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1 R Pe Towe, Rod No, otctos Ae, Bistupu, Jmshedpu 800, Tel (067)89, IIT JEE 0 Mthemtics Ppe I PART III MATHEMATIS SETION I (Totl Mks : ) (Sigle oect Aswe Type) This sectio cotis 7 multiple choice questios. Ech questio hs fou choices (A), (B), () d (D) out of which ONLY ONE is coect. 7. Let P {θ : si θ cos θ cos θ} d Q {θ : si θ cos θ si θ} be two sets. The (A) P Q d Q P (B) Q P () P Q (D) P Q 7. (D) P : si θ cos θ cos θ t θ Q : si θ cos θ si θ t θ P Q. 8. Let the stight lie x b divide the e eclsoed by y ( x), y 0, d x 0 ito two pts R (0 x b) d R (0 x ) such tht R R. The b equls (A) (B) () (D) 8. (B) R R b ( x ) dx ( x ) dx 0 b ( b ) ( b ) b Let α d β be the oots of x 6x 0, with α > β. If α β fo, the the vlue of 0 8 is 9 (A) (B) () (D) 9. () Sice α 6α 0 α 6α d β 6β 0 β 6β ( α β ) ( α β ) α ( α ) β ( β ) α. 6 α β. 6 β ( α β ) ( α β ) ( α β ) IIT JEE 0 (0 Ap ) Questio & Solutios Ppe I www. peclsses.com

2 0. A stight lie L though the poit (, ) is iclied t gle 60 to the lie x y. If L lso itesects the x xis, the the equtio of L is (A) y x 0 (B) y x 0 () y x 0 (D) y x 0 0. (B) If equied lie hs slope m m t 60 m m 0, Sice, the lie itesects the x xis equtio is, (y ) (x ) y x 0.. Let (x 0, y 0 ) be the solutio of the followig equtios (x) l (y) l l x l y The x 0 is (A) / 6 (B) / () / (D) 6. () Sice (l ) (l l x) (l ) (l l y) d l x. l l y. l 0 (l ). (l x) l y. l (l ) (l )... (i) d l x. l l y. l 0... (ii) Usig (i) l (ii) l, we get l x. {(l ) (l ) } (l ) {(l ) (l ) } l x l x /. l x si x. The vlue of dx is si x si(l 6 x ) l (A) ( / ) l ( / ) (B) ( / ) l ( / ) () l ( / ) (D) ( / 6) l ( / ) l. (A) I x si x dx si x si(l 6 x ) l x t x dx dt I l si t dt si t si(l 6 t ) l b b Usig f ( x ) dx ( f ( x ) f ( b x )) dx IIT JEE 0 (0 Ap ) Questio & Solutios Ppe I www. peclsses.com

3 l si t si(l 6 t ) I dt [sice l l l 6] si t si(l 6 t ) si(l 6 t ) si t l l l I dt [ t ] l l. l. Let i ˆ j ˆ k ˆ, b i ˆ j ˆ k ˆ, d c i ˆ j ˆ k ˆ be thee vectos. Avecto v i the ple of d b, whose pojectio o c is / is give by (A) i ˆ j ˆ k ˆ (B) i ˆ j ˆ k ˆ () i ˆ j ˆ k ˆ (D). () v λ( i ˆ j ˆ k ˆ ) µ ( i ˆ j ˆ k ˆ ) v. c (λ µ) (λ µ) (λ µ) µ λ c v i ˆ (λ ) j ˆ k ˆ (λ ) Oly optio () stisfies this oe. i ˆ j ˆ k ˆ IIT JEE 0 (0 Ap ) Questio & Solutios Ppe I www. peclsses.com

4 SETION II (Totl Mks : 6) (Multiple oect hoice Type) This sectio cotis multiple choice questios. Ech questio hs fou choices (A), (B), () d (D) out of which ONE OR MORE my be coect.. Let f : R R be fuctio such tht f (x y) f (x) f (y), x, y R. If f (x) is diffeetible t x 0, the (A) f (x) is diffeetible oly i fiite itevl cotiig zeo (B) f (x) is cotiuous x R () f ' (x) is costt x R (D) f (x) is diffeetible except t fiitely my poits. (B)() f ( x h ) f ( x ) f ( x ) f ( h ) f ( x ) f ( h ) f ( 0 ) f ( x ) Lim Lim Lim h 0 h h 0 h h 0 h Put x y 0, f (0) 0 f ' (x) f ' (0) f (x) f ' (0) x c f (x) kx c, c 0 s f (0) 0 Fuctio is cotiuous. f ' (x) is costt. x y. Let the ecceticity of the hypebol be ecipocl to tht of the ellipse x b y. If the hypebol psses though focus of the ellipse, the (A) the equtio of the hypebol is y (B) focus of the hypebol is (, 0) () the ecceticity of the hypebol is (D) the equtio of the hypebol is x y. (B)(D) x y,, b, e (b / ) e / ecceticity of hypebol /. Focus of ellipse (± e, 0) (±, 0) x y psses though focus (±, 0) b b (e ) x y Equtio of hypebol : b x y Focus of hypebol (± e, 0) (±, 0). x y IIT JEE 0 (0 Ap ) Questio & Solutios Ppe I www. peclsses.com

5 6. Let M d N be two o sigul skew symmetic mtices such tht MN NM. If P T deotes the tspose of P, the M N (M T N) (MN ) T is equl to (A) M (B) N () M (D) MN 6. () Sice M T M, N T N M N (M T N) (MN ) T M N ( MN) (MN ) T M N (N. ( M) ). (N T ). ( M) M N ( M). ( N). ( M) M (MN) (MN) ( M) M. Note tht the sttemet which is give i the poblem is icoect, s skew mtix of odd ode c t be ivetible. 7. The vecto(s) which is/e copl with vectos i ˆ j ˆ k ˆ d i ˆ j ˆ k ˆ, d pepedicul to the vecto i ˆ j ˆ k ˆ is / e (A) j ˆ k ˆ (B) i ˆ j ˆ () i ˆ j ˆ (D) j ˆ k ˆ 7. (A)(D) Let λ (i ˆ j ˆ k ˆ ) µ (i ˆ j ˆ k ˆ ) (λ µ)i ˆ (λ µ) j ˆ (λ µ) k ˆ. (i ˆ j ˆ k ˆ ) 0 λ µ λ ( j ˆ k ˆ ) Tkig λ, j ˆ k ˆ λ, j ˆ k ˆ. IIT JEE 0 (0 Ap ) Questio & Solutios Ppe I www. peclsses.com

6 IIT JEE 0 (0 Ap ) Questio & Solutios Ppe I 6 www. peclsses.com SETION III ( Totl Mks : ) (Pgph Type) This Sectio cotis pgphs. Bsed upo oe of the pgphs multiple choice questios d bsed o othe pgph multiple choice questios hve to be sweed. Ech of these hs fou choices (A), (B), () d (D) out of which ONLY ONE is coect. Pgph fo questio Nos 8 d 9 Let U d U be two us such tht U cotis white d ed blls, d U cotis oly white bll. A fi coi is tossed. If hed ppes the bll is dw t dom fom U d put ito U. Howeve, if til ppes the blls e dw t dom fom U d put ito U. Now bll is dw t dom fom U. 8. The pobbility of the dw bll fom U beig white is (A) / 0 (B) / 0 () 9 / 0 (D) / 0 8. (B) Fo ppeig til Fo ppeig hed P / Give tht the dw bll fom U is white, the pobbility tht hed ppeed o the coi is (A) 7 / (B) / () / (D) / 9. (D) Usig Byes theoem, P /.

7 Pgph fo questio Nos 60 d 6 Let, b d c be thee el umbes stisfyig 9 7 [ b c ] [ ]... (E) 60. If the poit P(, b, c), with efeece to (E), lies o the ple x y z, the the vlue of 7 b c is (A) 0 (B) () 7 (D) (D) D system hs o tivil solutio. If c k, b 6 / 7 k, k / 7 Q b c (k / 7) (6k / 7) k k 7 P (, b, c) (, 6, 7) 7 b c Let be solutio of x 0 with Im() > 0. If with b d c stisfyig (E), the the vlue of is equl to b c (A) (B) () (D) 6. (A), k c, b, ( ) ( ) ( ) 6. Let b 6, with d c stisfyig (E). If α d β e the oots of the qudtic equtio x bx c 0, the is 0 α β (A) 6 (B) 7 () 6 / 7 (D) 6. (B) b 6, k 7, c 7, Q.E. becomes x 6x 7 0 ; b 6, b 7 α 0 αβ β IIT JEE 0 (0 Ap ) Questio & Solutios Ppe I 7 www. peclsses.com

8 SETION IV (Totl Mks : 8) (Itege Aswe Type) This sectio cotis 7 questios. The swe to ech of the questios is sigle digit itege, gig fom 0 to 9. The bubble coespodig to the coect swe is to be dkeed i the ORS. 6. Let f : [, ) [(, ) be diffeetible fuctio such tht f (), if 6 X f ( t ) dt x f ( x ) x fo ll x, the the vlue of f () is Diffeetitig the give eltio, f ( x ) 6 f(x) f(x) x f ' (x) x f ' (x) x, which is i lie fom with I.F x x Hece, f(x) x cx with c. f (x) x x. f () 6. Note tht f () is / fom the give eltio, which is cotdictio. 6. If z is y complex umbe stisfyig z i, the the miimum vlue of z 6 i is 6.. Expessio Z i which is equivlet to double the distce betwee complex umbe z d (, / ) which lies o oe the dimete x of cicle give. Hece miimum vlue of expessio. 6. Let,, be ithmetic pogessio with d S p i, p 00. p i Fo y itege with 0, let m. If S S m does ot deped o, the is S S m {. ( ) d } {. ( ) d } ; ( 6 d d ) S m λ [ λ ] ( 6 d d ) S λ idepedet of whe d 6 d 9. IIT JEE 0 (0 Ap ) Questio & Solutios Ppe I 8 www. peclsses.com

9 66. oside the pbol y 8x. Let be the e of the tigle fomed by the ed poits of its ltus ectum d the poit P, o the pbol, d be the e of the tigle fomed by dwig tgets t P d t the ed poits of the ltus ectum. The 66.. Extemities of L.R. e (, ) & (, ) 6 Poits of itesectio of tgets t these poits e (, 0), (, ) & (, ). is 67. Let ( θ ) is. si si θ π π d f t, whee < θ < cos. The the vlue of ( f ( θ )) θ d (t θ ) 67.. Put k si θ t t k cos θ si θ cos θ t k sec k si θ si cos ec θ cos ec θ sec k θ si k cos ec cot θ θ t θ d ( f ( θ )) si k t θ f(θ) si k t θ Hece. d (t θ ) 68. The miimum vlue of the sum of el umbes,,,, 8 d 0 with > 0 is Usig A.M. G.M., Miimum vlue of equied expessio 8 (fo ). 69. The positive itege vlue of > stisfyig the equtio is π π π si si si {whee x (π / )} si x si x si x o si x si x si x. si x si x o si x. cos x si x. si x si si x si x [ Q si x 0] x x (ot possible, Q x 0) o x π x 7x π 7. (π / ) π 7. x IIT JEE 0 (0 Ap ) Questio & Solutios Ppe I 9 www. peclsses.com

PROBLEMS 05 - ELLIPSE Page 1

PROBLEMS 05 - ELLIPSE Page 1 PROBLEMS 0 ELLIPSE Pge 1 ( 1 ) The edpoits A d B of AB re o the X d Yis respectivel If AB > 0 > 0 d P divides AB from A i the rtio : the show tht P lies o the ellipse 1 ( ) If the feet of the perpediculrs