FIITJEE AIEEE 2004 (MATHEMATICS)

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1 FIITJEE AIEEE 4 (MATHEMATICS) Impotat Istuctios: i) The test is of hous duatio. ii) The test cosists of 75 questios. iii) The maimum maks ae 5. iv) Fo each coect aswe you will get maks ad fo a wog aswe you will get - mak.. Let R = {(, ), (4, ), (, 4), (, ), (, )} be a elatio o the set A = {,,, 4}. The elatio R is () a fuctio () efleive () ot symmetic (4) tasitive. The age of the fuctio f() = P is () {,, } () {,,, 4, 5} () {,,, 4} (4) {,,, 4, 5, 6} 7. Let z, w be comple umbes such that z+ iw= ad ag zw =. The ag z equals () () () 4 (4) 4. If z = i y ad z = p+ iq, the y + p q ( p + q ) is equal to () () - () (4) - 5. If z = z +, the z lies o () the eal ais () a ellipse () a cicle (4) the imagiay ais. 6. Let A =. The oly coect statemet about the mati A is () A is a zeo mati () A = I () A = I, whee I is a uit mati A does ot eist (4) ( )

2 AIEEE-PAPERS Let A = ( ) B = 5 α. If B is the ivese of mati A, the α is () - () 5 () (4) - 8. If a, a, a,...,a,... ae i G.P., the the value of the detemiat loga loga+ loga+ loga+ loga+ 4 loga+ 5, is loga loga loga () () - () (4) 9. Let two umbes have aithmetic mea 9 ad geometic mea 4. The these umbes ae the oots of the quadatic equatio () = () 8 6 = () = (4) 8+ 6 =. If ( p) is a oot of quadatic equatio ( ) (), () -, (), - (4) -, + p+ p =, the its oots ae. Let S(K) = ( K ) = + K. The which of the followig is tue? () S() is coect () Piciple of mathematical iductio ca be used to pove the fomula () S(K) S(K + ) (4) S(K) S(K + ). How may ways ae thee to aage the lettes i the wod GARDEN with the vowels i alphabetical ode? () () 48 () 6 (4) 4. The umbe of ways of distibutig 8 idetical balls i distict boes so that oe of the boes is empty is () 5 () 8 C 8 () (4) 4. If oe oot of the equatio + p+ = is 4, while the equatio oots, the the value of q is () 49 4 () 4 () (4) + p+ q = has equal FIITJEE Ltd. ICES House, Savapiya Viha (Nea Hauz Khas Bus Tem.), New Delhi - 6, Ph : , , 685 4, Fa : 6594

3 AIEEE-PAPERS-- 5. The coefficiet of the middle tem i the biomial epasio i powes of of ( +α) 4 ad of ( α ) 6 is the same if α equals 5 () () 5 () (4) 6. The coefficiet of i epasio of ( )( ) + is () ( ) () ( ) ( ) () ( ) ( ) (4) ( ) 7. If S = = C ad t = = C, the t S is equal to () () () (4) 8. Let T be the th tem of a A.P. whose fist tem is a ad commo diffeece is d. If fo some positive iteges m,, m, T m = ad T = m, the a d equals () () () (4) + m m 9. The sum of the fist tems of the seies is whe is eve. Whe is odd the sum is ( + ) () ( ) + () 4 ( + ) () + (4) ( ) ( + ). The sum of seies is! 4! 6! ( e ) () ( e ) () e ( e () ) e ( e ) (4) e FIITJEE Ltd. ICES House, Savapiya Viha (Nea Hauz Khas Bus Tem.), New Delhi - 6, Ph : , , 685 4, Fa : 6594

4 AIEEE-PAPERS--4. Let α, β be such that < α - β <. If siα + siβ = α β value of cos is () () () 6 65 (4) 6 65 ad cosα + cosβ = Ifu = a cos θ+ b si θ + a si θ+ b cos θ, the the diffeece betwee the maimum ad miimum values of u is give by a + b () a + b () ( ) () ( a+ b) (4) ( a b), the the. The sides of a tiagle ae siα, cosα ad + siαcosα fo some < α <. The the geatest agle of the tiagle is () 6 o () 9 o () o (4) 5 o 4. A peso stadig o the bak of a ive obseves that the agle of elevatio of the top of a tee o the opposite bak of the ive is 6 o ad whe he eties 4 mete away fom the tee the agle of elevatio becomes o. The beadth of the ive is () m () m () 4 m (4) 6 m 5. If f : R S, defied by f() = si cos +, is oto, the the iteval of S is () [, ] () [-, ] () [, ] (4) [-, ] 6. The gaph of the fuctio y = f() is symmetical about the lie =, the () f( + )= f( ) () f( + ) = f( ) () f() = f(-) (4) f() = - f(-) ( ) 7. si The domai of the fuctio f() = is 9 () [, ] () [, ) () [, ] (4) [, ) 8. a b If lim + + = e, the the values of a ad b, ae () a R, b R () a =, b R () a R, b = (4) a = ad b = 4 FIITJEE Ltd. ICES House, Savapiya Viha (Nea Hauz Khas Bus Tem.), New Delhi - 6, Ph : , , 685 4, Fa : 6594

5 AIEEE-PAPERS--5 ta 9. Let f() =,,, 4 4. If f() is cotiuous i,, the f 4 is () () () (4) - y +...to y+ e. If = e, >, the dy d is () + () () (4) + y. A poit o the paabola abscissa is () (, 4) () (, -4) (), 8 (4), 8 = 8 at which the odiate iceases at twice the ate of the. A fuctio y = f() has a secod ode deivative f () = 6( ). If its gaph passes though the poit (, ) ad at that poit the taget to the gaph is y = 5, the the fuctio is () ( ) () ( ) () ( + ) (4) ( + ). The omal to the cuve = a( + cosθ), y = asiθ at θ always passes though the fied poit () (a, ) () (, a) () (, ) (4) (a, a) 4. If a + b + 6c =, the at least oe oot of the equatio () (, ) () (, ) () (, ) (4) (, ) a + b + c = lies i the iteval 5. lim e = is () e () e () e (4) e + 6. si If d = A + Blogsi( α ) + C, the value of (A, B) is si( α) () (siα, cosα) () (cosα, siα) () (- siα, cosα) (4) (- cosα, siα) 7. d is equal to cos si FIITJEE Ltd. ICES House, Savapiya Viha (Nea Hauz Khas Bus Tem.), New Delhi - 6, Ph : , , 685 4, Fa :

6 AIEEE-PAPERS--6 () log ta + C 8 () log ta + C 8 () (4) log cot + C log ta + + C 8 8. The value of () 8 () 7 d is () 4 (4) 9. The value of I = 4. If / (si + cos ) + si d is () () () (4) / f(si ) d = A f(si ) d, the A is () () () 4 (4) f(a) 4. e I If f() =, I = + e g{( )}d ad I = g{( )}d the the value of I f( a) f( a) () () () (4) f(a) is 4. The aea of the egio bouded by the cuves y =, =, = ad the -ais is () () () (4) 4 4. The diffeetial equatio fo the family of cuves + y ay =, whee a is a abitay costat is () ( y )y = y () ( + y )y = y ()( y )y = y (4) ( + y )y = y 44. The solutio of the diffeetial equatio y d + ( + y) dy = is () = C () + log y = C y y () + log y = C (4) log y = C y 6 FIITJEE Ltd. ICES House, Savapiya Viha (Nea Hauz Khas Bus Tem.), New Delhi - 6, Ph : , , 685 4, Fa : 6594

7 AIEEE-PAPERS Let A (, ) ad B(, ) be vetices of a tiagle ABC. If the cetoid of this tiagle moves o the lie + y =, the the locus of the vete C is the lie () + y = 9 () y = 7 () + y = 5 (4) y = 46. The equatio of the staight lie passig though the poit (4, ) ad makig itecepts o the co-odiate aes whose sum is is y y y y () + = ad + = () = ad + = y y y y () + = ad + = (4) = ad + = 47. If the sum of the slopes of the lies give by cy 7y = is fou times thei poduct, the c has the value () () () (4) 48. If oe of the lies give by 6 y + 4cy = is + 4y =, the c equals () () () (4) 49. If a cicle passes though the poit (a, b) ad cuts the cicle the locus of its cete is () a + by + (a + b + 4) = () () a by + (a + b + 4) = (4) + y = 4 othogoally, the a + by (a + b + 4) = a by (a + b + 4) = 5. A vaiable cicle passes though the fied poit A (p, q) ad touches -ais. The locus of the othe ed of the diamete though A is ()( p) = 4qy () ( q) = 4py ()(y p) = 4q (4) (y q) = 4p 5. If the lies + y + = ad y 4 = lie alog diametes of a cicle of cicumfeece, the the equatio of the cicle is () + y + y = () + y y = () + y + + y = (4) + y + y = 5. The itecept o the lie y = by the cicle AB as a diamete is () + y y = () () + y + + y = (4) + y = is AB. Equatio of the cicle o + y + y = + y + y = 5. If a ad the lie b + cy + 4d = passes though the poits of itesectio of the paabolas y = 4a ad = 4ay, the () d + (b+ c) = () d + (b + c) = () d + (b c) = (4) d + (b c) = FIITJEE Ltd. ICES House, Savapiya Viha (Nea Hauz Khas Bus Tem.), New Delhi - 6, Ph : , , 685 4, Fa :

8 AIEEE-PAPERS The ecceticity of a ellipse, with its cete at the oigi, is. If oe of the diectices is = 4, the the equatio of the ellipse is () + 4y = () + 4y = () 4 + y = (4) 4 + y = 55. A lie makes the same agle θ, with each of the ad z ais. If the agle β, which it makes with y-ais, is such that si β= si θ, the cos θ equals () () 5 () 5 (4) Distace betwee two paallel plaes + y + z = 8 ad 4 + y + 4z + 5 = is () () 5 () 7 (4) A lie with diectio cosies popotioal to,, meets each of the lies = y + a = z ad + a = y = z. The co-odiates of each of the poit of itesectio ae give by () (a, a, a), (a, a, a) () (a, a, a), (a, a, a) () (a, a, a), (a, a, a) (4) (a, a, a), (a, a, a) 58. If the staight lies = + s, y = λs, z = + λs ad = t, y = + t, z = t with paametes s ad t espectively, ae co-plaa the λ equals () () () (4) 59. The itesectio of the sphees + y + z + 7 y z = ad + y + z + y+ 4z = 8 is the same as the itesectio of oe of the sphee ad the plae () y z = () y z = () y z = (4) y z = 6. Let a, b ad c be thee o-zeo vectos such that o two of these ae colliea. If the vecto a+ b is colliea with c ad b+ c is colliea with a (λ beig some o-zeo scala) the a+ b+ 6c equals () λa () λb () λc (4) 6. A paticle is acted upo by costat foces 4i ˆ+ ˆj kˆ ad i ˆ+ ˆj kˆ which displace it fom a poit ˆ i + j ˆ + k ˆ to the poit 5i ˆ+ 4j ˆ+ kˆ. The wok doe i stadad uits by the foces is give by 8 FIITJEE Ltd. ICES House, Savapiya Viha (Nea Hauz Khas Bus Tem.), New Delhi - 6, Ph : , , 685 4, Fa : 6594

9 AIEEE-PAPERS--9 () 4 () () 5 (4) 5 6. If a, b, c ae o-coplaa vectos ad λ is a eal umbe, the the vectos a+ b+ c, λ b+ 4c ad (λ )c ae o-coplaa fo () all values of λ () all ecept oe value of λ () all ecept two values of λ (4) o value of λ 6. Let u, v, w be such that u =, v =, w =. If the pojectio v alog u is equal to that of w alog u ad v, w ae pepedicula to each othe the u v + w equals () () 7 () 4 (4) Let a, b ad c be o-zeo vectos such that(a b) c = b c a. If θ is the acute agle betwee the vectos b ad c, the si θ equals () () () (4) 65. Coside the followig statemets: (a) Mode ca be computed fom histogam (b) Media is ot idepedet of chage of scale (c) Vaiace is idepedet of chage of oigi ad scale. Which of these is/ae coect? () oly (a) () oly (b) () oly (a) ad (b) (4) (a), (b) ad (c) 66. I a seies of obsevatios, half of them equal a ad emaiig half equal a. If the stadad deviatio of the obsevatios is, the a equals () () () (4) 67. The pobability that A speaks tuth is 4 5, while this pobability fo B is. The pobability that 4 they cotadict each othe whe asked to speak o a fact is () () 7 () 5 (4) A adom vaiable X has the pobability distibutio: X: p(x): FIITJEE Ltd. ICES House, Savapiya Viha (Nea Hauz Khas Bus Tem.), New Delhi - 6, Ph : , , 685 4, Fa :

10 AIEEE-PAPERS-- Fo the evets E = {X is a pime umbe} ad F = {X < 4}, the pobability P (E F) is ().87 ().77 ().5 (4) The mea ad the vaiace of a biomial distibutio ae 4 ad espectively. The the pobability of successes is () 7 () () With two foces actig at a poit, the maimum effect is obtaied whe thei esultat is 4N. If they act at ight agles, the thei esultat is N. The the foces ae ()( + )N ad ( )N () ( + )N ad ( )N () + N ad N (4) + N ad N 7. I a ight agle ABC, A = 9 ad sides a, b, c ae espectively, 5 cm, 4 cm ad cm. If a foce F has momets, 9 ad 6 i N cm. uits espectively about vetices A, B ad C, the magitude of F is () () 4 () 5 (4) 9 7. Thee foces P, Q ad R actig alog IA, IB ad IC, whee I is the icete of a ABC, ae i equilibium. The P : Q : R is A B C A B C () cos : cos : cos () si : si : si A B C A B C () sec : sec : sec (4) co sec : co sec : co sec 7. A paticle moves towads east fom a poit A to a poit B at the ate of 4 km/h ad the towads oth fom B to C at the ate of 5 km/h. If AB = km ad BC = 5 km, the its aveage speed fo its jouey fom A to C ad esultat aveage velocity diect fom A to C ae espectively (4) 8 56 () 7 4 km/h ad 4 km/h () 4 km/h ad 7 4 km/h () 7 9 km/h ad 9 km/h (4) 9 km/h ad 7 9 km/h 74. A velocity 4 m/s is esolved ito two compoets alog OA ad OB makig agles ad 45 espectively with the give velocity. The the compoet alog OB is () 8 m/s () ( ) m/s 4 () 4 m/s (4) ( 6 ) m/s 8 FIITJEE Ltd. ICES House, Savapiya Viha (Nea Hauz Khas Bus Tem.), New Delhi - 6, Ph : , , 685 4, Fa : 6594

11 AIEEE-PAPERS If t ad t ae the times of flight of two paticles havig the same iitial velocity u ad age R o the hoizotal, the t + t is equal to u 4u () () g g () u g (4) FIITJEE Ltd. ICES House, Savapiya Viha (Nea Hauz Khas Bus Tem.), New Delhi - 6, Ph : , , 685 4, Fa : 6594

12 AIEEE-PAPERS-- FIITJEE AIEEE 4 (MATHEMATICS) ANSWERS FIITJEE Ltd. ICES House, Savapiya Viha (Nea Hauz Khas Bus Tem.), New Delhi - 6, Ph : , , 685 4, Fa : 6594

13 AIEEE-PAPERS--. (, ) R but (, ) R. Hece R is ot symmetic. FIITJEE AIEEE 4 (MATHEMATICS) SOLUTIONS. 7 f() = P 7 7, ad =, 4, 5 Rage is {,, }.. Hee ω = z i z ag z. = i ag(z) ag(i) = ag(z) = z = ( p + iq) = p ( p q ) iq( q p ) y = p q & = q p p q 5. ( ) z z = + ( )( ) p y + q ( p + q ) 4 =. z z = z + z + z + z + zz = z+ z = R (z) = z lies o the imagiay ais. 6. A.A = = I. 7. AB = I A( B) = I α 5 5 α = α = 5+α loga loga+ loga+ loga+ loga+ 4 loga+ 5 loga loga loga C C C, C C C loga = loga+ log log log log = (whee is a commo atio). loga log log + 6 if α = Let umbes be a, b a + b = 8, ab = 4 ab = 6, a ad b ae oots of the equatio FIITJEE Ltd. ICES House, Savapiya Viha (Nea Hauz Khas Bus Tem.), New Delhi - 6, Ph : , , 685 4, Fa : 6594

14 AIEEE-PAPERS =.. () p + p p + p = (sice ( p) is a oot of the equatio + p + ( p) = ) ( ) ( ) ( ) ( p)( p+ p+ ) = ( p) = ( p) = p = sum of oot is α+β= p ad poduct αβ = p = (whee β = p = ) α+ = α = Roots ae,. S( k) = ( k ) = + k S(k + )= (k ) + (k + ) + k + k + = k + k + 4 [fom S(k) = + k ] = ( ) = + (k + k + ) = + (k + ) = S (k + ). Although S (k) i itself is ot tue but it cosideed tue will always imply towads S (k + ).. Sice i half the aagemet A will be befoe E ad othe half E will be befoe A. Hece total umbe of ways = 6! = 6.. Numbe of balls = 8 umbe of boes = Hece umbe of ways = 7 C =. 4. Sice 4 is oe of the oot of + p + = 6 + 4p + = p = 7 ad equatio + p + q = has equal oots D = 49 4q = q = Coefficiet of Middle tem i ( ) α = t = C α Coefficiet of Middle tem i ( α ) = t = C ( α ) 4 6 Cα = C. α 6 = α α = 6. Coefficiet of i ( + )( ) = ( + )( C C ( ) C + ( ) C ) = ( ) C + ( ) C ( ) ( ) =. 7. t = = = ( Q C = C ) t C C C = = = + = = C = C = t = = S C = t S = 8. ( ) 4 Tm = = a+ m d...() ad T = = a+ ( ) d...() m FIITJEE Ltd. ICES House, Savapiya Viha (Nea Hauz Khas Bus Tem.), New Delhi - 6, Ph : , , 685 4, Fa : 6594

15 AIEEE-PAPERS--5 fom () ad () we get Hece a d = a =, d= m m ( ) ( ) + 9. If is odd the ( ) is eve sum of odd tems = + =.. α α 4 6 e + e α α α = ! 4! 6! α α 4 6 e + e α α α = ! 4! 6! put α =, we get ( ) e = e! 4! 6!. si α + si β = ad cos α + cos β = Squaig ad addig, we get 7 + cos (α β) = (65) cos α β 9 α β = cos = α β Q < <.. u = a cos θ+ b si θ + a si θ+ b cos θ a b a b a b b a = + + cosθ cosθ a + b a b u = a + b + cos θ mi value of u = a + b + ab u = a + b ma value of ( ) ( ) u u = a b. ma mi. Geatest side is + siαcosα, by applyig cos ule we get geatest agle = ο. h 4. ta = 4 + b h= 4+ b..() ta6 = h/b h = b.() b = m 6 4 b h 5. si cos si cos+ age of f() is [, ]. Hece S is [, ]. FIITJEE Ltd. ICES House, Savapiya Viha (Nea Hauz Khas Bus Tem.), New Delhi - 6, Ph : , , 685 4, Fa :

16 AIEEE-PAPERS If y = f () is symmetic about the lie = the f( + ) = f( ) > ad [, ) a b + a b a b a b + a lim lim + + e a, b R = + + = = ta ta f() = lim = 4 4 y+ e y +... y+ e 4 y e + = e = l = y dy = =. d 9. Ay poit be t,9t ; diffeetiatig y = 8 dy = 9 = = (give) t=. d y t Poit is 9 9, 8. f () = 6( ) f () = ( ) + c ad f () = c = f () = ( ) + k ad f () = k = f () = ( ).. Elimiatig θ, we get ( a) + y = a. Hece omal always pass though (a, ). 4. Let f () = a + b + c f() = f() ( a b 6c 6d) a b + + c + d = + + +, Now f() = f() = d, the accodig to Rolle s theoem 6 f () = a + b + c = has at least oe oot i (, ) 5. lim e = = ed = (e ) 6. Put α = t si( α+ t) dt = si α cot tdt + cos α dt si t = cosα( α ) + siα l si t + c A = cos α, B = siα 6 FIITJEE Ltd. ICES House, Savapiya Viha (Nea Hauz Khas Bus Tem.), New Delhi - 6, Ph : , , 685 4, Fa : 6594

17 AIEEE-PAPERS d cos si = d cos + 4 = sec + d 4 = log ta + + C 8 8. ( ) d+ ( ) d+ ( ) d = + + = ( si + cos ) d = ( si + ) ( si + cos ) cos d = cos + si =. 4. Let I = f(si )d = ( )f(si)d = f(si)d I (sice f (a ) = f ()) I = / f(si)d A =. 4. f(-a) + f(a) = I = I = f(a) g{( )}d = ( ) g{( )}d f( a) f(a) f( a) f(a) Q ( ) = ( + ) f( a) g{( )}d = I I / I =. b a b f d f a b d a 4. Aea = ( )d + ( )d =. y= y = 4. + yy - ay = a = + yy (elimiatig a) y ( y )y = y. 45. y d + dy + y dy =. d(y) dy y + y = + log y = C. y 45. If C be (h, k) the cetoid is (h/, (k )/) it lies o + y =. locus is + y = 9. FIITJEE Ltd. ICES House, Savapiya Viha (Nea Hauz Khas Bus Tem.), New Delhi - 6, Ph : , , 685 4, Fa :

18 AIEEE-PAPERS y + = whee a + b = - ad 4 + = a b a b a =, b = - o a = -, b =. y y Hece = ad + =. c 47. m + m = ad m m = 7 7 m + m = 4m m (give) c =. 48. m + m = 4c, m m = 6 4c ad m =. 4 Hece c = Let the cicle be + y + g + fy + c = c = 4 ad it passes though (a, b) a + b + ga + fb + 4 =. Hece locus of the cete is a + by (a + b + 4) =. 5. Let the othe ed of diamete is (h, k) the equatio of cicle is ( h)( p) + (y k)(y q) = Put y =, sice -ais touches the cicle (h + p) + (hp + kq) = (h + p) = 4(hp + kq) (D = ) ( p) = 4qy. 5. Itesectio of give lies is the cete of the cicle i.e. (, ) Cicumfeece = adius = 5 equatio of cicle is + y + y =. 5. Poits of itesectio of lie y = with + y = ae (, ) ad (, ) hece equatio of cicle havig ed poits of diamete (, ) ad (, ) is + y y =. 5. Poits of itesectio of give paabolas ae (, ) ad (4a, 4a) equatio of lie passig though these poits is y = O compaig this lie with the give lie b + cy + 4d =, we get d = ad b + c = (b + c) + d =. 54. Equatio of diecti is = a/e = 4 a = b = a ( e ) b = Hece equatio of ellipse is + 4y =. 55. l = cos θ, m = cos θ, = cos β cos θ + cos θ + cos β = cos θ = si β = si θ (give) cos θ = / Give plaes ae + y + z 8 =, 4 + y + 4z + 5 = + y + z + 5/ = d d 8 5/ 7 Distace betwee plaes = = =. a + b + c FIITJEE Ltd. ICES House, Savapiya Viha (Nea Hauz Khas Bus Tem.), New Delhi - 6, Ph : , , 685 4, Fa : 6594

19 AIEEE-PAPERS--9 y+ a z 57. Ay poit o the lie = = = t (say) is (t, t a, t ) ad ay poit o the lie + a y z = = = t ( say) is (t a, t, t ). Now diectio cosie of the lies itesectig the above lies is popotioal to (t a t, t t + a, t t ). Hece t a t = k, t t + a = k ad t t = k O solvig these, we get t = a, t = a. Hece poits ae (a, a, a) ad (a, a, a). y z y z 58. Give lies + s ad = = = = = = t ae coplaa the pla λ λ / passig though these lies has omal pepedicula to these lies a - bλ + cλ = ad a + b c = (whee a, b, c ae diectio atios of the omal to the pla) O solvig, we get λ = Requied plae is S S = whee S = + y + z + 7 y z = ad S = + y + z + y + 4z 8 = y z =. a+ b = t c.() ad b+ c = ta.() () () a ( + t) + c( t 6) = + t = t = -/ & t = -6. Sice a ad c ae o-colliea. Puttig the value of t ad t i () ad (), we get a+ b+ 6c =. 6. ( ) 6. Wok doe by the focesf ad F is (F + F ) d, whee d is displacemet Accodig to questio F + F = (4i ˆ+ ˆj k) ˆ + (i ˆ+ ˆj k) ˆ = 7i ˆ+ j ˆ 4kˆ ad d = (5i ˆ+ 4j ˆ+ k) ˆ (i ˆ+ j ˆ+ k) ˆ = 4i ˆ+ j ˆ kˆ. Hece (F + F ) d is Coditio fo give thee vectos to be coplaa is λ 4 = λ =, /. λ Hece give vectos will be o coplaa fo all eal values of λ ecept, /. 6. Pojectio of v alog u ad w alog uis v u ad w u espectively u u Accodig to questio v u w = u v u = w u. ad v w = u u u v+ w = u + v + w u v+ u w v w= 4 u v+ w = 4. FIITJEE Ltd. ICES House, Savapiya Viha (Nea Hauz Khas Bus Tem.), New Delhi - 6, Ph : , , 685 4, Fa :

20 AIEEE-PAPERS ( ) a b c = b c a ( a c) b ( b c) a = b c a ( a c) b = b c + ( b c) a a c = ad b c + ( b c ) = b c + cosθ = cosθ = / siθ =. 65. Mode ca be computed fom histogam ad media is depedet o the scale. Hece statemet (a) ad (b) ae coect. 66. i = a fo i =,,..., ad i = a fo i =,..., S.D. = ( i ) = i i = Sice i i = = i= = a a = 67. E : evet deotig that A speaks tuth E :evet deotig that B speaks tuth Pobability that both cotadicts each othe = PE ( E) PE ( E) 68. P(E F) = P(E) + P(F) P ( E F) = =.77 + = 4 + = Give that p = 4, p q = q = / p = /, = 8 p( = ) = C = P + Q = 4, P + Q = 9 P = + N ad Q = N. 7. F. si θ = 9 F. 4 cos θ = 6 F = 5. 4cosθ θ C A θ B F siθ 7. By Lami s theoem A B C P:Q:R = si 9 + :si 9 :si cos A : cos B : cos C. B A 9+C/ 9+B/ 9+A/ C FIITJEE Ltd. ICES House, Savapiya Viha (Nea Hauz Khas Bus Tem.), New Delhi - 6, Ph : , , 685 4, Fa : 6594

21 AIEEE-PAPERS-- 7. Time T fom A to B = 4 = hs. T fom B to C = 5 5 = hs. Total time = 4 hs. C 5 Aveage speed = 7 4 km/ h. A B Resultat aveage velocity = 4 km/h. si si(45 + ) Compoet alog OB = 4 = ( 6 ) m/s. 75. t = t u si α, t = g 4u t + =. g u siβ whee α + β = 9 g FIITJEE Ltd. ICES House, Savapiya Viha (Nea Hauz Khas Bus Tem.), New Delhi - 6, Ph : , , 685 4, Fa : 6594