STUDY PACKAGE. Subject : Mathematics Topic : DIFFRENTIATION Available Online :

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1 fo/u fopkjr Hk# tu] uga vkjehks ke] foifr ns[k NksM+s rqjar e/;e eu j ';kea iq#"k flag layi j] lgrs foifr vus] ^cuk^ u NksM+s /;s; ks] j?kqcj jk[ks VsAA jfpr% ekuo /kez izksrk ln~q# J jknksm+nklt egkjkt STUDY PACKAGE Subject : Matematics Topic : DIFFRENTIATION Available Online : wwwmatsbsuagcom R Ine Teor Sort Revision Eercise (E + 5 6) Assertion & Reason 5 Que from Compt Eams 6 9 Yrs Que from IIT-JEE(Avance) 7 5 Yrs Que from AIEEE (JEE Main) Stuent s Name : Class Roll No : : Aress : Plot No 7, III- Floor, Near Patiar Stuio, Above Bon Classes, Zone-, MP NAGAR, Bopal : , , WatsApp wwwtekoclassescom wwwmatsbsuagcom

2 FREE Downloa Stu Package from website: wwwtekoclassescom & wwwmatsbsuagcom Get Solution of Tese Packages & Learn b Vieo Tutorials on wwwmatsbsuagcom Differentiation A First Principle Of Differentiation Te erivative of a given function f at a point a on its omain is efine as: Limit f(a ) f(a) 0, provie te limit eists & is enote b f (a) ie f (a) Limit f() f(a) a, provie te limit eists a If an + belong to te omain of a function f efine b f(), ten Limit f( ) f() 0 if it eists, is calle te Derivative of f at & is enote b f () or ie, f () Limit 0 f( ) f() Tis meto of ifferentiation is also calle ab-initio meto or first principle Solve Eample # Fin erivative of following functions b first principle (i) f() (ii) f() tan (iii) f() e sin Solution (i) f() lim 0 ( ) (ii) f() lim tan( ) tan 0 lim tan( )[ tan tan( )] 0 sin ( ) sin (iii) f() lim e e 0 sin ( ) sin e esin lim 0 sin( ) sin e sin lim sin( ) sin 0 Differentiation of some elementar functions f() f() lim 0 sin( ) sin lim e sin cos n n n ( R, n R) a a n a n log a n a 5 sin cos 6 cos sin 7 sec sec tan 8 cosec cosec cot 9 tan sec 0 tan ( + tan ) sec 0 cot Basic Teorems cosec (f ± g) f() ± g() (k f()) k f() (f() g()) f() g() + g() f() f() g() f() f()g () g() g () 5 (f(g())) f(g()) g() Tis rule is also calle te cain rule of ifferentiation an can be written as z z Note tat an important inference obtaine from te cain rule is tat Teko Classes, Mats : Suag R Karia (S R K Sir), Bopal Pone : , page OF 8 Successful People / Replace te wors like; "wis", "tr" & "soul" wit "I Will" Ineffective People on't

3 FREE Downloa Stu Package from website: wwwtekoclassescom & wwwmatsbsuagcom Get Solution of Tese Packages & Learn b Vieo Tutorials on wwwmatsbsuagcom anoter wa of epressing te same concept is b consiering f() an g() as inverse functions of eac oter f() an g() g() f() Solve Eample # Fin te ifferential of te following functions wit respect to (i) f() e sin (ii) f() sin( ) (iii) f() Solution (i) f() e sin f() e sin (sin ) (ii) f() sin ( ) e sin cos cos( ) (sin ( + )) sin ( ) sin ( ) (iii) f() ( ) () f() ( ) ( ) (iv) f() sin f() cos + sin Solve Eample # If f() sin ( + tan) ten fin value of f(0) Solution f() cos ( + tan) ( + sec ) f(0) Self Practice Problems : Fin te erviative of following functions using first principle (i) f() sin (ii) f() sin Ans (i) cos + sin (ii) sin cos Evaluate if f(5) 7, ten f(5 t) f(5 t) t 0 t Differentiate te following functions Ans 7 (i) ( + ) ( ) (ii) (iv) (vii) sin cos ( ) ( )( ) (iii) (v) cos sin (vi) e sin (viii) n (sin cos ) Ans (i)6 (5 + ) (ii) ( ) ( ) (v) cos cos sin B (vi) e ((sin + cos ) + sin ) (vii) (iii) sec Derivative Of Inverse Trigonometric Functions sin sin (iv) (iv) / / ( ) ( ) (viii) f() sin cos sin sin cos cos cos < < sin Note ere tat cos sin, rater cos ± sin but for values of,, cos is alwas positive an ence te result similarl let us fin erivative of oter inverse trigonometric functions Let tan tan sec + tan + ( R) Also if sec [0, ] sec sec tan tan sec Teko Classes, Mats : Suag R Karia (S R K Sir), Bopal Pone : , page OF 8 Successful People Replace te wors like; "wis", "tr" & "soul" wit "I Will" Ineffective People on't

4 FREE Downloa Stu Package from website: wwwtekoclassescom & wwwmatsbsuagcom Get Solution of Tese Packages & Learn b Vieo Tutorials on wwwmatsbsuagcom sec (, ) (, ) sec results for te erivative of inverse trigonometric functions can be summarize as : f() f() sin ; < cos ; < tan ; R cot ; R sec ; > cosec - ; > Solve Eample # If f() n (sin ) fin f() Solution f() (sin ) ( ) (sin ) Solve Eample # 5 If f() sec cosec () ten fin f( ) Solution f() sec () + f( ) sec ( ) f( ) + C Metos Of Differentiation Logritmic Differentiation Te process of taking logaritm of te function first an ten ifferentiate is calle Logaritmic Differentiation It is useful if (i) a function is te prouct or quotient of a number of functions OR (ii) a function is of te form [f()] g() were f & g are bot erivable, Solve Eample # 6 If fin Solution n n + n ( + n ) Solve Eample # 7 If (sin ) n, fin Solution n n n (sin ) n (sin ) + n cos sin / ( ) Solve Eample # 8 If / / 5 ( ) ( ) / (sin ) n fin Solution n n + n ( ) n ( ) 5 n ( ) n sin cot ( ) ( ) 5 ( ) 9 6 ( ) ( ) 5 ( ) Implicit ifferentiation If f(, ) 0, is an implicit function ten in orer to fin /, we ifferentiate eac term wrt regaring as a functions of & ten collect terms in / Solve Eample # 9 If + fin Solution Differentiation bot sies wrt, we get + + Note tat above result ols onl for points were 0 n Teko Classes, Mats : Suag R Karia (S R K Sir), Bopal Pone : , page OF 8 Successful People Replace te wors like; "wis", "tr" & "soul" wit "I Will" Ineffective People on't

5 FREE Downloa Stu Package from website: wwwtekoclassescom & wwwmatsbsuagcom Get Solution of Tese Packages & Learn b Vieo Tutorials on wwwmatsbsuagcom Solve Eample # 0 If e, ten fin Solution Taking log on bot sies n ( ) (i) ifferentiating wrt, we get + ln Solve Eample # If + ten fin u v Solution u + v + 0 were u & v n u n & n v n u u + n & v n v n + ( n ) u v n & n n + n n 0 n Self Practice Problems Differentiate te following functions : (i) sec ( ) (ii) tan (iii) (iv) e (v) (ln ) + () sin Fin if (i) cos ( + ) (ii) / + / a / (iii) n ( ) n If e, ten prove tat ( n ) If log Ans a, prove tat (i) (ii) (iii) n (iv) e (n + ) (v) sin n ( n) n (n ) + sin cos n sin( ) ( ) (i) (ii) (iii) sin( ) ( ) Differentiation using substitution Following substitutions are normall use to sumplif tese epression / (i) a a tan or a cot (ii) a a sin or a cos (iii) a a sec or a cosec a (iv) a cos a Solve Eample # : Differentiate tan Solution Let tan tan ;, sec tan [ sec sec, ] tan Teko Classes, Mats : Suag R Karia (S R K Sir), Bopal Pone : , page 5 OF 8 Successful People Replace te wors like; "wis", "tr" & "soul" wit "I Will" Ineffective People on't

6 FREE Downloa Stu Package from website: wwwtekoclassescom & wwwmatsbsuagcom Get Solution of Tese Packages & Learn b Vieo Tutorials on wwwmatsbsuagcom cos tan sin tan tan [tan (tan) for, ] tan ( ) Solve Eample # : Fin were tan Solution cos cos () ; [0, ] tan tan cos cos tan tan cos cos cos Note tat cos cos tan cos cos sin sin Solve Eample # 5 If + a( ), ten prove tat Solution Put sin sin () sin sin () cos + cos a (sin sin) Successful People Replace te wors like; "wis", "tr" & "soul" wit "I Will" Ineffective People on't but for 0,, Also tan (tan ) for, Solve Eample # If f() sin ten fin (i) f() (ii) f Solution tan tan () ; < < (iii) f() cos cos sin (sin ) tan f() tan f() ( tan ) ( ) 8 (i) f() (ii) f (iii) f( 5 + ) & f( ) + 5 f() oes not eist Aliter Above problem can also be solve witout an substution also, but in a little teious wa f() sin f() f() ( ( ( ( ) ) ) ) ( ) {( ) } ( ( ( ) ) ) tus f() Teko Classes, Mats : Suag R Karia (S R K Sir), Bopal Pone : , page 6 OF 8

7 FREE Downloa Stu Package from website: wwwtekoclassescom & wwwmatsbsuagcom Aliter Get Solution of Tese Packages & Learn b Vieo Tutorials on wwwmatsbsuagcom cos cos a cos sin cot (a) sin sin cot (a) ifferentiating wrt to Using implicit ifferention a 0 a a + ( ) ( )( ) ( )( ) ( ) Hence prove a a ( )( ) ( )( ) Parametric Differentiation If f() & g() were is a parameter, ten Solve Eample # 6 If a cos t an a sin t Fin / t / t a sin acos t cost t sint tan t Solve Eample # 7 If a cos t an a (t sint) fin te value of at t asint) a( cos t) t 5 Derivative of one function wit respect to anoter Let f(); z g() ten / f'( ) z z / g'( ) Solve Eample # 8 Fin erivative of n wit respect to z e / z z / e Self Practice Problems : Fin wen (i) a (cos t + t sin t) & a (sin t t cos t) (ii) a t t & b t t ( t )b Ans (i) tan t (ii) at If sin a a ten prove tat a If tan ten prove tat ( ) cot a / / Teko Classes, Mats : Suag R Karia (S R K Sir), Bopal Pone : , page 7 OF 8 Successful People Replace te wors like; "wis", "tr" & "soul" wit "I Will" Ineffective People on't

8 FREE Downloa Stu Package from website: wwwtekoclassescom & wwwmatsbsuagcom Get Solution of Tese Packages & Learn b Vieo Tutorials on wwwmatsbsuagcom u u If u sin (m cos ) an v cos (m sin ) ten prove tat v v D Derivatives of Higer Orer Let a function f() be efine on an open interval (a, b) It s erivative, if it eists on (a, b) is a certain function f () [or (/) or ] & is calle te first erivative of w r t If it appens tat te first erivative as a erivative on (a, b) ten tis erivative is calle te secon erivative of w r t & is enote b f () or ( / ) or Similarl, te r orer erivative of w r t, if it eists, is efine b or Solve Eample # 9 If n ten an Solution n + n + 6 n n n + Solve Eample # 0 If ten fin () Solution n n wen ( + n ) ( + n ) (i) again iff wrt to, ( + n ) ( + ln ) (using (i)) () 0 It must be carefull note tat in case of parametric functions / t altoug but / t wic on appling cain rule can be resolve as t / t t / t t t t t t / t / t rater t t t t t Solve Eample # If t + an t + t ten fin Solution t + t ; t t t + t + 6t t (t + t ) t It is also enote b f () / t / t Solve Eample # If cos t cos t an sin t sin t ten fin value of Solution cos t cos t sin t sin t t t t cost cost sin t sin t t tan t sin sin t t cos sin t t tan at t t t tan t Teko Classes, Mats : Suag R Karia (S R K Sir), Bopal Pone : , page 8 OF 8 Successful People Replace te wors like; "wis", "tr" & "soul" wit "I Will" Ineffective People on't

9 FREE Downloa Stu Package from website: wwwtekoclassescom & wwwmatsbsuagcom Get Solution of Tese Packages & Learn b Vieo Tutorials on wwwmatsbsuagcom t sec (sint sint) t Solve Eample # Fin secon orer erivative of sin wit respect to z e / cos Solution z z / e cos cos z z e e z e sin cos e (e ) e (sin cos) z e Solve Eample # : f() an g() are inverse functions of eac oter tan epress g() an g() in terms of erivative of f() Solution f() an g() g() f() (i) again ifferentiating wrt to g() f () f () f() g() f() f() g() f() (ii) wic can also be remembere as Solve Eample # 5 sin (sin) ten prove tat + (tan) + cos 0 Solution Suc epression can be easil prove using implict ifferention cos (sin ) cos sec cos (sin ) again ifferentiating wrt, we can get sec + sec tan sin (sin ) cos tan cos +(tan) + cos 0 Self Practice Problems : n n If ten fin Ans Prove tat + tan satisfies te ifferentiation equation cos + 0 sec If a (cos + sin ) an a(sin cos) ten fin Ans a sin cos Fin secon erivative of n wit respect to sin Ans cos 5 if e (A cos + B sin ), prove tat Solve Eample # 6 If (tan ) ten prove tat ( + ) Solution + ( + ) tan ( + ) tan () ( + ) + ( ) ( + ) + ( + ) Successful People Replace te wors like; "wis", "tr" & "soul" wit "I Will" Ineffective People on't Teko Classes, Mats : Suag R Karia (S R K Sir), Bopal Pone : , page 9 OF 8

10 FREE Downloa Stu Package from website: wwwtekoclassescom & wwwmatsbsuagcom Get Solution of Tese Packages & Learn b Vieo Tutorials on wwwmatsbsuagcom f() l() If F() u() f'() l() u() g'() m() v() g() m() v() '() n() w() () n(), were f, g,, l, m, n, u, v, w are ifferentiable functions of ten F () w() f() l'() + u() L Hospital s Rule: g() m'() v() () n'() w() If f() & g() are functions of suc tat: (i) (ii) (iii) Limit a f() 0 a + f() l() u'() g() m() v'() () n() w'() Limit g() OR Limit a f() Limit a g() & Bot f() & g() are continuous at a & Bot f() & g() are ifferentiable at a & (iv) Bot f () & g () are continuous at a, Ten f() f'() f"() Limit a Limit g() a Limit g'() a & so on till ineterminant form vanises g"() QUESTION BANK ON METHOD OF DIFFERENTIATION Select te correct alternative : (Onl one is correct) Q If g is te inverse of f & f () ten g () 5 + [g()] 5 (C) 5 [ g( )] [ g( )] e 5 n Q If tan ne + tan n ten 6 n (C) 0 (D) Q If f & f () tan ten 5 6 tan tan tan 5 6 (C) f ( 5 6) tan 5tan 6 Q If sin & + p, ten p ( ) 0 sin (C) sin of tese Q5 If f & f () sin ten sin sin (C) Q6 Let g is te inverse function of f & f () 5 a 0 0 a Q7 If sin () + cos () 0 ten 0 (C) sin If g() a ten g () is equal to a 0 a (C) (D) 0 a a (D) Teko Classes, Mats : Suag R Karia (S R K Sir), Bopal Pone : , page 0 OF 8 Q8 If sin ten is : Successful People Replace te wors like; "wis", "tr" & "soul" wit "I Will" Ineffective People on't

11 FREE Downloa Stu Package from website: wwwtekoclassescom & wwwmatsbsuagcom Get Solution of Tese Packages & Learn b Vieo Tutorials on wwwmatsbsuagcom 5 5 (C) 5 Q9 Te erivative of sec wrt at is : / (C) Q0 If P(), is a polnomial of egree, ten equals : Q P () + P () P () P () (C) P () P () (D) a constant Let f() be a quaratic epression wic is positive for all real If g() f() + f () + f (), ten for an real, wic one is correct g() < 0 g() > 0 (C) g() 0 (D) g() 0 Q If p q ( + ) p + q ten is : inepenent of p but epenent on q epenent on p but inepenent of q (C) epenent on bot p & q (D) inepenent of p & q bot Q Let f() g ( ) cos if 0 were g() is an even function ifferentiable at 0, passing troug te 0 if 0 origin Ten f (0) : is equal to is equal to 0 (C) is equal to (D) oes not eist Q If n m p m + m n p n + m p n p ten at emnp is equal to: e mnp e mn/p (C) e np/m Q5 log cos sin Lim 0 as te value equal to log cos sin (C) of tese Q6 If f is ifferentiable in (0, 6) & f () 5 ten Limit f f ( ) c 5 5/ (C) 0 (D) 0 Q7 Let l Lim 0 m (ln ) n were m, n N ten : l is inepenent of m an n l is inepenent of m an epens on m (C) l is inepenent of n an epenent on m (D) l is epenent on bot m an n cos Q8 Let f() sin Ten Limit f( ) 0 tan (C) (D) cos sin cos Q9 Let f() cos sin cos ten f cos sin cos 0 (C) (D) Q0 People living at Mars, instea of te usual efinition of erivative D f(), efine a new kin of erivative, D*f() b te formula f ( ) f ( ) D*f() Limit were f () means [f()] If f() ln ten 0 D * f ( ) e as te value e e (C) e Q If f() g() ; f () 9 ; g () 6 ten Limit f ( ) g( ) is equal to : (C) 0 Q If f() is a ifferentiable function of ten Limit f ( ) f ( ) 0 f () 5f () (C) 0 Teko Classes, Mats : Suag R Karia (S R K Sir), Bopal Pone : , page OF 8 Successful People Replace te wors like; "wis", "tr" & "soul" wit "I Will" Ineffective People on't

12 FREE Downloa Stu Package from website: wwwtekoclassescom & wwwmatsbsuagcom Get Solution of Tese Packages & Learn b Vieo Tutorials on wwwmatsbsuagcom Q If + e ten is : e e e (C) (D) e e e Q If + ten te value of at te point (, ) is : 5 (C) 8 Q5 If f(a), f (a), g(a), g (a) ten te value of Limit g( ) f ( a) g( a) f ( ) a is: a 5 /5 (C) 5 Q6 If f is twice ifferentiable suc tat f ( ) f( ), f ( ) g( ) ( ) f( ) g( ) an ( 0), ( ) ten te equation () represents : a curve of egree a curve passing troug te origin (C) a straigt line wit slope (D) a straigt line wit intercept equal to Q7 Te erivative of te function, f()cos - ( cos sin ) UVW + sin ( cos sin ) UVW wrt at is : RST 5 (C) 0 (D) 0 Q8 Let f() be a polnomial in Ten te secon erivative of f(e ), is : f (e ) e + f (e ) f (e ) e + f (e ) e (C) f (e ) e (D) f (e ) e + f (e ) e Q9 Te solution set of f () > g (), were f() (5 + ) & g() 5 + (ln 5) is : > 0 < < (C) 0 (D) > 0 Q0 If sin + sec, > ten is equal to : (C) 0 (D) Q If ten a b a b a b a b a b (C) (D) ab a ab b ab b ab a Q Let f () be a polnomial function of secon egree If f () f ( ) an a, b, c are in AP, ten f '(a), f '(b) an f '(c) are in GP HP (C) AGP (D) AP Q If sin m ten te value of 5 (were subscripts of sows te orer of erivatiive) is: inepenent of but epenent on m epenent of but inepenent of m (C) epenent on bot m & (D) inepenent of m & Q If + R (R > 0) ten k were k in terms of R alone is equal to (C) (D) R R R R Q5 If f & g are ifferentiable functions suc tat g (a) & g(a) b an if fog is an ientit function ten f (b) as te value equal to : / (C) 0 (D) / Q6 Given f() + sin 5 a sin a sin a 5 arc sin (a 8a + 7) ten : f() is not efine at sin 8 f (sin 8) > 0 Successful People Replace te wors like; "wis", "tr" & "soul" wit "I Will" Ineffective People on't RST Teko Classes, Mats : Suag R Karia (S R K Sir), Bopal Pone : , page OF 8

13 FREE Downloa Stu Package from website: wwwtekoclassescom & wwwmatsbsuagcom Get Solution of Tese Packages & Learn b Vieo Tutorials on wwwmatsbsuagcom (C) f () is not efine at sin 8 (D) f (sin 8) < 0 Q7 A function f, efine for all positive real numbers, satisfies te equation f( ) for ever > 0 Ten te value of f () (C) / (D) cannot be etermine Q8 Given : f() 6 cos a + sin a sin 6a + n a a ten : f() is not efine at / f (/) < 0 (C) f () is not efine at / (D) f (/) > 0 Q9 If (A + B) e m + (m ) e ten m + m is equal to : e e m (C) e m (D) e ( m) Q0 Suppose f () e a + e b, were a b, an tat f '' () f ' () 5 f () 0 for all Ten te prouct ab is equal to 5 9 (C) 5 (D) 9 Q Let () be ifferentiable for all an let f () (k + e ) () were k is some constant If (0) 5, ' (0) an f ' (0) 8 ten te value of k is equal to 5 (C) (D) Q Let e f() ln If g() is te inverse function of f() ten g () equals to : e e + (C) ( e e ) (D) e ( + ln ) Q Te equation e 9e efines as a ifferentiable function of Te value of for an is 5 9 (C) (D) 5 5 an g() ten : f () an g () f () an g () (C) f () an g () 0 (D) f () an g () Q5 Te function f() e +, being ifferentiable an one to one, as a ifferentiable inverse f () Te value of (f ) at te point f(l n) is (C) n log sin cos Q6 If f () for < 0 logsin cos for 0 ten, te number of points of iscontinuit of f in, is 0 (C) (D) Q7 If ( a ) a ( b ) b ten werever it is efine is equal to : a b Q Let f() ( a b) ( a ) ( b) ( a b) ( a ) ( b) (C) ( a b) ( a ) ( b) Successful People Replace te wors like; "wis", "tr" & "soul" wit "I Will" Ineffective People on't (D) ( a b) ( a ) ( b) Q8 If is a function of ten + 0 If is a function of ten te equation becomes : (C) 0 (D) 0 Q9 A function f () satisfies te conition, f () f () + f () + f () + were f () is a ifferentiable function inefinitel an as enotes te orer of erivative If f (0), ten f () is : e / e (C) e (D) e cos6 6cos 5cos 0 Q50 If, ten cos5 5cos 0cos sin + cos sin (C) cos (D) sin Q5 If + K ten te value of K is equal to (C) (D) 0 Teko Classes, Mats : Suag R Karia (S R K Sir), Bopal Pone : , page OF 8

14 FREE Downloa Stu Package from website: wwwtekoclassescom & wwwmatsbsuagcom Get Solution of Tese Packages & Learn b Vieo Tutorials on wwwmatsbsuagcom sin sin ( ) were 0, ten f ' () as te value equal to Q5 If f() ( ) zero (C) ( ) (D) e if 0 Q5 Let f() 0 if 0 Ten wic of te following can best represent te grap of f()? (C) (D) m n mn m n mn mn n m Q5 Diffrential coefficient of wrt is 0 (C) (D) b mn Q55 Let f () be iffrentiable at ten Lim g f ( ) f ( ) is equal to f() + f '() f() + f '() (C) f() + f '() (D) f() f '() Q56 If at + bt + c an t a + b + c, ten equals a (at + b) a (a + b) (C) a (at + b) (D) a (a + b) Q57 Limit 0 a arc tan b arctan a b as te value equal to a b 0 (C) ( a b ) (D) a b 6a b a b Q58 Let f () be efine for all > 0 & be continuous Let f() satisf f f () f () for all, & f(e) Ten : f() is boune f 0 as 0 (C) f() as 0 (D) f() ln Q59 Suppose te function f () f () as te erivative 5 at an erivative 7 at Te erivative of te function f () f () at, as te value equal to 9 9 (C) 7 (D) Q60 If 5 cot 8 an a + b ten te value of a + b is equal to 5 cot Successful People Replace te wors like; "wis", "tr" & "soul" wit "I Will" Ineffective People on't 5 (C) tan Q6 Suppose tat () f () g() an F() 5 (D) tan 8 f g(), were f () ; g() 5 ; g'() ; f '() an f '(5), ten F'() '() F'() '() (C) F'() '() Q6 Let f () wic one of te properties of te erivative enables ou to conclue tat f () as an inverse? f ' () is a polnomial of even egree f ' () is self inverse (C) omain of f ' () is te range of f ' () (D) f ' () is alwas positive Q6 Wic one of te following statements is NOT CORRECT? Te erivative of a iffrentiable perioic function is a perioic function wit te same perio If f () an g () bot are efine on te entire number line an are aperioic ten te function F() f () g () can not be perioic (C) Derivative of an even ifferentiable function is an o function an erivative of an o ifferentiable function is an even function (D) Ever function f () can be represente as te sum of an even an an o function Select te correct alternatives : (More tan one are correct) Q6 If tan tan tan ten as te value equal to : Teko Classes, Mats : Suag R Karia (S R K Sir), Bopal Pone : , page OF 8

15 FREE Downloa Stu Package from website: wwwtekoclassescom & wwwmatsbsuagcom Get Solution of Tese Packages & Learn b Vieo Tutorials on wwwmatsbsuagcom sec tan tan + sec tan tan + sec tan tan (cosec + cosec + cosec 6) (C) sec sec sec (D) sec + sec + sec Q65 If e e ten equals e e e e (C) (D) Q66 If ten ln ( ln + ) (C) ( ln + ) (D) ln e Q67 Let ten (C) (D) Q68 If + + ten as te value equal to : (C) (D) Q69 Te functions u e sin ; v e cos satisf te equation : v u u v u + v u v (C) v u of tese Q70 Let f () ten : Q7 f (0) f (/) (C) omain of f () is Two functions f & g ave first & secon erivatives at 0 & satisf te relations, f(0) g( 0) (0) g (0) g (0), g (0) 5 f (0) 6 f(0) ten : if () f ( ) g( ) (0) 5 if k() f() g() sin ten k (0) (C) Limit 0 Q7 If (C) g ( ) f ( ) n( n) ( n) n, ten is equal to : n n n n n (ln )ln (ln ) ( ln (ln ) + ) ((ln ) + ln (ln )) (D) n ( ln (ln ) + ) n ANSWER KEY Q A Q C Q B Q D Q5 B Q6 B Q7 B Q8 C Q9 A Q0 C Q B Q D Q B Q D Q5 C Q6 D Q7 A Q8 B Q9 C Q0 C Q A Q B Q B Q B Q5 C Q6 C Q7 C Q8 D Q9 D Q0 C Q D Q D Q D Q B Q5 D Q6 D Q7 B Q8 D Q9 A Q0 C Q C Q C Q D Q D Q5 B Q6 C Q7 B Q8 C Q9 A Q50 B Q5 D Q5 B Q5 C Q5 B Q55 A Q56 D Q57 D Q58 D Q59 A Q60 B Q6 B Q6 D Q6 B Q6 A,B,C Q65 A,C Q66 C,D Q67 A,C,D Q68 A,B,C,D Q69 A,B,C Q70 A,B Q7 A,B,C Q7 B,D Teko Classes, Mats : Suag R Karia (S R K Sir), Bopal Pone : , page 5 OF 8 Successful People Replace te wors like; "wis", "tr" & "soul" wit "I Will" Ineffective People on't

16 FREE Downloa Stu Package from website: wwwtekoclassescom & wwwmatsbsuagcom Get Solution of Tese Packages & Learn b Vieo Tutorials on wwwmatsbsuagcom EXERCISE - Part : Onl one correct option If f() an f( ) ten at is (C) (D) If ten n ( n + ) (C) ( n + ) (D) n e tan sin If f() e, ten f(0) If a b a a ab a ten b ab b (C) (D) If f(), g(), () are polnomials in of egree an F() f g f g, ten F() is equal to 0 (C) (D) f() g() () If sin m ten te value of 5 (were settings of sows te orer of erivative) is: Successful People Replace te 6 wors 7 like; 8 "wis", "tr" & "soul" wit "I Will" Ineffective People on't (C) a ab b 5 Let f() sin ; g() & () log e & F() [g(f())] ten F is equal to: cosec cot ( ) cosec ( ) (C) cot (D) cosec 6 If ( + ) ( + ) ( + ) ( + n ), ten at 0 is (C) 0 (D) n 7 If sin 0 8 If e t were t sin 9 If sin an ( ) + p, ten p (D) b ab a (C) sin (D) ten : (C) (D) + sec, > ten is equal to: 0 Te ifferential coefficient of sin t t t (C) (C) 0 (D) wrt cos Differentiation of tan wrt tan is: tan tan (C) t t is: tan (D) ( tan ) Let f() be a polnomial in Ten te secon erivative of f(e ), is: f (e ) e + f (e ) f (e ) e + f (e ) e (C) f (e ) e (D) f (e ) e + f (e ) e f g Teko Classes, Mats : Suag R Karia (S R K Sir), Bopal Pone : , page 6 OF 8

17 FREE Downloa Stu Package from website: wwwtekoclassescom & wwwmatsbsuagcom Get Solution of Tese Packages & Learn b Vieo Tutorials on wwwmatsbsuagcom inepenent of but epenent on m epenent of but inepenent of m (C) epenent on bot m & (D) inepenent of m & 5 7 ten Limit f ( 5 t) f ( 5 t) t0 If f (5) t (C) 7 (D) Let e f() ln If g() is te inverse function of f() ten g () equals to: e e + (C) 7 If u a + b ten n n u [f(u)] n n [f(a + b)] is equal to: 8 If f & f () sin ten (C) sin sin n e e (D) e + ln a n u [f(u)] (C) an n u [f(u)] (D) an n [f(u)] Successful People Replace te wors like; "wis", "tr" & "soul" wit "I Will" Ineffective People on't n sin 9 If P(), is a polnomial of egree, ten equals: P () + P () P () P () (C) P () P () (D) a constant Part : Ma ave more tan one options correct 0 Two functions f & g ave first & secon erivatives at 0 & satisf te relations, f(0) g( 0) (0) g (0) g (0), g (0) 5 f (0) 6 f(0) ten: if () f ( ) g( ) (0) 5 if k() f() g() sin ten k (0) (C) Limit g ( ) 0 f ( ) f n () If f n () e for all n N an f o (), ten {fn ()} is equal to: f n () {fn ()} f n () f n () (C) f n () f n () f () f () of tese If f is twice ifferentiable suc tat f() f() an f() g() If () is a twice ifferentiable function suc tat () [f()] + [g()] If (0), (), ten te equation () represents: a curve of egree a curve passing troug te origin (C) a straigt line wit slope (D) a straigt line wit intercept equal to Given f() + sin 5 a sin a sin a 5 sin (a 8a + 7) ten: f() + sin6 sin sin8 f (sin 8) > 0 (C) f () is not efine at sin 8 (D) f (sin 8) < 0 If f() + f() + f() for all R ten f(0) + f() f() f(0) + f() 0 (C) f() + f() f() of tese 5 If f() (a + b) sin + (c + ) cos, ten te values of a, b, c an suc tat f() cos for all are a b 0 (C) c 0 (D) b c EXERCISE - If A e kt cos (p t + c) ten prove tat + k + n 0, were n p + k t t Evaluate te following limits using L ospitale rule as oterwise Limit log (tan ) 0 If f () tan (a) (b) (c) (a) (b) (c) ten f () (a) (b) (c) (a) (b) (c) Fin te value of n Teko Classes, Mats : Suag R Karia (S R K Sir), Bopal Pone : , page 7 OF 8

18 FREE Downloa Stu Package from website: wwwtekoclassescom & wwwmatsbsuagcom Get Solution of Tese Packages & Learn b Vieo Tutorials on wwwmatsbsuagcom If a t & b t, were t is a parameter, ten prove tat sina 5 If sin sin (a + ), sow tat cosa 8b 7 7a t 6 F" f" g" c F f g If F() f() g() & f () g () c, prove tat & F f g fg F f g 7 If be a repeate root of a quaratic equation f() 0 & A(), B(), C() be te polnomials of egree, & 5 A() B() C() A( ) respectivel, ten sow tat A'( ) 8 Sow tat R / B( ) B'( ) C( ) is ivisible b f(), were as enotes te erivative C' ( ) can be reuce to te form R / Also sow tat, if a sin ( + cos ) & a cos ( cos ) ten te value of R equals to a cos 9 Differentiate te following functions wit respect to (i) n e (i) Eercise # sin cos sin cos (iii) tan tan cos cos A C A D 5 D 6 B 7 D 8 B 9 C 0 A C D B D 5 C 6 C 7 C 8 B 9 C 0 ABC AC CD AD ABC 5 ABC Eercise # 9 (i) e ( n + + n ) (ii) sincos ( ) (iii) sec For 9 Years Que from IIT-JEE(Avance) & 5 Years Que from AIEEE (JEE Main) we istribute a book in class room Teko Classes, Mats : Suag R Karia (S R K Sir), Bopal Pone : , page 8 OF 8 Successful People Replace te wors like; "wis", "tr" & "soul" wit "I Will" Ineffective People on't