevelopment.

Bhoodan, Sarvodaya, National Integration andWelfare State, Basic teachings of Mahatma Gandhi.

Forces shaping the modern World; RenaissanceExploration 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, Earthquakesand Volcanoes.

Ocean Current and Tides.

Atmosphere and its composition; Temperature andAtmospheric Pressure, Planetary winds, Cyclones and

Anti-cyclones; Humidity; Condensation and Precipita-tion; 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.

Important Sea Ports and main sea, land and airroutes of India. Main items of Imports and Exports ofIndia.

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 Interna-tional including those connected with cultural activitiesand 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 *DescriptionTest (PP & DT). The candidates will be shortlistedbased on combination of performance in QIR Testand PP and DT.

(b) Stage-II comprises of Interview, Group TestingOfficer Tasks, Psychology Tests and theConference. These tests are conducted over 4 days.The details of these tests are given on the websitewww.joinindianarmy. 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 intoconsideration the performance of the candidateholistically in all the test. In addition, marks forConference are also allotted based on the initialperformance of the Candidate in the three techniquesand decision of the Board. All these have equalweightage.

The various tests of IO, GTO and Psych aredesigned to bring out the presence/absence of OfficerLike Qualities and their trainability in a candidate.candidates are Recommended or NotRecommended at the SSB

N DAPractice Sets

Practice Set–1Paper-I

Mathematics1. Let Z and ω be the two non-zero complex numbers

such that |Z| = |ω| and arg Z + arg ω = π. Then Z isequal to—(A) ω (B) – ∞

(C) –ω (D) – –ω

2. The vector Z = 3 – 4i is terned anticlockwise throughan angle of 180° and streched 2·5 times. The complexnumber corresponding to the newly obtained vectoris—

(A)152

– 10i (B) – 152

+ 10i

(C) – 152

– 10i (D) ± 152

– 10i

3. If A = {1, 2}; B = {2, 5}, C = {5, 7}, then (A × B)∩ (A × C) is equal to—

(A) {(2, 5) (1, 5)} (B) {(2, 2) (5, 5)}

(C) {(2, 7), (1, 5)} (D) {(2, 5), (2, 7)}

4. The points z1, z2, z3, z4 in the complex plane are thevertices of a parallelogram taken in order if andonly if—

(A) z1 + z4 = z2 + z3 (B) z1 + z3 = z2 + z4

(C) z1 + z2 = z3 + z4 (D) None of these

5. If z(3 + 4i) = 2 + 3i, then the value of z is—

(A) (18 – i)/25 (B) (9 + 9i)/16

(C) (18 + i)/25 (D) (5 + 2i)/9

6. If x, y, z are in H.P., then the value of log (x + 3) +log (x – 2y + z) is equal to—

(A) 0 (B) log (x + z)2

(C) log (x – z)2 (D) None of these

7. The 5th term of a H.P. is 145

and 11th term is 169

,

then its 16th term will be—

(A) 1/89 (B) 1/85(C) 1/80 (D) 1/79

8. If 56Pr + 6 : 54Pr + 3 = 30800 : 1, then the value of ris—

(A) 30 (B) 37

(C) 41 (D) None of these

9. How many different words can be formed from theword DAUGHTER so that ending and beginningletters are consonant ?(A) 7200 (B) 14400(C) 360 (D) None of these

10. The sum of first n terms of the given series 12 + 2·22

+ 32 + 2·42 + 52 + 2·62 + …… is n (n + 1)2

2 when n is

even. When n is odd, the sum will be—

(A)n (n + 1)2

2(B)

12 n2 (n + 1)

(C) n (n + 1)2 (D) n2 (n + 1)

11. Which Venn diagram show Y ∩ (X ∪ Z)—

(A) X Y

Z

(B) X Y

Z

(C) X Y

Z

(D) X Y

Z

4A | N.P.W.B.

12. If in a set A the relation R is equivalence, then R – 1

is—(A) Reflexive (B) Symmetric(C) Transitive (D) All of these

13. Let a, b, c be real numbers a ≠ 0. If α is a root ofa2x2 + bx + c = 0, β is a root a2x2 – bx – c = 0 and0 < α < β, then the equation a2x2 + 2bx + 2c = 0 hasa root γ that always satisfies—

(A) γ = α + β

2(B) γ = α +

β2

(C) γ = α (D) α < γ < β

14. The fourth term in the expansion of ⎝⎜⎛

⎠⎟⎞

2x – 13

8 is—

(A) 8C5·x5 2227

(B) – 8C5 · x5 2732

(C) – 8C5 · x5 3227

(D) None of these

15. If [1 x 1]

⎣⎢⎢⎢⎢⎡

⎦⎥⎥⎥⎥⎤2 3 2

0 5 1

0 3 2

⎣⎢⎢⎢⎢⎡

⎦⎥⎥⎥⎥⎤1

1

x

= 0, then the value

of x are—(A) 1, 8 (B) – 1, 8(C) – 1, – 8 (D) 1, – 8

16. If J =

⎣⎢⎢⎢⎢⎡

⎦⎥⎥⎥⎥⎤1 1 1

1 1 1

1 1 1

, then J2 is equal to—

(A) 3J (B) 2J

(C) 4J (D)12 J

17. If α1, α 2 and β1, β2 are the roots of the equationsax2 + bx + c = 0 and px2 + qx + r = 0 respectivelyand the system of equations α1y + α2z = 0 and β1y +β2z = 0 has a non-zero solution, then—(A) α2qc = p2br (B) b2pr = q2ac(C) c2ar = r2pb (D) None of these

18. m men and n women are to be seated in a row sothat no two women sit together. If m > n, then thenumber of ways in which they can be seated—

(A)m! (m + 1)!(m – n + 1)!

(B)m! (m – 1)!(m – n + 1)

(C)(m – 1)! (m + 1)!

(m – n + 1)!(D) None of these

19. If A = ⎣⎢⎢⎡

⎦⎥⎥⎤1 2

2 1 and B =

⎣⎢⎢⎡

⎦⎥⎥⎤2 1

1 2 , then A (BA)

is equal to—

(A) ⎣⎢⎢⎡

⎦⎥⎥⎤14 14

13 13 (B)

⎣⎢⎢⎡

⎦⎥⎥⎤14 13

13 14

(C) ⎣⎢⎢⎡

⎦⎥⎥⎤13 14

14 13 (D)

⎣⎢⎢⎡

⎦⎥⎥⎤14 13

14 13

20. If | z – 1 | = 2, then the value of z –z – z –

–z is—

(A) 4 (B) 2(C) 1 (D) 3

21. If a1, a2, a3,…a24 are in Arithmetic progression anda1 + a5 + a10 + a15 + a20 + a24 = 225, then a1 + a2 +a3 +… a24 is—

(A) 909 (B) 75(C) 750 (D) 900

22. Given that fourth term in the expansion of ⎝⎜⎛

⎠⎟⎞

px + 1x

n

is 52, then the value of p is—

(A) 6 (B) 3(C) 1/2 (D) None of these

23. The value of sin ⎝⎜⎛

⎠⎟⎞π

10 sin ⎝⎜⎛

⎠⎟⎞13π

10 is equal to—

(A) 1/2 (B) – 1/2(C) – 1/4 (D) 1

24. If tan α = m

m + 1 and tan β =

1(2m + 1)

, then α + β

is equal to—

(A)π2

(B)π3

(C)π4

(D) None of these

25. 2 sin2 θ = 3 cos θ, 0 ≤ θ ≤ 2π, then the value of θis—(A) π/3 and 5π/3 (B) π/3 and 2π/3(C) 2π/3 and 4π/3 (D) None of these

26. A ladder rests against a wall at an angle α tohorizontal. Its foot is pulled away from the wallthrough a distance a, so that it slides a distance bdown the wall making an angle β with the

horizontal, then tan 12 (α + β) is equal to—

(A)ab

(B)ba

(C)a – ba + b

(D)a + ba – b

27. From the top of the light house 60 metre high withits base at the sea level, the angle of depression ofboat is 15°. The distance of the boat from the foot ofthe light house is—

(A) 60(√⎯ 3 – 2) (B) 120 (2 – √⎯ 3)

(C) 60(√⎯ 3 + 2) (D) 40(√⎯ 3 + 2)

28. rth term in the expansion of (a + 2x)n is—

(A)n(n + 1) … (n – r + 1)

r! an – r + 1 (2x)r

(B)n(n – 1) …… (n – r + 2)

(r – 1)! an – r + 1 (2x)r – 1

N.P.W.B. | 5A

(C)n(n + 1) …… (n – r)

(r + 1)! an – r (n)r

(D) None of the above

29. At an election three wards of a town are convassedby 4, 5 and 8 men respectively. If 20 men volunteerin, how many ways can they be allotted to thedifferent wards ?(A) 20P4 × 16C5

(B) 20C4 × 20C5 × 20C8

(C) 20C4 + 16C5 + 11C8

(D) 20C4 × 16C5 × 11C8

30. The system ⎣⎢⎢⎡

⎦⎥⎥⎤x + 2y 6

0 2x – y =

⎣⎢⎢⎡

⎦⎥⎥⎤3

0 · [ – 1, 2]

has the solution—

(A) x = – 65 , y = –

35

(B) x = – 35, y = –

65

(C) x = 35 , y = –

65

(D) x = 35 , y =

65

31. If

⎣⎢⎢⎢⎢⎡

⎦⎥⎥⎥⎥⎤6i –3i 1

4 3i – 1

20 3 i

= x + iy, then (x, y) is—

(A) (3, 1) (B) (1, 3)(C) (0, 3) (D) (0, 0)

32. The value of cot–1 21 + cot–1 13 + cot–1 (– 8) isequal to—(A) 0 (B) cot – 1 – 1(C) ∞ (D) None of these

33. If sin A = sin B and cos A = cos B, then the value ofA – B is equal to—(A) 2nπ (B) nπ(C) nπ/2 (D) None of these

34. If for positive integers r > 1, n > 2, the coefficient ofthe (3r)th and (r + 2)th powers of x in the expansionof (1 + x)2n are equal, then—(A) n = 2r (B) n = 3r(C) n = 2r + 1 (D) n = 4r

35. The number line

− 5 − 4 − 3 − 2 − 1 0 1 2 3 4 5 6

represent—(A) {x : 1 < x < 5 and x ∈ R}(B) {x : 1 ≥ x ≥ 5 and x ∈ R}(C) {x : 1 ≤ x < 5 and x ∈ R}

(D) None of these

36. The sides of a triangle are 7 cm, 4√⎯ 3 cm and

√⎯⎯13 cm, then the smallest angle is—

(A) 15° (B) 30°(C) 45° (D) None of these

37. If the distance between (x, 4) and (5, 0) is 5, then xis equal to—(A) – 2, 8 (B) 2, 8(C) 2, 6 (D) – 2, – 6

38. The value of loge ( )1 + ax2 + a2 + ax2 is—

(A) a ( )x2 – 1x2 –

a2

2 ( )x4 –

1x4

+ a3

3 ( )x6 –

1x6 – ……

(B) a ( )x2 + 1x2 –

a2

2 ( )x4 +

1x4

+ a3

3 ( )x6 +

1x6 – ……

(C) a ( )x2 + 1x2 +

a2

2 ( )x4 +

1x4

+ a3

3 ( )x6 +

1x6 + ……

(D) a ( )x2 – 1x2 +

a2

2 ( )x4 –

1x4

+ a3

3 ( )x6 –

1x6 + ……

39. The latus rectum of the hyperbola 16x2 – 9y2 = 144is—(A) 16/3 (B) 32/3(C) 8/3 (D) 4/3

40. ∫x

√⎯⎯⎯x – 1 dx is equal to—

(A)23 (x – 1)3/2 + 2(x – 1)– 1/2 + C

(B)23 (x – 1)3/2 + 2(x – 1)1/2 + C

(C)23 (x – 1)3/2 – 2(x – 1)– 1/2 + C

(D)23

(x – 1)– 3/2 + 2(x – 1)– 1/2 + C

41. If a, b, c are three unequal numbers such that a, b, care in A.P. and b – a, c – b, a are in G.P., then a : b: c is—(A) 1 : 2 : 3 (B) 1 : 3 : 5(C) 2 : 3 : 4 (D) 1 : 2 : 4

42. The following system of equations3x – 2y + z = 0, λx – 14y + 15z = 0, x + 2y – 3z = 0has a solution other than x = y = z = 0 for λ equalto—(A) 1 (B) 2(C) 3 (D) 5

43. ∫cot x

√⎯⎯⎯⎯sin x dx is equal to—

(A) 2 √⎯⎯⎯sin x (B) – 2 √⎯⎯⎯sin x

(C)2

√⎯⎯⎯sin x(D)

– 2

√⎯⎯⎯sin x

6A | N.P.W.B.

44 If x = cy + bz, y = az + cx, z = bx + ay where, x, y, zare not all zero, then—(A) a2 + b2 + c2 – 2abc = 0(B) a2 + b2 + c2 + 2abc = 0(C) a2 + b2 + c2 + 2abc = 1(D) a2 + b2 + c2 – 2abc = 1

45. In a triangleABC cosec A (sin B· cos C + cos B· sin C)

is equal to—(A) 1 (B) bc(C) a/c (D) None of these

46. Degree and order of the differential equation

2 ⎝⎜⎛

⎠⎟⎞dy

dx 3 – y =

⎝⎜⎛

⎠⎟⎞d2y

dx2

3/2 are respectively of—

(A) order 2, degree 3 (B) order 1, degree 3(C) order 3, degree 2 (D) order 3, degree 3

47. If y = tan – 1 √⎯⎯⎯⎯1 – x2

x ‚ then

dydx

is equal to—

(A) √⎯⎯⎯⎯1 – x2 (B)1

√⎯⎯⎯⎯1 – x2

(C)– 1

√⎯⎯⎯⎯1 – x2(D) None of these

48. If →u = i × (

→a × i ) + j × (

→a × j ) + k × (

→a × k),

then—

(A)→u = 0 (B)

→u = 2

→a

(C)→u =

→a (D)

→u = i + j + k

49. If ƒ (x) =

1 x 1 + x

2x x(x – 1) (1 + x)x

3x(x – 1) x(x – 1) (x – 2) (1 + x) x(x – 1)

,

then ƒ (100) equal to—

(A) 0 (B) 1

(C) 100 (D) None of these

50. If A =

⎣⎢⎢⎢⎢⎡

⎦⎥⎥⎥⎥⎤2 0 0

0 2 0

0 0 2

, then A5 equal to—

(A) 10A (B) 4A

(C) 5A (D) 16A

51. For all values of A, B, C and P, Q, R the value of

⎪⎪⎪⎪⎪

⎪⎪⎪⎪⎪cos (A – P)

cos (B – P)cos (C – P)

cos (A – Q)cos (B – Q)cos (C – Q)

cos (A – R)cos (B – R)cos (C – R)

is—

(A) 0 (B) cos A.cos B.cos C

(C) sin A.sin B.sin C (D) cos P.cos Q.cos R

52. If →a = i + j – k,

→b = i – j + k,

→c = i – j – k , then

→a × (

→b ×

→c ) is—

(A) i – j + k (B) 2i – 2j

(C) 3 i – j + k (D) 2 i + 2j – k

53. In a triangle ABC2 cos A

a +

cos Bb

+ 2 cos C

c =

abc

+ bca

, then the value

of the angle A is—(A) 30° (B) 60°(C) 45° (D) 90°

54. The value of tan ⎣⎢⎢⎡

⎦⎥⎥⎤cos – 1

⎝⎜⎛

⎠⎟⎞4

5 + tan – 1 ⎝⎜⎛

⎠⎟⎞2

3 is equal

to—

(A)617

(B)716

(C)176

(D) None of these

55. ∫π/2

0

sin8 x · dx is equal to—

(A)21π128

(B)35π256

(C)63π512

(D) None of these

56. The area of the parallelogram whose adjacent sidesare i – 2j + 3k and 2i + j – 4k is—

(A) √⎯⎯14 (B) √⎯⎯21

(C) 5 √⎯ 6 (D) 7 √⎯ 6

57. For any 2 × 2 matrix A, if A (adj A) = [ ]10 0

010 ,

then | A | is equal to—(A) 0 (B) 10(C) 20 (D) 100

58. The equation of the locus of a point equidistantfrom the points (a1, b1) and (a2, b2) is (a1 – a2)x +(b1 – b2)y + c = 0, then the value of c is—

(A) a12 – a2

2 + b12 – b2

2

(B) √⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯a12 + b1

2 – a22 – b2

2

(C)12 (a1

2 + a22 + b1

2 + b22)

(D)12 (a2

2 + b22 – a1

2 – b12)

59. If →a = i + 2j + 3k,

→b = – i + 2j + k and

→c =

3 i + j , then (→a + t

→b ) is ⊥r to

→c , the value of t

is—

(A) 3 (B) 4(C) 5 (D) 8

N.P.W.B. | 7A

60. A bag contains 5 brown and 4 white socks. A manpulls out two socks. The probability that they are ofthe same colour is—

(A) 1/6 (B) 5/108

(C) 4/9 (D) 5/18

61. The equation of a circle is 9x2 + y2 = 4 (x2 – y2 – 2x),then its centre is—

(A) ⎝⎜⎛

⎠⎟⎞

– 45 ‚ 0 (B)

⎝⎜⎛

⎠⎟⎞4

5 ‚ 0

(C) ⎝⎜⎛

⎠⎟⎞

– 45 ‚

35 (D)

⎝⎜⎛

⎠⎟⎞4

5 ‚ – 35

62. A differential equation dydx

= y tan x – 2 sin x has an

integrating factor—(A) cos x(B) e ∫ tan x · dx

(C) e – ∫ sin x · dx

(D) e ∫ cos x · dx

63. There are 3 Mathematics, 4 Physics and oneChemistry books and they are placed in a self. Theprobability that the books of the same subject isplaced together is—

(A)140

(B)3

140

(C)970

(D) None of these

64. The equation of the projection of the line x – 1

2

= y – 2

1 =

z – 33

on the plane x + y + z – 1 = 0 are—

(A) x + y + z – 1 = 0 = 2x – y – z + 3(B) x + y – z – 1 = 0 = x + 2y – z + 3(C) 2x – y + 3z – 1 = 0 = x + y + z + 1(D) x + 2y – 3z = 0 = x + y + z + 1

65. If dydx

= x √⎯⎯⎯⎯⎯2x2 + 3 and t = 2x2 + 3, then dydt

is equal

to—

(A)t2

(B)t4

(C)t2

4(D) None of these

66. From the top of a light house, 60 metres high withits base at the sea level, the angle of depression of aboat is 15°. The distance of the boat from the foot ofthe light house is—

(A) ⎝⎜⎜⎛

⎠⎟⎟⎞√⎯ 3 – 1

√⎯ 3 + 1 60 m (B)

⎝⎜⎜⎛

⎠⎟⎟⎞√⎯ 3 + 1

√⎯ 3 – 1 60 m

(C)√⎯ 3 + 1

√⎯ 3 – 1 m (D) None of these

67. If 3 is the mean and 32 is the S.D. of a binomial

distribution, the distribution is—

(A) ⎝⎜⎛

⎠⎟⎞1

4 + 34

12(B)

⎝⎜⎛

⎠⎟⎞1

3 + 23

12

(C) ⎝⎜⎛

⎠⎟⎞1

5 + 45

12(D) None of these

68. If Aα = ⎣⎢⎢⎡

⎦⎥⎥⎤cos α sin α

– sin α cos α , then Aα + β equals—

(A) Aα + Aβ (B) Aα – Aβ

(C) Aα·Aβ (D) None of these

69. sin 12°. sin 48°. sin 54° is equal to—

(A) 1 (B)14

(C)18

(D) None of these

70. The mode and median of the following data 82, 98,73, 71, 43, 82 and 90 are respectively—

(A) 82, 80 (B) 82, 78

(C) 82, 82 (D) None of these

71. If the function ƒ (x) = x

x – 1 , express ƒ (3x) in term

of ƒ (x)—

(A)3 ƒ (x)

3 ƒ (x) – 1(B)

3 ƒ (x)3 ƒ (x) – 3

(C)3 ƒ (x)

2 ƒ (x) + 1(D) 3 ƒ (x) – 1

72. If limx → 1

ƒ (x) – 2ƒ (x) + 2

= 0, then limx → 1

(x) is equal to—

(A) 1 (B) – 1

(C) – 2 (D) 2

73. The maximum value of 1

√⎯ 2 (sin x – cos x) is—

(A) 1 (B) √⎯ 2

(C)1

√⎯ 2(D) 3

74. At a point on a level plane a tower subtends anangle θ and a flag-staff a feet in length at top of thetower subtends an angle φ. The height of the toweris—

(A)a sin θ.cos φcos (θ + φ)

(B)a sin θ cos (θ + φ)

sin φ

(C)a cos (θ + φ)sin θ.sin φ

(D) None of the above

8A | N.P.W.B.

75. The standard deviation of the set of first n naturalnumbers is—

(A)n2 – 1

12(B)

n2 – 14n

(C)n2 + 1

4n(D)

n2 + 112

76. Directrix of the parabola y2 + 4x + 4y – 3 = 0 is—(A) 4x = 11y (B) 4x = 11(C) 11x = 4 (D) x = y

77. The distance between the planes2x – 2y + z + 3 = 0 and 4x – 4y + 2z + 5 = 0 is—

(A)13

(B)12

(C)16

(D)15

78. ∫cos x ·dx

(1 + sin x) (2 + sin x) is equal to—

(A) log 2 + sin x1 + sin x

(B) log (1 + sin x) (2 + sin x)

(C) log 1 + sin x2 + sin x

(D) None of the above

79. If A, B and C are independent events such that P(A)= 0·2, P(B) = 0·1 and P(C) = 0·4, then P(A ∪ B ∪C) equals—(A) 0·9 (B) 0·008(C) 0·560 (D) 0·568

80. The coefficient of x in the equation x2 + px + q = 0was taken as 17 in the place of 13 and its roots werefound to be – 2 and – 15. The roots of the originalequation are—(A) 2, 15 (B) 10, 3(C) – 10, – 3 (D) – 2, 15

81. The angle between the vectors→a = 2i + j – 3k,

→b = 3i – 2j – k is—

(A) π (B)π2

(C)π3

(D) None of these

82. A population grows at the rate of 8% per year. Howlong does it take for the population to triple usedifferential equation for it ?

(A)252

log 9 (B)252

log 3

(C) 25 log 3 (D)253

log 3

83. The conversion of (11101)2 in decimal system is—

(A) 31 (B) 28(C) 29 (D) None of these

Directions—(Q. 84) The people of various religionof a city are represent by the following Pie chart—

Total number of people of the city is 28800. Answerthe question on the basis of the Pie chart.

84. How many number of Muslim people is less thanthe number of Hindu people—

(A) 3,200 (B) 2,400

(C) 4,800 (D) None of these

85. If loge ⎝⎜⎛

⎠⎟⎞a + b

2 = 12 (loge a + loge b), then relation

between a and b will be—

(A) a = b (B) a = b2

(C) 2a = b (D) a = b3

86. Equation of the circle, which passes through theorigin and cuts orthogonally each of the circles x2 +y2 – 8y + 12 = 0 and x2 + y2 – 4x – 6y – 3 = 0, is—

(A) x2 + y2 + 6x + 3y = 0

(B) x2 + y2 + 3x – 6y = 0

(C) x2 + y2 + 6x – 3y = 0

(D) x2 + y2 – 3x + 6y = 0

87. ∫∞

– ∞

1

x2 + 1 · dx is equal to—

(A) ∞ (B)π2

(C) 0 (D) π

88. ∫2

1

1x2 e

– 1/x · dx is equal to—

(A)1

√⎯ e +

1e

(B)1

√⎯ e –

1e

(C)1

√⎯ e –

14

(D) None of these

89. The unit vector which is perpendicular to the

vectors →a = 2i + j + k and

→b = 3i + 4j – k is—

(A)i + j + k

√⎯ 3(B)

– i + j – k

√⎯ 3

(C)i – j + k

√⎯ 3(D)

– i + j + k

√⎯ 3

N.P.W.B. | 9A

90. The angle of intersection of the sphere x2 + y2 + z2 +2x – 2y + 6z + 2 = 0 and x2 + y2 + z2 = 4

(A)π2

(B)π3

(C) cos – 1 13

(D) cos – 1 16

91. Find the linear inequation for which the shaded areain given figure is the solution set is—

0 1

2

4

52 4

6

8

10

Y

3X

(A) 4x + y ≤ 9 (B) 4x + y ≤ 9

3x + 2y ≤ 12 3x + 2y ≥ 12

x , y ≤ 0 x, y ≥ 0

(C) 4x + y ≤ 9 (D) 4x + y ≤ 9

3x + 2y ≤ 12 3x + 2y ≥ 12

x, y ≥ 0 x, y ≤ 0

92. An office has 5 computers that are connectedtogether in a linear network (i.e., they will bearranged in 5 positions on one straight connectingcable, see figure below) communications can goeither way along the cable. In how many differentways they can be arranged ?

A B C D E

(A) 40 (B) 50

(C) 60 (D) 80

93. ∫π/2

0

cos (log x)

x · dx is equal to—

(A) 2 sin ⎝⎜⎛

⎠⎟⎞

log π2 (B) cos

⎝⎜⎛

⎠⎟⎞

log π2

(C) sin ⎝⎜⎛

⎠⎟⎞

log π2 (D) sin

π2 – log

π2

94. If →a ,

→b ,

→c ,

→d be the position vectors of

vertices to a parallelogram, then—

(A) →a +

→c =

→b +

→d

(B) →a +

→b =

→c +

→d

(C) →a +

→d =

→b +

→c

(D) →b –

→a =

→d –

→c

95. limx → 0 √⎯⎯⎯⎯1 + x – √⎯⎯⎯1 – x

x is given by—

(A) 0 (B) – 1

(C) 1 (D)12

96. Solution of cos x · dydx

+ y sin x = 1 is—

(A) y sec x · tan x = C(B) y sec x = tan x + C(C) y tan x = sec x + C(D) y tan x = sec x · tan x + C

97. The value of limn → ∞ ⎝⎜⎛

⎠⎟⎞en

π 1/n

is—

(A) 1 (B) 1/π(C) e (D) ∞

98. Chord of curvature of the curve y = a log sec ( )xa

parallel to y-axis is—

(A) a sec xa

(B)12 a

(C) a cos xa

(D) 2a

99. The standard deviation of a set of values is 2·5. Now16 is added to each of the given set of value. Thestandard deviation of the new set of values is—(A) 2·5 (B) 6·5(C) 18·5 (D) None of these

100. →a × (

→b +

→c ) +

→b × (

→c +

→a ) +

→c × (

→a +

→b ) is

equal to—

(A) 2→a · (

→b ×

→c ) (B) 3

→a × (

→b ×

→c )

(C) [→a ·

→b ·

→c ] (D) 0

101. Out of 50 student in a class, 10 were of the age of10 years, 14 were of the age of 11 years and theremaining of 12 years. Then A.M. of student is—(A) 11·5 (B) 11·24(C) 11·32 (D) None of these

102. The variance of the following number 15, 25, 5,10, 30 is—(A) 50·20 (B) 9·27(C) 430 (D) 86

103. If the probabilities that A and B will die within ayear are p and q respectively, then the probabilitythat only one of them will be alive at the end of theyear is—(A) p + q (B) p + q – 2pq(C) p + q – pq (D) p + q + pq

104. If n + 1C3 = 2nC2, then n =

(A) 3 (B) 4

(C) 5 (D) 6

Practice Work Book - NDA & NAExamination

Publisher : Upkar Prakashan ISBN : 9788174823007Author : Dr. H. P. Sharma &Dr. M. B. Lal

Type the URL : http://www.kopykitab.com/product/11851

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