Mathematics Paper- II
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1 R Prerna Tower, Road No -, Conracors Area, Bisupur, Jamsedpur - 8, Tel - (65789, Maemaics Paper- II Jee Advance PART III - MATHEMATICS SECTION - : (One or more opions correc Type Tis secion conains muliple coice quesions. Eac quesion as four coices (A, (B, (C and (D, ou of wic ONLY ONE opion is correc.. Te quadraic equaion p( = wi real coefficiens as purely imaginary roos.ten p(p( = as (A only purely imaginary roos (B all real roos (C wo real and wo purely imaginary roos (D neier real nor purely imaginary roos. (D Since, b = for p( = a + b + c, as roos are pure imaginary. ( c i c wic are clearly neier pure real nor pure imaginary, as c. a. Tree boys and wo girls sand in a queue. Te probabiliy, a e number of boys aead of every girl is a leas one more an e number of girls aead of er, is (A / (B / (C / (D /. (A Possible cases : B, G, G, B, B G, G, B,B, B G, B, G, B, B G, B, B, G, B B, G, B, G, B n(e = 5 = 6 P(E = 6 / 5! = /.. Si cards and si envelopes are numbered,,,, 5, 6 and cards are o be placed in envelopes so a eac envelope conains eacly one card and no card is placed in e envelope bearing e same number and moreover e card numbered is always placed in envelope numbered. Ten e number of ways i can be done is (A 6 (B 65 (C 5 (D 67. (C Toal derrangemen of 6 cards = D(6 = = A( + A( + A( + A(5 + A(6 [Cases wen envelope (or A as or or or 5 or 6 numbered card all are equally likely] 65 = 5 A( A( = 5. JEE Advanced (5-May- Quesion & Soluions Paper-II -- www. prernaclasses.com
2 . In a riangle e sum of wo sides is and e produc of e same wo sids is y. If c = y, were c is e ird side of e riangle, en e raio of e in-radius o e circum-radius of e riangle is ( c (A (B c( c (C ( c. (B a + b =, ab = y, c = y a b c c S (D c( c c = y (a + b c a b c = ab / ab cos c = / c = /.. a. b. sin r / s R abc / abc. s c. y. c ( c. c 5. Te common angens o e circle + y = and e parabola y = 8 ouc e circle a e poins P, Q and e parabola a e poins R, S. Ten e area of e quardilaeral PQRS is (A (B 6 (C 9 (D 5 5. (D P Q Equaion of angen PR, P(, S y m, m Wic is also angen o circle + y =. m m = m Ps of conac R & S are (, and (, and P, Q are (, & (, Area of quardilaeral PQRS = ½ [ + 8] = 5. JEE Advanced (5-May- Quesion & Soluions Paper-II -- www. prernaclasses.com
3 6. Te funcion y = f ( is e soluion of e differenial equaion saisfying f ( =. Ten / f ( d is / dy y in (, d (A ( / ( / (B ( / ( / (C ( / 6 ( / (D ( / 6 ( / dy y 6. (B I.F. = d e Sol n 5 is y d ( / 5 c / 5 ( / 5 f ( =, c = f ( d ( / ( /. / / 7. Le f : [, ] R be a funcion wic is coninuous on [, ] and is differeniable on (, wi / f ( =. Le f ( f ( d for [, ]. If F' ( = f ' ( for all (,, en F( equals (A e (B e (C e (D e 7. (B F' ( = f (. f ' ( = f (. F( e d e. f ( e wi = 8. Coefficien of in e epansion of ( + ( + 7 ( + is (A 5 (B 6 (C (D 8. (C C. 7 C. C + C. 7 C. 6 C + 7 C C + 7 C C =. 9. For (,, e equaion sin + sin sin = as (A infiniely many soluions (B ree soluions (C one soluion (D no soluion 9. (D (sin sin + sin = sin (cos cos = cos cos = sin Clearly cos cos /, equaliy old wen cos = ½ sin sin Hence no soluion in (, JEE Advanced (5-May- Quesion & Soluions Paper-II -- www. prernaclasses.com
4 5. Te following inegral / 7 (cosec d is equal o / (A log( ( e u e u 6 du (B log( ( e u e u 7 du log( (C ( e u e u 7 du (D log( 5. (A Pu cosec = e u + e u co = e u e u I ln( ln( ( e u u 7 du u u 6 ( e e ( e e du. cosec u e u SECTION - (Paragrap Type Tis secion conains paragraps eac describing eory, eperimens, daa ec. Si quesions relae o ree paragraps wi wo quesions on eac paragrap. Eac quesion as only one correc answer among e four coices (A, (B, (C and (D. Paragrap for Quesions 5 and 5 Bo conains ree cards bearing numbers,,; bo conains five cards bearing numbers,,,, 5; and bo conains seven cards bearing numbers,,,, 5, 6, 7. A card is drawn from eac of e boes. Le i be e number on e card drawn from e i bo, i =,,. 5. Te probabiliy a + + is odd, is (A 9 / 5 (B 5 / 5 (C 57 / 5 (D / 5. Te probabiliy a,, are in an arimeic progression, is (A 9 / 5 (B / 5 (C / 5 (D 7 / 5 Sol. 5. (B 5. (C 5. n(s = 5 7 = : odd Case I : all ree odd =. Case II : wo even and one odd = + + n(e = 5 P(E = 5 / 5. 5.,, in A.P. AP wi cd =, (,,, (,, (,, 5 cd =, (,, 5, (,,6 (, 5, 7 cd =, (,, 7 cd =, (,,, (,, (,, cd =, (,, P(E = / 5. 6 du JEE Advanced (5-May- Quesion & Soluions Paper-II -- www. prernaclasses.com
5 Paragrap for quesion 5 and 5 Le a, r, s, be non zero real numbers. Le P(a, a, Q, R(ar, ar and S(as, as be disinc poins on e parabola y = a. Suppose a PQ is e focal cord and lines QR and PK are parallel, were K is e poin (a,. 5. Te value of r is (A / (B (C / (D 5. If s =, en e angen a P and e normal a S o e parabola mee a a poin wose ordinae is ( (A Sol. 5. (D 5. (B 5. m QR = m PK a( (B ar (a / a ar ( a / a a a( (C r = 5. Tangen a P : y = + a y s = a... (i Normal a s : y + s = as + as... (ii a( (i + (ii y = Given a for eac a (,,. a( (D ParParagrap for quesion 55 and 56agrap for quesion 5 and 5 Lim a a given a e funcion g(a is differeniable on (,. ( d eiss. Le is limi be g(a. In addiion, i is 55. Te value of g( / is (A (B (C / (D / 56. Te value of g' ( / is (A / (B (C / (D Sol. 55. (A 56. (D d 55. g(/ Lim Lim sin (( sin ( ( = sin ( sin ( =. a 56. g' ( a Lim g'. ( a. ln d b b Lim / / ( /. (. ln d [Using d a a / / Lim (.. ln d f ( f ( a b d ] JEE Advanced (5-May- Quesion & Soluions Paper-II -5- www. prernaclasses.com
6 g' ( / =. SECTION - : Macing Lis Type (Only One Opion Correc Tis secion conains four quesions, eac aving wo macing liss. Coices for e correc combinaion of elemens from Lis - I and Lis - II are given as opions (A, (B, (C and (D, ou of wic one is correc. 57. Lis I Lis II P. Te number of polynomials f ( wi non -negaive. 8 ineger coefficiens of degree, saisfying f ( = and f ( d, is Q. Te number of poins in e inerval [, ] a wic. f ( = sin ( + cos ( aains is maimum value, is R. S. ( e d equals. / cos log d / equals. / cos log d Codes P Q R S (A (B (C (D 57. (D (P f ( = a + b [ f ( = ] f ( d (a / + (b / = a + b = 6 Hence, a =, b = or a =, b = Hence only wo polynomials (Q f ( = sin ( + ( / f ' ( = cos ( + ( / = Hence, = or + ( / = (n + ( / Values of in [, ] are, ± /, ± /, ± /, ± / Hence poins of maima. (R I d d d 6 I = 8. e e JEE Advanced (5-May- Quesion & Soluions Paper-II -6- www. prernaclasses.com
7 / (S f ( cos log is an odd funcion. Hence f ( d. 58. Lis I Lis II P. Le y( = cos ( cos, [, ], ± /.. / d y( dy( Ten ( y( d d equals Q. Le A, A,..., A n (n > be e verices of a regular. polygon of n sides wi is cenre a e origin. Le a be e posiion vecor of e poin A k, k =,,..., n. k n n If a k ak a k. ak k k, en e minimum value of n is R. If e normal from e poin P(, on e ellipse. 8 y 6 is perpendicular o e line + y = 8, en e value of is S. Number of posiive soluions saisfying e equaion. 9 an an an Codes P Q R S (A (B (C (D 58. (A (P y( = cos ( = cos = y( is ( y'' y' ( ( 9 n n (Q a k ak a k. ak k k (n sin ( / n = (n cos ( / n an ( / n = minimum n = 8. (R Normal a, 6 sec y cosec = (6 sec / ( cosec = an = / Hence, normal y = n =. (S. =, /. JEE Advanced (5-May- Quesion & Soluions Paper-II -7- www. prernaclasses.com
8 Hence, number of posiive soluions =. 59. Le f : R R, f : [, R, f : R R and f : R [, be defined by if ( f ; f e if ( = sin if ; ( f( f( if f and f( if f( f( if Lis I Lis II P. f is. ono bu no one-one Q. f is. neier coninuous nor one-one R. f of is. differeniable bu no one-one S. f is. coninuous and one-one Codes P Q R S (A (B (C (D 59. (D f ( : f ( : f ( : f of ( e y = y = sin y = e y = f ( : No sarp corner Differeniable bu no one-one y = Ono bu no one-one,, N.D. y = e y = JEE Advanced (5-May- Quesion & Soluions Paper-II -8- www. prernaclasses.com
9 neier coninuous nor one-one. 6. k k Le z k cos i sin ; k =,,..., 9. Lis I Lis II P For eac z k ere eiss a z j suc a z k. z j =. True Q Tere eiss a k {,,..., 9} suc a. False z. z = z k as no soluion z in e se of comple numbers. R. z z... z9 equals. 9 k S. cos equals. k Codes P Q R S (A (B (C (D 6. (C Clearly, z k = k were z k For (P : z k. z j = e i ( / (k + j =, if k + j = muliple of i.e. possible for eac k. For (Q : z. z = z k is clearly incorrec. For (R : Epression = z Lim z z 9 k For (S + z k = cos k Epression =. JEE Advanced (5-May- Quesion & Soluions Paper-II -9- www. prernaclasses.com
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