Single Correct Type. cos z + k, then the value of k equals. dx = 2 dz. (a) 1 (b) 0 (c)1 (d) 2 (code-v2t3paq10) l (c) ( l ) x.

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1 IIT JEE/AIEEE MATHS y SUHAAG SIR Bhopl, Ph. (755)3 Qusion. & Soluion. In. Cl. Pg: of 6 TOPIC = INTEGRAL CALCULUS Singl Corr Typ Qu.. L f () = sin + sin + + sin + hn h primiiv of f() w.r.. is 3 3 3sin 3 3os 3 () + C () + C () sin 3 + C (d) os3 + C whr C is n rirry onsn. (od-vt3paq5) Qu.. If h dpndn vril y is hngd o z y h susiuion y = n z hn h diffrnil quion + = + + is hngd o d z d = dz os z + k, hn h vlu of k quls d d y ( y) dy d y d () () () (d) (od-vt3paq) l n d is qul o (od-vt5paq) Qu. 3. () + C (). n Qu.. Th vlu of h dfini ingrl l () ( l ) 8 8 (8) + n + C (d) Non f '() + f '( ) d quls (od-vt5paq3) () f (8) + f ( 8) () f (8) f ( 8) () (d) f ( 8) f (8) Qu. 5. ln + d ln quls (od-vt5paq) () () () (d) Qu. 6. If g() os d, = hn g () g() + g( ) () g() g( ) () g()g + quls (od-vt5paq5) (d) [ g() g ] Qu. 7. L f posiiv funion. L = ( ) = ( ) I Thn is I k I f d; I f d, k k () k () / () (d) k whr k >. (od-vt5paq7) Qu. 8. d 6 hs h vlu qul o (od-vt5paq8) () C r s () r s + C () C 6 6 (d) C

2 IIT JEE/AIEEE MATHS y SUHAAG SIR Bhopl, Ph. (755)3 Qusion. & Soluion. In. Cl. Pg: of 6. d = d, hn (od-vt5paq) Qu. 9. If () =, =, = () =, =, = () =, =, = (d) =, =, = Qu.. If n - d= - ln hn h vlu of h dfini ingrl ( + ) n d quls () l n () +l n () l n (d) l n (od-vt5paq) Qu.. n d n n lim n quls (od-vt5paq) () () () (d) Qu.. If f is oninuous funion nd F() = ( + 3). f (u)du d hn F"() is qul o () 7f () () 7f '() () 3f '() (d) 7f () (od-vt5paq5) os l n sin d is qul o (od-vt5paq6) sin Qu. 3. ( + + ) () / () () (d) Qu.. L f :[, ) R oninuous srily inrsing funion, suh h (od-vt5paq7) 3 f () =.f ()d for vry. Th vlu of f (6) is () () 6 () (d) 36 Qu. 5. If h vlu of dfini ingrl / + o d, is qul o / 6 / + hn ( + ) quls sin / 6 () () + () (d) 3 (od-vt5paq8) n Qu. 6. L J = l ln d 3 nd K = d + hn + () J + K = () J K = () J + K < (d) non (od-vt5paq)

3 IIT JEE/AIEEE MATHS y SUHAAG SIR Bhopl, Ph. (755)3 Qusion. & Soluion. In. Cl. Pg: 3 of 6 Qu. 7. Th vlu of > sisfying h quion l n d =, is (od-vt5paq) () () () (d) Qu. 8. If + u F() = f ()d whr f() = du hn h vlu of F"() quls (od-vt5paq3) u () 7 7 () 5 7 () 57 (d) Qu. 9. L f oninuous funion on [, ]. If F() = f ()d f ()d ( ( + ) ) som (,) hn hr is suh h (od-vt8paq6) () f ()d = f ()d () f ()d f ()d = f ()( + ) () f ()d f ()d = f ()( ( + ) ) (d) f ()d + f ()d = f ()( ( + ) ) Qu.. Th vlu of h dfini ingrl = + os sin / I os sin os sin sin sin d, / () ( + ) () ( os sin) os / () os sin is / / (d) [ os+ sin ] / () ( + ) () ( os+ sin) os / () / / (d) [ os+ sin ] + (odvtpaq) Qu.. A nk wih piy of lirs originlly onins gms of sl dissolvd in lirs of wr. Bginning im = nd nding im = minus, wr onining gm of sl pr lirs nrs h nk h of lirs pr minu, nd h wll mid soluion is drind from h nk r of lir/minu. Th diffrnil quion for h moun of sl y in h nk im is () dy = y d + dy = () d ( ) () dy = y d + y + (d) dy = y d 5 + (od-vtpaq)

4 Qu.. L y IIT JEE/AIEEE MATHS y SUHAAG SIR Bhopl, Ph. (755)3 Qusion. & Soluion. In. Cl. Pg: of 6 y = y() soluion o h diffrnil quion y' + y =, hn lim is (od-vtpaq) () zro () () (d) Non isn. Qu. 3. Th r of h rgion oundd low y y = sin, ov y y = os nd on h lf y y-is, is () () () + (d) (od-vtpaq7) Qu.. d is qul o (od-vt3paq) () () () (d) Qu. 5. If. d hs h vlu qul o k hn h vlu of k quls (od-vtpaq) Qu. 6. L () () () 8 (d) d + p + q = n + C 8 r N nd nd no disin, hn h vlu of ( p + q + r) quls (od-vtpaq5) () 6 () 6 () 6 (d) 6 / sin l n(sin ) o d is (od-vt7paq7) Qu. 7. ( + ) () () () (d) Indrminn Qu. 8. L () y = l n + os hn h vlu of d () + os d y () + quls (od-vt7paq8) y / ( + os ) (d) ( + os ) d Qu. 9. Th vlu of dfini ingrl is (od-vt8paq) ( + )( + ) () / () / () / 8 (d) /6 Qu. 3. Th prssion 3 d y y on h llips 3 + y = is qul o (od-vt9paq5) d () 9 () 9 () 9 (d) 9 Qu. 3. Th vlu of h dfini ingrl ( N ) sin d n is qul (od-vt9paq6) + os n () n l n () n l n () nl n (d) n ln

5 IIT JEE/AIEEE MATHS y SUHAAG SIR Bhopl, Ph. (755)3 Qusion. & Soluion. In. Cl. Pg: 5 of 6 Comprhsion Typ # Prgrph for Q. o Q. 3 A urv in rprsnd prmrilly y h quions = os nd y = sin is prmr. Thn. Th rlion wn h prmr nd h ngl α wn h ngn o h givn urv nd h -is is givn y, quls (od-vtpaq,,3) () α () + α () α d y (d) α. Th vlu of h poin whr = is d () () () (d) 3 F() y d h poin h vlu of F F() is 3. If = ( + ) / () () () (d) # Prgrph for Q. o Q. 6 L f () is drivl funion sisfying f() f() d = + n( + ) l wih f () =l n. L g() = f '() hn (od-vtpaq,5,6,). Rng of g () is () [, ) () [, ) () [, ) (d) [, ) 5. For h funion f whih on of h following is orr? () f is nihr odd nor vn () f is rnsndnl () f is injiv (d) f is symmri w.r.. origin. 6. f () d quls () l n ( 3+ ) () l n ( + ) () l n + (d) # 3 Prgrph for Q. 7 o Q. 9 Suppos nd r posiiv rl numrs suh h =. L for ny rl prmr, h disn from h origin h lin + y = dnod y D() hn (od-vt6paq.5.6) 7. Th vlu of h dfini ingrl d I = is qul o ( D() ) + () + () + 8. Th vlu of whih I is minimum, is () 9. Minimum vlu of I is () () () () + (d) (d) () () (d) + r + +

6 IIT JEE/AIEEE MATHS y SUHAAG SIR Bhopl, Ph. (755)3 Qusion. & Soluion. In. Cl. Pg: 6 of 6 Assrion & Rson Typ In his sion h qu. onins STATEMENT- (Assrion) & STATEMENT-(Rson).Eh qusion hs hois (A), (B), (C) nd (D), ou of whih only on is orr. Bul (A) STATEMENT- is ru, STATEMENT- is Tru; STATEMENT- is orr plnion for STATEMENT-. Bul (B) STATEMENT- is Tru, STATEMENT- is Tru; STATEMENT- is NOT orr plnion for STATEMENT-. Bul (C) STATEMENT- is Tru, STATEMENT- is Fls. Bul (D) STATEMENT- is Fls, STATEMENT- is Tru. Qu.. Smn : L f () = + d is odd funion nd g() = f '() is n vn funion. us (od-vtpaq8) Smn : For diffrnil funion f () if f '() is n vn funion hn f() is n odd funion. Qu.. Smn : Th soluion of ( y d dy) o = ny d n is sin y = y us Smn : Suh yp of diffrnil quions n only solvd y h susiuion = vy. Qu. 3. Considr h following smns (od-vt6paq) (od-vtpaq) ) Smn : us 3 3 d = = = 3 3 Smn : If f is oninuous on [,] hn f ()d = F() F() whr F is ny nidriviv of f, h is F' = f. Mor hn On My Corr Typ Qu.. Whih of h following dfini ingrl(s) hs/hv hir vlu qul o h vlu of ls on of h rmining hr? (od-vtpaq3) () / 6 + sin.os d sin () / 6 + os / sin / os / sin / os d () / 6 os + sin d Qu.. Whih of h following dfini ingrl vnishs? (d) 3/ d 3 (od-vtpaq6) n () ( n+ ) / d n N + () log ( log ) ln d () \ sin d (d) os m.sin n d, whr ( m, n I) nd ( m n) is vn ingr.

7 IIT JEE/AIEEE MATHS y SUHAAG SIR Bhopl, Ph. (755)3 Qusion. & Soluion. In. Cl. Pg: 7 of 6 Qu. 3. Th funion f is oninuous nd hs h propry f ( f ()) = for ll [,] hn nd J = f ()d () 3 f + f = () h vlu of J qul o (od-vt8paq) () f.f = 3 3 (d) / sin d 3 hs h sm vlu s J. ( sin + os ) Qu.. Th diffrnil quion orrsponding o h fmily of urvs y = A os( B + D ), is () of ordr 3 () of ordr () dgr (d) dgr (od-vtpaq3) d Qu. 5. quls (od-vt7paq3) + + () + n + C 3 3 () n n + C () + n + C 3 3 (d) n + n + C whr C is n rirry onsn. Qu. 6. If h indpndn vril is hngd o y hn h diffrnil quion d y dy dy d d d + = is hngd o d dy d = k dy whr k quls (od-vt7paq6) () () () (d) d dy n d Qu. 7. L L = lim n whr R hn L n (od-vt9paq9) n + () () () (d) Sujiv Typ ( Up o digi) 7 Qu.. If h vlu of h dfini ingrl ( ) (od-vt8pdq) 7 C7. d is qul o k whr k N. Find k.

8 Qu.. D. No h IIT JEE/AIEEE MATHS y SUHAAG SIR Bhopl, Ph. (755)3 Qusion. & Soluion. In. Cl. Pg: 8 of 6 [SOLUTION] Singl Corr Typ sin + sin + + sin + = sin + sin + + sin + = sin 3 ( + + = = 3) 3 os 3 sin 3 d C. = + Qu.. D. Givn y = n z dy dz s z. d = d... () Now d y d z dz d = s z. +. (s z) [using produ rul] d d d d d z dz d dz = s z. +. (s z). d d dz d s z. d y d z dz = +.s z.n z d d d...() Now ( ) ( + ) y n z + = +.s z. = + + n z.s z. + y d s z d d dy dz dz dz dz = + s z + n z.s z d d From () nd (3) w hv RHS of() = (3)...(3) d z dz s z. = + s z d d d z dz os z k. = + = d d d n d d n Qu. 3. A. I = ( ln ) d = ( + ln ) d L = = l = ( + l ) = = + = + I d C C Qu.. B. f '() + f '( ) I = d; us King nd dd Rsul + Qu. 5. C. I = + ln d ln d n = n + d = d I = = pu l ( l ) Qu. 6. A. + + g( + ) = os d = os d + os d = g() + os d = g() + g. k Qu. 7. D. = ( ) = ( ) I f d;i f d k k k Using King k k k I = ( k)f ( ) d I = f ( ) d I = f ( ) d = I =. I k k k I

9 IIT JEE/AIEEE MATHS y SUHAAG SIR Bhopl, Ph. (755)3 Qusion. & Soluion. In. Cl. Pg: 9 of 6 Qu. 8. C. d d d pu = d = = L 6 = u ; 3 du du; d = du 6 u Qu. 9. A. u du u 6 = C C. 6 = + = + u 6 6. d = d diffrniing oh sids, w g. = = ( + ( ) + ) =,( ) =, = =, =, =. Qu.. A. I = n ( + )d = o d n d = = n d n d = + ( ) = n d n ( ) d n d n n. + = = = l l Qu.. B. Qu.. A. n+ n+ n n lim. = lim. lim. =. n + n + + / n n n n n n F'() = ( + 3) f (u)du f ''() = ( + 3)f () + f (u)du. F''() = 7f () + 7f (). sin Qu. 3. C. Ingrnd is (.)' = = = = sin sin..... / Qu.. B. Givn 3 f () =.f ()d diffrniing, 3f ()f '() = f () f () f '() = ; 3 f () = + C Bu f () = C = f (6) = 6. 6 / os o os d; pu = ; d = d Qu. 5. A. ( + ) / 6 / / ( + ) = ( ) os () o().os () d os () o().os () d / 6 / 6 / 6 6 / 6 = os () = + + =.

10 IIT JEE/AIEEE MATHS y SUHAAG SIR Bhopl, Ph. (755)3 Qusion. & Soluion. In. Cl. Pg: of 6 ( + ) ln ln d Qu. 6. A. J + K = d = 3 + sr = J + K = ( J + K) J + K =. + Qu. 7. A. l n I = ln d = ln.. d n = l = = ` ln = [ ln ] = ( s > ) ln = =. Qu. 8. C. Now f '() = =... () F() = f ()du F'() = f () F''() = f ''() F''() = f '() Form () f '() = 56 + = 57. Qu. 9. B. Givn F() = f ()d f () d ( d ( + ) )... () s f is oninuous hn F() is lso oninuous. Also pu =. F() = f () d = f () d nd pu = hn F () = F() hn Roll s horm is pplil o F () som (, ) suh h F'() = F'() = f ()d f () d + ( ( + )) f () + f () = Now [ ] F'() f () d f () d = f () ( + ) [ ] F = f () d ( ) Qu.. A. / + os os I = os(sin ) + sin(sin ) d / = + + / [{ os(sin ) sin(sin ) } os { os(sin ) sin(sin ) }] d f () f '() = os + sin. Qu.. B. Qu.. B. dy y y ()() = = d + + dy y y ()() = = d + +

11 IIT JEE/AIEEE MATHS y SUHAAG SIR Bhopl, Ph. (755)3 Qusion. & Soluion. In. Cl. Pg: of 6 Qu. 3. (B) / / os y dy + / = sin y] os y] sin y dy / / / = =. Qu.. D. = sin θ d = 8sin θ.os θ dθ / sin θ I = 8sin θ.os θdθ = 8 os θ / sin θdθ / sin θdθ = / θ sin θ = =. Qu. 5. D. / Us Wlli s formul o g. I =. d pu = sin θ 8 sin θos θd θ. Qu. 6. C d + I = = d = d ( + ) + 7 ( ) 7 k ln ln + = ln l + = = ln + C p + q + r = Qu. 7. C. / / d I = sin d = sin = lim sin = = d Qu. 8. A. l ( + os ) os sin ( sin ) sin y = n + os y y = + os + os n( os ) os l + + y / =. y = = = + =. y / ( + os ) ( + os ) ( + os ) Qu. 9. B. ( + )( + ) d I =...() = d. + + (using King) d I =...() dding () nd () ( + )( + ) ( + ) d d d ( + )( + ) ( + ) ( + ) I = = = d I = = n = / ( + ) [onvr i ino vlu of dfini ingrl T is sm s]

12 Qu. 3. B. IIT JEE/AIEEE MATHS y SUHAAG SIR Bhopl, Ph. (755)3 Qusion. & Soluion. In. Cl. Pg: of 6 Diffrniing impliiy w hv 6 + 8yy' = nd hn y' = 3 ; yy'' + ( y' ) = 3 y diffrniing gin nd susiu for y w hv 3+ ( y' ) + yy'' = nd hn yy'' = y muliplying y y, 9 3 d y 3y + + y = d 3y y + y y '' = y y'' = u 3 + y = nd hn n Qu. 3. A. I = d... sin + os y y''' = vry poin on h llips] 3 9 or ( n ) n sin I = d... + os dd () nd () n n sin sin I = n d I = n d U sin g f d n f d + os = + os / / / n sin d sin d os d I =. = n = n + os + os + sin = n.ln + sin ] = n ln Ans. / Comprhsion Typ # Prgrph for Q. o Q C.. - B C. (i) dy d y = sin = [os + sin ] = os = [os sin ] d d dy os + sin = = n α d os sin n + = n α + α = α. (ii) (iii) s d y + d y = =. d (os sin ) d = / / F() = (os + sin ) d = sin + C F F() = + C =.

13 . - B B A. IIT JEE/AIEEE MATHS y SUHAAG SIR Bhopl, Ph. (755)3 Qusion. & Soluion. In. Cl. Pg: 3 of 6 # Prgrph for Q. o Q. 6 (I) f () f ()d = + n ( + ) l diffrniing f '() f () = + + ( + ) f '() = + rng of g() = f '() is [,). + f '() = + f '() is odd f() is vn. Ingring (), i.. d d f '() = f () = f () n () I, + = l + d whr I = ; pu = n θ d = s dθ + s θdθ + I = = os θdθ = n ( os θ o θ ) = n + C n θs θ l l + f () ln ln + C f () = ln ln + C + + l f () = lm C; pu =,f () ln f () = ln C = f () = n + +. Now f ()d =. ln d + II I ingring y prs f ().] f '()d f () d = + = ln + ln + + = ln + ln + + { } = ln + n + {} = n + = n + l l l.

14 IIT JEE/AIEEE MATHS y SUHAAG SIR Bhopl, Ph. (755)3 Qusion. & Soluion. In. Cl. Pg: of 6 # 3 Prgrph for Q. 7 o Q C. 8. D. 9. B. (i) D() = = = + ( ) + ( ) + ( D() ) I = ( + ) d = = ( ) ( ) ( ) + ( ) = = = +. (ii) Now pu I = I = + = + I is minimum if = =. = = (iii) nd Imin = =. Qu.. C. If f () is odd f '() Qu.. C. Qu. 3. D. Assrion & Rson Typ is vn u onvrs is no ru.g. If f '() = sin hn f () = sin os + C ; f ( ) = sin + os + C f () + f ( ) = onsn whih nd no o zro For S-: f () = + d; g() = + f ( ) = + d; = y f ( ) + y dy f () + f ( ) = f is odd nd g is oviously vn. 3 d dos no is. Mor hn On My Corr Typ Qu.. A,B,C,D. No h h ingrnd in A,B nd C ll rdus o (+sin ) 3 I = ( sin ) d os ] + = = ( ) os = / 6 Now, D = d 3 Pu = os θ + 3sin θ d = sin θos θ dθ 3/

15 whn IIT JEE/AIEEE MATHS y SUHAAG SIR Bhopl, Ph. (755)3 Qusion. & Soluion. In. Cl. Pg: 5 of 6 3 = hn sin θ = θ = 6 whn = hn sin θ = θ = / / sin θ I =.sin θ os θ dθ = os θ dθ = θ sin θ = θ sin θ os θ Qu.. A,B,C,D. / 6 / = = + 3 A,B,C, D. 6 (A). Pu = / nd dd o g rsul / ] / 6 / / 6 (B). n n l l d if f () = f '() = l l l l n n n n I = ln = ln = n n n l l l (C). = d = d / / I = sin d = sin d = sin d = I I = I =. / ln Alrnivly for (C); pu ( ) = I sin d = (odd funion) ln (D). sin n.os m d sin(n m) sin(n m) d = + + os(n + m) os(n m) = + = + = n + m n m n + m n m n + m n m Qu. 3. A,B,D. Givn f ( f ()) = + rpling f () ( ) f ( ) = f () + Now = = ( ) J f ()d f d f f f () = f () + f () + f =... () (A) (Using King) J = ( f () + f ( ) ) d; J = d = J =. Qu.. A,D. y = A[ os B os D sin B sin D] y = C os B + C sin B... () ( A os D C ; A sin D C ) = = = + = + y BC sin B BC os B y BC sin B BC os B y = B ( C os B + C sin B ) y = B y = B y d y dy d y yy3 yy = y =. 3 d d d

16 IIT JEE/AIEEE MATHS y SUHAAG SIR Bhopl, Ph. (755)3 Qusion. & Soluion. In. Cl. Pg: 6 of 6 d d + / + Qu. 5. B,C. I =. n n = = = + + / 3/ / = n + C. 3 3 Alrnivly : ( ) d d ( + + ) ( + ) ( + + ) ( + + )( + ) ( + + )( + ) I = = = d d d d = / 3 / + / 3 / ( ) d + + = = Qu. 6. A,C,D. n n n n C. Qu. 7. A, B, C Considr nd I = =.n ( n n) = n n n n + n if < = = = if > L lim n n / if A, B, C 7 Qu.. 8 L = ( ) 7 7 I C d II Sujiv Typ ( Up o digi) I I = C. +. d = C.. d zro I.B.P. gin 6 mor ims ( 7 )! 7! 7! = C. d = !! ! 7! =. = = k = 8 7!7! 8 8 k Ans. ]

(A) 1 (B) 1 + (sin 1) (C) 1 (sin 1) (D) (sin 1) 1 (C) and g be the inverse of f. Then the value of g'(0) is. (C) a. dx (a > 0) is

(A) 1 (B) 1 + (sin 1) (C) 1 (sin 1) (D) (sin 1) 1 (C) and g be the inverse of f. Then the value of g'(0) is. (C) a. dx (a > 0) is [STRAIGHT OBJECTIVE TYPE] l Q. Th vlu of h dfii igrl, cos d is + (si ) (si ) (si ) Q. Th vlu of h dfii igrl si d whr [, ] cos cos Q. Vlu of h dfii igrl ( si Q. L f () = d ( ) cos 7 ( ) )d d g b h ivrs