PART - III : MATHEMATICS

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1 JEE(Advnced) 4 Finl Em/Pper-/Code-8 PART - III : SECTION : (One or More Thn One Options Correct Type) This section contins multiple choice questions. Ech question hs four choices (A), (B), (C) nd (D) out of which ONE or MORE THAN ONE re correct. 4. Let ƒ : [, b] [, ) be continuous function nd let g : R R be defined s ì if < g() = íò ƒ(t) dt if b. b > îò ƒ(t)dt if b Then (A) g() is continuous but not differentible t (B) g() is differentible on R (C) g() is continuous but not differentible t b (D) g() is continuous nd differentible t either or b but not both. 4. Ans. (A,C) Given tht ƒ : [, b] [, ) ì < g() = íò ƒ(t)dt, b b ƒ(t)dt, > b îò Now g( ) = = g( + ) = g() Þ Now g(b ) = g(b + ) = g(b) = g is continuous " Î R ì : < g'() = íƒ() : < < b î : > b b ò ƒ(t)dt g'( ) = but g'( + ) = ƒ() ³ Þ g is non differentible t = nd g'(b + ) = but g'(b ) = ƒ(b) ³ Þ g is non differentible t = b 4. For every pir of continuous function ƒ,g : [, ] R such tht m{ƒ() : Î [, ]} = m{g() : Î [,]}, the correct sttement(s) is(re) : (A) (ƒ(c)) + 3ƒ(c) = (g(c)) + 3g(c) for some c Î [,] (B) (ƒ(c)) + ƒ(c) = (g(c)) + 3g(c) for some c Î [,] (C) (ƒ(c)) + 3ƒ(c) = (g(c)) + g(c) for some c Î [,] (D) (ƒ(c)) = (g(c)) for some c Î [,] CODE-8 / JEE(Advnced) 4 Finl Em/Pper-/Held on Sundy 5th My, 4

2 JEE(Advnced) 4 Finl Em/Pper-/Code-8 Sol. Ans. (A,D) ƒ, g [,] R we tke two cses. Let ƒ & g ttin their common mimum vlue t P. Þ ƒ(p) = g(p) where p Î [,] let ƒ & g ttin their common mimum vlue t different points. Þ ƒ() = M & g(b) = M Þ ƒ() g() > & ƒ(b) g(b) < Þ ƒ(c) g(c) = for some c Î [,] s 'ƒ' & 'g' re continuous functions. Þ ƒ(c) g(c) = for some c Î [,] for ll cses....() Option (A) Þ ƒ (c) g (c) + 3 (ƒ(c) g(c)) = which is true from () Option (D) Þ ƒ (c) g (c) = which is true from () Now, if we tke ƒ() = & g() = " Î [,] options (B) & (C) does not hold. Hence option (A) & (D) re correct. 43. Let M be symmetric mtri with integer entries. Then M is invertible if (A) the first column of M is the trnspose of the second row of M (B) the second row of M is the trnspose of the first column of M (C) M is digonl mtri with nonzero entries in the min digonl (D) the product of entries in the min digonl of M is not the squre of n integer Sol. Ans. (C,D) Let é bù M = ê b c ú ë û é (A) Given tht ù é b ù ê = b ú ê c ú Þ = b = c = (let) ë û ë û é ù Þ M = ê úþ M = Þ Non-invertible ë û (B) Given tht [b c] = [ b] Þ = b = c = (let) gin M = Þ Non-invertible (C) As given Þ M is invertible é ù M = ê c ú ë û Þ M = c ¹ (Q & c re non zero) (D) é bù M = ê Þ M = c- b ¹ b c ú ë û Q c is not equl to squre of n integer \ M is invertible rr r p 44. Let,y nd z be three vectors ech of mgnitude nd the ngle between ech pir of them is. 3 If r is nonzero vector perpendiculr to r r r r r nd y z nd b is nonzero vector perpendiculr to y nd r r z, then r r (A) r r r r r r r r b = (b.z)(z -) (B) = (.y)(y -z) r r r r r r r r r r r (C).b =-(.y)(b.z) (D) = (.y)(z -y) / JEE(Advnced) 4 Finl Em/Pper-/Held on Sundy 5th My, 4 CODE-8

3 JEE(Advnced) 4 Finl Em/Pper-/Code-8 Sol. Ans. (A,B,C) r r r Given tht = y = z = nd ngle between ech pir is 3 p rr rr rr \.y = y.z = z. =.. = Now r is ^ to r r r & (y z) r r r r Let =l ( (y z)) rrr rrr r r =l((.z)y -(.y)z) =l(y-z).y r r =l(y.y rr - y.z) rr =l( - ) =l r r r r r Þ = (.y)(y -z) r r r r r r Now let b =m ( y (z ) ) =m(z -) r r b.z =m(- ) =m r r Þ r r r b = (b.z)(z -) r r r r r r r r r r Now.b = (.y)(y -z).(b.z)(z -) r r rr rr rr rr rr = (.y)(b.z)(y.z -y. - z.z + z.) r r r r = (.y)(b.z)( -- + ) r r r r = -(.y)(b.z) 45. From point P(l,l,l), perpendiculrs PQ nd PR re drwn respectively on the lines y =, z = nd y =, z =. If P is such tht ÐQPR is right ngle, then the possible vlue(s) of l is(re) (A) (B) (C) (D) Sol. Ans. (C) Line L given by y = ; z = cn be epressed s L : y z - = = Similrly L (y = ; z = ) cn be epressed s y z+ L : = = - Let ny point Q (,,) on L nd R(b, b, ) on L Given tht PQ is perpendiculr to L Þ (l ). + (l ). + (l ). = Þ l = \ Q(l,l,) Similrly PR is perpendiculr to L (l b). + (l + b)( ) + (l + ). = Þ b = \ R(,, ) Now s given uuur uuur Þ PR.PQ=.l +.l + (l ) (l + ) = l ¹ s P & Q re different points Þ l = CODE-8 3/ JEE(Advnced) 4 Finl Em/Pper-/Held on Sundy 5th My, 4 Q( l,l,) P( l,l,) R(,, )

4 JEE(Advnced) 4 Finl Em/Pper-/Code Let M nd N be two 3 3 mtrices such tht MN = NM. Further, if M ¹ N nd M = N 4, then (A) determinnt of (M + MN ) is (B) there is 3 3 non-zero mtri U such tht (M + MN )U is zero mtri (C) determinnt of (M + MN ) ³ (D) for 3 3 mtri U, if (M + MN ) U equls the zero mtri then U is the zero mtri Sol. Ans. (A,B) (A) (M N ) (M + N ) = O...() (\ MN = N M) Þ M N M + N = Cse I : If M + N = \ M + MN = Cse II : If M + N ¹ Þ M + N is invertible from () (M N )(M + N )(M + N ) = O Þ M N = O which is wrong (B) (M + N )(M N ) = O pre-multiply by M Þ (M + MN )(M N ) = O...() Let M N = U Þ from eqution () there eist sme non zero 'U' (M + MN )U = O 47. Let ƒ : (, ) R be given by Sol. ƒ() = æ ö - ç t t dt e è + ø ò. t Then (A) ƒ() is monotoniclly incresing on [, ) (B) ƒ() is monotoniclly decresing on [, ) æ ö (C) ƒ() + ƒç =, for ll Î [, ) è ø (D) ƒ( ) is n odd function of on R Ans. (A,C,D) ƒ() = ò e æ ö - ç t+ è t ø t dt æ ö æ ö - ç + - ç + è ø è ø e æ-öe ƒ '() =. - ç è ø / æ ö æ ö æ ö - ç + - ç + - ç + è ø è ø è Xø e e e = + = \ ƒ() is monotoniclly incresing on (, ) Þ A is correct & B is wrong. æ ö æ ö - ç t+ - ç t+ è tø è tø Now ƒ() + æö e /e ƒç = dt + dt / t t è ø ò ò 4/ JEE(Advnced) 4 Finl Em/Pper-/Held on Sundy 5th My, 4 CODE-8

5 = " Î (, ) æ ö - ç t+ è ø Now let = =ò t e g() ƒ( ) dt - t æ ö - ç t+ è tø g( ) = ƒ( - ) = ò e dt =-g() t \ ƒ( ) is n odd function. JEE(Advnced) 4 Finl Em/Pper-/Code-8 æ p pö 48. Let ƒ : ç -, R be given by è ø Sol. ƒ() = (log (sec + tn )) 3 Then :- (A) ƒ() is n odd function (C) ƒ() is n onto function Ans. (A,B,C) ( ) 3 ( ) = l ( + ) ƒ n sec tn ( ) ( l ( + )) ( + ) ( sec + tn ) 3 n sec tn sec tn sec ƒ' = > ƒ() is n incresing function lim ƒ p - ( ) - limƒ & ( ) p Rnge of ƒ() is R nd onto function 3 æ æ öö ƒ( - ) = ( ln( sec - tn ) ) =ç ln ç è è sec + tn øø ƒ( ) = (ln(sec + tn)) 3 (B) ƒ() is one-one function (D) ƒ() is n even function 3 p/ p/ ƒ() + ƒ( ) = Þ ƒ() is n odd function. 49. A circle S psses through the point (, ) nd is orthogonl to the circles ( ) + y = 6 nd + y =. Then :- () rdius of S is 8 (B) rdius of S is 7 (3) centre of S is ( 7, ) (D) centre is S is ( 8, ) Sol. Ans. (B,C) Let circle is + y + g + ƒy + c = Put (,) + ƒ + c =...() orthogonl with + y 5 = g( ) = c 5 Þ c = 5 g...() orthogonl with + y = c =...(3) Þ g = 7 & ƒ = centre is ( g, ƒ) º ( 7,) rdius = g + ƒ - c = = 7 CODE-8 5/ JEE(Advnced) 4 Finl Em/Pper-/Held on Sundy 5th My, 4

6 5. Let Î R nd let ƒ : R R be given by Sol. ƒ() = Then (A) ƒ() hs three rel roots if > 4 (B) ƒ() hs only one rel roots if > 4 (C) ƒ() hs three rel roots if < 4 (D) ƒ() hs three rel roots if 4 < < 4 Ans. (B,D) f() = \ = 5 5 \ f() hs only one rel root if > 4 or < 4 4 f() hs three rel roots if 4 < < 4 SECTION : (One Integer Vlue Correct Type) JEE(Advnced) 4 Finl Em/Pper-/Code-8 This section contins questions. Ech question, when worked out will result in one integer from to 9 (both inclusive). 5. The slope of the tngent to the curve (y 5 ) = ( + ) t the point (,3) is Sol. Ans. 8 (y 5 ) = ( + ) Þ (y 5 ædy 4ö ) ç - 5 = ( + ) + ( + ). èd ø Put =, y = 3 dy d = 8 5. Let ƒ : [,4p] [,p] be defined by ƒ() = cos (cos). The number of points Î [,4p] stisfying the eqution ( ) Sol. Ans. 3 y= p p/ y - ƒ = is (,) p/ p 3p/ p 5p/ 3p y= y=cos (cos) = from bove figure it is cler tht of solutions = 3 - y = - nd y = cos (cos ) intersect t 3 distinct points, so number 6/ JEE(Advnced) 4 Finl Em/Pper-/Held on Sundy 5th My, 4 CODE-8

7 JEE(Advnced) 4 Finl Em/Pper-/Code The lrgest vlue of the non-negtive integer for which Sol. Ans. (+ ) ( - ) ìsin( - ) + (-) ü - lim í ý î ( - ) + sin( - ) þ + ìsin( -) ü - ( -) lim í sin( ) ý = î ( -) þ æ-ö Þ ç = è ø 4 Þ ( ) = Þ = or = 4 ( ) ( ) - sin ü- ì lim í ý = î + sin - - þ 4 but for = bse of bove limit pproches - nd eponent pproches to nd since bse cnnot be negtive hence limit does not eist. 54. Let ƒ : R R nd g : R R be respectively given by ƒ() = + nd g() = +. Define h : R R by { ( ) ( )} ( ) ( ) ì m ƒ,g if, h( ) =í î min{ ƒ,g } if >. The number of points t which h() is not differentible is Sol. Ans. 3 y O + + is h() is not differentible t = ± & 55. For point P in the plne, let d (P) nd d (P) be the distnces of the point P from the lines y = nd + y = respectively. The re of the region R consisting of ll points P lying in the first qudrnt of the plne nd stisfying < d (P) + d (P) < 4, is Sol. Ans. 6 Let P(,y) is the point in I qud. Now - y + y y + + y 4 CODE-8 7/ JEE(Advnced) 4 Finl Em/Pper-/Held on Sundy 5th My, 4

8 JEE(Advnced) 4 Finl Em/Pper-/Code-8 Cse-I : ³ y (- y) + (+ y) 4 Þ Î[, ] Cse-II : < y Ö < y y = y- + (+ y) 4 y Î[, ] Ö > y - A = ( ) ( ) = 6 Ö Ö 56. Let n < n < n 3 < n 4 < n 5 be positive integers such tht n + n + n 3 + n 4 + n 5 =. The the number of such distinct rrngements (n,n,n 3,n 4,n 5 ) is Sol. Ans. 7 s n ³,n ³,n3 ³ 3,n4 ³ 4,n5 ³ 5 Let n = ³, n = ³... n 5 5 = 5 ³ Þ New eqution will be = Þ = 5 = 5 Now So 7 possible cses will be there. ò ì 5 ü 4 ýd d î þ is 3 d 57. The vlue of í ( - ) Sol. Ans. using integrtion by prt ò ( - ) 3 5 " 4 æ ö ç d è ø ' ù (( ) ) ú ò ( ) ' ( ) = d û using integrtion by prt 8/ JEE(Advnced) 4 Finl Em/Pper-/Held on Sundy 5th My, 4 CODE-8

9 JEE(Advnced) 4 Finl Em/Pper-/Code-8 (( ) )) ò ( ) é ù 5 5 = dú êë û 5.. ( ) d ò = - Let = t Þ d = dt - ò = 4 t 5 æ dt ö ç - è ø r 58. Let,b r ò = 5 t dt =, nd c r be three non-coplnr unit vectors such tht the ngle between every pir of them is 3 p. r r r r r r r p + q + r If b + b c = p + qb + rc, where p,q nd r re sclrs, then the vlue of q Sol. Ans. 4 rr rr rr..b.c r r r r r r r rr We know é b cù ë û = b. b.b b.c rr rr rr c. c.b c.c 5 3 = = - = 4 4 r r r \ é b cù ë û =...() r r r r r r r s given b + b c = p + qb + rc tke dot product with r r r r r r r r r r rr q r Þ. ( b) +. ( b c) = p + qb. + rc. Þ + p = () r Now, tke dot product with b & c r p r = + q+...(3) p q & r = (4) is eqution () eqution (4) Þ p - r = Þ p= rþ p+ q= by eqution (3) \ p + q + r q p + p + p = = 4 p CODE-8 9/ JEE(Advnced) 4 Finl Em/Pper-/Held on Sundy 5th My, 4

10 JEE(Advnced) 4 Finl Em/Pper-/Code Let,b,c be positive integers such tht b rithmetic men of,b,c is b +, then the vlue of Sol Ans. 4 Let,b,c re, r, r where r Î N is n integer. If,b,c re in geometric progression nd the is lso + b + c = b + 3 Þ + r + r = 3(r) + 6 Þ r r + = 6 Þ (r ) = 6 Q 6 must be perfect squre & Î N \ cn be 6 only. Þ r = ± Þ r = & = = 4 Ans Let n > b n integer. Tke n distinct points on circle nd join ech pir of points by line segment. Colour the line segment joining every pir of djcent points by blue nd the rest by red. If the number of red nd blue line segments re equl, then the vlue of n is Sol. Ans. 5 Numbr of red line segments = n C n Number of blue line segments = n \ n C n = n ( - ) n n = nþ n = 5 Ans. / JEE(Advnced) 4 Finl Em/Pper-/Held on Sundy 5th My, 4 CODE-8

15 - TRIGONOMETRY Page 1 ( Answers at the end of all questions )

15 - TRIGONOMETRY Page 1 ( Answers at the end of all questions ) - TRIGONOMETRY Pge P ( ) In tringle PQR, R =. If tn b c = 0, 0, then Q nd tn re the roots of the eqution = b c c = b b = c b = c [ AIEEE 00 ] ( ) In tringle ABC, let C =. If r is the inrdius nd R is the