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1 Home Assignment #

2 STRAIGHT OBJCTIV TYP J-Mathematics. Assume that () = and that for all integers m and n, (m + n) = (m) + (n) + (mn ), then (9) = () = {} + { + } + { + }...{ + 99}, then [ ( ) ] where {.} denotes fractional art function & [.] denotes the greatest integer function = If () = /( + ) and n+ = o n for n =,,,..., then n () is - n () = (n+ )+ f () n n+ n lim = 7 does not eist æ 5. Let () is even and g() is an odd function which satisfies () ö ç èø () + () + () + ()= f 6. If () be a function such that ( + ) = () -, " Î N and () = then (999) is - f () + = g(), then NOD6\_NOD6 ()\DATA\\IIT-J\TARGT\MATHS\HOM ASSIGNMNT (Q.BANK)\NG\HOM ASSIGNMNT # 7. If () = + 5 and h() = + +, then function g such that og = h is none of these 8. The rincile value of cos æ 7 ö ç -sin è 6 ø is tan sin ( )( ln( sin)) lim = ln ln ln FILL TH ANSWR HR none of these. A B C D. A B C D. A B C D. A B C D 5. A B C D 6. A B C D 7. A B C D 8. A B C D 9. A B C D

3 J-Mathematics éa+ < ê. If () = ê = ê ëb+ > is continuous for all Î R, then A,B are : (, ) (, ) (, ) (, ). æ ö ç è ø = + lim ( + + ) e e. ( lim ) / + = 9 infinity. æ -æaöö æ -æa öö cosç + cos ç + cos - cos b ç ç b è è øø è è øø is equal to - a+ b ± ± a. The value of b a+ b æ n æ - + r öö tan lim tan n ç å is - r = ç + r ( r+ ) è è øø. A B C D. A B C D. A B C D. A B C D. A B C D 5. A B C D 6. A B C D 7. A B C D a + b b none of these none of these 5. The comlete solution of the equation [] =, where [. ] = the greatest integer less than or equal to, are - = n +, nî N = n -, nîn = n +, nî I n< < n+, nîi 6. If ()= 5 log 5 then (a b) where a, b Î R is equal to - (a) ƒ (b) - ƒ ( a) - ƒ ( b) K ƒ( a-b) n n(+ K) 7. If (n) = limå l n e - ƒ(), g(n) = lim K= å, then lim = K= g() ƒ( a) -ƒ( b) does not eist data inadequate NOD6\_NOD6 ()\DATA\\IIT-J\TARGT\MATHS\HOM ASSIGNMNT (Q.BANK)\NG\HOM ASSIGNMNT #

4 8. If ƒ () = limn( /n ) n -, >, then ƒ(y) is - ƒ()ƒ(y) ƒ() + ƒ(y) ƒ() ƒ(y) ƒ() ƒ(y) J-Mathematics 9. The number of oints where ƒ () = [sin cos] is not continuous in [, ] are (where [ ] denotes. Let the greatest integer function) ì ï, ³ ƒ() = í be continuous and differentiable everywhere. Then a and b are - ï îa + b, < -,, -,,. The comlete solution set of the inequality [tan ] 8[tan ] + 6, where [.] denotes the greatest. integer function is - (, tan] [tan, tan] [tan, tan5) no solution limsin sin cot - - æö ç is equal to - è ø non-eistent æö -. Let f() = ( + cos ), ¹. If f() is continuous at =, then the value of f ç è ø is - e e NOD6\_NOD6 ()\DATA\\IIT-J\TARGT\MATHS\HOM ASSIGNMNT (Q.BANK)\NG\HOM ASSIGNMNT #. If f() = n lim ( + ) ( + ) ( Set of oints where ƒ () = 8. A B C D 9. A B C D. A B C D. A B C D. A B C D. A B C D. A B C D 5. A B C D 6. A B C D ) ( + )...( + is differentiable, is - n ) where <, then - + (, ) È (, ) (, ) È (, ) (, ) (, ) 6. If y = + e, then at =, d dy -e e ( + e) is equal to - -e ( + e) -e ( + e)

5 J-Mathematics ì æ ö ï(- )sin ç, ¹ 7. Let ƒ() = í è-ø, which of the following statements is true?! ï î, = ƒ is differentiable at = but not at = ƒ is neither differentiable at = nor at = ƒ is differentiable at = and at = ƒ is differentiable at = but not at = 8. Let ƒ & g be differentiable functions satisfying g'(m) = & g(m) = b & ƒog() be an identity function then ƒ'(b) is - / / / 9. If ƒ () = + +, g() = e +. The value of ƒ () so that ƒ() = and h() =, then ƒ '(h'(g'()) = 5 may be continuous at = is - + æ ln ln ö ç èe ø 5. If I = d +, then I equals to - 5/ / (+ ) + (+ ) + c 9 l n c l n c 9 / / (+ ) - (+ ) + c. If ƒ () = ln( + ( + + ) + c + ), then ƒ ''()d is equal to A B C D 8. A B C D 9. A B C D. A B C D. A B C D. A B C D. A B C D - + c. Let ¹ n, n Î N. Then the value of l n sec( + ) + c c ln( + + ) sin( + ) - sin ( + ) sin( ) sin ( ) d is æ + ö ln secç + c è ø l n sec( + ) + c ln c sec( + ) + NOD6\_NOD6 ()\DATA\\IIT-J\TARGT\MATHS\HOM ASSIGNMNT (Q.BANK)\NG\HOM ASSIGNMNT #

6 J-Mathematics. sec.cosec d cot - sec cosec l n sec + tan + c ln sec+ cosec + c l n sec + tan + c ln sec + cosec + c sin( -[]) - 5. If l = lim - and m = lim, where [.] denotes greatest integer function, then m l + + l n( ) d is equal to - + l n 6. If ƒ() is an even function which is also eriodic with the eriod T and a ƒ()d = and NOD6\_NOD6 ()\DATA\\IIT-J\TARGT\MATHS\HOM ASSIGNMNT (Q.BANK)\NG\HOM ASSIGNMNT # T/ -T/ ƒ ()d = 8, then a+ 5T ƒ ()d is equal to - -a If antiderivative of which asses through (, ) is ( / + ) ( - ) + n. Then value of + m m+n is equal to If ƒ() + 6 = ƒ() + +, then ƒ() is necessarily non-negative in - (- 6, 6) (, ) È (, ) [, ] none of these 9. If ƒ(8 t) = ƒ(t) and ƒ( a)da= 8, then 8 ƒ( g )dg is If ƒ () ³ " Î R and area bounded by the curve y = ƒ (), =, = a and -ais is tan a, then the number of solutions of the equation ƒ () = tan is - infinitely many. If ƒ () = æ ö ç tan( n) + d è l ø & ƒ () =, then ƒ (e/) is - - e / - e / / (+ e ) -. A B C D 5. A B C D 6. A B C D 7. A B C D 8. A B C D 9. A B C D. A B C D. A B C D 5 / e +

7 J-Mathematics. If t ì z ï tan z + -tan z üï = e í dz ý ïî zsec z ï & þ t ì z ï-tan z - tan z üï y = e í dz ý ïî zsec z ïþ. Then the inclination of the tangent to the curve at t = is - æ ö. Let ƒ() = sin ç è+ ø, is - - æ- ö g() = cos ç è+ ø. The derivative of ƒ() with resect to g() at =. tan lim is equal to - - I= d = ƒ + c, then ƒ '() is equal to - 5. ( ) (- ) + n - 6. Which of the following reresents the grah of the function ƒ() = lim n n +? y =. A B C D. A B C D. A B C D 5. A B C D 6. A B C D 7. A B C D 8. A B C D 9. A B C D 5. A B C D 5. A B C D y = 6 7. If the non-negative solution set of the equation [] = [ + 6] is given by [a, b), then the value of a + b is ([.] is greatest integer function) If L = lim (cosec ) + / ln, then the value of l nl is - none of these /n n æ ö ç Õ(r + n) r= 9. If = lim è ø, then log n n (e) is equal to - 5. If D*ƒ () = ƒ () D(ƒ ()), then (D*(ln)) = e is - e e 5. If ƒ () = + + 6, then ƒ ( ) is - does not eist NOD6\_NOD6 ()\DATA\\IIT-J\TARGT\MATHS\HOM ASSIGNMNT (Q.BANK)\NG\HOM ASSIGNMNT #

8 J-Mathematics 5. The comlete solution set of the inequality [cosec ] > [sec ], where [.] is greatest integer function, is - [, cosec] [, sec] [cosec, sec] none of these ì cos cot (cot ).(cos ), ¹ ï 5. If ƒ() = í ï k, = ïî is continuous at =, then k = 5. If ƒ () = ( - ), then ƒ '() at =.5 is (sec ) sec + - d is equal to - lnsec + c æ lnsec + c ö ç ( n) + n + d è l l ø is equal to - l nsec+ c lnsec+ c ö ç ( l n) - + c è ø æ n + n ( e) 57. If ƒ () = l l d ln ( e) ( n- ) + c l ( ln) ƒ ; ƒ () = and g() = ( ) + c ln+ c. Then the domain of g() is - (, ) (,) (, ) È æ ö æ ö ç, È ç, è eø èe ø (, ) NOD6\_NOD6 ()\DATA\\IIT-J\TARGT\MATHS\HOM ASSIGNMNT (Q.BANK)\NG\HOM ASSIGNMNT # 58. If ƒ( ) cos d = ƒ ( ) æ ö + C, where ƒ() is not a cosntant function & ƒ ç =, then the eriod è ø ƒ of g() = ( ) ƒ( ) is - not defined - ì k + k - ï if > 59. Let for k >, ƒ() =í, if ƒ () is continuous at =, then k is equal to - ï îln(k -)- if e or or e or e only e 5. A B C D 5. A B C D 5. A B C D 55. A B C D 56. A B C D 57. A B C D 58. A B C D 59. A B C D 7

9 J-Mathematics é 6 ù ê ú 6. Let A= ê -5 ú and ê ú ë û 6. Let equal to - é ù B= ê ú. If a function is defined as ƒ () = tr(ab), then ê ú êë 8úû n c l n 5 c l n c l ln + c + - æ ö F() = e - d è - ø sin ç and F() =. If æ ö k e F 6 ç =, then k is equal to - è ø / / d is ƒ() 6. If 9- (9- ) 9 / d = k. + c, then the value of 'k' is - 6. The value of n 6. A B C D 6. A B C D 6. A B C D 6. A B C D 6. A B C D 65. A B C D 66. A B C D é ù êlim å n n r= r ú, where [.] reresents greatest integer function, is - æ ö ê + cosç ú êë è n øúû 6. If ƒ() =, then ,, 9/ d is equal to - + 7/ 7 ( ) 5 e æ ö ƒç is discontinuous at = è - ø 8,, log c c 7,, æ + ö ç + c è ø log - / + + ( ) sin( ln) d is equal to - log c,, 7 NOD6\_NOD6 ()\DATA\\IIT-J\TARGT\MATHS\HOM ASSIGNMNT (Q.BANK)\NG\HOM ASSIGNMNT #

10 67. The value of ma(-,, + )d is J-Mathematics 68. If for a continuous function ƒ, ƒ () = ƒ () = & ƒ '() = and g() = ƒ (e ). e ƒ (), then g'() is equal to cos{log(ƒ ()) + log(g())} {ƒ (). g'() + g(). ƒ ' ()} d is equal to - ƒ().g() sin{log (ƒ (). g())} + C ƒ() sin {log (ƒ (). g())} + C g() d is equal to - (+ ) 7. /5 /5 / g() sin {log (ƒ (). g())} + C ƒ() none of these K 5 5 K + + æ ö 5 5 ç + + K è ø K d is equal to / + / + ì ü lim í ý is equal to - n î n n n n þ NOD6\_NOD6 ()\DATA\\IIT-J\TARGT\MATHS\HOM ASSIGNMNT (Q.BANK)\NG\HOM ASSIGNMNT # 7. If - cos (B()) cosec 9 A() d =- + c 9 9, then number of solution of the equation A() {} [, ] is (where {.} reresents fractional art function) - 7. If g() = is - l n(sec t tan t - sec t + )dt, then set of value of in -, B() = in æ ö ç for which g() is increasing, è ø æ ö æ ö æ ö ç -, ç, ç -, è 6 6ø è ø è ø f 67. A B C D 68. A B C D 69. A B C D 7. A B C D 7. A B C D 7. A B C D 7. A B C D 7. A B C D 9

11 J-Mathematics 75. ( - )sin cos d is equal to cos d is equal to - 6 cos + cos ( -) 77. If + y = 5 and y'' + ky =, then k is equal to The value of é [] ù ê d -ë+ ú û, where [.] denotes the greatest integer function, is - none of these 79. If ƒ () = e, g() = e & h() = ƒ(g()), then the value of ln h'() is equal to - none of these 8. If d = ƒ () 8 6 cos + cos + cos + cos bounded & eriodic unbounded & eriodic ƒ() tan ç è ø - æ ö + bounded & aeriodic c, then ƒ () is - unbounded & aeriodic 8. If sina and cosa are the roots of the equation a b + c =, a ¹, then cos (a + ac b ) is equal to - 8. If æ+ ö ƒç = -, then ƒ()d is equal to - è-ø ln( ) + c ln ( ) + c ln( ) 6 + c ln ( ) + c 8. The differential coefficient of ƒ (ln), with resect to ln, where ƒ () = ln, is - ln 8. If ƒ '() = and ln 75. A B C D 76. A B C D 77. A B C D 78. A B C D 79. A B C D 8. A B C D 8. A B C D 8. A B C D 8. A B C D 8. A B C D ln (ƒ() + ƒ ''())cos d =, then ƒ '() is equal to - ln NOD6\_NOD6 ()\DATA\\IIT-J\TARGT\MATHS\HOM ASSIGNMNT (Q.BANK)\NG\HOM ASSIGNMNT #

12 J-Mathematics 85. ƒ'() Let ƒ () be a differentiable function satisfying the equation = ƒ() e " Î R. If ƒ '() =, then the number of solutions of the equation ƒ () = ƒ '() is - none of these 86. æ n+ n+ n+ 6 5 ö If L = lim ç n èn + n + n + n + n + n + 9 7n ø, then el is equal to e 87. Let ƒ () = a + b + c, where b, c Î R, a >. If ƒ () = has two real and different ositive roots a and b(a < b), then the value of ƒ ( ) + ƒ ( ) d is - b -b NOD6\_NOD6 ()\DATA\\IIT-J\TARGT\MATHS\HOM ASSIGNMNT (Q.BANK)\NG\HOM ASSIGNMNT # a ƒ ()d b ƒ ()d a 85. A B C D 86. A B C D 87. A B C D 88. A B C D 89. A B C D 9. A B C D 9. A B C D 9. A B C D b ƒ ()d 88. Let ƒ () be a differentiable function such that ƒ () + ƒ () =, then 89. If 9. If equals - - ƒ() + c - ƒ() =, then + (- ) - cot d k - ƒ() - + c - dƒ () d - (- ) =, then k equals - is equal to - ƒ() + + c (- ) ƒ() + ƒ() d (ƒ() + )( -ƒ()) + ƒ() + c - (- ) -/ / / 9. If ƒ () = ( + ) d, then ƒ (7) ƒ () equals Let y = ƒ() be a differentiable curve satisfying then / 9 ƒ() d equals - -/ cos ƒ (t)dt t ƒ (t)dt, + = +

13 J-Mathematics 9. If y = ƒ() is a linear function satisfying the relation ƒ(y) = ƒ().ƒ(y) ",yî R, then the curve 9. If y + (sin t + a t + bt)dt = a, aî R + cuts y = ƒ () at - no oint eactly one oint atleast two oints infinite oints I n e n (log)d() e n =, then In+ I is equal to - n - e e MULTIPL OBJCTIV TYP 95. The value of a for which equation (t - 8t + )dt = sin a has a solution, is (are) If (g()) = and g( ()) = then which of the following may be the functions () & g() - () = g() = (7 / ) () = ¹- ; g() = ¹ ì g() = () =, Î í Q î -, Ï Q () = log( ), > ; g() = e +, Î R 97. æ ö Let () = lncos sinç + è ø then - æ 8 ö ç è 9 ø = æ5 ö æ l n ç 8 ö ç è 8 ø è 9 ø = l æ ö n ç è 8 ø æ-7ö ç è ø = n æ ö l ç () = è ø Identify the incorrect statement(s) - tan- æln( + ) ö lim = limç = e / è ø lim = lim + - = 99. sin e The value(s) of for which ƒ() = - -9 is continuous, is (are) - 5 all Î (, ] È [, ) 9. A B C D 9. A B C D 95. A B C D 96. A B C D 97. A B C D 98. A B C D 99. A B C D NOD6\_NOD6 ()\DATA\\IIT-J\TARGT\MATHS\HOM ASSIGNMNT (Q.BANK)\NG\HOM ASSIGNMNT #

14 J-Mathematics æ ö. One of the values of satisfying tan(sec ) = sincos ç è 5 ø is - 5 æ - ö. If < < then tan is equal to - ç + è ø cos cos. Which of the following limits vanishes? - - sin - + tan - æ ö Lim ç - è tan ø æ + ö - Lim ç è - ø æ æ öö Limç tan ç + è è 8 øø + tan - + Lim - NOD6\_NOD6 ()\DATA\\IIT-J\TARGT\MATHS\HOM ASSIGNMNT (Q.BANK)\NG\HOM ASSIGNMNT # -n æ æ öæ ö æ öö. If L = lim n ç (n + ) ç n + ç n +... n n ç + n- è øè ø è è øø k is (are) -. A B C D. A B C D. A B C D. A B C D. A B C D 5. A B C D n & k = lnl, then the ossible value(s) of. Which of the following statement(s) are correct - 5. If If ƒ() º , then the equation ƒ '() = must have a real root. If ( ) is a factor of the olynomial P() (degrees 5) reeated times, then is the root of the equation P'() = reeated times If ƒ() is a differentiable function, then if its grah is symmetric about origin, then the grah of ƒ'() will be symmetric about y-ais. If y = sin (cos(sin )) + cos (sin(cos )) then dy d n I n= ( - ) d I, then - n = I n+ - n n I n = n n! (n + ) is indeendent of. I n =..6...n (n + ) n+ I = I n n n-

15 J-Mathematics 6. Identify the correct statement(s) - - æ7 ö tan tan ç è 7 ø is negative cos (cos( + sin)) = + sin for all Î R -æ ö -æ ö sin ç sin + cos ç cos = tan è 5 ø è 5 > cot ø -æ-ö - 7. tan ç + tan =, then is equal to - è ø 8. Which of the following is true = " Î- - - cos sin [, ] 6. A B C D 7. A B C D 8. A B C D 9. A B C D. A B C D. A B C D. A B C D 9 + = " Î- - - cos cos [, ] - - sin = - cos " Î- [, ] - - sin = - sin " Î- [, ] d ln ( ) is equal to - d( ln) (ln). - cos (cos )d. If ( n ) e l (ln)( ln ). (ln )( ln ) (ln ) is equal to - - æ æ öö cos cos + d ç ç è è øø nt ƒ() = l dt, then - + t æ ö lnt ƒ ç =- dt è ø t(+ t) æö ç èø / - 8 sin (sin )d ƒ( ) + ƒ = l n () ( ). Let ƒ( ) ì ï (5 + - t )dt, if >, then ƒ () is - =í ï + î5, if - sin (sin )d æ ö lnt ƒ ç = dt è ø t(+ t) æö ƒ =-ƒ ç è ø discontinuous at = not differentiable at = continuous at = differentiable at = NOD6\_NOD6 ()\DATA\\IIT-J\TARGT\MATHS\HOM ASSIGNMNT (Q.BANK)\NG\HOM ASSIGNMNT #

16 . Which of the followings has the value equal to the integral lim cos t dt sin tan cot t dt dt + + t t(+ t )? /e /e J-Mathematics ì nü lim ítan tan tan...tan ý n î n n n n þ /n RASONING TYP. Let : R [, /) defined by () = tan ( + + a) then - Statement- : The set of values of a for which () is onto is and é ö ê, ë ø. NOD6\_NOD6 ()\DATA\\IIT-J\TARGT\MATHS\HOM ASSIGNMNT (Q.BANK)\NG\HOM ASSIGNMNT # Statement- : Minimum value of + + a is a. Statement- is True, Statement- is True ; Statement- is a correct elanation for Statement-. Statement- is True, Statement- is True; Statement- is NOT a correct elanation for Statement-. Statement- is True, Statement- is False. Statement- is False, Statement- is True. 5. Statement- : If and Statement- : If lim lim a a ( ().g()) eists then () and lim a. A B C D. A B C D 5. A B C D 6. A B C D lim 5 a () and g() eists finitely then lim a g() eists finitely lim ().g() = lim a a (). lim Statement- is True, Statement- is True ; Statement- is a correct elanation for Statement-. Statement- is True, Statement- is True; Statement- is NOT a correct elanation for Statement-. Statement- is True, Statement- is False. Statement- is False, Statement- is True. 6. Statement- : Let a, a and a be the three real roots of the equation a + b + c + d = such that a, a, a, ad > and cos a + cos a + cos a = then the given cubic equation has eactly three negative real roots. and Statement- : If Þ cos If < Þ < cos Statement- is True, Statement- is True ; Statement- is a correct elanation for Statement-. Statement- is True, Statement- is True; Statement- is NOT a correct elanation for Statement-. Statement- is True, Statement- is False. Statement- is False, Statement- is True. a g()

17 J-Mathematics 7. Statement- : The curves æ ö ç -, è ø. and sin n(cos) = = l intersects eactly at one oint in the interval y &y Statement- : () ³ g() Þ a () ³ a g(), a Î R +. Statement- is True, Statement- is True ; Statement- is a correct elanation for Statement-. Statement- is True, Statement- is True; Statement- is NOT a correct elanation for Statement-. Statement- is True, Statement- is False. Statement- is False, Statement- is True. 8. Let () is a bijective function. Then Statement- : () = () Þ () =. and Statement- : () = Þ () = (). Statement- is True, Statement- is True ; Statement- is a correct elanation for Statement-. Statement- is True, Statement- is True; Statement- is NOT a correct elanation for Statement-. Statement- is True, Statement- is False. Statement- is False, Statement- is True. ìa- B - ï 9. Let ƒ () = í + A + B Î- (, ] ï î > Statement- : ƒ() is continuous at all if A =, B =. and Statement- : Polynomial function is always continuous. Statement- is True, Statement- is True ; Statement- is a correct elanation for Statement-. Statement- is True, Statement- is True; Statement- is NOT a correct elanation for Statement-. Statement- is True, Statement- is False. Statement- is False, Statement- is True.. Let ƒ : R R ƒ() = " Î Q and ƒ() be continuous function. Statement- : ƒ() = and Statement- : ƒ() is many-one into function. Statement- is True, Statement- is True ; Statement- is a correct elanation for Statement-. Statement- is True, Statement- is True; Statement- is NOT a correct elanation for Statement-. Statement- is True, Statement- is False. Statement- is False, Statement- is True. 7. A B C D 8. A B C D 9. A B C D. A B C D 6 NOD6\_NOD6 ()\DATA\\IIT-J\TARGT\MATHS\HOM ASSIGNMNT (Q.BANK)\NG\HOM ASSIGNMNT #

18 J-Mathematics NOD6\_NOD6 ()\DATA\\IIT-J\TARGT\MATHS\HOM ASSIGNMNT (Q.BANK)\NG\HOM ASSIGNMNT #. Statement- : Number of oints in (, 5), where ƒ() = ( ) + + ( ) + tan is non-differentiable is. and Statement- : A function is non-differentiable at any oint if it is discontinuous or its grah ossesses a shar corner at that oint. Statement- is True, Statement- is True ; Statement- is a correct elanation for Statement-. Statement- is True, Statement- is True; Statement- is NOT a correct elanation for Statement-. Statement- is True, Statement- is False. Statement- is False, Statement- is True.. Consider ƒ : R R Statement- : If ƒ (a) = and and lim ƒ() Þ ƒ () = has finite number of solutions. Statement- : If lim ƒ() Þ ƒ is aeriodic function. Statement- is True, Statement- is True ; Statement- is a correct elanation for Statement-. Statement- is True, Statement- is True; Statement- is NOT a correct elanation for Statement-. Statement- is True, Statement- is False. Statement- is False, Statement- is True.. Statement- : and Statement- : 99 æ ö cosec ç - d = / è ø. a ƒ()d = if ƒ ( ) = ƒ (). -a Statement- is True, Statement- is True ; Statement- is a correct elanation for Statement-. Statement- is True, Statement- is True; Statement- is NOT a correct elanation for Statement-. Statement- is True, Statement- is False. Statement- is False, Statement- is True.. Consider ƒ : R R be a function satisfying ƒ ( ) = ƒ ( + ) and ƒ ( ) = ƒ () " Î R. Statement- : If and ƒ()d =, then 5 ƒ()d = 9. Statement- : If ƒ () is eriodic with eriod T, then -9. A B C D. A B C D. A B C D. A B C D 7 a+ nt T ƒ()d = n ƒ()d, a Î R & n Î I. a Statement- is True, Statement- is True ; Statement- is a correct elanation for Statement-. Statement- is True, Statement- is True ; Statement- is NOT a correct elanation for Statement-. Statement- is True, Statement- is False. Statement- is False, Statement- is True.

19 J-Mathematics 5. Statement- : and n d e lim =, nîn, where {.} denotes fractional art function. e- n -{} e Statement- : nt T ƒ ()d = n ƒ ()d, n Î N, where ƒ ( + T) = ƒ (), T > " Î R. Statement- is True, Statement- is True ; Statement- is a correct elanation for Statement-. Statement- is True, Statement- is True ; Statement- is NOT a correct elanation for Statement-. Statement- is True, Statement- is False. Statement- is False, Statement- is True. COMPRHNSION Paragrah for Question 6 to 8 The function whose values at any number is the smallest integer greater than or equal to is called the least integer function or the integer ceiling function. It is denoted by é ù. for eamle é ù=, é ù=, é- ù=-, éù= Answer the following questions. 6. éù - éù+ = then belongs to - [, ) (, ) È (, ) {, } (, ] 7. éù + éê ùú+ éê ùú éê ùú = 6 9 none of these 8. The ossible value(s) of éù -[ ] where [] is greatest integer function is (are) - {} {, } {,, } {, } ì+, if If () = í î5 -, if > ì, if g() = í î -, if > then answer the following questions : 9. The range of () is - Paragrah for Question 9 to (, ) (, 5) R (, ]. If Î (, ) then g( ()) is equal to Number of negative integral solutions of g( ()) + = are - 5. A B C D 6. A B C D 7. A B C D 8. A B C D 9. A B C D. A B C D. A B C D 8 NOD6\_NOD6 ()\DATA\\IIT-J\TARGT\MATHS\HOM ASSIGNMNT (Q.BANK)\NG\HOM ASSIGNMNT #

20 J-Mathematics Consider the functions, Paragrah for Question to ƒ () = cos sin g() = sec cosec & h() = m On the basis of above informations, answer the following questions.. Number of solution(s) of the equation ƒ () = tan. If the equation g() = h() has eactly solutions then the range of m - [, ) (, ] (, ] [, ). Which of the following best reresents the grah of y = sin(ƒ ()) - (, ) (, ) NOD6\_NOD6 ()\DATA\\IIT-J\TARGT\MATHS\HOM ASSIGNMNT (Q.BANK)\NG\HOM ASSIGNMNT # Paragrah for Question 5 to 7 K() is a function such that K(ƒ ()) = a + b + c + d, where ì if ƒ () is even ï a = í - if ƒ () is odd ï î if ƒ () is niether even nor odd ì if ƒ () is eriodic b = í î if ƒ () is aeriodic ì5 if ƒ () is one one c = í î6 if ƒ () is many one ì7 if ƒ () is onto d = í î8 if ƒ () is into A = {, e, sin, } all the functions in set A are defined from R to R B = {8, 9, 6, 7} æe + e + ö h : R R, h() = ç èe - e + ø æ ö f : ç -, R, f() = tan è ø On the basis of above informations, answer the following questions. 5. K(f()) = A B C D. A B C D. A B C D 5. A B C D 9 (, )

21 J-Mathematics 6. K(h()) = If K() is a function such that K : A B, y = K() where Î A, y Î B then K() is - one one onto one one into many one into many one onto Consider the function y = f() Paragrah for Question 8 to f : R {} R The functional rule for the function y = f() is same as that of the functional rule for hyotenuse 'h' of the right triangle with area 5 (units) eressed as a function of its erimeter. On the basis of above information, answer the following : 8. The function y = f() is - one-one onto one-one into many-one onto many-one into 9. The value of [cos cos ƒ ( (log. log log 5... log ) + )], where [ ] denotes the greatest integer function -. The sum of all the values of at which fof() is discontinuous - Paragrah for Question to Consider the following functions ì ƒ () = -, í î -, < g() = + + h() = On the basis of above informations, answer the following questions :. The function goƒ () is - discontinuous at = continuous but not derivable at = continuous and derivable at = non derivable at more that one oint in [, ]. The function hoƒ () is - discontinuous at = continuous but not derivable at = continuous and derivable at = increasing in (, ). The domain of the function ƒoƒogoh() is - [, ] [, ] [, ] f 6. A B C D 7. A B C D 8. A B C D 9. A B C D. A B C D. A B C D. A B C D. A B C D NOD6\_NOD6 ()\DATA\\IIT-J\TARGT\MATHS\HOM ASSIGNMNT (Q.BANK)\NG\HOM ASSIGNMNT #

22 J-Mathematics NOD6\_NOD6 ()\DATA\\IIT-J\TARGT\MATHS\HOM ASSIGNMNT (Q.BANK)\NG\HOM ASSIGNMNT #. 5. Paragrah for Question to 6 é ì ü êma, íg(t), t, ý ê î þ ê Let g() = 6sin 8sin and ƒ() = ê-- < < ê ê - sin (sin( - )) 6 ê êë On the basis of above informations, answer the following questions : lim ƒ() - is equal to - / ƒ ()d is equal to ƒ () + ƒ (6) + ƒ (6) is equal to Paragrah for Question 7 to 9 If ƒ () = + +, then On the basis of above information, answer the following : d is - 7. ƒ( e ) 8. If tan - æe + ö ç + c è ø - æe + ö tan ç + c è ø - ƒ(tan )d = - tan (g()) + c, then for Î -, æ ö ç -, + è ø æ ö -, ç - + è ø e + - æ ö tan ç + c è ø - æ + ö tan ç + c è ø e æ ö ç è ø (-, ) none of these range of g() is -. A B C D 5. A B C D 6. A B C D 7. A B C D 8. A B C D

23 J-Mathematics 9. If æ ö æö - + ƒ( )d+ ƒ d =l n + c l, then l is - ç ç è ø è ø + + MATCH TH COLUMN 5. Column-I Column-II Solution set of the inequality (P) - ³ - (Q) 6 - ( - ) - cosec cosec ( cosec ) Solution set of the inequality sin ³ cos é -ù é ù ê-, È, ú ê ú ë û ë û æ ç è ù, ú û Domain of the function (R) (, ] È [, ) ƒ () = - ln(sin ) + ln-ln Domain of the function (S) [, ] ƒ () = sin (sin(sin (sin(sin (sin(sin(sin ))))))) 5. Column-I Column-II æ 5 ö If ƒ () = maç -, è then minimum (P) ø value of ƒ () is Let ƒ () be a function such that (Q) 5/6 ƒ ( + y) = ƒ ()ƒ (y) ", y Î R. If ƒ () is not identically zero then f()f( ) = Sum of the squares of all the solution(s) of (R) 8 sin ( + ) = cos ( + ) Let P and Q be olynomials such that P() (S) and Q(P(Q())) have the same roots. If the degree of P is 8 then degree of Q is 9. A B C D NOD6\_NOD6 ()\DATA\\IIT-J\TARGT\MATHS\HOM ASSIGNMNT (Q.BANK)\NG\HOM ASSIGNMNT #

24 J-Mathematics NOD6\_NOD6 ()\DATA\\IIT-J\TARGT\MATHS\HOM ASSIGNMNT (Q.BANK)\NG\HOM ASSIGNMNT # 5. Column-I Column-II e If () = cos -- tan for ¹ is continuous (P) at = then () must be (Q) / ( l l ) lim n- n( -+ ) e - - sin tan lim = (R) (S) 5. Column-I Column-II Number of integers in the range of (P) the function ƒ() = cos cos is The eriod of the function (Q) cos + -[] æ æö æöö ƒ() = ç sec ç -tan ç is è è ø è øø where [ ] denotes greatest integer function (R) not defined If the number of solution of the equation sin (sin ) m = is 5, then 7m equals to (where 'm' is ositive number) The eriod of the function ƒ : (, ) R (S) n - ƒ() = å é ër + e + r-ù û is r= where [ ] reresents greatest integer function 5. Column-I Column-II The number of the values of in (, ), where the function (P) tan + cot tan -cot ƒ() = - is continuous but not differentiable is (Q) The number of oints where the function ƒ () = min{, +, +} is non-derivable The number of oints where ƒ () = ( + ) / is non-differentiable is (R) Consider 5. ì æ. ö - ln +, < ï ç ƒ() è ø =í ï - sin sin, < < ïî æ ö Number of oints in ç,, where ƒ() is non-differentiable is è ø (S)

25 J-Mathematics 55. Column-I Column-II If range of ƒ() = cos + + is [a, + b], (P) then a + b is equal to Length of the tangent drawn from a oint on the circle + y = 5 (Q) to the circle + y = is If 6 -[] e.d = (e -q), where [.] denotes greatest integer function, (R) 7 then + q is Minimum ossible number of ositive roots of the quadratic equation (S) ( + b) + b =, is 56. Column-I Column-II e Period of the function ƒ () = 6 [6 + 7] + cos - [], e (P) 5 where [.] denotes greatest integer function, is ( - cos )sin 5 If ƒ () = d, then tan æ ö limç ƒ'() è ø is equal to (Q) æö If ƒ () = l n(8cos - 6cos ), then ƒ' ç is equal to (R) non-eistent è ø (S) 57. Column-I Column-II d is equal to (P) ( )( ) æ (.+ n) (. + n) (.+ n) (.n + n) ö lim ç è + n.+ n + n. + n + n. + n n ø n (Q) does not eist is equal to lim sin - t dt is equal to (R) ln The set of values of satisfying the inequality (S) - æ ç ö è ø is equal to 55. sec + (8 + - ) 56. is [a, b] {c} then a + b + c 57. NOD6\_NOD6 ()\DATA\\IIT-J\TARGT\MATHS\HOM ASSIGNMNT (Q.BANK)\NG\HOM ASSIGNMNT #

26 J-Mathematics 58. Column-I Column-II æsin - sin + ö Range of the function ƒ () = log ç is [m, M] èsin + sin + ø (P) then m + M is equal to If the tangent at every oint to the curve ƒ () = + a (Q) 5 is inclined with ositive direction of -ais at non-zero acute angle then number of integral values of 'a' is If æ ö cot ç >, n Î N, èø 6 - n then maimum value of n is (R) If ƒ () = l n, where ƒ : [, e] R and the maimum value of (S) ƒ () is M, then lnm is 59. Column I Column II If b d = aln - -l n - + C (P) then a + b is equal to æ - + ö ç è ø - If tan - - ç sin + cos = sin ( tan ) (Q) 5 where is a ositive real number then 7 5 is equal to If f() = ln, where Î é eù ê, e ú, then range of (R) 6 ë û NOD6\_NOD6 ()\DATA\\IIT-J\TARGT\MATHS\HOM ASSIGNMNT (Q.BANK)\NG\HOM ASSIGNMNT # ƒ () is If é ù ê -,b a ú. The value of a+b is ë û æ ö tan d ç = è ø a tan b l n (+ )+C (S) then a + b is equal to

27 J-Mathematics 6. Column I Column II Number of solutions of the equation (P) ( ) n n ( å sin + å cos ) = in (, ) n= n= If the range of m for which the equation cosec = m (Q) æ lù has eactly two solutions is ç, ú, then l is equal to è û Sum of integral solutions of the equality [ ] = [ + 6] is, (R) where [.] reresent greatest integer function The number of solutions to the equation (S) æ ( ( - ) d æ 9 9 öö cos ec sin sin ) = ç ç èdè 8 7 øø 6. Column-I Column-II = 9 If tan æ ö l l = limç sin t dt, then is equal to (P) - è ø If 6. n æ n æ ö ö ç + ç ƒ () = lim ç è ø n n-, then ç n- æ ƒ ()cos d is equal to (Q) ö ç + ç è è ø ø / æ dq ö ç - is equal to (R) è tan q+ cot q+ cosecq+ secq ø cos sin d 5 is equal to 6 6. (S) - NOD6\_NOD6 ()\DATA\\IIT-J\TARGT\MATHS\HOM ASSIGNMNT (Q.BANK)\NG\HOM ASSIGNMNT #

28 J-Mathematics INTGR TYP / SUBJCTIV TYP + m+ n 6. If ƒ be a function such that ƒ : R R, ƒ () = + m + n. and the range of ƒ () is [, ) then find 6. If æ ö a ç = + è ø 7 b tan cos sin 8 tan tan c. where a, b & c are co-rime numbers. Then find the value of a + b + c. 6. If y = sin + cos & d dy = = a+ b c, a,b,cî I, " a+ b+ c= If é ùé ù é ù ê 6 úê ú ê a b ú ê úê ú = ê + ú êë5 úê ûë úû êë5+ c + úû " Î R and ƒ () is a differentiable function satisfying ƒ () + ƒ æ + y ö (y) = ƒç è - y ø for all, y Î R, (y ¹ ) and ƒ() lim =, then find the value of where [.] denotes greatest integer function. é a + b+ c ù ê d ú, ë ƒ () û 66. Let.g(ƒ()).ƒ '(g()).g'()=ƒ(g()).g'(ƒ()).ƒ '() " Î R.ƒ is nonnegative & g is ositive. Also a -a e ƒ (g())d = - " a ÎR. Given that g(ƒ ()) =, then the value of ln(g(ƒ ())) is equal to... NOD6\_NOD6 ()\DATA\\IIT-J\TARGT\MATHS\HOM ASSIGNMNT (Q.BANK)\NG\HOM ASSIGNMNT #

29 J-Mathematics ANSWR KY. D. C. A. A 5. B 6. D 7. B 8. C 9. B. C. D. B. C. A 5. C 6. B 7. B 8. B 9. A. A. D. A. C. A 5. C 6. B 7. D 8. C 9. C. D. D. B. B. A 5. D 6. C 7. A 8. C 9. C. B. D. D. B. A 5. C 6. C 7. C 8. B 9. C 5. B 5. A 5. A 5. C 5. B 55. B 56. D 57. C 58. B 59. C 6. A 6. D 6. A 6. B 6. C 65. D 66. A 67. B 68. C 69. A 7. B 7. D 7. D 7. A 7. D 75. D 76. D 77. C 78. B 79. C 8. C 8. C 8. A 8. C 8. A 85. B 86. C 87. A 88. B 89. A 9. C 9. C 9. C 9. C 9. D 95. A,C 96. A,B,C,D 97. B,C 98. A,C 99. A,B. B. A,B,C. A,B,C. A,C. B,C,D 5. A,B,C 6. A,B,C,D 7. A,D 8. A,C 9. A,B. A,B,C. B,C. B,C. A,C,D. D 5. D 6. D 7. C 8. D 9. B. B. A. D. B. D 5. B 6. D 7. A 8. B 9. A. B. C. C. A. D 5. A 6. D 7. C 8. C 9. C. A. B. A. D. A 5. B 6. D 7. D 8. A 9. C 5. (R); (P); (Q); (S) 5. (Q); (P); (S); (P) 5. (Q); (Q); (R) 5. (Q), (P), (S), (R) 5. (R), (P), (S), (Q) 55. (S), (Q), (R), (P) 56. (S), (P), (R) 57. (S), (R), (Q), (P) 58. (S), (R), (Q), (P) 59. (Q); (P); (S); (R) 6. (R); (S); (P); (Q) 6. (S); (P); (Q); (R) NOD6\_NOD6 ()\DATA\\IIT-J\TARGT\MATHS\HOM ASSIGNMNT (Q.BANK)\NG\HOM ASSIGNMNT #