AIEEE 2004 (MATHEMATICS)

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1 AIEEE 00 (MATHEMATICS) Importat Istructios: i) The test is of hours duratio. ii) The test cosists of 75 questios. iii) The maimum marks are 5. iv) For each correct aswer you will get marks ad for a wrog aswer you will get - mark.. Let R = {(, ), (, ), (, ), (, ), (, )} be a relatio o the set A = {,,, }. The relatio R is () a fuctio () refleive () ot symmetric () trasitive. The rage of the fuctio f() = P is () {,, } () {,,,, 5} () {,,, } () {,,,, 5, 6} 7. Let z, w be comple umbers such that z + iw = 0 ad arg zw =. The arg z equals () () 5 () (). If z = i y ad z p iq + p y q = +, the ( p + q ) is equal to () () - () () - 5. If z = z +, the z lies o () the real ais () a ellipse () a circle () the imagiary ais Let A = 0 0. The oly correct statemet about the matri A is 0 0 () A is a zero matri () A = I () A = I, where I is a uit matri A does ot eist () ( )

2 AIEEE-PAPERS-- 7. Let A = ( 0) B = 5 0 α. If B is the iverse of matri A, the α is () - () 5 () () - 8. If a, a, a,...,a,... are i G.P., the the value of the determiat loga loga loga + + loga loga loga loga loga loga , is () 0 () - () () 9. Let two umbers have arithmetic mea 9 ad geometric mea. The these umbers are the roots of the quadratic equatio () = 0 () 8 6 = 0 () = 0 () = 0 0. If ( p) is a root of quadratic equatio ( ) () 0, () -, () 0, - () -, + p + p = 0, the its roots are. Let S(K) = ( K ) = + K. The which of the followig is true? () S() is correct () Priciple of mathematical iductio ca be used to prove the formula () S(K) S(K + ) () S(K) S(K + ). How may ways are there to arrage the letters i the word GARDEN with the vowels i alphabetical order? () 0 () 80 () 60 () 0. The umber of ways of distributig 8 idetical balls i distict boes so that oe of the boes is empty is () 5 () 8 C () 8 (). If oe root of the equatio + p + = 0 is, while the equatio roots, the the value of q is () 9 () () () + p + q = 0 has equal

3 AIEEE-PAPERS-- 5. The coefficiet of the middle term i the biomial epasio i powers of of ( + α ) ad of ( α ) 6 is the same if α equals 5 () () 5 () 0 () 0 6. The coefficiet of i epasio of ( ) ( ) + is () ( ) () ( ) ( ) () ( ) ( ) () ( ) 7. If S = ad t C r = 0 r r =, the C r = 0 r t S is equal to () () () () 8. Let Tr be the rth term of a A.P. whose first term is a ad commo differece is d. If for some positive itegers m,, m, Tm ad T m () 0 () () () + m m ( + ) is 9. The sum of the first terms of the series whe is eve. Whe is odd the sum is ( + ) () ( + ) () () () ( + ) ( + ) 0. The sum of series is!! 6! ( e ) () ( e ) () e () ( e ) e ( ) () e e

4 AIEEE-PAPERS--. Let α, β be such that < α - β <. If siα + siβ = ad cosα + cosβ = 7, the the α β value of cos is () () 0 0 () 6 6 () If u = a cos θ + b si θ + a si θ + b cos θ, the the differece betwee the maimum ad miimum values of u is give by () ( a b ) + () a + b () ( a + b) () ( a b). The sides of a triagle are siα, cosα ad + si α cos α for some 0 < α <. The the greatest agle of the triagle is () 60 () 90 ()0 () 50. A perso stadig o the bak of a river observes that the agle of elevatio of the top of a tree o the opposite bak of the river is 60 ad whe he retires 0 meter away from the tree the agle of elevatio becomes0. The breadth of the river is () 0 m () 0 m () 0 m () 60 m 5. If f : R S, defied by f() = si cos +, is oto, the the iterval of S is () [0, ] () [-, ] () [0, ] () [-, ] 6. The graph of the fuctio y = f() is symmetrical about the lie =, the () f( + )= f( ) () f( + ) = f( ) () f() = f(-) () f() = - f(-) 7. The domai of the fuctio f() = ( ) si 9 () [, ] () [, ) () [, ] () [, ) a b 8. If lim + + = e, the the values of a ad b, are () a R, b R () a =, b R () a R, b = () a = ad b = is

5 AIEEE-PAPERS--5 ta 9. Let f() =,, 0,. If f() is cotiuous i 0,, the f is () () () () - 0. y +...to If y+ e = e, > 0, the dy d is () + () () () +. A poit o the parabola y = 8 at which the ordiate icreases at twice the rate of the abscissa is () (, ) () (, -) (), 8 (), 8. A fuctio y = f() has a secod order derivative f () = 6( ). If its graph passes through the poit (, ) ad at that poit the taget to the graph is y = 5, the the fuctio is () ( ) () ( ) () ( + ) () ( + ). The ormal to the curve = a( + cosθ), y = asiθ at θ always passes through the fied poit () (a, 0) () (0, a) () (0, 0) () (a, a). If a + b + 6c =0, the at least oe root of the equatio a + b + c = 0 lies i the iterval () (0, ) () (, ) () (, ) () (, ) 5. lim r = e r is () e () e () e () e + 6. si If d = si( α ) A + B log si( α ) + C, the value of (A, B) is () (siα, cosα) () (cosα, siα) () (- siα, cosα) () (- cosα, siα) 7. d is equal to cos si 5

6 AIEEE-PAPERS--6 () log ta + C 8 () log ta + C 8 () () log cot + C log ta C 8. The value of () 8 () 7 d is () () 9. The value of I = 0. If / 0 (si + cos ) + si d is () 0 () () () / f(si ) d = A f(si ) d, the A is 0 0 () 0 () () () e. If f() = + e, I = f(a) g{( )}d ad I = f( a) f(a) I g{( )}d the the value of I f( a) () () () () is. The area of the regio bouded by the curves y =, =, = ad the -ais is () () () (). The differetial equatio for the family of curves + y ay = 0, where a is a arbitrary costat is () ( y )y = y () ( + y )y = y ()( y )y = y () ( + y )y = y. The solutio of the differetial equatio y d + ( + y) dy = 0 is () = C () + log y = C y y () log y C y + = () log y = C 6

7 AIEEE-PAPERS Let A (, ) ad B(, ) be vertices of a triagle ABC. If the cetroid of this triagle moves o the lie + y =, the the locus of the verte C is the lie () + y = 9 () y = 7 () + y = 5 () y = 6. The equatio of the straight lie passig through the poit (, ) ad makig itercepts o the co-ordiate aes whose sum is is y y y y () + = ad + = () = ad + = y y y y () + = ad + = () = ad + = 7. If the sum of the slopes of the lies give by cy 7y = 0 is four times their product, the c has the value () () () () 8. If oe of the lies give by 6 y + cy = 0 is + y = 0, the c equals () () () () 9. If a circle passes through the poit (a, b) ad cuts the circle the locus of its cetre is () a + by + (a + b + ) = 0 () () a by + (a + b + ) = 0 () a + by (a + b + ) = 0 a by (a + b + ) = 0 + y = orthogoally, the 50. A variable circle passes through the fied poit A (p, q) ad touches -ais. The locus of the other ed of the diameter through A is ()( p) = qy () ( q) = py ()(y p) = q () (y q) = p 5. If the lies + y + = 0 ad y = 0 lie alog diameters of a circle of circumferece 0, the the equatio of the circle is () + y + y = 0 () + y y = 0 () + y + + y = 0 () 5. The itercept o the lie y = by the circle AB as a diameter is () + y y = 0 () () + y + + y = 0 () + y + y = 0 + y = 0 is AB. Equatio of the circle o + y + y = 0 + y + y = 0 5. If a 0 ad the lie b + cy + d = 0 passes through the poits of itersectio of the parabolas y = a ad = ay, the () d + (b + c) = 0 () () d + (b c) = 0 () d + (b + c) = 0 d + (b c) = 0 7

8 AIEEE-PAPERS The eccetricity of a ellipse, with its cetre at the origi, is. If oe of the directrices is =, the the equatio of the ellipse is () + y = () () + y = () + y = + 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 () Distace betwee two parallel plaes + y + z = 8 ad + y + z + 5 = 0 is () () 5 () 7 () A lie with directio cosies proportioal to,, meets each of the lies = y + a = z ad + a = y = z. The co-ordiates of each of the poit of itersectio are give by () (a, a, a), (a, a, a) () (a, a, a), (a, a, a) () (a, a, a), (a, a, a) () (a, a, a), (a, a, a) 58. If the straight lies = + s, y = λs, z = + λs ad = t, y = + t, z = t with parameters s ad t respectively, are co-plaar the λ equals () () () () The itersectio of the spheres + y + z + 7 y z = ad + y + z + y + z = 8 is the same as the itersectio of oe of the sphere ad the plae () y z = () y z = () y z = () y z = 60. Let a, b ad c be three o-zero vectors such that o two of these are colliear. If the vector a + b is colliear with c ad b + c is colliear with a (λ beig some o-zero scalar) the a + b + 6c equals () λ a () λ b () λ c () 0 6. A particle is acted upo by costat forces i ˆ + ˆj kˆ ad i ˆ + ˆj kˆ which displace it from a poit ˆ i + j ˆ + k ˆ to the poit 5i ˆ + ˆj + kˆ. The work doe i stadard uits by the forces is give by 8

9 AIEEE-PAPERS--9 () 0 () 0 () 5 () 5 6. If a, b, c are o-coplaar vectors ad λ is a real umber, the the vectors a + b + c, λ b + c ad (λ )c are o-coplaar for () all values of λ () all ecept oe value of λ () all ecept two values of λ () o value of λ 6. Let u, v, w be such that u =, v =, w =. If the projectio v alog u is equal to that of w alog u ad v, w are perpedicular to each other the u v + w equals () () 7 () () 6. Let a, b ad c be o-zero vectors such that (a b) c = b c a. If θ is the acute agle betwee the vectors b ad c, the si θ equals () () () () 65. Cosider the followig statemets: (a) Mode ca be computed from histogram (b) Media is ot idepedet of chage of scale (c) Variace is idepedet of chage of origi ad scale. Which of these is/are correct? () oly (a) () oly (b) () oly (a) ad (b) () (a), (b) ad (c) 66. I a series of observatios, half of them equal a ad remaiig half equal a. If the stadard deviatio of the observatios is, the a equals () () () () 67. The probability that A speaks truth is 5, while this probability for B is. The probability that they cotradict each other whe asked to speak o a fact is () 0 () 5 () 7 0 () A radom variable X has the probability distributio: X: p(x):

10 AIEEE-PAPERS--0 For the evets E = {X is a prime umber} ad F = {X < }, the probability P (E F) is () 0.87 () 0.77 () 0.5 () The mea ad the variace of a biomial distributio are ad respectively. The the probability of successes is () 7 56 () 8 56 () 9 56 () With two forces actig at a poit, the maimum effect is obtaied whe their resultat is N. If they act at right agles, the their resultat is N. The the forces are ()( + )N ad ( )N () ( + )N ad ( )N () + N ad N () + N ad N 7. I a right agle ABC, A = 90 ad sides a, b, c are respectively, 5 cm, cm ad cm. If a force F has momets 0, 9 ad 6 i N cm. uits respectively about vertices A, B ad C, the magitude of F is () () () 5 () 9 7. Three forces P, Q ad R actig alog IA, IB ad IC, where I is the icetre of a ABC, are i equilibrium. 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 () co sec : co sec : co sec 7. A particle moves towards east from a poit A to a poit B at the rate of km/h ad the towards orth from B to C at the rate of 5 km/h. If AB = km ad BC = 5 km, the its average speed for its jourey from A to C ad resultat average velocity direct from A to C are respectively () 7 km/h ad km/h () km/h ad 7 km/h () 7 9 km/h ad 9 km/h () 9 km/h ad 7 9 km/h 7. A velocity m/s is resolved ito two compoets alog OA ad OB makig agles 0 ad 5 respectively with the give velocity. The the compoet alog OB is () 8 m/s () ( ) m/s () m/s () ( 6 ) m/s 8 0

11 AIEEE-PAPERS If t ad t are the times of flight of two particles havig the same iitial velocity u ad rage R o the horizotal, the t + t is equal to () u g u () g () () u g

12 AIEEE-PAPERS-- ANSWERS SHEET

13 AIEEE-PAPERS-- SOLUTIONSs. (, ) R but (, ) R. Hece R is ot symmetric.. 7 f() = P , ad =,, 5 Rage is {,, }.. Here ω = z i z arg z. = i. z = ( p + iq) = p ( p q ) iq ( q p ) y = p q & = q p p q 5. z ( z ) = + ( ) ( ) arg(z) arg(i) = arg(z) =. y + p q =. ( p + q ) z z = z + z + z + z + zz = 0 z + z = 0 R (z) = 0 z lies o the imagiary ais A.A = 0 0 = I AB = I A(0 B) = 0 I α α = α = α 0 0 loga loga loga loga loga loga loga loga loga C C C, C C C loga logr logr = loga logr logr + loga logr logr + 6 = 0 (where r is a commo ratio). if α = Let umbers be a, b a + b = 8, ab = ab = 6, a ad b are roots of the equatio = 0.

14 AIEEE-PAPERS-- 0. () ( ) ( ) ( ) ( p) ( p + p + ) = 0 p + p p + p = 0 (sice ( p) is a root of the equatio + p + ( p) = 0) ( p) = 0 ( p) = 0 p = sum of root is α + β = p ad product α β = p = 0 (where β = p = 0) α + 0 = α = Roots are 0,. S ( k ) = ( k ) = + k S(k + )= (k ) + (k + ) = ( ) + k + k + = k + k + [from S(k) = + k ] = + (k + k + ) = + (k + ) = S (k + ). Although S (k) i itself is ot true but it cosidered true will always imply towards S (k + ).. Sice i half the arragemet A will be before E ad other half E will be before A. Hece total umber of ways = 6! = 60.. Number of balls = 8 umber of boes = Hece umber of ways = 7 C =.. Sice is oe of the root of + p + = p + = 0 p = 7 ad equatio + p + q = 0 has equal roots D = 9 q = 0 q = Coefficiet of Middle term i ( ) + α = t = C α 6 6 Coefficiet of Middle term i ( α ) = t = C ( α ) 6 C α = C. α 6 = 0α α = 6. Coefficiet of i ( + )( ) = ( + )( C 0 C ( ) C + ( ) C ) = ( ) C + ( ) = C ( ) ( ). 0 r r r 7. t = = = ( Q Cr = C r ) t C C C r = 0 r r = 0 r r = 0 r r + r = = C r = 0 Cr r = 0 r t = = S C r = 0 r t S = Tm = = a + m d...() ad T = = a + ( ) d...() m from () ad () we get a =, d = m m 8. ( )

15 AIEEE-PAPERS--5 Hece a d = 0 ( ) ( ) + 9. If is odd the ( ) is eve sum of odd terms = + =. 0. α α 6 e + e α α α = !! 6! α α 6 e + e α α α = !! 6! put α =, we get ( ) e = e!! 6!. si α + si β = ad cos α + cos β = Squarig ad addig, we get 70 + cos (α β) = (65) cos α β = cos = 0 0 α β 9 α β Q < <.. u = a cos θ + b si θ + a si θ + b cos θ = a + b a b a + b b a + cos θ + + cos θ a + b a b u = a + b + cos θ mi value of u = a + b + ab ma value of u = ( a + b ) ( ) u u = a b. ma mi. Greatest side is + si α cos α, by applyig cos rule we get greatest agle = 0 ο. h. ta0 = 0 + b h = 0 + b..() ta60 = h/b h = b.() b = 0 m b h 5. si cos si cos + rage of f() is [, ]. Hece S is [, ]. 6. If y = f () is symmetric about the lie = the f( + ) = f( ). 5

16 AIEEE-PAPERS > 0 ad [, ) a b + 8. a b a b a b a lim lim e a, b R = + + = = ta ta f() = lim = y + e y +... y+ e y = e = e + l = y dy = =. d 9. Ay poit be t, 9t ; differetiatig y = 8 dy = 9 = = (give) t =. d y t 9 9 Poit is, 8. f () = 6( ) f () = ( ) + c ad f () = c = 0 f () = ( ) + k ad f () = k = 0 f () = ( ).. Elimiatig θ, we get ( a) + y = a. Hece ormal always pass through (a, 0).. Let f () = a + b + c f() = f() ( a b 6c 6d) a b + + c + d = + + +, Now f() = f(0) = d, the accordig to Rolle s theorem 6 f () = a + b + c = 0 has at least oe root i (0, ) 5. r = r lim e = 0 e d = (e ) 6. Put α = t si( α + t) dt = si α cot tdt + cos α dt si t cos α α + si α l si t + c = ( ) A = cos α, B = si α 6

17 AIEEE-PAPERS d cos si = cos + d = sec d + = log ta C 8. ( ) d + ( ) d + ( ) d = + + = ( si + cos ) d = ( ) 0 ( si + cos ) si + 0 cos d = cos si 0 + =. 0. Let I = f(si )d = ( )f(si )d = f(si )d I (sice f (a ) = f ()) I = 0 / f(si )d A = 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. Area = ( )d + ( )d =. y= y =. + yy - ay = 0 a = + yy (elimiatig a) y ( y )y = y. 5. y d + dy + y dy = 0. d(y) dy 0 y + y = + log y = C. y 5. If C be (h, k) the cetroid is (h/, (k )/) it lies o + y =. locus is + y = 9. 7

18 AIEEE-PAPERS y a + b = where a + b = - ad + = a b a =, b = - or a = -, b =. y y Hece = ad + =. c 7. m + m = ad m m = 7 m + m = m m (give) c =. 8. m + m = c, m m = 6 c ad m =. Hece c = Let the circle be + y + g + fy + c = 0 c = ad it passes through (a, b) a + b + ga + fb + = 0. Hece locus of the cetre is a + by (a + b + ) = Let the other ed of diameter is (h, k) the equatio of circle is ( h)( p) + (y k)(y q) = 0 Put y = 0, sice -ais touches the circle (h + p) + (hp + kq) = 0 (h + p) = (hp + kq) (D = 0) ( p) = qy. 5. Itersectio of give lies is the cetre of the circle i.e. (, ) Circumferece = 0 radius r = 5 equatio of circle is + y + y = Poits of itersectio of lie y = with + y = 0 are (0, 0) ad (, ) hece equatio of circle havig ed poits of diameter (0, 0) ad (, ) is + y y = Poits of itersectio of give parabolas are (0, 0) ad (a, a) equatio of lie passig through these poits is y = O comparig this lie with the give lie b + cy + d = 0, we get d = 0 ad b + c = 0 (b + c) + d = Equatio of directri is = a/e = a = b = a ( e ) b = Hece equatio of ellipse is + y =. 55. l = cos θ, m = cos θ, = cos β cos θ + cos θ + cos β = cos θ = si β = si θ cos θ = /5. (give) 56. Give plaes are + y + z 8 = 0, + y + z + 5 = 0 + y + z + 5/ = 0 d d 8 5 / 7 Distace betwee plaes = = =. a + b + c + + 8

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 directio cosie of the lies itersectig the above lies is proportioal 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 are (a, a, a) ad (a, a, a). y + z y z 58. Give lies = = = s ad = = = t are coplaar the pla λ λ / passig through these lies has ormal perpedicular to these lies a - bλ + cλ = 0 ad a + b c = 0 (where a, b, c are directio ratios of the ormal to the pla) O solvig, we get λ = Required plae is S S = 0 where S = + y + z + 7 y z = 0 ad S = + y + z + y + z 8 = 0 y z =. a + b = t c.() ad b + c = ta.() () () a ( + t ) + c ( t 6) = 0 + t = 0 t = -/ & t = -6. Sice a ad c are o-colliear. Puttig the value of t ad t i () ad (), we get a + b + 6c = ( ) 6. Work doe by the forcesf ad F is (F + F ) d, where d is displacemet Accordig to questio F + F = (i ˆ + ˆj k) ˆ + (i ˆ + ˆj k) ˆ = 7i ˆ + ˆj kˆ ad d = (5i ˆ + ˆj + k) ˆ (i ˆ + j ˆ + k) ˆ = i ˆ + j ˆ kˆ. Hece (F + F ) d is Coditio for give three vectors to be coplaar is 0 λ = 0 λ = 0, /. 0 0 λ Hece give vectors will be o coplaar for all real values of λ ecept 0, /. 6. Projectio of v alog u ad w alog u is v u u ad w u u Accordig to questio v u w = u v u = w u. ad v w = 0 u u respectively u v + w = u + v + w u v + u w v w = u v + w =. 9

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 = 0 ad + ( c ) = 0 b c + cos θ = 0 cosθ = / siθ =. 65. Mode ca be computed from histogram ad media is depedet o the scale. Hece statemet (a) ad (b) are correct. 66. i = a for i =,,..., ad i = a for i =,..., i = i S.D. = ( ) = = i i Sice 0 i = i= a = a = 67. E : evet deotig that A speaks truth E : evet deotig that B speaks truth Probability that both cotradicts each other = P ( E E ) P ( E E ) 68. P(E F) = P(E) + P(F) P ( E F) = = = + = Give that p =, p q = q = / p = /, = 8 p( = ) = C = P + Q =, P + Q = 9 P = + N ad Q = N. 7. F. si θ = 9 F. cos θ = 6 F = 5. cosθ θ C A θ B F siθ 7. By Lami s theorem A B C P : Q : R = si 90 + : si 90 : si cos A : cos B : cos C. B A 90+C/ 90+B/ 90+A/ C 0

21 AIEEE-PAPERS-- 7. Time T from A to B = = hrs. C T from B to C = 5 5 = hrs. 5 Total time = hrs. Average speed = 7 km/ hr. A B Resultat average velocity = km/hr. si0 si(5 + 0 ) 8 7. Compoet alog OB = = ( 6 ) m/s. 75. t = usi α g t u t + =. g, t = usi β g where α + β = 90 0

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