JEE-(Advanced) 2016 PART I : MATHEMATICS SECTION-III [SINGLE CORRECT CHOICE TYPE] and I be the identity matrix of order 3.

Size: px

Start display at page:

Download "JEE-(Advanced) 2016 PART I : MATHEMATICS SECTION-III [SINGLE CORRECT CHOICE TYPE] and I be the identity matrix of order 3."

Polly Parker
5 years ago
Views:

1 CODE : 9 PAPER- PART I : MATHEMATICS SECTION-III [SINGLE CORRECT CHOICE TYPE] Q. to Q.6 hs four choices (A), (B), (C), (D) out of which ONLY ONE is correct Let P d I be the idetity mtrix of order 3. If Q = [q ij ] is mtrix such tht P 50 Q = I, the q q q 3 3 equls ekuk fd 0 0 P vksj I rhu dksfv dk rrled vko;wg gsa ;f Q = [q ij ],d vko;wg bl izdkj gs fd P 50 Q = I gs] rc q q q 3 3 dk eku gsa (A) 5 (B) 03 (C) 0 (D) 05 B Let P I A A I A A A = 0 3 Give : P 50 = Q + I I + Q = (I + A) 50 = I + 50A A + 0 Q = q q 3 = 50 4 = 00, q = 50 4 = 00 : Opposite Relice Petrol Pump, Piprli Rod, Sikr Ph , Mob ,

2 q q q Are of the regio (x, y) : y x 3,5y x 9 5 {ks=k (x, y) : y x 3,5y x 9 5 is equl to dk {ks=kqy gs & (A) 6 (B) 4 3 (C) 3 (D) 5 3 C Shiftig origi to ( 3, 0) (-,) C E B(4,) (-,0)D A(4,0) (9,0) Are x, yr : y x,5y x 6 5 Desired Are = Are of trpezium ABCDA [Are DCOD + Are OBAO] x dx x dx The vlue of 3 k (k ) k si si is equl to k (k ) k si si dk eku gsa : Opposite Relice Petrol Pump, Piprli Rod, Sikr Ph , Mob ,

3 (A) 3 3 C (B) 3 3 (C) 3 (D) 3 k si (k ) k cot (k ) cot k si si si (k ) k k cot cot cot cot ( ( 3)) ( 3) Let b i > for i =,,..., 0. Suppose log e b, log e b,..., log e b 0 re i Arithmetic Progressio (A.P.) with the commo differece log e. Suppose,,... 0 re i A.P. such tht = b d 5 = b 5. If t = b + b b 5 d s = , the (A) s > t d 0 > b 0 (B) s > t d 0 < b 0 (C) s < t d 0 d b 0 (D) s < t d 0 < b 0 ekuk fd i =,,..., 0 ds fy, b i > gsa eku yhft, fd log e b, log e b,..., log e b 0 lkozvrj log e okyh lekrj Js.kh es gsa eku yhft, fd,,... 0 lekrj Js.kh es bl izdkj gs fd = b rfkk 5 = b 5. ;f t = b + b b 5 rfkk s = gs] rks (A) s > t vksj 0 > b 0 (B) s > t vksj 0 < b 0 (C) s < t vksj 0 d b 0 (D) s < t vksj 0 < b 0 B If log e b, log e b... log e b 0 A.P. the b, b,... b 0 G.P. Give = b d 5 = b 5,, , 5 re i Arithmetic progressio. Here, We c sy, 3, rearithmetic mes betwee d 50. By A.M. > G.M. > b, 3 > b 3,... s > t Also, 5, 0 re i Arithmetic Progressio d b, b 5, b 0 re i Geometric Progressio. b 0 > 0 ( = b, 5 = b 5 ) : Opposite Relice Petrol Pump, Piprli Rod, Sikr Ph , Mob ,

4 PHYSICS 4. The vlue of x cos x x dx e is equl to x cos x x dx e dk eku gsa (A) A 4 (B) b b Apply Let I f x dx f b x dx / x cos x / e x dx (C) 4 / / e e (D) I x cos x dx x si x x cos x si x / x cos x / x cos x I dx dx / x / x e e 4. Let P be the imge of the poit (3,, 7) with respect to the ple x y + z = 3. The the equtio of the ple pssig through P d cotiig the stright lie x y z is ekuk fd fcuq (3,, 7) dk] lery x y + z = 3 ds lkis{k izfrfcec P gsa rc fcuq P ls xqtjus okyh vksj ljy js[kk x y z dks /kkj.k djus okys lery dk lehdj.k gsa (A) x + y 3z = 0 (B) 3x + z = 0 (C) x 4y + 7z = 0 (D) x y = 0 C Get imge of P(,3,7) with respect to give ple i.e. Q(,5,3) OQ ˆ i j ˆ k ˆ Norml vector = = ˆi 4j ˆ 7kˆ Equtio of ple will be x 4y + 7z = Poit (0, 0, 0) will lie i the ple So the required ple is x 4y + 7z = 0 : Opposite Relice Petrol Pump, Piprli Rod, Sikr Ph , Mob ,

5 [MULTIPLE CORRECT CHOICE TYPE] Q. to Q.6 hs four choices (A), (B), (C), (D) out of which ONLY ONE is correct. 43. Let,bd f : be defied by f(x) = cos ( x 3 x ) + b x si( x 3 + x ). The f is (A) differetible t x = 0 if = 0 d b = (B) differetible t x = if = d b = 0 (C) NOT differetible t x = 0 t = d b = 0 (D) NOT differetible t x = if = d b = ekuk fd, b rfkk f :, f(x) = cos ( x 3 x ) + b x si( x 3 + x ) ls ifjhkkf"kr gsa rc f (A) x = 0 ij vodyuh; gs ;f = 0 vksj b = (B) x = ij vodyuh; gs ;f = vksj b = 0 (C) x = 0 ij vodyuh; ugh gs ;f = vksj b = 0 (D) x = ij vodyuh; ugh gs ;f = vksj b = AB f(x) = cos(x 3 x) + bxsi(x 3 + x) x R Which is compositio d sum of differetible fuctios, therefore f(x) is lwys cotiuous d differetible. (x )(x )...(x ) 44. Let f(x) lim x!(x )(x )...(x ) 4 x, for ll x > 0. The ekuk fd lhkh x > 0 ds fy, (x )(x )...(x ) f(x) lim x!(x )(x )...(x ) 4 x gsa rc & (A) f f () BC (B) f f 3 3 (C) f ' () 0 (D) f '(3) f '() f (3) f () f x Lim x r r r x r r r x r l f x x L im log x. log. log x. r r / r r r / t t l f x x log 0 x dt log t dt log x dt 0 0 : Opposite Relice Petrol Pump, Piprli Rod, Sikr Ph , Mob ,

6 tx l f x x log dt 0 t x put tx = u x u l f x log du 0 u x u f x log du e0 u 45. Let f : (0, ) d g : be twice differetible fuctios such tht f" d g" re cotiuous fuc- f (x)g(x) tios of. Suppose f ' () = g(), f "() 0 d g ' () 0. If lim x, the f '(x)g '(x) (A) f hs locl miimum t x = (B) f hs locl mximum t x = (C) f " () > f() (D) f(x) f " (x) = 0 for t lest oe x ekuk fd f : (0, ) vksj g :,sls ks ckj vodyuh; Qyu gs fd ij f " vksj g " lrr Qyu gsa eku f (x)g(x) yhft;s fd f ' () = g(), f "() 0 vksj g ' () 0 gsa ;f lim x gs] rc & f '(x)g '(x) (A) x = ij f dk LFkkuh; fueure gs (B) x = ij f LFkkuh; mppre gsa (C) f " () > f() (D) de ls de,d x ds fy, f(x) f " (x) = 0 AD f x g x Lim x f ' x g ' x f ' xgx f '' xg ' x f x g ' x f x x Lim f ' x g '' x f() = f ''() = +ve Hece f hs locl mximum t x =. ( co-domi (0, )) f x Lim x f '' x 46. Let û u ˆ ˆ ˆ i u j u3k be uit vector i 3 ˆ ˆ ˆ d ˆ (i j k). Give tht there exists vector 6 i 3 such tht û d ˆ.(u ˆ ). Which of the followig sttemet(s) is(re) correct? (A) There is exctly oe choice for such (B) There re ifiitely my choices for such (C) If û lies i the xy-ple th u = u (D) If û lies i the xz-ple th u = u 3 ekuk fd 3 es û u ˆ ˆ ˆ i u j u3k,d ek=kd lf'k gs vksj ˆ (i ˆ ˆ j k) ˆ gsa f;k gqvk gs fd 3 6 es lf'k dk vflrro bl izdkj gs fd û vksj ˆ.(u ˆ ) gsa fueufyf[kr es ls dksu lk ¼ls½ dfku lgh gs ¼gS½ \ (A) bl izdkj ds ds fy, Bhd p;u lhko gsa (B) bl izdkj ds ds fy, vuur p;u lhko gsa (C) ;f û, xy-lery ij gs rc u = u gsa : Opposite Relice Petrol Pump, Piprli Rod, Sikr Ph , Mob ,

7 (D) ;f û, xz-lery ij gs rc u = u 3 gsa BC w. u v w ˆ uˆ v cos cos = wˆ uˆ d ŵ v s it is give there exist vector v ˆ ˆ ŵ must be to û hece ifiite my such v exists. û u ˆi u ˆj u.w 0 (u u ) 0 if u = u if u u ˆ ˆ i u3k u.w 0 u + u 3 = 0 u = u Let P be the poit o the prbol y = 4x which is t the shortest distce from the ceter S of the circle x + y 4x 6y + 64 = 0. Let Q be the poit o the circle dividig the lie segmet SP iterlly. The (A) SP 5 (B) SQ : QP 5 : (C) the x-itercept of the orml to the prbol t P is 6 (D) the slope of the tget to the circle t Q is ekuk fd ijoy; y = 4x ij P,d,slk fcuq gs tks o`r x + y 4x 6y + 64 = 0 ds dsuz fcuq S ls U;wure wjh ij gsa ekuk fd o`r ij fcuq Q,slk gs fd og js[kk[km SP dks vkrfjd fohkkftr djrk gsa rc (A) SP 5 (B) SQ : QP 5 : (C) ijoy; ds fcq P ij vfhkyec dk x vr% [km 6 gs (D) o`r ds fcuq Q ij Li'kZ js[kk dh <ky gsa ACD : Opposite Relice Petrol Pump, Piprli Rod, Sikr Ph , Mob ,

8 48. Let, b d + b 0. Suppose S z : z, t, t 0 ibt, where i. If z = x + i y d z S, the (x, y) lies o (A) the circle with rdius d cetre,0 for > 0, b 0 (B) the circle with rdius = (C) the x-xis for 0, b = 0 (D) the y-xis for = 0, b 0 d cetre ekuk fd, b vksj + b 0 gsa eku yhft, fd ;f z = x + i y vksj z S gs] rc (x, y),0 for < 0, b 0 (A) ml o`r ij gs ftldh f=kt;k vksj dsz fcuq,0 gs tc > 0, b 0 S z : z, t, t 0 ibt, tgk i gsa (B) ml o`r gs fd ftldh f=kt;k (C) x-v{k ij gs tc 0, b = 0 (D) y-v{k ij gs tc = 0, b 0 ACD ibt x iy b t x b t y bt b t, vksj dsuz fcuq,0 for < 0, b 0 : Opposite Relice Petrol Pump, Piprli Rod, Sikr Ph , Mob ,

9 (C) for b = 0 (D) for = 0 x = 0, x, y 0 (A) Elimitig t we get y bt x y 0 x 49. Let,,. Cosider the system of lier equtios x + y = 3x y = Which of the followig sttemet(s) is(re) correct? (A) If = 3, the the system hs ifiitely my solutios for ll vlues of d (B) If 3, the the system hs uique solutio for ll vlues of d (C) If = 0, the the system hs ifiitely my solutios for = 3 (D) If = 0, the the system hs o solutio for = 3 ekuk fd,, gsa bu jsf[kd lehdj.kks ds fudk; (system of lier equtios) ij fopkj dhft,a x + y = 3x y = fueufyf[kr es ls dksu lk (ls) dfku lgh gs (gs)? (A) ;f = 3, rc vksj ds lhkh ekuks ds fy, fudk; ds vuur (ifiitely my) gy gs (B) ;f 3, rc vksj ds lhkh ekuks ds fy, fudk; dk vf}rh; (uique) gy gs (C) ;f = 0, rc = 3 ds fy, fudk; ds vuur gy gs (D) ;f = 0, rc = 3 ds fy, fudk; dk dksbz gy ugh gs BCD System hs uique solutio for 3 system hs ifiitely my solutio for 3 d o solutio for Let f :, d g", be fuctios defied by f(x) = [x 3] d g(x) = x f(x) + 4x 7 f(x), where [y] deotes the gretest iteger less th or equl to y for y. The (A) f is discotiuous exctly t three poits i, (B) f is discotiuous exctly t four poits i, (C) g is NOT differetible exctly t four poits i, : Opposite Relice Petrol Pump, Piprli Rod, Sikr Ph , Mob ,

10 (D) g is NOT differetible exctly t five poit i, ekuk fd Qyu f :, vksj g",, f(x) = [x 3] vksj g(x) = x f(x) + 4x 7 f(x) ls ifjhkkf"kr gs, tgk y ds fy, y ls de ;k y ds cjkcj ds egùke iw.kkzd (gretest iteger less th or equl to y) dks [y] }kjk 'kkz;k x;k gsa rc (A), es f Bhd rhu (exctly three) fcuqvks ij vlrr (discotiuous) gs (B), es f Bhd pkj (exctly four) fcuqvks ij vlrr gs (C), es g Bhd pkj (exctly four) fcuqvks ij vodyuh; (differetible) ugh gs (D), es g Bhd ik p (exctly five) fcuqvks ij vodyuh; (differetible) ugh gs BC Clerly from the grph f is discotiuous t four poits. g(x) = f(x)( x + 4x 7 ) f(x) is o-differetible t x =,, 3 & x + 4x 7 is o-differetible t x = 0, 7 4 But f(x) = 0 x 3, Hece g(x) is o-differetible t x = 0,,, 3 [PARAGRAPH TYPE] Q.3 to Q.6 hs four choices (A), (B), (C), (D) out of which ONLY ONE is correct. Prgrph for questio os. 5 to 5 : Opposite Relice Petrol Pump, Piprli Rod, Sikr Ph , Mob ,

11 Footbll tems T d T hve to ply two gmes gist ech other. It is ssumed tht the outcomes of the gmes re idepedet. The probbilities of T wiig, drwig d losig gme gist T d, 6 d 3, respectively. Ech tem gets 3 poits for wi, poit for drw d 0 poit for loss i gme. Let X d Y deote the totl poits scored by tems T d T, respectively, fter two gmes. QqVckWy yks T vksj T dks,d wljs ds fo:) ks [ksy (gmes) [ksyus gsa ;g eku fy;k x;k gs fd ksuks [ksyks ds ifj.kke,d wljs ij fuhkzj ugh djrsa y T ds y T ds fo:),d [ksy es thrus] cjkcj gksus vksj gkjus dh izkf;drk Øe'k%, 6 vksj gsa izr;sd y thrus ij 3 vd] cjkcjh ij vd gkjus ij 0 vd vftzr djrk gsa ekuk fd ks [kksyks 3 ds i'pkr y T vksj y T ds }kjk vftzr dqy vd Øe'k% X vksj Y gsa 5. P(X > Y) is : P(X > Y) dk eku gs (A) 4 B 5. P(X = Y) is : P(X = Y) dk eku gs (B) 5 (A) 36 (B) 3 C Let W T wis P(X > Y) P(X = Y) L T wis D drw (C) (C) 3 36 = P(WW) + P(WD) + P(PW) = = P(WL) + P(LW) + P(DD) = (D) 7 (D) Prgrph for questio os. 53 to 54 x y Let F (x, 0) d F (x, 0) for x < 0 d x > 0, be the foci of the ellipse 9 8. Suppose prbol hvig vertex t the origi d focus t F itersects the ellipse t poit M i the first qudrt d t poit N i the fourth qudrt. : Opposite Relice Petrol Pump, Piprli Rod, Sikr Ph , Mob ,

12 x y ekuk fd F (x, 0) vksj F (x, 0) (ftles x < 0, x > 0) h?kzo`ùk (ellipse) dh ukfhk;k (foci) gsa ekuk 9 8 fd,d ijoy; (prbol) ftldk 'kh"kz (vertex) ewyfcuq (origi) ij vksj ukfhk (focus) F ij gs] h?kzo`ùk dks izfke prqfkkz'k (first qudrt) es M ij vksj prqfkz prqfkkz'k (fourth qudrt) es N ij izfrpnsfr djrk gsa 53. The orthocetre of the trigle F MN is f=khkq F MN dk yecdsuz (orthocetre) gs 9 (A),0 0 A (B),0 3 9 (C),0 0 (D), 6 3 x y Ellipse : () foci of ellipse re (±, 0) Equtio of prbol hvig vertex (0, 0) d focus (, 0) is y = 4x...() from equtio () & () 4x 3 x + 9x 8 = 0 x, 6 (rejected) 9 8 x 3 M, 6 d 3 N, 6 3 Equtio of ltitude from vertex M, 6 put y = 0 we get 9 orthoceer of F MN is,0 0 is 5 3 y 6 x 3 9 x x If the tgets to the ellipse t M d N meet t R d the orml to the prbol t M meets the x-xis t Q, the the rtio of re of the trigle MQR to re of the qudrilterl MF NF is ;f h?kzo`ùk ds fcuqvks M vksj N ij Li'kZ js[kk, (tgets) R ij feyrh gs vksj ijoy; ds fcuq M ij vfhkyec x- v{k dks Q ij feyrk gs, rc f=khkqt MQR ds {ks=kqy vksj prqhkqzt (qudrilterl) MF NF ds {ks=kqy dk vuqikr (rtio) gs (A) 3 : 4 (B) 4 : 5 (C) 5 : 8 (D) : 3 C 3 Equtio of tget t poit M, 6 to the ellipse is x(3 / ) y : Opposite Relice Petrol Pump, Piprli Rod, Sikr Ph , Mob ,

13 3 Equtio of orml to the prbol t poit M, 6 7 put y = 0 Q is, Are of MQR 5 Now Are of qudrilterl MF NF is 6 3 y 6 x : Opposite Relice Petrol Pump, Piprli Rod, Sikr Ph , Mob ,

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).

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). 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