NDA/NA National Defence Academy/NavalAcademy PracticeSet1 Instructions (i) This practice set contains 270 questions in all comprising Paper 1 (Mathematics 120 questions) and Paper 2 (Part-A English 50 questions) and (Part-BGeneralAbility100questions) (ii) Candidate/Studentshouldattemptallquestions. (iii) Each question is followed by four alternative suggested answers or completions. In each case, you are required to select the one that correct the answer. (iv) For each questions for which a wrong answer has been given by the candidate, only third (0.33) of the marks assigned to the questions will be deductedaspenalty. S.No. Contents No.ofQuestions MaxMarks TimeAllotted Paper1 Mathematics 120 300 2:30hrs Paper2 EnglishandGeneralAbility 150 600 2:30hrs Paper 1 Mathematics 1. The number of solutions to the equation z2 + z = 0 6. Thedegreeofthedifferentialequation is dz dz 4z= y + sin is (a) 1 (b) 2 (c) 3 (d) 4 dy dy 2. If|z2−1 | = | z 2 | + 1,then z liesona (a) 0 (a) realaxis (b) imaginaryaxis (b) 1 (c) non-realaxis (d) Noneofthese (c) 2 3. If three complex number are in arithmetic (d) Cannotbedetermined progression,thentheylieonincomplexplane. dy 3x− 4 y − 2 7. Thesolutionoftheequation = is (a) circle (b) parabola dx 3x− 4 y − 3 (c) straightline (d) Noneofthese (a) (x− y )2 + C = log(3 x − 4 y + 1 ) 4. What is the number of solutions of the equation (b) x− y + C =log(3 x − 4 y + 4 ) 2 − + = x5| x | 6 0? (c) x− y + C =log(3 x − 4 y − 3 ) (a) 2 (b) 0 (c) 1 (d) 4 (d) x− y + C =log(3 x − 4 y + 1 ) dx 5. Thedifferentialequation y =y − 1, y( )0= 1has 8. The orthogonal trajectory of the family of dy parabolas x2 =4 µ y, µ beingparameteris (a) uniquesolutions (b) twosolutions (a) x2+ y 2 = C (b) 2x2+ y 2 = C (c) infinite number of solutions (d) no solutions (c) x2+2 y 2 = C (d) x2−2 y 2 = C 2 PracticeSet1 10PracticeSets for NDA/NAExam 2− 3 19. If roots of the equation ax2 + bx + c = 0 are in the 9. Theinversematrixof is − λ 4 2 ratio λ :1,thenthevalueof is 1 2 3 1 2 4 ()λ + 1 2 (a) − (b) − bc ca 8 4 2 8 3 2 (a) (b) a b 1 2 3 1 2 4 ca bc (c) (d) (c) (d) 8 4 2 8 3 2 b2 a2 10. If A isasquarematrix,then()AA+ T is 20. If x=3 + i, then the value of the expression (a) nullmatrix (b) anidentitymatrix x3−3 x 2 − 8 x + 15 is (c) asymmetricmatrix (d) askew-symmetricmatrix (a) −15 (b) −1 11. A matrix which is symmetric and skew-symmetric (c) 8 (d) 3 21. 2 − + = is If log10 (x6 x 45 ) 2,thenthevalueof x (a) orthogonalmatrix (b) idempotentmatrix (a) 6,9 (b) 10,5 (c) nullmatrix (d) Noneofthese (c) 9,5 (d) 11, − 5 12. 64 Whatisthenumberofdigitsin( )20 ? 22. If ABC, and are three sets, then ABC×() ∪ is = (if log10 2 0301. ) equalto (a) 81 (b) 82 (c) 83 (d) 84 (a) ()()ABAC× ∩ × (b) ()()ABAC∩ × ∩ 13. Which one of the following statement is correct? (c) ()()ABAC× ∪ × (d) Noneofthese Thenumbers log6 7, log42 7, log294 7 in 23. The relation ‘less than’ in the set of natural number (a) AP (b) HP is (c) GP (d) Noneofthese (a) symmetric (b) transitive (c) reflexive (d) equivalencerelation 2 − 1+x − 1 14. If y = tan 1 ,then 24. If R′ is the set of all straight lines in a plane and the x relation R is defined by aRb if a⊥ b for a, b∈ R ′, then dy = dy = 1 (a) 1 (b) (a) R isreflexive (b) R issymmetric dx = dx = 2 x 0 x 0 (c) R isanti-symmetric (d) Noneofthese dy = (c) 0 (d) Noneofthese 25. If the AM and HM of two numbers are 25 and 4, dx = x 0 thentheirGMis 2 (a) 10 (b) 9 −1 2 d y 15. If x= sin ( t ) and y=log (1 − t ), then is (c) 16 (d) Noneofthese dx 2 cosx− sin x equalto 26. lim isequalto π π 2 − 2 x → − (a) (b) 4 x 1−t 2 1−t 2 4 1 1 (c) (d)Noneofthese (a) −1 (b) − (c) 2 (d) − 2 1−t 2 2 π /2 16. d x+ a 27. Thevalueof ∫ log tan x dx ()a x isequalto 0 dx π x a π − − − a x (a) loge 2 (b) loge 2 (a) x()() ax1+ a x a 1 (b) + 2 loga log x (c) 0 (d) e x a (c) alog a+ x (log x ) (d) Noneofthese dx 28. ∫ isequalto d 2 2+ 2 2 17. (cosx° ) isequalto acos x b sin x dx −1 a + 1 −1 b + π (a) tan tan x C (b) tan cot x C (a) −sin x ° (b) −sin x ° b ab a 180 1 − b 180 (c) tan1 tan x + C (d) Noneofthese (c) −sin x ° (d) Noneofthese π ab a 18. 1 3 2 If the sum of ‘n’ terms of a series is a quadratic 29. If A = and A− KA −5 I = 0, then K is equal expressionin‘n’,thentheseriesis 3 4 (a) GP (b) HP to (c) AP (d) Noneofthese (a) 5 (b)3 (c)7 (d)4 10PracticeSets for NDA/NAExam PracticeSet 1 3 30. Thevalueofthedeterminant x 1 3 −a2 − ab ac 40. If the determinant 0 0 1 = 0, then x is equal to 2 ba− b bc is 1x 4 − 2 − − ac bc c (a) 2or 2 (b) 3or 3 (c) 1or −1 (d) 3or4 2 2 2 (a) abc (b) a b c 41. The class marks of a distribution are 6, 10, 14, 18, (c) 1 (d) 0 22,26,30,thentheclasssizeis 31. Solutionofthedifferentialequation (a) 4 (b) 2 sec2x tan y dx+ sec 2 y tan x dy = C is (c) 5 (d) 7 42. (a) tanx+ tan y = C (b) tanx tan y= C A graph representing cummulative frequency is (c) tanx− tan y = C (d) tanx sec y= C termedas (a) ogive (b) barcharts 32. Thesolutionofthedifferentialequation (c) piecharts (d) histogram dy y + = x n is 43. A committee of 3 is to be chosen from a group dx x consisting of 4 men and 5 women. If the selection y (a) =C (b) y+ x = C is made at random, then the probability that two of x + themembersofcommitteearemen,isgivenby x n 2 x n (c) yx = + C (d) y = + C (a) 3/14 (b) 5/14 n + 2 n + 2 (c) 1/21 (d) 8/21 2 33. The necessary condition for a maximum or 44. − x2 dx Thevalueoftheintegral ∫− 1| | minimumofafunction f() x is 2 df dn f (a) 2 (b) 4 (a) ≠ 0 (b) = + ve (c) 0 (d) 1 dx dx 2 45. d2 f df The value of the scalar triple product (c) = − ve (d) = 0 []a b+ c a +b + c is dx 2 dx (a) 0 (b) 1 2 2 π /2 sinx− cos x (c) |a ||b || c | (d) Noneofthese 34. Thevalueof ∫ dx is 0 sin3x+ cos 3 x 46. Theareaofthetrianglewithvertices (a) 0 (b) 1 i+ j , i+ k , j+ k is 1 1 (c) (d) 3 2 3 (a) 3 (b) 2 35. 2− + 2 + − + = 1 2 The conic 17x 12 xy 8 y 46 x 28 y 170 (c) (d) represents 3 3 (a) circle (b) ellipse 47. Distancebetweentheplanes (c) parabola (d) hyperbola 2x− 2 y + z + 1 = 0 and 4x− 4 y + 2 z + 3 = 0 is 36. Theexpression (a) 1/2 (b) 1/3 (c) 1/6 (d) 0 cos(ABABABAB− )cos( + ) − sin( − )sin( + ) 48. Themaximumvalueof sinx+ cos x is isequalto (a) 1 (b) − 2 (c) 2 (d) 1+ 2 (a) 2sinAB cos (b) sin 2A (c) cos 2A (d) 2 cosAB sin 49. Thevalueof[,,]a− b b − c c − a 1 1 (a) 0 (b) 1 37. If tan A = and tan B = , then what is the value (c) 2 (d) Noneofthese 2 3 of tan (2AB+ )? 50. The number of vectors of unit length perpendicular (a) 1 (b) 2 (c) 3 (d) 4 to the vector a = (,,)1 1 0 and b = (,,)0 1 1 is 38. ω (a) one (b) two If is a complex cube root of unity, then the value (c) three (d) Noneofthese of ω99+ ω 100 + ω 101 is 51. A tower of height 15 m stands vertically on the (a) 1 (b) −1 ground. From a point on the ground the angle of (c) 3 (d) 0 elevation of the top of the tower is found to be 30°. 39. Thenumber1753inbinarynotationis What is the distance of the point from the foot of the tower? (a) ()11011011101 2 (b) ()11011111001 2 (a) 15 3 m (b) 10 3 m (c) 5 3 m (d) 30m (c) ()11011011001 2 (d) Noneofthese 4 PracticeSet1 10PracticeSets for NDA/NAExam 52. A husband and wife appear in an interview for two 63. If x=1 + a + a2 +… ∞,()a < 1, vacancies at the same post. The probability of = + +2 +… ∞ < 1 y1 b b,() b 1, husband selection is and that of wife selection is 5 thenthevalueof1 +ab + a2 b 2 + … ∞ is 1 xy xy xy xy . What is the probability that only one of them will (a) (b) (c) (d) 3 x+ y −1 x+ y + 1 x− y −1 x− y + 1 beselected. 1 2 64. The sum of n terms of an AP be 3n2 + 5, then the (a) (b) 5 5 numberoftermswhichisequalto159? 3 4 (c) (d) (a) 13 (b) 21 5 5 (c) 27 (d) Noneofthese 53.