STUDY PACKAGE. Subject : Mathematics ENJOY MATHEMATICS WITH SUHAAG SIR. Student s Name : Class Roll No.

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1 fo/u fopkjr Hk# tu] uga vkjehks dke] foifr ns[k NksM+s rqjar e/;e eu dj ';kea iq#"k flag ladyi dj] lgrs foifr vusd] ^cuk^ u NksM+s /;s; dks] j?kqcj jk[ks VsdAA jfpr% ekuo /kez izksrk ln~q# J jknksm+nklt egkjkt STUDY PACKAGE Subject : Matematics Topic: Differentiation Inde Teor Question Bank Eercise Que from Compt Eams 5 8 Yrs Que from AIEEE 6 Yrs Que from IIT-JEE R ENJOY MATHEMATICS WITH SUHAAG SIR Student s Name : Class Roll No : : Head Office: -B, III- Floor, Near Hotel Arc Manor, Zone-I MP NAGAR, Main Road, Bopal :(0755) , Free Stu Package download from website : wwwiitjeeiitjeecom, wwwtekoclassescom

2 FREE Download Stu Package Views of students available at website: wwwiitjeeiitjeecom, wwwtekoclassescom Differentiation A First Principle Of Differentiation Te derivative of a given function f at a point a on its domain is defined as: Limit f(a + ) f(a) 0, provided te limit eists & is denoted b f (a) ie f (a) Limit f() f(a) a, provided te limit eists a If and + belong to te domain of a function f defined b f(), ten Limit f( + ) f() 0 if it eists, is called te Derivative of f at & is denoted b f () or ie, f () Limit d 0 f( + ) f() Tis metod of differentiation is also called ab-initio metod or first principle Solved Eample # Find derivative of following functions b first principle (i) f() (ii) f() tan (iii) f() e sin Solution (i) f () lim 0 (ii) f () lim 0 ( + ) lim (iii) f () lim 0 tan( + ) tan tan( + )[ + tan tan( + )] 0 sin ( + ) sin 0 e lim e [ ] sin ( + ) sin e esin sin( + ) sin sin( + ) sin e sin lim 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 cot Basic Teorems cosec d d (f ± g) f () ± g () d d (k f()) k d d f() d d (f() g()) f() g () + g() f () d f() g() f () f()g () d g() g () 5 d d (f(g())) f (g()) g () Tis rule is also called te cain rule of differentiation and can be written as 0 tan ( + tan ) sec d dz dz d d Note tat an important inference obtained from te cain rule is tat d TEKO CLASSES, MATHS BY SUHAAG SIR PH: (0755) , of 8 d d /

3 FREE Download Stu Package Views of students available at website: wwwiitjeeiitjeecom, wwwtekoclassescom anoter wa of epressing te same concept is b considering f() and g() as inverse functions of eac oter d f () and d g () g () f () Solved Eample # Find te differential of te following functions wit respect to (i) f() e sin (ii) f() sin( + ) (iii) f() (iv) f() sin + Solution (i) f() e sin f () e sin d d (sin ) e sin cos (ii) f() sin ( + ) d cos( + ) (sin ( + )) sin ( + ) d sin ( + ) (iii) f() + f () (+ ) () (+ ) (+ ) (iv) f() sin f () cos + sin Solved Eample # If f() sin ( + tan) ten find value of f (0) Solution f () cos ( + tan) ( + sec ) f (0) Self Practice Problems : Find te derviative 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 lim f(5 + t) f(5 t) t 0 t Differentiate te following functions Ans 7 (i) ( + ) ( ) (ii) ( ) ( )( ) (iii) + (iv) + (v) cos sin (vi) e sin (vii) sin + cos (viii) n (sin cos ) Ans (i)6 (5 + ) + + (ii) (iii) (iv) / / ( ) ( ) + ( + ) ( ) (v) cos cos sin (vi) e ((sin + cos ) + sin ) (vii) cos + sin sec (viii) sin cos B Derivative Of Inverse Trigonometric Functions sin π π sin d cos d cos < < sin d Note ere tat cos sin, rater cos ± π π sin but for values of,, cos is alwas positive and ence te result similarl let us find derivative of oter inverse trigonometric functions Let tan tan d d sec + tan + d ( R) + π Also if sec [0, π] sec d sec tan d tan d ± sec TEKO CLASSES, MATHS BY SUHAAG SIR PH: (0755) , of 8

4 FREE Download Stu Package Views of students available at website: wwwiitjeeiitjeecom, wwwtekoclassescom sec > d (, ) (, ) d sec < results for te derivative of inverse trigonometric functions can be summarized as : f() f () sin ; < cos ; < tan + ; R cot + ; R sec ; > cosec - ; > Solved Eample # If f() n (sin ) find f () Solution f () (sin ) ( ) (sin ) Solved Eample # 5 If f() sec cosec () ten find f ( ) Solution f () sec () + f ( ) sec ( ) + π 5 + f ( ) + C Metods Of Differentiation Logritmic Differentiation Te process of taking logaritm of te function first and ten differentiate is called Logaritmic Differentiation It is useful if (i) a function is te product or quotient of a number of functions OR (ii) a function is of te form [f()] g() were f & g are bot derivable, Solved Eample # 6 If find d Solution n n Solved Eample # 7 If (sin ) n, find d Solution n n n (sin ) n (sin ) + n d + n ( + n ) d d cos sin / ( ) Solved Eample # 8 If / / 5 ( ) ( ) / d (sin ) n find d n ( sin ) + cot n Solution n n + n ( ) n ( ) n ( ) d ( ) ( ) + 5 ( ) d ( ) ( ) 5 ( ) Implicit differentiation If f(, ) 0, is an implicit function ten in order to find /d, we differentiate eac term wrt regarding as a functions of & ten collect terms in /d Solved Eample # 9 If + find d Solution Differentiation bot sides wrt, we get + + d d d Note tat above result olds onl for points were 0 TEKO CLASSES, MATHS BY SUHAAG SIR PH: (0755) , of 8

5 FREE Download Stu Package Views of students available at website: wwwiitjeeiitjeecom, wwwtekoclassescom Solved Eample # 0 If e, ten find d Solution Taking log on bot sides n ( ) (i) differentiating wrt, we get + ln d d Solved Eample # If + ten find d du dv Solution u + v + 0 d d were u & v n u n & n v n u du + n & d d v d + n dv n + d d d (+ n ) du dv + n & d d d n + d + n + d n + n + d 0 d n + Self Practice Problems Differentiate te following functions : (i) sec ( ) (ii) + tan (iii) + (iv) e (v) (ln ) + () sin Find if d (i) cos ( + ) (ii) / + / a / (iii) n ( ) n If e, ten prove tat d ( + n ) a If log d + Ans (i) (ii) (iii) n (iv) e ( n + ) (v) sin n ( n) + n ( n ) + sin + cos n / sin( + ) ( ) (i) + sin( + ) (ii) (iii) ( + ) Differentiation using substitution Following substitutions are normall used to sumplif tese epression (i) + a a tan θ or a cot θ (ii) a a sin θ or a cos θ (iii) a a sec θ or a cosec θ (iv) + a a a cos θ Solved Eample # : Differentiate tan + Solution Let tan θ θ tan ; π π θ, sec θ tan π π [ secθ secθ θ, ] tanθ TEKO CLASSES, MATHS BY SUHAAG SIR PH: (0755) , of 8

6 FREE Download Stu Package Views of students available at website: wwwiitjeeiitjeecom, wwwtekoclassescom cos θ θ tan tan sinθ tan θ π π [tan (tan) for, ] tan d ( + ) Solved Eample # : Find were tan d Solution cosθ θ cos () ; θ [0, π] tan tan + cosθ + cosθ + θ tan θ + tan cosθ cosθ π cos Note tat + cosθ θ cos but for θ tan π 0,, π θ d θ cos θ cos θ cos + cos θ Also tan π π (tan ) for, Solved Eample # If f() sin ten find + (i) f () (ii) f (iii) f () Solution tan θ π π θ tan () ; < θ < sin (sin θ) θ sin θ sin π π θ < θ < π > π tan > + π π θ θ f() tan < < f () + ( π + tan ) < π < ( π + θ) π < θ < + 8 (i) f () (ii) f (iii) f ( 5 + ) & f ( ) + 5 f () does not eist Aliter Above problem can also be solved witout an substution also, but in a little tedious wa f() sin + f () f () (+ ( + ( ( + ) ) ) ) (+ ) {(+ ) } ( ( (+ ) ) ) tus f () + + Solved Eample # 5 If + a( ), ten prove tat d Solution Put sin α α sin () sin β β sin () cosα + cos β a (sinα sinβ) < > TEKO CLASSES, MATHS BY SUHAAG SIR PH: (0755) , of 8

7 FREE Download Stu Package Views of students available at website: wwwiitjeeiitjeecom, wwwtekoclassescom Aliter α + β α β α + β α β cos cos a cos sin α β cot (a) sin sin cot (a) differentiating wrt to 0 d d Using implicit differention a d d a d a + d ( ) + ( )( ) + d ( )( ) + ( ) + d Hence proved + + a + d a ( )( ) ( )( ) Parametric Differentiation If f(θ) & g(θ) were θ is a parameter, ten Solved Eample # 6 If a cos t and a sin t Find d / dt a sin t cos t tan t d d / dt acos t sint π Solved Eample # 7 If a cos t and a (t sint) find te value of at t d asint) d a( cos t) d π t 5 Derivative of one function wit respect to anoter Let f(); z g() ten / d f'( ) dz dz / d g'( ) Solved Eample # 8 Find derivative of n wit respect to z e / d dz dz / d e Self Practice Problems : Find wen d (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 d + a If tan ten prove tat d ( ) + α β cot a /dθ d d/dθ TEKO CLASSES, MATHS BY SUHAAG SIR PH: (0755) , of 8

8 FREE Download Stu Package Views of students available at website: wwwiitjeeiitjeecom, wwwtekoclassescom du u If u sin (m cos ) and v cos (m sin ) ten prove tat dv v D Derivatives of Higer Order Let a function f() be defined on an open interval (a, b) It s derivative, if it eists on (a, b) is a certain function f () [or (/d) or ] & is called te first derivative of w r t If it appens tat te first derivative as a derivative on (a, b) ten tis derivative is called te second derivative of w r t & is denoted b f () or (d /d ) or d d Similarl, te rd order derivative of w r t, if it eists, is defined b d d It is also denoted b f () d d or Solved Eample # 9 If n ten and Solution n + n + 6 n n n + Solved Eample # 0 If ten find () Solution n n wen ( + n ) ( + n ) (i) again diff wrt to, ( + n ) ( + ln ) (using (i)) () 0 It must be carefull noted tat in case of parametric functions / dt d d / dt d d / dt altoug but d d / dt rater d d / dt d d d / dt wic on appling cain rule can be resolved as d d d d d / dt dt d dt dt dt dt dt d dt d / dt d d d d dt d d d d dt dt dt dt d d dt Solved Eample # d If t + and t + t ten find d Solution d t + t ; dt dt d d dt t + t d (t + t ) d dt d d d + 6t d Solved Eample # If cos t cos t and sin t sin t ten find value of d π at t Solution d cos t cos t sin t sin t dt dt t t sin sin cos t cos t d sint sint t t cos sin d t tan d d d d t d tan d d dt t tan dt d TEKO CLASSES, MATHS BY SUHAAG SIR PH: (0755) , of 8

9 FREE Download Stu Package Views of students available at website: wwwiitjeeiitjeecom, wwwtekoclassescom t sec d d d (sint sint) d π t Solved Eample # Find second order derivative of sin wit respect to z e Solution / d cos dz dz / d e d d cos d d cos d dz dz e d d e dz e sin cos e (e ) e d (sin + cos) dz e Solved Eample # : f() and g() are inverse functions of eac oter tan epress g () and g () in terms of derivative of f() Solution d f () and d g () g () f () (i) again differentiating wrt to d g () f () d d d f () f () g () f () f () g () f () (ii) wic can also be remembered as d d d d Solved Eample # 5 sin (sin) ten prove tat + (tan) + cos 0 Solution Suc epression can be easil proved using implict differention cos (sin ) cos sec cos (sin ) again differentiating wrt, we can get sec + sec tan sin (sin ) cos tan cos +(tan) + cos 0 Self Practice Problems : n d n If ten find Ans d Prove tat + tan satisfies te differentiation equation d cos d + 0 d sec θ If a (cosθ + θ sin θ) and a(sin θ θ cosθ) ten find Ans d aθ sin cos Find second derivative of n wit respect to sin Ans cos 5 if e (A cos + B sin ), prove tat d d d d Solved Eample # 6 If (tan ) ten prove tat ( + ) d Solution + ( + ) d tan d + ( + ) tan () d d ( + ) + d d ( + ) d ( + ) d + ( + ) d TEKO CLASSES, MATHS BY SUHAAG SIR PH: (0755) , of 8

10 FREE Download Stu Package Views of students available at website: wwwiitjeeiitjeecom, wwwtekoclassescom 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 differentiable 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 differentiable at a & (iv) Bot f () & g () are continuous at a, Ten f() f'() f"() Limit a g() Limit a g'() Limit a g"() & so on till indeterminant form vanises 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 () [g()] 5 (C) (D) none [()] g + [()] g n e Q If tan ne + tan + n ten 6 n d (C) 0 (D) Q If f + & f () tan ten d tan tan + tan (C) f ( 5 + 6) tan (D) none 5tan + 6 Q If sin + & d + p, ten p ( ) 0 sin (C) sin (D) none of tese Q5 If f & f () sin ten + d + ( + ) + sin + sin + + (C) sin + + (D) none + ( ) ( ) Q6 Let g is te inverse function of f & f () ( + ) 5 + a 0 0 a Q7 If sin () + cos () 0 ten d Q8 If sin ten + d is : 0 (C) ( ) If g() a ten g () is equal to 0 a + a (C) (D) 0 + a a (D) TEKO CLASSES, MATHS BY SUHAAG SIR PH: (0755) , of 8

11 FREE Download Stu Package Views of students available at website: wwwiitjeeiitjeecom, wwwtekoclassescom 5 5 (C) 5 (D) none Q9 Te derivative of sec wrt at is : / (C) (D) none Q0 If d P(), is a polnomial of degree, ten d equals : d Q P () + P () P () P () (C) P () P () (D) a constant Let f() be a quadratic 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 d is : independent of p but dependent on q dependent on p but independent of q (C) dependent on bot p & q (D) independent of p & q bot g ( )cos if Q Let f() 0 were g() is an even function differentiable at 0, passing troug te 0 if 0 origin Ten f (0) : is equal to is equal to 0 (C) is equal to (D) does not eist Q If n m p m + m n p n + m p n p ten at emnp is equal to: d e mnp e mn/p (C) e np/m (D) none Q5 log cos sin Lim 0 as te value equal to log cos sin (C) (D) none of tese Q6 If f is differentiable in (0, 6) & f () 5 ten Limit f( ) fc 5 5/ (C) 0 (D) 0 Q7 Let l Lim 0 + m (ln ) n were m, n N ten : l is independent of m and n l is independent of m and depends on m (C) l is independent of n and dependent on m (D) l is dependent on bot m and n Q8 cos Let f() sin Ten Limit f () 0 tan (C) (D) Q9 cos sin cos π Let f() cos sin cos ten f cos sin cos Q0 0 (C) (D) People living at Mars, instead of te usual definition of derivative D f(), define a new kind of derivative, 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 (D) none Q If f() g() ; f () 9 ; g () 6 ten Limit f( ) g( ) is equal to : (C) 0 (D) none Q If f() is a differentiable function of ten Limit f ( + ) f ( ) 0 f () 5f () (C) 0 (D) none TEKO CLASSES, MATHS BY SUHAAG SIR PH: (0755) , of 8

12 FREE Download Stu Package Views of students available at website: wwwiitjeeiitjeecom, wwwtekoclassescom Q If + e ten d is : e ( + e ) e e (C) ( + e ) (D) ( + e ) Q If + ten te value of at te point (, ) is : d 5 (C) (D) none 8 Q5 If f(a), f (a), g(a), g (a) ten te value of Limit g ( ) fa ( ) ga ( ) f ( ) a is: a 5 /5 (C) 5 (D) none Q6 If f is twice differentiable suc tat f ( ) f( ), f ( ) g( ) ( ) [ f( ) ] + [ g( ) ] and ( 0), ( ) ten te equation () represents : a curve of degree a curve passing troug te origin (C) a straigt line wit slope (D) a straigt line wit intercept equal to Q7 Te derivative of te function, f()cos - ( cos sin ) UVW + sin ( cos + sin ) UVW wrt RST + at is : 5 (C) 0 (D) 0 Q8 Let f() be a polnomial in Ten te second derivative 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 RST 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 : + d (C) 0 (D) Q If ten a + b + a + b + a + b + d a b a b (C) (D) ab + a ab + b ab + b ab + a Q Let f () be a polnomial function of second degree If f () f ( ) and a, b, c are in AP, ten f '(a), f '(b) and f '(c) are in GP HP (C) AGP (D) AP Q If sin m ten te value of 5 (were subscripts of sows te order of derivatiive) is: independent of but dependent on m dependent of but independent of m (C) dependent on bot m & (D) independent 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 differentiable functions suc tat g (a) & g(a) b and if fog is an identit 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 defined at sin 8 f (sin 8) > 0 TEKO CLASSES, MATHS BY SUHAAG SIR PH: (0755) , of 8

13 FREE Download Stu Package Views of students available at website: wwwiitjeeiitjeecom, wwwtekoclassescom Q7 (C) f () is not defined at sin 8 (D) f (sin 8) < 0 A function f, defined for all positive real numbers, satisfies te equation f( ) for ever > 0 Ten te value of f () (C) / (D) cannot be determined Q8 Given : f() 6 cos a + sin a sin 6a + n( a a ) ten : f() is not defined at / f (/) < 0 (C) f () is not defined at / (D) f (/) > 0 Q9 If (A + B) e m + (m ) e ten m d d + m is equal to : e e m (C) e m (D) e ( m) Q0 Suppose f () e a + e b, were a b, and tat f '' () f ' () 5 f () 0 for all Ten te product ab is equal to 5 9 (C) 5 (D) 9 Q Let () be differentiable for all and let f () (k + e ) () were k is some constant If (0) 5, ' (0) and 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 defines as a differentiable function of Te value of for d and is 5 9 (C) (D) 5 5 Q Let f() ( ) ( ) ten : and g() f () and g () f () and g () (C) f () and g () 0 (D) f () and g () Q5 Te function f() e +, being differentiable and one to one, as a differentiable inverse f () Te value of d d (f ) at te point f(l n) is n (C) (D) none logsin cos Q6 If f () for < π 0 logsin cos for 0 ten, te number of points of discontinuit of f in π π, is 0 (C) (D) Q7 If ( a ) a ( b ) b ten werever it is defined is equal to : a + b d + ( a+ b) ( a+ b) ( a+ b) + ( a+ b) (C) (D) ( a ) ( b) ( a ) ( b) ( a ) ( b) ( a ) ( b) Q8 If is a function of ten d + 0 If is a function of ten te equation becomes : d d d + d 0 d d d d + 0 (C) 0 (D) d d 0 Q9 A function f () satisfies te condition, f () f () + f () + f () + were f () is a differentiable function indefinitel and das denotes te order of derivative If f (0), ten f () is : e / e (C) e (D) e cos6+ 6cos+ 5cos+ 0 Q50 If, ten cos5+ 5cos+ 0cos d sin + cos sin (C) cos (D) sin d Q5 If + d K ten te value of K is equal to d (C) (D) 0 TEKO CLASSES, MATHS BY SUHAAG SIR PH: (0755) , of 8

14 FREE Download Stu Package Views of students available at website: wwwiitjeeiitjeecom, wwwtekoclassescom were 0, ten f ' () as te value equal to ( ) zero (C) ( ) (D) π Q5 If f() sin + sin ( ( )) 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 m+ n m n+ m n m n n m Q5 Diffrential coefficient of wrt is 0 (C) (D) b mn + g f( ) f( ) Q55 Let f () be diffrentiable at ten Lim is equal to f() + f '() f() + f '() (C) f() + f '() (D) f() f '() Q56 If at + bt + c and t a + b + c, ten d equals a (at + b) a (a + b) (C) a (at + b) (D) a (a + b) Q57 Limit 0 + aarc tan barctan a b as te value equal to a b 0 (C) ( a b ) (D) a b 6a b a b Q58 Let f () be defined for all > 0 & be continuous Let f() satisf f f () f () for all, & f(e) Ten : f() is bounded f 0 as 0 (C) f() as 0 (D) f() ln Q59 Suppose te function f () f () as te derivative 5 at and derivative 7 at Te derivative of te function f () f () at, as te value equal to 9 9 (C) 7 (D) + Q60 If and a + b ten te value of a + b is equal to + + d 5π 5π 5π 5π cot cot (C) tan (D) tan 8 8 Q6 Suppose tat () f () g() and F() f ( g() ), were f () ; g() 5 ; g'() ; f '() and f '(5), ten F'() '() F'() '() (C) F'() '() (D) none Q6 Let f () wic one of te properties of te derivative enables ou to conclude tat f () as an inverse? f ' () is a polnomial of even degree f ' () is self inverse (C) domain of f ' () is te range of f ' () (D) f ' () is alwas positive Q6 Wic one of te following statements is NOT CORRECT? Te derivative of a diffrentiable periodic function is a periodic function wit te same period If f () and g () bot are defined on te entire number line and are aperiodic ten te function F() f () g () can not be periodic (C) Derivative of an even differentiable function is an odd function and derivative of an odd differentiable function is an even function (D) Ever function f () can be represented as te sum of an even and an odd function Select te correct alternatives : (More tan one are correct) Q6 If tan tan tan ten d as te value equal to : TEKO CLASSES, MATHS BY SUHAAG SIR PH: (0755) , of 8

15 FREE Download Stu Package Views of students available at website: wwwiitjeeiitjeecom, wwwtekoclassescom 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 d equals e e e e (C) (D) + Q66 If ten d ln ( ln + ) (C) ( ln + ) (D) ln e Q67 Let ten d (C) (D) Q68 If + + ten as te value equal to : d ( ) (C) (D) ( ) Q69 Te functions u e sin ; v e cos satisf te equation : v du d u dv d u + v du d v (C) dv u d (D) none of tese Q70 Let f () ten : Q7 f (0) f (/) (C) domain of f () is Two functions f & g ave first & second derivatives at 0 & satisf te relations, (D) none f(0) g( 0) (0) g (0) g (0), g (0) 5 f (0) 6 f(0) ten : if () f ( ) g ( ) ten (0) 5 if k() f() g() sin ten k (0) (C) Limit g ( ) 0 f ( ) (D) none n( n) ( n) Q7 If, ten is equal to : d ( n n + n n ( n) ) (ln )ln (ln ) ( ln (ln ) + ) (C) ((ln ) n + ln (ln )) (D) ( ln (ln ) + ) n 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, MATHS BY SUHAAG SIR PH: (0755) , of 8

16 FREE Download Stu Package Views of students available at website: wwwiitjeeiitjeecom, wwwtekoclassescom EXERCISE - Part : Onl one correct option If f () and f( ) ten at is d (C) (D) If ten d n ( n + ) (C) ( n + ) + (D) + n e tan sin If f() e, ten f (0) If a + b+ a + a ab + a ten d b ab + b (C) (D) (C) a ab + b 5 Let f() sin ; g() & () log e & F() [g(f())] ten d F is equal to: d cosec cot ( ) cosec ( ) (C) cot (D) cosec 6 If ( + ) ( + ) ( + ) ( + n ), ten at 0 is d (C) 0 (D) n 7 If sin + and d ( ) + p, ten p (D) b ab + a 0 (C) sin (D) 8 If + e t were t sin ten + d : + (C) + (D) If sin + sec +, > ten is equal to: + d (C) 0 (D) 0 Te differential coefficient of sin + t wrt cos + t t is: t (C) + t (D) none Differentiation of tan wrt tan is: + tan + tan (C) ( + tan ) (D) ( + tan ) Let f() be a polnomial in Ten te second derivative 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 f g If f(), g(), () are polnomials in of degree and F(), ten F () is equal to f g 0 (C) (D) f() g() () If sin m ten te value of (were settings of sows te order of derivative) is: TEKO CLASSES, MATHS BY SUHAAG SIR PH: (0755) , of 8

17 FREE Download Stu Package Views of students available at website: wwwiitjeeiitjeecom, wwwtekoclassescom independent of but dependent on m (C) dependent on bot m & dependent of but independent of m (D) independent of m & 5 7 tenlimit f( 5 + t) f( 5 t) t 0 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) e (D) e + ln 7 If u a + b ten d d n d n du [f(u)] n n [f(a + b)] is equal to: 8 If f & f () sin ten + d + + sin + ( ) + (C) + ( ) sin + n e + a d n du [f(u)] (C) d an n du [f(u)] (D) d a n n d [f(u)] + + (D) none n ( ) ( ) sin + 9 If P(), is a polnomial of degree, ten d d equals: d 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 & second derivatives 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 ( ) ten (0) 5 if k() f() g() sin ten k (0) (C) Limit g ( ) 0 f ( ) (D) none f n () If f n () e d for all n N and f o (), ten {fn ()} is equal to: d d f n () {fn d ()} f n () f n () (C) f n () f n () f () f () (D) none of tese If f is twice differentiable suc tat f () f() and f () g() If () is a twice differentiable function suc tat () [f()] + [g()] If (0), (), ten te equation () represents: a curve of degree 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 defined 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() (D) none of tese 5 If f() (a + b) sin + (c + d) cos, ten te values of a, b, c and d suc tat f () cos for all are a d b 0 (C) c 0 (D) b c EXERCISE - If A e d d kt cos (p t + c) ten prove tat + k + n 0, were n p + k dt dt 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 () λ ( a) ( b) ( c) ( a) ( b) ( c) Find te value of λ n TEKO CLASSES, MATHS BY SUHAAG SIR PH: (0755) , of 8

18 FREE Download Stu Package Views of students available at website: wwwiitjeeiitjeecom, wwwtekoclassescom d If a t & b t, were t is a parameter, ten prove tat d 8b 7 7a t 5 d sina If sin sin (a + ), sow tat d cosa+ 6 If F() f() g() & f () g () c, prove tat " f" g" c F f g + + & + F f g fg F f g 7 If α be a repeated root of a quadratic equation f() 0 & A(), B(), C() be te polnomials of degree, & 5 A() B() C() A( α) respectivel, ten sow tat A'( α) + d 8 Sow tat R d / B( α) B'( α) C( α) is divisible b f(), were das denotes te derivative C'( α) can be reduced to te form R / + d d d d Also sow tat, if a sin θ ( + cos θ) & a cos θ ( cos θ) ten te value of R equals to a cos θ 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) sin+ cos ( ) (iii) sec for Yrs Que of IIT-JEE & 8 Yrs Que of AIEEE we ave distributed alrea a book TEKO CLASSES, MATHS BY SUHAAG SIR PH: (0755) , of 8