1. If * is the operation defined by ab = a b for a, b N, then (2 3) * 2 is equal to (A) 81 (B) 512 (C) 216 (D) 64 (E) 243 ANSWER : D

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Horace King
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1 . If * is the opertion defined by *b = b for, b N, then ( * ) * is equl to (A) 8 (B) 5 (C) 6 (D) 64 (E) 4. The domin of the function ( 9)/( ),if f( ) = is 6, if = (A) (0, ) (B) (-, ) (C) (-, ) (D) (, ) (E) (-, ). Let f( ) = nd g( ) =. The vlues of such tht g(f()) = f (g()) re (A) 0, (B), (C) 0, ± (D), ± (E) 0, ± 4. + If, f =, ϵ N, then the vlue of is equl to f () is equl to (A) (B) 4 (C) (D) (E) 5 5. If A\B = {, b}, B\A = {c, d} nd A B = {e, f}, then the set B is equl to (A) {, b, c, d} (B) {e, f, c, d} (C) {, b, e, f} (D) {c, d,, e} (E) {d, e,, b} 6. The function f : A B is given by f () =, ϵa, is one to one but not onto. Then (A) B A (B) A = B (C) A' B' (D) A B (E) A B = φ 7. + sin + i cos The principl rgument of the comple number z = + sin i cos is (A) (B) (C) 6 (D) (E) 4 ( + i)( + i)( 4 i) 8. If = + ib, then + b = ( i)( i)( + 4i) (A) (B) 5 (C) 44 (D) 8 (E) 9. Let z, w be two nonzero comple numbers. If z + iw = 0 nd rg (zw) =, then rg Z = (A) (B) (C) (D) (E) 4 6 8

2 0. i If z =, then Re(z ) + lm(z ) is equl to i (A) (B) - (C) (D) - (E). If z + < z, then z lies (A) On the -is (B) On the y-is (C) In the region < 0 (D) In the region y > 0 (E) In the region > y. If z =, then the gretest vlue of z is z (A) (B) (C) (D) 4 (E) 5. If the roots of the qudrtic eqution m n + k = 0 re tn 0 nd tn 0 then the vlue of m+ n+ k m is equl to (A) 0 (B) (C) (D) (E) 4 4. If α nd β re the roots of = 0, then β = (A) (B) (C) (D) 4 α α α α (E) α 5. If α nd α re the roots of the eqution c = 0, then the positive vlue of c is (A) (B) (C) 4 (D) 9 (E) 8 6. If one of the roots of the qudrtic eqution - b + = 0 is 6, then vlue of (A) 6 (B) 6 (C) 7 6 (D) 6 7. If the eqution (+) + 8 = 0 hs equl roots, then one of the vlues of is (A) -9 (B) -5 (C) - (D) (E) 9 8. If 6 th term of G.P. is, then the product of first terms of the G.P. is equl to (A) 5 (B) 04 (C) 048 (D) 56 (E) 9. If the produce of five consecutive terms of G.P. is 4 (E) 6 7, then the middle term is b is equl to

3 (A) (B) (C) 4 (D) 4 (E) 0. If,,, 4 re in A.P., then (A) 4 4 (D) (B) (E) (C) 4 =. If,,, 0 re in A.P. nd + 0 = 45, then,,, 0 is equl to (A) 90 (B) 900 (C) 50 (D) 450 (E) 70. Sum of the series () + ( + ) + ( + +5) + 4 ( ) ( ) is equl to (A) 85 (B) 05 (C) 5 (D) 05 (E) 05. In n A.P., the 6 th term is 5 nd th term is. Then the common difference is equl to (A) 4 (B) 0 (C) (D) 8 (E) 6 4. If the coefficients of nd 4 in the epnsion of ( + k) 9 re equl, then the vlue of k is (A) (B) (C) (D) (E) 5. The totl number of 7 digit positive integrl numbers with district digits tht cn be formed using the digits 4,, 7,,, 0, 5 is (A) 40 (B) 440 (C) 40 (D) 40 (E) If n P 4 = 5 ( n P 4 ), then the vlue of n is equl to (A) 5 (B) 6 (C) 7 (D) 8 (E) 9 7. The reminder when 06 is divided by 6, is (A) (B) 8 (C) 7 (D) (E) 6 8. If n C + n C = 6 C nd n C = n C,, then the vlue of is equl to (A) 5 (B) 4 (c) (D) 6 (E) ANSWER :C

4 8 8 k 9. If =, then the vlue of is equl to C 8 8 k= 0 Ck k= 0 k (A) (B) 9 (C) 6 (D) 8 (E) 6 0. If the squre of the mtri b is the unit mtri, then b is equl to b (A) + (B) (C) + (D) (E) +. If [ ] 0 5 = 0, then the vlues of re 0 0 (A), 5 (B) -, -5 (C), 6 (D) -, -6 (E), If A = 5, then the vlue of A is equl to (A) 0 (B) 6 (C) 64 (D) 400 (E) If A= 0 nd det(a) =, then the vlue of is (A) - (B) (C) (D) -8 (E) - 4. The coefficient of in the epnsion of the determinnt is (A) -0 (B) -8 (C) - (D) -6 (E) 8 i 5. Let A =. Then A 00 = i (A) 00 (B) 99 A (C) 98 A (D) A (E) A

5 The lest integer stisfying < is (A) (B) (C) (D) 4 (E) 5 7. If + 8, then the vlues of lie in the intervl (A) (-, -) (B) (-, 6) (C) (-, 7) (D) (-, ) (E) (6, ) 8. Let p : 57 is n odd prime number, q: 4 is divisor of. r : 5 is the LCM of nd 5 Be three simple logicl sttements. Which one of the following is true? p q r p q r (C) ( p qv ) (A) ( ) (B) ( ) (D) ( p q) r (D) p ( q r) 9. Let p, q, r be three simple sttements. Then ( p q) ( p r) (A) (~ p) ( q r) (B) (~ p) ( q r) (C) p ( q r) (D) p ( q r) (E) ( p q) r 40. If p : is prime number nd q ; one plus one is three, then the compound sttement It is not tht is prime number or it is not tht one plus one is three is (A) p q (B) ( p q) (C) p q (D) p q (E) p q 4. The vlue of sin + sin +sin 5 + sin 7 is equl to (A) 8 (B) 4 (C) (D) (E) 4. The vlue of sin5 cos5 0 0 is equl to (A) 4 (B) (C) (D) 4. If sin + cos =, then sin cos = (A) (B) (E) (C) (D) (E)

6 44. If tn θ = nd tn φ =, then tn (θ + φ)= (A) 4 (B) 4 (C) (D) (E) 45. The vlue of stisfying the eqution tn - + tn - = 7 tn- 4 is equl to (A) (B) - (C) (D) - (E) 46. If tn A tn B = nd cot B cot A = y, then cot(a-b) is (A) (B) (C) y y + y + (D) y (E) + y 47. If tn - + tn - y =, then cot - + cot - y is equl to (A) (B) (C) (D) (E) 48. If the orthocenter, centroid, incentre nd circumcentre concide in tringle ABC, nd if the length of side AB is 75 units, then the length of the ltitude of the tringle through the verte A is (A) units (B) units (C) 5 units (D) 5 units (E) 5 units 49. If A (, 4) nd B ( 6, 0) re two fied points nd if point P moves so tht APB is lwys right ngle, then the locus of P is (A) + y y + 5 = 0 (B) + y y - 5 = 0 (C) + y + 8-4y + 5 = 0 (D) + y - 8-4y - 5 = 0 (E) + y - 8-4y + 5 = The points (-, 0) nd (-, ) re the two etremities of digonl of prllelogrm. If (-6, 5) is the third verte, then the fourth verte of the prllelogrm is (A) (, -6) (B) (, -5) (C) (, -4) (D) (-, 4) (E) (, -5)

7 y 5. The slope of the stright line 0 4 = is (A) 5 (B) 5 (C) 5 (D) 5 (E) 4 5. If y-intercept of the line 4 y = 8 is thrice its -intercept, then the vlue of is equl to (A) 4 (B) 4 (C) - 4 (D) - 4 (E) - 5. The eqution of one of the stright lines pssing through the point (0, ) nd is t distnce of 5 units from the origin is (A) 4 + y = (B) + y = (C) +y = (D) 5 + 4y = 4 (D) y = The nerest point on the line + y - = 0 from the point (, -) is (A) (, 5) (B) ( 4, ) (C), -5) (D) (5, -) (E) 5, -) 55. The imge of the origin with respect to the line 4 + y = 5, is (A) (4, ) (B) (, 4) (C)(6, 8) (D) 4, 6) (E) (8, 6) 56. If the re of the circle 4 + 4y +8-6y + λ = 0 is 9 sq. units, then the vlue of λ is (A) 4 (B) -4 (C) 6 (D) -6 (E) The rdius of the circle pssing through the points (, ), (, 7) nd (5, ) is (A) 5 (B) 4 5 (C) (D) (E) If dimeter of the circle + y - -6y+6 = 0 is chord of nother circle C hving centre (, ), then the rdius of the circle C is (A) (B) (C) (D) 5 (E) 5 ANSWER :C 59. In the fmily of concentric circles ( + y ) = k, the rdius of the circle pssing through (, ) is (A) (B) 4 (C) (D) (E)

8 60. Let P be point on n ellipse t distnce of 8 units from focus. If the eccentricity is the distnce of the point P from the directri is (A) 5/8 (B) 8/5 (C) 5 (D) 8 (E) 0 4 5, then 6. If (-, 0) is the verte nd y-is is the directri of prbol, then its focus is t the point (A) (0, -6) (B) (-6, 0) (C) (6, 0) (D) (0, 0) (E) (, 0) 6. The foci o the ellipse 4 + 9y = re 5 (A) ±,0 (B) ±,0 (C) ± 5,0 (D) ± 5,0 6 (E) ± 5, The directri of prbol is + 8 = 0 nd its focus is t (4,). Then the length of the ltusrectum of the prbol is (A) 5 (B) 9 (C) 0 (D) (E) 4 ANSWER :E 64. If the eccentricity of the ellipse + 4y = 4, ( <4) is /, then its semi minor is is equl to (A) (B) (C) (D) (E) 65. y The hyperbol = psses through the point ( b 6,) nd the length of the ltus rectum is 8/5. Then the length of the trnsverse is is equl to (A)5 (B) 4 (C) (D) (E) 66. The ngle between nd b is 5 /6 nd the projection of on b 9 is then is equl to (A) (B) 8 (C) 0 (D) 4 (E) The direction cosines of the stright line given by the plnes = 0 nd z= 0 re (A),0,0 (B) 0,0, (C),,0 (D) 0,,0 (E) 0,, ANSWER :D 68. = ˆ i ˆj mkˆ 4 nd b = iˆ ˆj+ kˆ re colliner, then the vlue of m is equl to 7 7 (A) 7 (B) (C) (D) 7 (E)

9 69. Let = iˆ+ 5ˆj 7 kˆ, b = iˆ+ ˆj+ 5 kˆ..then ( 5 b).(4 5 b) = (A) 7 (B) 0 (C) (D) (E) 8 70 If + b c = 0 nd b + b c + c = λ b, then the vlue of λ is equl to (A)5 (B)4 (C) (D) (E) 4 7. If b. = 0 nd + b (A)0 (B) b (C) is equl to (D) b (E) b nd mkes n ngle of 60 o with b then 7. If + b nd b re perpendiculr nd b = iˆ 4ˆj+ kˆ, then is equl to (A) 4 (B) 9 (C) 9 (D) 9 (E) 7. The stright line r = ( iˆ+ ˆj+ kˆ) + α(iˆ ˆj+ 4 kˆ) meets the y plne t the point (A)(,, 0 (B)(,4,0) (C),,0 4 7 (D),,0 4 (E) 5,, The eqution of the plne pssing through (, 5, 7 ) nd prllel to the plne 5y + 7z + = 0, is (A).(ˆ 5 ˆ 7 ˆ r i j k) + 76 = 0 (B) r.(iˆ 5 ˆj+ 7 kˆ ) + 76 = 0 (C).(ˆ 5 ˆ 7 ˆ r i j+ k) + 75 = 0 (D) r.(iˆ 5 ˆj+ 7 kˆ ) + 65 = 0 (E) r.(iˆ 5 ˆj 7 kˆ ) + 55 = The ngle subtended t the point (,,) by the points P(,4,5) nd Q (,,), is (A) 90 o (B) 60 o (C) 0 o (D) 0 o (E) 45 o 76. If the two lines y z y z = = nd = = 4 4 re perpendiculr, then the vlue of is equl to (A) 4 (B) 5 (C) 5 (D) 4 (E)

10 + y+ z+ 77. If the line = = meets the plne + y + z = 4 t P, then the distnce between P 4 nd the origin is (A) 4 (B) 5 (C) (D) (E) The point of the intersection of the stright lines ˆ ˆ ˆ ˆ ˆ ˆ y+ 4 z 5 r = (i 4 j+ 5 k) + λ( i j+ k) nd = = is 7 (A) (, 4, 5) (B) (, 4, 5) (C) (, 4, 5) (D) (, 4, 5) (E) (, 4, 5) 79. The vector of the stright line (A) r = iˆ+ kˆ+ t( iˆ+ ˆj+ kˆ) (C) r = iˆ+ kˆ+ ( iˆ ˆj+ kˆ) (E) r = iˆ+ kˆ+ ( iˆ ˆj kˆ) y z = = is (B) r = iˆ kˆ+ t( iˆ ˆj kˆ) (D) r = iˆ ˆj+ t( iˆ ˆj kˆ ) 80. The stright line r = ( iˆ ˆj+ 5 k) + t(iˆ+ 5ˆj+ kˆ ) is prllel to the plne r.(iˆ+ ˆj kˆ ) = 5 Then the distnce the stright line nd the plne is (A)9/ 4 (B)8/ 4 (C)7/ 4 (D)6/ 4 (E)5/ Two fir dice re rolled. Then the probbility of getting composite number s the sum of the fce vlues is equl to (A) 7/ (B) 5/ (C) / (C)/4 (D)/4 (E) / 8. If the men of the numbers, b, 8,5,0 is 6 nd their vrince is 6.8, then b is equl to (A) 6 (B) 7 (C) (D) 4 (E) 5 8. In clss, in n emintion in Mthemtics, 0 students scored 00 mrks ech, students scored zero nd the verge of the remining students is 7 mrks. I f the clss verge is 76, then the number of students in the clss is (A) 44 (B) 40 (C) 8 (D) 4 (E) 84. A bg contins red, 4 white nd 5 blue blls. If two blls re drwn t rndom, then the probbility tht they re different colours is (A) 47/66 (B) / (C) 47/ (D) 47/ (D) 70/

11 85. There re 5 positive numbers nd 6 negtive numbers. Three numbers re chosen t rndom nd multiplied. The probbility tht the product being negtive number is (A) /4 (B) 7/ (C) 6/5 (D) 5/44 (E) 6/ cot The vlue of lim is equl to 0 cos ec (A) 4/ (B) ¾ (C) / (D) / (E) Let f() = cos if 0 cos if < 0 which one of the following sttements is not true? (A) f() is continuous t = (B) f() is continuous t = (C) f() is continuous t = (D) f() is continuous t = (E) f() is continuous t = 0 ANSWER :E n n P C 88. The vlue of lim is equl to n n (A) - 5/6 (B)5/6 (C) /6 (D) / 6 (E) / 89. If f() = + 5 nd g() =, then (f o g) ( ) is equl to (A) (B) (C) (D) (E) The period of the function f() = (4 ) is (A) (B) / (C) (D) /4 (E) /4 9. If + y = +y, then the vlue of dy t(,) is equl to d (A) (B) (C) 0 (D) (E) sin 9. If f() = then the vlue of ( ) f ( ) f( ) is (A) 0 (B) (C) (D) (D) 4

12 0 9. (0) f () f () f () f () If f()=, then f () is equl to 0 (A) (B) 0 (C) (D) 5 (E) f If f (4) = 5, g (4) =, f (4) g(4) = nd g(4) = 6, then (4) = g (A) 5/6 (B) /8 (C) /6 (D) /8 (E) 9/6 95. If the derivtive of ( 5)e t = 0 is -, then the vlue of is equl to (A) 8 (B) 5 (C) 5 (D) (E) ' 96. Let y = tn - (sec + tn). Then dy d = (A) /4 (B) / (C) sec + tn (D) sec (E) tn 97. If s = sec - ds nd t, then t is = = dt (A) (B) (C) (D) 4 (E) The minimum vlue of is (A) 4 (B) 5 (C) 6 (D) 7 (E) The slope of the curve y = e cos, (, ) is mimum t (A) = (B) = (C) = 4 (D) = 0 (E) = 00. If y = f() is continuous on [0,6], different on (0,6), f(0) = - nd f(6) = 6, then t some point between = 0 nd = 6, f ( ) must be equl to (A) 8 (B) (C) (D) 4 (E) 8 0. The eqution of the tngent to the curve y = t (,) is (A) 6 y = 0 (B) 6 y = 0 (C) 6 + y + = 0 (D) 6 y + = 0 (E) 6y = 0

13 0. Let f() = 5 4 +,. The point t which the tngent to the curve is prllel to the is, is (A) (, 4) (B) (, 9 ) (C) (, 4) (D) (, ) (E) ( 5) 0. Two sides of tringle re 8 m nd 5 m in length. The ngle between them is incresing t the rte 0.08 rd/sec. When the ngle between the sides of fied length is /, the rte t which the re of the tringle is incresing is (A) 0.4 m /sec (B) 0.8n /sec (C) 0.6m /sec (D) 0.04m /sec (E) 0.08 md /sec 04. If y = is strictly decresing function in the intervl (A) ( 5, 6) (B)(, ) (C) 5,6) (D)(, ) (E),) 05. m (sec ) (tn + tn ) d is equl to (A) sec m+ + C (B) tn m+ + C (C) (D) m+ tn + C m + (E) m + sec + C m + m+ sec + C m sin + 0 d is equl to 7 7 (A) cos C 7 7 (D) 7cos C 7 (B) cos C 7 7 (E) cos ( + 70) + C (C) cos C d is equl to + (A) log + + C (B) log + (D) log + + C (E) log + + C (C) log + C C + +

14 e cos( e ) d is equl to 5 (A) sin 5 ( e ) + C (B) sin 5 ( e ) + C (C) sin ( e ) (D) sin ( e ) + C (E) sin( e ) + C + C sin d is equl to + cos (A) + log tn + C (B) log tn + C (C) tn + C (D) log cos + C (E) log sin + C d is equl to sin cos (A) log tn + C (B) log sin + C (C) log sec + C (D) log cos + C (E) log sin + C. d is equl to 8sin + (A) sin - (tn ) + C (D) tn - (tn ) + C sin (tn ) (B) + C (C) (E) sin - ( tn ) + C tn (tn ) + C. / 0 cos log d is equl to sin (A) / (B) /4 (C) (D) (E)0 The vlue of 4 d is equl to (A) 7 (B) 6 (C) 5 (D) 4 (E) 4. The re of the region bounded by y = 6, y = 0, = 0 in the first qudrnt is (in squre units) (A) 8 (B) 6 (C) (D) 4 (E) /

15 5. The vlue of 4 ( )( )( 4) d is equl to (A) / (B) (C) (D) / (E) 0 6. The re bounded by the lines y =, y = 4 nd the y-is is equl to (in squre units) (A) (B) 4 (C) 0 (D) (E) 7. The generl solution of the differentil eqution ( + y + ) dy = is d (A) + y + = Ce y (B) + y + 4 = Ce y (C) + y + = Ce -y (D) + y + 4 = Ce -y (E) + y + 4e y = C 8. The differentil eqution representing the fmily of curves y = (+b) where nd b re rbitrry constnts, is of (A) order, degree (B) order, degree (C) order, degree (D) order, degree 4 (E) order, degree dy y 9. The solution of the differentil eqution d =0 is y (A) sin y 5 = C (B) sin y = 0 + C (D) sin y = 0 + C (E) sin y + 5 = C (C) y 5 C = + 0. The generl solution of the differentil eqution dy y d = y d is y (A) y = (B) = (C)y = (C+ ) () C C+ y (D) y = (E) = C+ C

16 KEAM ANSWER KEY-06 Qn. No. BOOK LET CODE BOOK LET CODE BOOK LET CODE Qn. No. Qn. No. B B B B4 B B B B4 B B B B4 D D E B 4 E E C D 8 A C D E C E D E 4 A D B B 8 A E A C E A A A 4 B E C E 8 D A C D 4 D A B B 44 D B C C 84 A D B E 5 B B D D 45 A A A A 85 E E B C 6 D C E B 46 E C C C 86 B A D C 7 A B D B 47 C A E C 87 E B E D 8 E B E A 48 C E A D 88 A E A A 9 B A B D 49 E E D A 89 B A A D 0 A D A E 50 C B E E 90 D A B E C E C B 5 C A A C 9 B D C D C A A D 5 D C B C 9 B C B C D B E E 5 A D E C 9 A E B C 4 A D E D 54 D A A B 94 D D A A 5 E A B C 55 A E A A 95 E B D E 6 C E A A 56 D B D D 96 B D E B 7 C C C B 57 C E C E 97 D B A D 8 C D D B 58 C A E C 98 E E B E 9 B E A E 59 B B D C 99 D C D B 0 A C E C 60 E D B A 00 C A A D D C B C 6 B B D D 0 D C E E E D E B 6 D B B A 0 B C C D C A A C 6 E A E C 0 B D D A 4 C D B C 64 B D C B 04 E A E B 5 A E D A 65 D E A B 05 C E C D 6 D D B C 66 E B C D 06 C C C E 7 A C B E 67 D D C E 07 B C D D 8 C C A A 68 A E D A 08 C C A E 9 B A D D 69 B D A A 09 C B D B 0 B E E E 70 D C E B 0 A A E A D B B A 7 E A C C C D D C E D D B 7 D B C B E E C A A E E E 7 E B C B A C C E 4 A B D A 74 B E B A 4 D C A E 5 C D C A 75 A C A D 5 E A E B 6 B E A D 76 C C D E 6 E D B A 7 B D B C 77 A B E A 7 A A D C 8 B A B E 78 E C C B 8 E C E D 9 A B E D 79 E C C D 9 A B B A 40 D D C B 80 E A A A 0 A B D E

MATHEMATICS PART A. 1. ABC is a triangle, right angled at A. The resultant of the forces acting along AB, AC

MATHEMATICS PART A. 1. ABC is a triangle, right angled at A. The resultant of the forces acting along AB, AC FIITJEE Solutions to AIEEE MATHEMATICS PART A. ABC is tringle, right ngled t A. The resultnt of the forces cting long AB, AC with mgnitudes AB nd respectively is the force long AD, where D is the AC foot