Time: 2 hours IIT-JEE 2006-MA-1. Section A (Single Option Correct) + is (A) 0 (B) 1 (C) 1 (D) 2. lim (sin x) + x 0. = 1 (using L Hospital s rule).

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1 IIT-JEE 6-MA- FIITJEE Solutios to IITJEE 6 Mthemtics Time: hours Note: Questio umber to crries (, -) mrks ech, to crries (5, -) mrks ech, to crries (5, -) mrks ech d to crries (6, ) mrks ech.. For >, lim ((si) / (/) si ) + is Sectio A (Sigle Optio Correct) (D) si / lim (si ) + lim si l + e (usig L Hospitl s rule).. d is equl to c + + c + + c (D) + + c (D) 5 d + Let + z dz z z c c.. Give isosceles trigle, whose oe gle is d rdius of its icircle. The the re of the trigle i sq. uits is (D) π b () FIITJEE Ltd. ICES House, Srvpriy Vihr (Ner Huz Khs Bus Term.), New Delhi - 6, Ph : , , 685, F : 659

2 IIT-JEE 6-MA- Also si si b b d s d s (+ b) ( + b) () From () d (), we get ( + 7 ).. If < θ < π, the the itervls of vlues of θ for which si θ 5 siθ + >, is π 5π π 5π,, π, , π π, π π (D), π si θ 5siθ + > (siθ ) (siθ ) > siθ < π 5π θ,, π w wz If w α + iβ, where β d z, stisfies the coditio tht is purely rel, the the set of vlues of z is z {z : z } {z : z z } {z : z } (D) {z : z, z } (D) w wz w wz z z (zz )(w w) zz z z. 6. Let, b, c be the sides of trigle. No two of them re equl d λ R. If the roots of the equtio + ( + b+ c) + λ (b + bc + c) re rel, the 5 λ< λ > 5 λ, 5 (D) λ, D ( + b + c) λ (b + bc + c) + b + c λ + (b + bc + c) Sice b < c + b b < c () b c < b + c bc < () c < b c + c < b () + b + c From (), () d (), we get <. b + bc + c Hece λ< + λ <. FIITJEE Ltd. ICES House, Srvpriy Vihr (Ner Huz Khs Bus Term.), New Delhi - 6, Ph : , , 685, F : 659

3 IIT-JEE 6-MA- 7. If f () f() d g() f () d F() f + g d give tht F(5) 5, the F() is equl to 5 (D) 5 f () f() d f () g() f (). f () + f(). f () f() + (f ()) c (f() + (g()) c F() c F() If r, s, t re prime umbers d p, q re the positive itegers such tht the LCM of p, q is r t s, the the umber of ordered pir (p, q) is (D) (D) Required umber of ordered pir (p, q) is ( ) ( 5 ) ( ). π 9. Let θ, d t (tθ) tθ, t (tθ) cotθ, t (cotθ) tθ d t (cotθ) cotθ, the t > t > t > t t > t > t > t t > t > t > t (D) t > t > t > t π Give θ,, the tθ < d cotθ >. Let tθ λ d cotθ + λ where λ d λ re very smll d positive. λ +λ the t ( λ ),t ( λ ) λ +λ +λ +λ t ( ) d t ( ) Hece t > t > t > t.. The is of prbol is log the lie y d the distce of its verte from origi is d tht from its focus is. If verte d focus both lie i the first qudrt, the the equtio of the prbol is ( + y) ( y ) ( y) ( + y ) ( y) ( + y ) (D) ( y) 8 ( + y ) (D) Equtio of directri is + y. Hece equtio of the prbol is + y ( ) + (y ) Hece equtio of prbol is ( y) 8( + y ).. A ple psses through (,, ) d is perpediculr to two ples y + z d y + z. The distce of the ple from the poit (,, ) is (D) (D) The ple is ( ) + b(y + ) + c(z ) where b + c d b + c b c So, the equtio of ple is + y + FIITJEE Ltd. ICES House, Srvpriy Vihr (Ner Huz Khs Bus Term.), New Delhi - 6, Ph : , , 685, F : 659

4 IIT-JEE 6-MA- Distce of the ple from the poit (,, ) Let ˆi+ j ˆ+ k, ˆ b ˆi ˆj+ kˆ d c ˆi+ ˆj kˆ. A vector i the ple of d b i ˆ ˆj + kˆ i ˆ+ ˆj kˆ i ˆ+ ˆj kˆ (D) i ˆ+ ˆj kˆ whose projectio o c is, is Vector lyig i the ple of d b is r λ +λb d its projectio o c is ˆi+ ˆj kˆ λ ˆ ˆ ˆ +λ i+ λ λ j+ λ +λ k r λ i ˆ+ ˆj+ λ kˆ ( ) ( ) ( ) λ λ ( ) ( ) Hece the required vector is i ˆ+ 5j ˆ k. ˆ Sectio B (My hve more th oe optio correct). The equtios of the commo tgets to the prbol y d y ( ) is/re y ( ) y y ( ) (D) y 5, Equtio of tget to y is y m m () Equtio of tget to ( ) y is y m( ) + m () () d () re ideticl. m or Commo tgets re y d y.. If f() mi {,, }, the f() is cotiuous R f () >, > f() is ot differetible but cotiuous R (D) f() is ot differetible for two vlues of, f() mi. {,, }, f(), > f() is cotiuous R d o-differetible t. y y y 5. A tget drw to the curve y f() t P(, y) cuts the -is d y-is t A d B respectively such tht BP : AP :, give tht f(), the dy equtio of curve is y d orml t (, ) is + y curve psses through (, /8) dy (D) equtio of curve is y d + FIITJEE Ltd. ICES House, Srvpriy Vihr (Ner Huz Khs Bus Term.), New Delhi - 6, Ph : , , 685, F : 659

5 IIT-JEE 6-MA-5, (D) Equtio of the tget is Y y dy (X ) d Give BP so tht AP d dy dy y y d + l l y l c l (l cy) cy. Give f() c y. dy, y d B P(, y) A y, dy / d y 6. If hyperbol psses through the focus of the ellipse + d its trsverse d cojugte es coicide with 5 6 the mjor d mior es of the ellipse, d the product of eccetricities is, the y y the equtio of hyperbol is the equtio of hyperbol is focus of hyperbol is (5, ) (D) focus of hyperbol is ( 5, ), Eccetricity of ellipse 5 Eccetricity of hyperbol 5 d it psses through (±, ) y its equtio 9 b b 5 where + b y d its foci re (±5, ) Iterl bisector of A of trigle ABC meets side BC t D. A lie drw through D perpediculr to AD itersects the side AC t E d the side AB t F. If, b, c represet sides of ABC the AE is HM of b d c AD bc cos A b + c EF bc si A (D) the trigle AEF is isosceles b + c,,, (D). We hve ABC ABD + ACD A A bcsia cadsi + b ADsi bc A AD cos b + c A A/ E Agi AE AD sec A bc b + c AE is HM of b d c. F B D C FIITJEE Ltd. ICES House, Srvpriy Vihr (Ner Huz Khs Bus Term.), New Delhi - 6, Ph : , , 685, F : 659

6 IIT-JEE 6-MA-6 EF ED + DF DE AD t A bc cos A t A b + c bc A si b + c As AD EF d DE DF d AD is bisector AEF is isosceles. Hece A, B, C d D re correct swers. 8. f() is cubic polyomil which hs locl mimum t. If f() 8, f() d f () hs locl miim t, the the distce betwee (, ) d (, f()), where is the poit of locl miim is 5 f() is icresig for [, 5 ] f() hs locl miim t (D) the vlue of f() 5, The required polyomil which stisfy the coditio is f() ( ) f() hs locl mimum t d locl miimum t Hece f() is icresig for, Let A be vector prllel to lie of itersectio of ples P d P through origi. P is prllel to the vectors j ˆ+ kˆ d j ˆ kˆ d P is prllel to ˆ j k ˆ d i ˆ+ j ˆ, the the gle betwee vectors A d i ˆ+ ˆj kˆ is π π π (D) π 6, (D) Vector AB is prllel to (i ˆ+ k) ˆ () k ˆ (j ˆ k) ˆ (i ˆ+ j) ˆ 5(j ˆ k) ˆ Let θ is the gle betwee the vector, the cosθ± ±.5 π π Hece θ,. e,. f() e, < d g() f() t dt, [, ] the g () hs e, < locl mim t + l d locl miim t e locl mim t d locl miim t o locl mim (D) o locl miim, e g() f() e < e < g (), whe + l d e e < g() < FIITJEE Ltd. ICES House, Srvpriy Vihr (Ner Huz Khs Bus Term.), New Delhi - 6, Ph : , , 685, F : 659

7 IIT-JEE 6-MA-7 g ( + l ) e l < hece t + l, g() hs locl mimum g (e) > hece t e, g() hs locl miimum. f() is discotiuous t, the we get locl mim t d locl miim t. Sectio C Comprehesio I There re urs ech cotiig + blls such tht the ith ur cotis i white blls d ( + i) red blls. Let u i be the evet of selectig ith ur, i,,, d w deotes the evet of gettig white bll.. If P(u i ) i, where i,,,, the lim P( w) is equl to (D) P(u i ) ki ΣP(u i ) k ( + ) lim P(w) lim lim i + + i ( + )( + ) ( ) ( ) 6. If P(u i ) c, where c is costt the P(u /w) is equl to + + (D) + c u + Σ i c ( + P w +.. If is eve d E deotes the evet of choosig eve umbered ur ( P(u i ) ), the the vlue of P( w/e ) is (D) + ( + ) + w P E ( + ) (+ ) FIITJEE Ltd. ICES House, Srvpriy Vihr (Ner Huz Khs Bus Term.), New Delhi - 6, Ph : , , 685, F : 659

8 IIT-JEE 6-MA-8 Comprehesio II Suppose we defie the defiite itegrl usig the followig formul f()d ( f() + f(b) ) c b c Whe c b c (, b) Fc ( ) ( f( ) f( c) ) ( f(b) f(c) ) b b b, for more ccurte result for + b, f()d ( f() + f(b) + f(c) ).. π / si d is equl to π π ( + ) ( + ) 8 π 8 π (D) π π π / + + si d π si() si si + + π ( + ) Dt could ot be retrieved. 6. If f () < (, b) d c is poit such tht < c < b, d (c, f(c)) is the poit lyig o the curve for which F(c) is mimum, the f (c) is equl to f( b) f( ) f ( ( b) f( ) ) b b f ( b) f ( ) (D) b (F (c) (b ) f (c) + f() f(b) F (c) f (c) (b ) < f(b) f() F (c) f(c). b Comprehesio III Let ABCD be squre of side legth uits. C is the circle through vertices A, B, C, D d C is the circle touchig ll the sides of the squre ABCD. L is lie through A. 7. If P is poit o C d Q i other poit o C, the PA PB PC PD QA + QB + QC + QD.75.5 (D).5 is equl to Let A, B, C d D be the comple umbers,, i d i respectively z + z+ + z+ i + z i PA PB PC PD QA QB QC QD z z z i z i z +. z + FIITJEE Ltd. ICES House, Srvpriy Vihr (Ner Huz Khs Bus Term.), New Delhi - 6, Ph : , , 685, F : 659

9 IIT-JEE 6-MA-9 8. A circle touches the lie L d the circle C eterlly such tht both the circles re o the sme side of the lie, the the locus of cetre of the circle is ellipse hyperbol prbol (D) prts of stright lie Let C be the cetre of the required circle. Now drw lie prllel to L t distce of r (rdius of C ) from it. Now CP AC C lies o prbol. C P C L A 9. A lie M through A is drw prllel to BD. Poit S moves such tht its distces from the lie BD d the verte A re equl. If locus of S cuts M t T d T d AC t T, the re of T T T is sq. uits sq. uits sq. uit (D) sq. uits AG AT T G [s A is the focus, T is the verte d BD is the directri of prbol]. Also T T is ltus rectum T T Are of T T T. M T D A T T G B C Comprehesio IV A, if U, U d U re colums mtrices stisfyig. AU AU, AU d U is mtri whose colums re U, U, U the swer the followig questios,. The vlue of U is / (D) Let U be y so tht z y z y z FIITJEE Ltd. ICES House, Srvpriy Vihr (Ner Huz Khs Bus Term.), New Delhi - 6, Ph : , , 685, F : 659

10 IIT-JEE 6-MA- Similrly U, U. Hece U d U.. The sum of the elemets of U is (D) Moreover dj U Hece U dju d sum of the elemets of U. U is 5 5/ (D) /. The vlue of [ ] U The vlue of [ ] [ ] [ ] Sectio D. If roots of the equtio c d re, b d those of b re c, d, the the vlue of + b + c + d is (, b, c d d re distict umbers) As + b c d c + d b d, cd b c d (b + d) 9( + c) c d c c b + c c (b + d) ( + c) () 9( + c) ( + c) or (rejected) + b + c + d. FIITJEE Ltd. ICES House, Srvpriy Vihr (Ner Huz Khs Bus Term.), New Delhi - 6, Ph : , , 685, F : 659

11 IIT-JEE 6-MA-. The vlue of 55 5 ( ) d 5 ( ) d is 5 55 ( ) d 5 ( ) d 5 5 I 55 I I ( )( ) d I 9 5 I ( ) d 5 5 ( ) ( ) I 55 I I 55 I I 5. If + + ( ) d b, the fid the miimum turl umber such tht b > > ( ) 7 + b > < 6 < 7 7 < 6 < 6 miimum turl umber If f() is twice differetible fuctio such tht f(), f(b), f(c), f(d), f(e), where < b < c < d < e, the the miimum umber of zeroes of g() (f ()) + f () f() i the itervl [, e] is d d to get the zero of g() we tke fuctio h() f(). f () betwee y two roots of h() there lies t lest oe root of h () g() h() g() ( f() f ()) FIITJEE Ltd. ICES House, Srvpriy Vihr (Ner Huz Khs Bus Term.), New Delhi - 6, Ph : , , 685, F : 659

12 IIT-JEE 6-MA- f() or f () f() hs miimum solutios f () miimum three solutio h() miimum 7 solutio h () g() hs miimum 6 solutios. Sectio E 7. Mtch the followig: Normls re drw t poits P, Q d R lyig o the prbol y which itersect t (, ). The (i) Are of PQR (ii) Rdius of circumcircle of PQR 5/ (iii) Cetroid of PQR (5/, ) (iv) Circumcetre of PQR (D) (/, ) As orml psses through (, ) m m m m m m, ± ( m + m + m) m ( m m) Cetroid + +,, m + m Circumcetre (mid poit of PR), (m + m ) (, ). m+ m Circum rdius ( ) ( ) Q m, m (, ) R m, m (,) uits. Are of PQR sq. uits. R QR si QPR si(t ) 5 si t 5 5 circumcetre.. 8. Mtch the followig π / cos si (i) ( si ) ( cos cot log ( si) ) d (ii) Are bouded by y d 5y (iii) Cosie of the gle of itersectio of curves y log d y is 6 l (iv) Dt could ot be retrieved. (D) / (i) π / cos si I (si ) (cos cot log(si ) )d π / d cos. d I (si) d FIITJEE Ltd. ICES House, Srvpriy Vihr (Ner Huz Khs Bus Term.), New Delhi - 6, Ph : , , 685, F : 659

13 IIT-JEE 6-MA- (ii) The poits of itersectio of y d 5y is (, ) d (, ) Hece required re ( 5y )dy y dy. (iii) The poit of itersectio of y log d y is (, ) Hece dy + log.log. d for y dy. d (, ) dy d (, ) If θ is the gle betwee the curve the tθ cosθ. (iv) dy d + y d y dy y/ y/ e y e dy + y + ke y/ e y/. 9. Mtch the followig (i) Two rys i the first qudrt + y d y itersects ech other i the itervl (, ), the vlue of is (ii) Poit (α, β, γ) lies o the ple + y + z. Let α ˆi+β ˆj+γkˆ, k ˆ (k ˆ ), the γ. / (iii) ( y ) dy + ( y ) dy (iv) If sia sib sic + cosa cosb, the the vlue of sic (D) (i) Solvig the two equtios of ry i.e. + y d y + we get > d y > + + whe + > ; we get >. (ii) We hve α ˆi+β ˆj+γkˆ kˆ γ Now; ˆ ˆ ˆ k (k ) ˆ (k )k ˆ (k ˆ k) ˆ γk ˆ ( α ˆi+β ˆj+γ k) ˆ α ˆi+β ˆj α β As α + β + γ γ. d + + d (iii) ( y )dy + (y )dy ( y )dy d + + d d / d. FIITJEE Ltd. ICES House, Srvpriy Vihr (Ner Huz Khs Bus Term.), New Delhi - 6, Ph : , , 685, F : 659

14 IIT-JEE 6-MA- (iv) sia sib sic + cosa cosb sia sib + cosa cosb cos(a B) cos(a B) cos(a B) sic.. Mtch the followig (i) t t i i, the t t (ii) Sides, b, c of trigle ABC re i AP d cosθ b + c, cosθ b + c, cosθ c + b, the θ θ t + t (iii) A lie is perpediculr to + y + z d psses through (,, ). The perpediculr distce of this lie from the origi is (iv) Dt could ot be retrieved. 5 (D) / (i) i t Now; i t i i t i + ( ) ( ) t i + t i (t t ) + (t 5 t ) + + t ( + ) t ( )... t t ( + ) t t + ( + ) π t t t + θ t (ii) We hve θ b+ c cosθ t θ + t b + c b+ c+ θ t Also, c θ + b c cosθ t θ + t + b + b+ c b t θ θ + t b (iii) Lie through (,, ) d perpediculr to ple + y + z is give by y z r. Let P(r, r +, r) be the foot of perpediculr o the stright lie the r + (r + ) + r r 9 5 Poit is give by,, Required perpediculr distce uits. 8 (iv) Dt could ot be retrieved. FIITJEE Ltd. ICES House, Srvpriy Vihr (Ner Huz Khs Bus Term.), New Delhi - 6, Ph : , , 685, F : 659

[Q. Booklet Number]

[Q. Booklet Number] 6 [Q. Booklet Numer] KOLKATA WB- B-J J E E - 9 MATHEMATICS QUESTIONS & ANSWERS. If C is the reflecto of A (, ) i -is d B is the reflectio of C i y-is, the AB is As : Hits : A (,); C (, ) ; B (, ) y A (,