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

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1 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 of the perpendiculr from A onto BC. The mgnitude of the resultnt is AB + AC () () (AB)(AC) (AB) (AC) AB+ AC () + () AB AC AD Mgnitude of resultnt AB + AC = AB + = AC AB AC = BC BC AB AC = AD BC = AD A C D B. Suppose popultion A hs observtions,,,, nd nother popultion B hs observtions 5, 5,, 5. If V A nd V B represent the VA vrinces of the two popultions, respectively, then VB is () () 9/ () /9 () / Ans. () d i σ =. (Here devitions re tken from the men) n Since A nd B both hs consecutive integers, therefore both hve sme stndrd devition nd hence the vrince. VA As d i is sme in both the cses. V = ( ) B. If the roots of the qudrtic eqution + p + q = re tn nd tn5, respectively then the vlue of + q p is () () () () Ans. () + p + q = tn + tn 5 = p tn tn 5 = q FIITJEE Ltd., ICES House, 9 A, Klu Sri, Srvpriy Vihr, New Delhi -, Ph 5599, 599, F: -59.

2 FIITJEE Solutions to AIEEE tn + tn5 p tn 5 = = = tn tn5 q p = q q p = + q p =.. The vlue of the integrl, d 9 + () / () / () () Ans. () I = I = I = 9 + d 9 d 9 + d = I =. 5. The number of vlues of in the intervl [, ] stisfying the eqution sin + 5sin = is () () () () Ans. () sin + 5 sin = (sin + ) ( sin ) = sin = In (, ), hs vlues. If ( b) c = ( b c), where, b nd c re ny three vectors such tht b, b c, then nd c re () inclined t n ngle of / between them () inclined t n ngle of / between them () perpendiculr () prllel ( b) c = ( b c), b, b c ( c) b ( b c) = ( c) b ( b) c ( b) c = ( b c) c 7. Let W denote the words in the English dictionry. Define the reltion R by : FIITJEE Ltd., ICES House, 9 A, Klu Sri, Srvpriy Vihr, New Delhi -, Ph 5599, 599, F: -59.

3 FIITJEE Solutions to AIEEE R = {(, y) W W the words nd y hve t lest one letter in common}. Then R is () not refleive, symmetric nd trnsitive () refleive, symmetric nd not trnsitive () refleive, symmetric nd trnsitive () refleive, not symmetric nd trnsitive Ans. () Clerly (, ) R W. So, R is refleive. Let (, y) R, then (y, ) R s nd y hve t lest one letter in common. So, R is symmetric. But R is not trnsitive for emple Let = DELHI, y = DWARKA nd z = PARK then (, y) R nd (y, z) R but (, z) R. 8. If A nd B re squre mtrices of size n n such tht A B = (A B) (A + B), then which of the following will be lwys true? () A = B () AB = BA () either of A or B is zero mtri () either of A or B is n identity mtri Ans. () A B = (A B) (A + B) A B = A + AB BA B AB = BA. 9. The vlue of k k sin + icos k= () i () () () i k k k k sin + i cos sin i cos = + k= k= k= = + i ( ) = i.. All the vlues of m for which both roots of the equtions m + m = re greter thn but less thn, lie in the intervl () < m < () m > () < m < () < m < Ans. () Eqution m + m = ( m) = ( m + ) ( m ) = = m, m + < m nd m + < FIITJEE Ltd., ICES House, 9 A, Klu Sri, Srvpriy Vihr, New Delhi -, Ph 5599, 599, F: -59.

4 m > nd m < < m <. FIITJEE Solutions to AIEEE. A prticle hs two velocities of equl mgnitude inclined to ech other t n ngle θ. If one of them is hlved, the ngle between the other nd the originl resultnt velocity is bisected by the new resultnt. Then θ is () 9 () () 5 () Ans. () u sin θ θ tn = u u+ cosθ θ θ θ sin + sin cosθ= sinθ cos θ θ θ θ sin = sin = sin sin θ θ sin = = or θ =. u R R θ/ θ/ θ/ u/ u. At telephone enquiry system the number of phone cells regrding relevnt enquiry follow Poisson distribution with n verge of 5 phone clls during -minute time intervls. The probbility tht there is t the most one phone cll during -minute time period is () () 5 e () 5 55 m r e m P (X = r) = r! P (X ) = P (X = ) + P (X = ) = e e 5 =. 5 e () 5 e. A body flling from rest under grvity psses certin point P. It ws t distnce of m from P, s prior to pssing through P. If g = m/s, then the height bove the point P from where the body begn to fll is () 7 m () 9 m () m () 8 m Ans. () FIITJEE Ltd., ICES House, 9 A, Klu Sri, Srvpriy Vihr, New Delhi -, Ph 5599, 599, F: -59.

5 FIITJEE Solutions to AIEEE We hve h= gt nd h + = g(t ) +. Subtrcting we get = 8g + gt t = 8 sec h = = m Desired height = + = 7 m. h m Q(t) P(t+). f(sin )d is equl to () I = f(cos)d () / () f(sin)d () = I = f(sin ) d = ( ) f(sin ) d f(sin)d I f(sin)d / I = f(sin)d = f(sin)d = / f(cos)d. f(sin)d / f(cos)d 5. A stright line through the point A(, ) is such tht its intercept between the es is bisected t A. Its eqution is () + y = 7 () y + 7 = () + y = () + y = 5 Ans. () The eqution of es is y = the eqution of the line is + y = + y =.. The two lines = y + b, z = cy + d; nd = y + b, z = c y + d re perpendiculr to ech other if () + cc = () + cc = () + c = c () + c = c FIITJEE Ltd., ICES House, 9 A, Klu Sri, Srvpriy Vihr, New Delhi -, Ph 5599, 599, F: -59.

6 FIITJEE Solutions to AIEEE Ans. () Eqution of lines b y z = = d c b z d = y = c Lines re perpendiculr + + cc =. 7. The locus of the vertices of the fmily of prbols y = + is 5 (!) y = () y = 5 () y = () y = 5 Ans. () Prbol: y = + Verte: (α, β) 8 + / + α = =, β = = / 5 5 = = 5 5 αβ = =. 8. The vlues of, for which the points A, B, C with position vectors i ˆ ˆj + k, ˆ ˆi j ˆ 5kˆ nd i ˆ j ˆ+ kˆ respectively re the vertices of right-ngled tringle with C = re () nd () nd () nd () nd Ans. () BA = ˆi j ˆ+ kˆ CA = ( )i ˆ+ j ˆ CB = ( )i ˆ kˆ CA CB = ( ) ( ) = =,. FIITJEE Ltd., ICES House, 9 A, Klu Sri, Srvpriy Vihr, New Delhi -, Ph 5599, 599, F: -59.

7 FIITJEE Solutions to AIEEE / cos d is equl to 9. ( + ) + ( + ) / () () () + () Ans. () I = / / Put + = t I = = ( ) cos ( ) d / / t cos t + dt = cos tdt / / ( + cos t) dt = +.. If is rel, the mimum vlue of () / () () () 7/7 Ans. () y = (y ) + 9(y ) + 7y 7 = D is rel 8(y ) ( y )( 7y 7) (y ) (y ) y.. In n ellipse, the distnce between its foci is nd minor is is 8. Then its eccentricity is is () 5 (B) (C) 5 Ans. () e = e = b = 8 b = b = ( e ) = e = + 9 = 5 = 5 e = = 5 (D) 5 FIITJEE Ltd., ICES House, 9 A, Klu Sri, Srvpriy Vihr, New Delhi -, Ph 5599, 599, F: -59.

8 FIITJEE Solutions to AIEEE. Let A = nd B =,, b N. Then b () there cnnot eist ny B such tht AB = BA () there eist more thn one but finite number of B s such tht AB = BA () there eists ectly one B such tht AB = BA () there eist infinitely mny B s such tht AB = BA A = B = b b AB = b BA = b = b b AB = BA only when = b. The function f() = + hs locl minimum t () = () = () = () = Ans. () + is of the form + & equlity holds for =. Angle between the tngents to the curve y = 5 + t the points (, ) nd (, ) is () () () () Ans. () dy 5 d = m = ( 5) (, ) =, m = ( 5) (, ) = m m = 5. Let,,, be terms of n A.P. If () () p p =,p q q q () 7 (), then equls FIITJEE Ltd., ICES House, 9 A, Klu Sri, Srvpriy Vihr, New Delhi -, Ph 5599, 599, F: -59.

9 FIITJEE Solutions to AIEEE p ( p ) d + p + ( p ) d ( q ) d q + ( + ) p = = q q d q p + d p = q q + d For, p =, q = =. The set of points where f() = is differentible is + () (, ) (, ) () (, ) (, ) () (, ) () (, ) Ans. (), <, < ( ) f( ) = f () =,, + ( + ) f () eist t everywhere. 7. A tringulr prk is enclosed on two sides by fence nd on the third side by stright river bnk. The two sides hving fence re of sme length. The mimum re enclosed by the prk is () () 8 () () Ans. () sin Are = θ Am = t sin, θ = θ = θ 8. At n election, voter my vote for ny number of cndidtes, not greter thn the number to be elected. There re cndidtes nd re of be elected. If voter votes for t lest one cndidte, then the number of wys in which he cn vote is () 5 () () 85 () Ans. () C + C + C + C = = 85 FIITJEE Ltd., ICES House, 9 A, Klu Sri, Srvpriy Vihr, New Delhi -, Ph 5599, 599, F: -59.

10 FIITJEE Solutions to AIEEE 9. If the epnsion in powers of of the function , then n is n n b () b n+ n+ b () b ( )( b) n n b () b b () b n+ n+ is ( ) ( b) = ( )( + b + b +...) n n n n n n coefficient of = b + b + b b + = n b = n+ n+ b b b n+ n+. For nturl numbers m, n if ( y) m ( + y) n = + y + y +, nd = =, then (m, n) is () (, 5) () (5, ) () (5, 5) () (5, 5) m n m m n n ( y) ( + y) = Cy + Cy... + Cy + Cy +... mm ( ) nn ( ) = + ( n m) + + mn y +... m + n m n mn = n m = nd = = So, n m = nd (m n) (m + n) = m + n = 8 m = 5, n = 5. The vlue of []f ()d, >, where [] denotes the gretest integer not eceeding is () f() {f() + f() + + f([])} () [] f() {f() + f() + + f([])} () [] f([]) {f() + f() + + f()} () f([]) {f() + f() + + f()} Ans. () Let = k + h, where [] = k nd h < k k+ h [ ] ( ) ( ) ( ) ( ) ( ) f ' d = f ' d + f ' d +... k d + kf ' d k k {f() f()} + {f() f()} + {f() f()}+.+ (k ) {f(k) f(k )} + k{f(k + h) f(k)} = f() f() f(). f(k) + k f(k + h) = [] f() {f() + f() + f() +. + f([])} FIITJEE Ltd., ICES House, 9 A, Klu Sri, Srvpriy Vihr, New Delhi -, Ph 5599, 599, F: -59.

11 FIITJEE Solutions to AIEEE. If the lines y 7 = nd y 5 = re two dimeters of circle of re 9 squre units, the eqution of the circle is () + y + y 7 = () + y + y = () + y + y = () + y + y 7 = Point of intersection of y 7 = nd y 5 = is (, ), which is the centre of the circle nd rdius = 7. Eqution is ( ) + (y + ) = 9 + y + y 7 =.. The differentil eqution whose solution is A + By =, where A nd B re rbitrry constnts is of () second order nd second degree () first order nd second degree () first order nd first degree () second order nd first degree A + By = () dy A+ By = d () dy dy A + By + B = d d () From () nd () d y dy dy By B By + = d d d dy dy dy y + y = d d d. Let C be the circle with centre (, ) nd rdius units. The eqution of the locus of the mid points of the chords of the circle C tht subtend n ngle of t its centre is () () + y = (B) + y = + y = (D) 7 h + k 9 cos = h + k = 5. If (, ) flls inside the ngle mde by the lines belongs to (), (), + y = 9 y =, > nd y =, >, then () (, ) (), FIITJEE Ltd., ICES House, 9 A, Klu Sri, Srvpriy Vihr, New Delhi -, Ph 5599, 599, F: -59.

12 FIITJEE Solutions to AIEEE Ans. () < nd > < <. The imge of the point (,, ) in the plne y = is 7 9 (),, () (5,, ) 7 9 (),, () (8,, ) α β+ If (α, β, γ) be the imge then = α β α β = 7 () α+ β γ nd = = () From () nd () 9 α=, β =, γ = 5 5 No option mtches. 7. If z + z + =, where z is comple number, then the vlue of z+ z z z z z z z is () 8 () 5 () () z + z + = z = ω or ω so, z+ = ω+ω =, z + = ω +ω=, z + = ω +ω = z z z 5 z + =, z + = nd z + 5 z z z = The given sum = = 8. If < < nd cos + sin =, then tn is () ( 7 ) (B) ( + 7) () ( 7) + () ( 7 ) Ans. () cos + sin = + sin = sin =, so is obtuse tn nd tn 8tn + tn = FIITJEE Ltd., ICES House, 9 A, Klu Sri, Srvpriy Vihr, New Delhi -, Ph 5599, 599, F: -59.

13 FIITJEE Solutions to AIEEE 8± ± 7 tn = = 7 tn < tn = 9. If,,, n re in H.P., then the epression n n is equl to () n( n ) () (n ) ( n ) () n n () (n ) n = =... = = d (sy) n n n n Then =, =,..., n n = d d d n n n = Also, = + ( n ) d d n = ( n ) d n n. If m n m n y = (+ y) +, then dy d is () y () + y y () y () y Ans. () m n m+ n.y = ( + y) mln + nln y = ( m + n) ln( + y) m n dy m + n dy m m + n m + n n dy + = + = y d + y d + y + y y d my n my n dy dy y = = ( + y) y( y) + d d FIITJEE Ltd., ICES House, 9 A, Klu Sri, Srvpriy Vihr, New Delhi -, Ph 5599, 599, F: -59.

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

1. If * is the operation defined by a*b = 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 . 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) (, )