MATHEMATICS Code No. 13 INSTRUCTIONS
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1 DO NOT OPEN THIS TEST BOOKLET UNTIL YOU ARE ASKED TO DO SO COMBINED COMPETITIVE (PRELIMINARY) EXAMINATION, 0 Serial No. MATHEMATICS Code No. A Time Allowed : Two Hours Maimum Marks : 00 INSTRUCTIONS. IMMEDIATELY AFTER THE COMMENCEMENT OF THE EXAMINATION, YOU SHOULD CHECK THAT THIS TEST BOOKLET DOES NOT HAVE ANY UNPRINTED OR TORN OR MISSING PAGES OR ITEMS, ETC. IF SO, GET IT REPLACED BY A COMPLETE TEST BOOKLET.. ENCODE CLEARLY THE TEST BOOKLET SERIES A, B, C OR D AS THE CASE MAY BE IN THE APPROPRIATE PLACE IN THE RESPONSE SHEET.. You have to eter your Roll Number o this Your Roll No. Test Booklet i the Bo provided alogside. DO NOT write aythig else o the Test Booklet. 4. This Booklet cotais 00 items (questios). Each item comprises four resposes (aswers). You will select oe respose which you wat to mark o the Respose Sheet. I case you feel that there is more tha oe correct respose, mark the respose which you cosider the best. I ay case, choose ONLY ONE respose for each item. 5. I case you fid ay discrepacy i this test booklet i ay questio(s) or the Resposes, a writte represetatio eplaiig the details of such alleged discrepacy, be submitted withi three days, idicatig the Questio No(s) ad the Test Booklet Series, i which the discrepacy is alleged. Represetatio ot received withi time shall ot be etertaied at all. 6. You have to mark all your resposes ONLY o the separate Respose Sheet provided. See directios i the Respose Sheet. 7. All items carry equal marks. Attempt ALL items. Your total marks will deped oly o the umber of correct resposes marked by you i the Respose Sheet. 8. Before you proceed to mark i the Respose Sheet the respose to various items i the Test Booklet, you have to fill i some particulars i the Respose Sheet as per istructios set to you with your Admit Card ad Istructios. 9. While writig Cetre, Subject ad Roll No. o the top of the Respose Sheet i appropriate boes use ONLY BALL POINT PEN. 0. After you have completed fillig i all your resposes o the Respose Sheet ad the eamiatio has cocluded, you should had over to the Ivigilator oly the Respose Sheet. You are permitted to take away with you the Test Booklet. DO NOT OPEN THIS TEST BOOKLET UNTIL YOU ARE ASKED TO DO SO EIJ-4986-A [Tur over
2 ROUGH WORK EIJ-4986-A
3 . If A = {, y) + y = 5} ad B = {, y) + 9y = 44} the A Bcotais : Oe poit Two poits Three poits (D) Four poits. The umber of subsets of a set cotaiig elemets is : (D). 0 teachers of a school either teach Maths or Physics. of them teach Maths while 4 teach both the subjects. The umber of teachers teachig Physics oly is : If a relatio R is defied o the set Z of itegers as follows : Domai (R) = {, 4, 5} {0,, 4, 5} { 0, ±, ± 4, ± 5}. The 5. If R is a relatio o a fiite set havig elemets, the the umber of relatios o A is : (D) 6. R is a relatio o the set Z of itegers ad it is give by The R is : (a,b)( Z, Zy) + (4 R i)z a y b. + Refleive (4 + = i) 5. Z ad + 5 Trasitive = 0 Refleive ad Symmetric Symmetric ad Trasitive (D) A equivalece relatio 7. The equatio represets a circle of radius : If Z, Z, Z are comple umbers such that : Z = + is : Z = Z = + + = the Z + Z Z Z Z Z Equal to Less tha Greater tha (D) Equal to 9. The locus of poit Z satisfyig Re(Z ) = 0 is : A pair of straight lies A rectagular hyperbola A circle EIJ-4986-A [Tur over
4 0. If Zr r r = cos + i si,r = 0,,,,4 thez0 Z Z Z Z If α, β, γ are the roots of the equatio = 0. The ( α +β) + ( β+ γ ) + ( γ + α) = 4 (D) 5. Let A, G ad H be the Arithmetic mea, Geometric mea ad Harmoic mea of two positive umbers a ad b. The quadratic equatio whose roots are A ad H is : A (A + G ) + AG = 0 A (A + H ) + AH = 0 H (H + G ) + HG = 0. G is a group uder 7 where G = {,,, 4, 5, 6}. If 5 7 = 4 the = (D) 5 4. I the group G = {,, 7, 9} uder multiplicatio module 0, ( 7 - ) - is equal to : (D) 5. The idetity elemet i the group multiplicatio is : M = 0 ad is real with respect to matri If a b = a + b, the the value of ( 4 5) is : (4 + 5 ) + (4 + 5) (D) ( ) EIJ-4986-A 4
5 7. I Z, the set of all itegers, the iverse of -7 with respect to defied by all a,b Z is : for 8. The uits of the field F = {0,, 4, 6, 8} uder are : {0} {, 4, 6, 8} F 9. ( Z,, ) is a field if ad oly if is : Eve Prime 0. The ideals of a field F are : Oly {0} Both {0} ad F. Every fiite itegral domai is : Not a field Vector space Odd Oly F Field. The order of i i the multiplicative group of fourth roots of uity is : 4 a.b b= ad = 0 a + b + 7 (D) 0 0. The o zero elemets a, b of a rig (R, +,.) are called zero divisors if : a.b = 0 (D) 4. If the rig R is a itegral domai the : R[] is a field R[] is ot a itegral domai R[] is a itegral domai 5. The product of a eve permutatio ad a odd permutatio is : Eve Odd Neither ever or odd EIJ-4986-A 5 [Tur over
6 6. If : 0 i i (D) Noe of the above 7. If AB = A ad BA = B where A ad B are square matrices the : A = A ad B = B A A ad B = B A = A ad B B (D) A A ad B B a If A = 0 a 0, the the value of adj A is : 0 0 a a 7 a 9 a 6 (D) a 9. If A =, the adj (adj A) is : (D) 4 cosθ siθ T 0. If A =, ad A + A= I si cos where A T is the traspose of A ad I θ θ is the Uit matri. The : θ =, Z θ = +, Z EIJ-4986-A 6
7 4. The matri A = 4 is ilpotet of ide : The rak of the matri A = 0 is : 0 4 (D) Idetermiate. For what value of λ, the system of equatios + y + z = 6 + y + z =0 + y + λz = is Icosistet? λ = λ = λ = (D) λ = 4. If A is a matri ad B is its adjoit such that B = 64, the A = 64 ± 64 ± 8,the + A + A + A +... (D) If A = 0, the + A + A equals : A ( A ) - ( + A ) - 6. If A = equals to : (D) 4 0 EIJ-4986-A 7 [Tur over
8 7. If s = a + b + c the the value of s + c a b = c s+ a b is : c a s+ b s s s (D) s 8. 4 lim is equal to : log 4 log 4 9. The value of lim ( + )( + ) is : l l 40. e d 0 lim 4 = e 0 (D) if 4. The fuctio f () = if Cotiuous ad differetiable at = Neither cotiuous or differetiable at = Cotiuous but ot differetiable at = (D) Noe of the above is : = EIJ-4986-A 8
9 si 4. Let f () = 5 K,, 0 = 0. If f() is cotiuous at = 0, the the value of K is : 5 (D) 0 f ( ) f ( ) 4. If f() is differetiable ad strictly icreasig fuctio, the the value of lim is : 0 f ( ) f (0) 0 (D) 44. The umber of poits at which the fuctio f() = + + does ot have a derivative i the iterval [ 4, 4 ] is : 45. If f() satisfies the coditios of Rolle s theorem i [, ] ad f() is cotiuous i [, ], the 5 f ( ) d is equal to : 0 (D) 46. Let f () = e, [0,], the a umber c of the Lagrage s mea value theorem is : log e (e ) log e (e+ ) 47. The maimum value of y subject to + y = 8 is : (D) The series < represets the fuctio : 4 si cos ( + ) (D) log ( + ) EIJ-4986-A 9 [Tur over
10 EIJ-4986-A Epasio of si i powers of is : The equatio of taget to the curve = t 4, y = t + at the poit where t = is : y 9 = 0 y + 9 = 0 + y 9 = 0 (D) + y + 6 = 0 5. If the ormal to the curve y = 5 at the poit (, ) is of the form a 5y + b = 0. The a ad b are : 4, 4 4, 4 4, 4 (D) 4, 4 5. The least value of 0, 8 () f > + = is : The radius of curvature for the curve b a r b a p + = is : b a p b p a p b a (D) a b p
11 54. The cetre of curvature of the curve y = at (0,0) is : 0,,, The radius of curvature of the curve r = a si θ at origi is : a a (D) a 56. The asymptote parallel to co-ordiate aes of the curve ( + y ) ay = 0 is : y a = 0 y + a = 0 a = 0 (D) + a = The asymptote of the curve y = e is give by : y = 0 = 0 y = e (D) = e a 58. For the curve y ( + ) = ( ), the origi is a : Node Cusp Cojugate poit 59. The curve y = has a poit of ifleio at : = = = (D) = 60. The curve y = log is : Cocave upwards i (0, ) Cocave dowwards i (0, ) Cocave upwards i (, ) (D) Cocave dowwards i (, ) 6. The poits of ifleio o the curve = (log y) are : (0, ) ad (8, e ) (, 0) ad (8, e ) (0, ) ad (e, 8) (D) (, 0) ad (e, 8) 6. The graph of Circle Cycloid t t =, y = is a : + t + t Ellipse EIJ-4986-A [Tur over
12 6. The umber of leaves i the curve r = a si 5θ are : Two Five Te u 64. If u = f (y+ a) + φ (ya) the = u y u a y u u a (D) a y y z z 65. If Z = log ( + y ) the + y = y 0 (D) 66. If y = si + si + dy si the ( y ) is give by : d si cos ta (D) cot 67. The series is : Coditioally Coverget Diverget 68. The series is : 4 Coditioally Coverget Oscillatory Absolutely Coverget (D) Noe of the above Absolutely Coverget (D) Noe of the above 69. The series = Coverget Oscillatory ( log ) is : Diverget EIJ-4986-A
13 70. The series = Coverget Oscillatory is : Diverget ( + ) 7. The series..... = is Coverget if : < < (D) 4 < 7. d = + 0 si 7. d = 0 si cos + < lim = (D) 74. log e log e 6 log e 75. The etire legth of the curve + y = a is : 8a 4 a 6a (D) 8a EIJ-4986-A [Tur over
14 76. The perimeter of r = a ( + cosθ) is : a 4a a (D) 8a 77. The legth of oe arch of Cycloid = a(θ + siθ) y = a( cosθ) is : a 4a 8a (D) a 78. The area bouded by the curve y =, & ais ad the ordiates =, = is equal to : 4 (D) 8 y 79. The area of the ellipse + = is : a b ab ab ab 80. The area bouded by the curve y = ad = y is give by : 0 (D) 8. The whole area of the curve r = a cosθ is : a a a (D) a 8. The lie y = + is revolved about -ais. The volume of solid of revolutio formed by revolvig the area covered by the give curve, -ais ad the lies = 0, = is : 9 (D) 7 EIJ-4986-A 4
15 8. The volume geerated by revolutio of the ellipse about major ais is [assume that a > b] : 4ab 4a b 4a b 84. The surface of the solid of revolutio about -ais of the area bouded by the curve y =, -ais ad the ordiates = 0 ad = is equal to : 4 9 (D) The value of si d = : y + = 6 a si b d (D) 86. = 0 cos cos (D) Does ot eist 87. Order ad degree of the differetial equatio / dy d y 4 d d + = order, degree order, degree order, degree (D) order, degree are respectively : EIJ-4986-A 5 [Tur over
16 dy 88. If P, Q are fuctios of, the solutio of differetial equatio + Py = Q is : d Pd = Pd ye Qe d + c Pd y = Qe d + C Pd Pd y = e Qe d + C 89. The differetial equatio of the form Auiliary equatio Clairaut s equatio dy d + Py = Qy where P ad Q are fuctios of, is called : Bessel s equatio (D) Beroulli s equatio 90. The solutio of (y cos + ) d + si dy = 0 is : y si = c y si = c y + si = c (D) + y si = c 9. If at every poit of a certai curve the slope of the taget equals, the curve is : y A straight lie A parabola A circle (D) A ellipse 9. The itegratig factor for the differetial equatio ( y y ) d ( y) dy is give by : y y (D) y 9. The geeral solutio of P = log (p y) is : y = c e c y + = log c y + c = e c (D) y + c = e 94. The geeral solutio of a differetial equatio of first order represets : A family of surfaces A pair of curves i y plae A family of curves i y plae EIJ-4986-A 6
17 dy 95. The sigular solutio of the differetial equatio P + P y = 0 is where P = : d 7y + 4 = 0 y = 4a + y = a 96. The orthogoal trajectory of the family of curves ay = is : y = costat + y = costat + y = costat (D) + y = costat d y dy 97. Solutio of + y = 0 is : d d c e + c e c e + c e c e + c e 98. The geeral solutio of the differetial equatio D (D + ) y = e is : y = c + c + (c + c 4 )e y = c + c e + (c + c 4 )e + 4 e y = c + c + (c + c 4 )e + 4 e 99. The particular itegral of the differetial equatio (D + ) (D ) y = e is : e e 8 e y z 00. The equatio of the cylider whose geerators are parallel to the lie = = ad whose guidig curve is + y =, z = 0 is give by : (z ) + (z + y) = 9 ( + z) + (y z) = 9 ( z) + (y + z) = 9 (D) (z + ) + (y ) = 9 EIJ-4986-A 7
18 ROUGH WORK EIJ-4986-A 8
19 ROUGH WORK EIJ-4986-A 9
20 ROUGH WORK EIJ-4986-A 0 75
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