TANGENT & NORMAL EXERCISE I Q. Find the equtions of the tngents drwn to the curve y 4y + 8 = 0 from the point (, ). Q. Find the point of intersection of the tngents drwn to the curve y = y t the points where it is intersected by the curve y = y. Q. Find ll the lines tht pss through the point (, ) nd re tngent to the curve represented prmetriclly s = t t nd y = t + t. Q.4 In the curve y b = K +b, prove tht the portion of the tngent intercepted between the coordinte es is divided t its point of contct into segments which re in constnt rtio. (All the constnts being positive). Q.5 A stright line is drwn through the origin nd prllel to the tngent to curve y = ln y y t n rbitry point M. Show tht the locus of the point P of intersection of the stright line through the origin & the stright line prllel to the -is & pssing through the point M is + y =. Q.6 Prove tht the segment of the tngent to the curve y = ln the y-is & the point of tngency hs constnt length. Q.7 A function is defined prmetriclly by the equtions f(t) = = t t sin if t 0 t 0 if t 0 nd g(t) = y = sin t t o contined between if t 0 if t 0 Find the eqution of the tngent nd norml t the point for t = 0 if eist. Q.8 Find ll the tngents to the curve y = cos ( + y),, tht re prllel to the line + y = 0. Q.9 () Find the vlue of n so tht the subnorml t ny point on the curve y n = n + my be constnt. (b) Show tht in the curve y =. ln (² ²), sum of the length of tngent & subtngent vries s the product of the coordintes of the point of contct. Q.0 Prove tht the segment of the norml to the curve = sin t + sin t cos t ; y = cos t contined between the co-ordinte es is equl to. Q. Show tht the normls to the curve = (cos t + t sin t) ; y = (sin t t cos t) re tngent lines to the circle + y =. Q. The chord of the prbol y = + 5 4 touches the curve y = t the point = nd is bisected by tht point. Find ''. Q. If the tngent t the point (, y ) to the curve + y = ( 0) meets the curve gin in (, y ) then y show tht =. y http://kj59.wordpress.com
Q.4 Determine differentible function y = f () which stisfies f ' () = [f()] nd f (0) =. Find lso the eqution of the tngent t the point where the curve crosses the y-is. Q.5 If p & p be the lengths of the perpendiculrs from the origin on the tngent & norml respectively t ny point (, y) on curve, then show tht p sin y cos dy p cos ysin where tn =. If in the d bove cse, the curve be / + y / = / then show tht : 4 p + p =. Q.6 The curve y = + b + c + 5, touches the - is t P (, 0) & cuts the y-is t point Q where its grdient is. Find, b, c. Q.7 The tngent t vrible point P of the curve y = meets it gin t Q. Show tht the locus of the middle point of PQ is y = 9 + 8 8. Q.8 Show tht the distnce from the origin of the norml t ny point of the curve = e sin cos & y = e cos sin is twice the distnce of the tngent t the point from the origin. Q.9 Show tht the condition tht the curves / + y / = c / & ( / ) + (y /b ) = my touch if c = + b. Q.0 The grph of certin function f contins the point (0, ) nd hs the property tht for ech number 'p' the line tngent to y = f () t p, f (p) intersect the -is t p +. Find f (). Q. A curve is given by the equtions = t & y = t. A vrible pir of perpendiculr lines through the origin 'O' meet the curve t P & Q. Show tht the locus of the point of intersection of the tngents t P & Q is 4y =. Q.() Show tht the curves y K b K = & K b y K = intersect orthogonlly. (b) Find the condition tht the curves y = & b y b = my cut orthogonlly. Q. Show tht the ngle between the tngent t ny point 'A' of the curve ln ( + y ) = C tn y line joining A to the origin is independent of the position of A on the curve. nd the Q.4 For the curve / + y / = /, show tht perpendiculr from (0, 0) to the tngent t (, y) on the curve. z + p = where z = + i y & p is the length of the Q.5 A nd B re points of the prbol y =. The tngents t A nd B meet t C. The medin of the tringlebc from C hs length 'm' units. Find the re of the tringle in terms of 'm'. http://kj59.wordpress.com
EXERCISE II RATE MEASURE AND APPROXIMATIONS Q. Wter is being poured on to cylindricl vessel t the rte of m /min. If the vessel hs circulr bse of rdius m, find the rte t which the level of wter is rising in the vessel. Q. A mn.5 m tll wlks wy from lmp post 4.5 m high t the rte of 4 km/hr. (i) how fst is the frther end of the shdow moving on the pvement? (ii) how fst is his shdow lengthening? Q. A prticle moves long the curve 6 y = +. Find the points on the curve t which the y coordinte is chnging 8 times s fst s the coordinte. Q.4 An inverted cone hs depth of 0 cm & bse of rdius 5 cm. Wter is poured into it t the rte of.5 cm /min. Find the rte t which level of wter in the cone is rising, when the depth of wter is 4 cm. Q.5 A wter tnk hs the shpe of right circulr cone with its verte down. Its ltitude is 0 cm nd the rdius of the bse is 5 cm. Wter leks out of the bottom t constnt rte of cu. cm/sec. Wter is poured into the tnk t constnt rte of C cu. cm/sec. Compute C so tht the wter level will be rising t the rte of 4 cm/sec t the instnt when the wter is cm deep. Q.6 Snd is pouring from pipe t the rte of cc/sec. The flling snd forms cone on the ground in such wy tht the height of the cone is lwys /6th of the rdius of the bse. How fst is the height of the snd cone incresing when the height is 4 cm. Q.7 An open Cn of oil is ccidently dropped into lke ; ssume the oil spreds over the surfce s circulr disc of uniform thickness whose rdius increses stedily t the rte of 0 cm/sec. At the moment when the rdius is meter, the thickness of the oil slick is decresing t the rte of 4 mm/sec, how fst is it decresing when the rdius is meters. Q.8 Wter is dripping out from conicl funnel of semi verticl ngle /4, t the uniform rte of cm /sec through tiny hole t the verte t the bottom. When the slnt height of the wter is 4 cm, find the rte of decrese of the slnt height of the wter. Q.9 An ir force plne is scending verticlly t the rte of 00 km/h. If the rdius of the erth is R Km, how fst the re of the erth, visible from the plne incresing t min fter it strted scending. Tke visible re A = R h R h Where h is the height of the plne in kms bove the erth. Q.0 A vrible ABC in the y plne hs its orthocentre t verte 'B', fied verte 'A' t the origin nd the 7 third verte 'C' restricted to lie on the prbol y = +. The point B strts t the point (0, ) t time 6 t = 0 nd moves upwrd long the y is t constnt velocity of cm/sec. How fst is the re of the tringle incresing when t = 7 sec. Q. A circulr ink blot grows t the rte of cm per second. Find the rte t which the rdius is incresing 6 fter seconds. Use =. 7 http://kj59.wordpress.com
Q. Wter is flowing out t the rte of 6 m /min from reservoir shped like hemisphericl bowl of rdius R = m. The volume of wter in the hemisphericl bowl is given by V = y (R y) when the wter is y meter deep. Find () At wht rte is the wter level chnging when the wter is 8 m deep. (b) At wht rte is the rdius of the wter surfce chnging when the wter is 8 m deep. Q. If in tringle ABC, the side 'c' nd the ngle 'C' remin constnt, while the remining elements re d db chnged slightly, show tht = 0. cos A cos B Q.4 At time t > 0, the volume of sphere is incresing t rte proportionl to the reciprocl of its rdius. At t = 0, the rdius of the sphere is unit nd t t = 5 the rdius is units. () Find the rdius of the sphere s function of time t. (b) At wht time t will the volume of the sphere be 7 times its volume t t = 0. Q.5 Use differentils to pproimte the vlues of ; () 5. nd (b) 6. EXERCISE III Q. Find the cute ngles between the curves y = nd y = t their point of intersection. [ REE '98, 6 ] Q. Find the eqution of the stright line which is tngent t one point nd norml t nother point of the curve, = t, y = t. [ REE 000 (Mins) 5 out of 00 ] Q. If the norml to the curve, y = f () t the point (, 4) mkes n ngle 4 with the positive is. Then f () = (A) (B) 4 (C) 4 (D) [JEE 000 (Scr.) out of 5 ] Q.4 The point(s) on the curve y + = y where the tngent is verticl, is(re) (A) 4, (B), (C) (0, 0) (D) 4, [JEE 00 (Scr.), ] Q.5 Tngent to the curve y = + 6 t point P (, 7) touches the circle + y + 6 + y + c = 0 t point Q. Then the coordintes of Q re (A) ( 6, ) (B) ( 9, ) (C) ( 0, 5) (D) ( 6, 7) [JEE 005 (Scr.), ] http://kj59.wordpress.com
EXERCISE I (MONOTONOCITY) Q. Find the intervls of monotonocity for the following functions & represent your solution set on the number line. 4 () f() =. e (b) f() = e / (c) f() = e (d) f () = ln Also plot the grphs in ech cse. Q. Let f () =. Find ll rel vlues of stisfying the inequlity, f () f () > f ( 5) Q. Find the intervls of monotonocity of the function () f () = sin cos in [0, ] (b) g () = sin + cos in (0 ). Q.4 Show tht, 9 + 0 is positive for ll vlues of > 4. Q.5 Let f () = + + nd g() = m{f (t) : 0 t } Discuss the conti. & differentibility of g() in the intervl (0,).,0, Q.6 Find the set of ll vlues of the prmeter '' for which the function, f() = sin 8( + )sin + (4 + 8 4) increses for ll R nd hs no criticl points for ll R. Q.7 Find the gretest & the lest vlues of the following functions in the given intervl if they eist. () f () = sin ln in, (b) y = in (0, ) (c) y = 5 5 4 + 5 + in [, ] Q.8 Find the vlues of '' for which the function f() = sin sin sin + increses throughout the number line. e Q.9 Prove tht f () = 9 cos ( n t) 5 cos(ln t) 7 l dt is lwys n incresing function of, R Q.0 If f() = + ( - ) + + is monotonic incresing for every R then find the rnge of vlues of. Q. Find the set of vlues of '' for which the function, f() = 4 + 5 + 7 is incresing t every point of its domin. Q. Find the intervls in which the function f () = cos 4 + 0 cos + 6 cos, 0 ; is monotoniclly incresing or decresing. Q. Find the rnge of vlues of '' for which the function f () = + ( + ) + ( + ) + 5 is monotonic in R. Hence find the set of vlues of '' for which f () in invertible. Q.4 Find the vlue of > for which the function http://kj59.wordpress.com
F () = t l n dt is incresing nd decresing. t Q.5 Find ll the vlues of the prmeter '' for which the function ; f() = 8 sin 6 7 sin 5 increses & hs no criticl points for ll R. Q.6 If f () = e e + ( + ) monotoniclly increses for every R then find the rnge of vlues of. 9 Q.7 Construct the grph of the function f () = nd comment upon the following () Rnge of the function, (b) Intervls of monotonocity, (c) Point(s) where f is continuous but not diffrentible, (d) Point(s) where f fils to be continuous nd nture of discontinuity. (e) Grdient of the curve where f crosses the is of y. Q.8 Prove tht, > ln > 4( ) ln for >. Q.9 Prove tht tn + 6 ln sec + cos + 4 > 6 sec for,. Q.0 If ² + (b/) c for ll positive where > 0 & b > 0 then show tht 7b 4c. Q. If 0 < < prove tht y = ln (²/) + (/) is function such tht d y/d > 0. Deduce tht ln > ( /) (/). Q. Prove tht 0 <. sin (/) sin² < (/) ( ) for 0 < < /. Q. Show tht ² > ( + ) [ln( + )] > 0. Q.4 Find the set of vlues of for which the inequlity ln ( + ) > /( + ) is vlid. Q.5 If b >, find the minimum vlue of ( ) + ( b), R. EXERCISE II Q. Verify Rolles throrem for f() = ( ) m ( b) n on [, b] ; m, n being positive integer. Q. Let f : [, b] R be continuous on [, b] nd differentible on (, b). If f () < f (b), then show tht f ' (c) > 0 for some c (, b). Q. Let f () = 4 +, use Rolle's theorem to prove tht there eist c, 0< c < such tht f(c) = 0. Q.4 Using LMVT prove tht : () tn > in 0,, (b) sin < for > 0 Q.5 Prove tht if f is differentible on [, b] nd if f () = f (b) = 0 then for ny rel there is n (, b) such tht f () + f ' () = 0. Q.6 For wht vlue of, m nd b does the function f () = m b stisfy the hypothesis of the men vlue theorem for the intervl [0, ]. 0 0 Q.7 Suppose tht on the intervl [, 4] the function f is differentible, f ( ) = nd f ' () 5. Find the bounding functions of f on [, 4], using LMVT. http://kj59.wordpress.com
Q.8 Let f, g be differentible on R nd suppose tht f (0) = g (0) nd f ' () g ' () for ll 0. Show tht f () g () for ll 0. Q.9 Let f be continuous on [, b] nd differentible on (, b). If f () = nd f (b) = b, show tht there eist distinct c, c in (, b) such tht f ' (c ) + f '(c ) =. Q.0 Let f () nd g () be differentible functions such tht f ' () g () f () g ' () for ny rel. Show tht between ny two rel solutions of f () = 0, there is t lest one rel solution of g () = 0. Q. Let f defined on [0, ] be twice differentible function such tht, f " () for ll [0, ] If f (0) = f (), then show tht, f ' () < for ll [0, ] Q. f () nd g () re differentible functions for 0 such tht f (0) = 5, g (0) = 0, f () = 8, g () =. Show tht there eists number c stisfying 0 < c < nd f ' (c) = g' (c). Q. If f,, re continuous in [, b] nd derivble in ], b[ then show tht there is vlue of c lying between & b such tht, f() () () f(b) (b) (b) f (c) (c) = 0 (c) Q.4 Show tht ectly two rel vlues of stisfy the eqution = sin + cos. Q.6 Let, b, c be three rel number such tht < b < c, f () is continuous in [, c] nd differentible in (, c). Also f ' () is strictly incresing in (, c). Prove tht (c b) f () + (b ) f (c) > (c ) f (b) Q.7 Use the men vlue theorem to prove, < ln <, > Q.8 Use men vlue theorem to evlute, Lim Q.9 Using L.M.V.T. or otherwise prove tht difference of squre root of two consecutive nturl numbers greter thn N is less thn. N Q.0 Prove the inequlity e > ( + ) using LMVT for ll R 0 nd use it to determine which of the two numbers e nd e is greter. Q. If f () = & g () = sin tn (A) both f () & g () re incresing functions (C) f () is n incresing function EXERCISE III, where 0 <, then in this intervl :. (B) both f () & g () re decresing functions (D) g () is n incresing function [ JEE '97 (Scr), ] Q. Let + b = 4, where < nd let g () be differentible function. If dg d 0 b 0 > 0 for ll, prove tht g( ) d g( ) d increses s (b ) increses. [JEE 97, 5] Q.() Let h() = f() (f()) + (f()) for every rel number. Then : (A) h is incresing whenever f is incresing (B) h is incresing whenever f is decresing (C) h is decresing whenever f is decresing (D) nothing cn be sid in generl. http://kj59.wordpress.com
(b) f() =, for every rel number, then the minimum vlue of f : (A) does not eist becuse f is unbounded (B) is not ttined even though f is bounded (C) is equl to (D) is equl to. [ JEE '98, + ] Q.4() For ll (0, ) : (A) e < + (B) log e ( + ) < (C) sin > (D) log e > (b) Consider the following sttements S nd R : S : Both sin & cos re decresing functions in the intervl (/, ). R : If differentible function decreses in n intervl (, b), then its derivtive lso decreses in (, b). Which of the following is true? (A) both S nd R re wrong (B) both S nd R re correct, but R is not the correct eplntion for S (C) S is correct nd R is the correct eplntion for S (D) S is correct nd R is wrong. (c) Let f () = e ( ) ( ) d then f decreses in the intervl : (A) (, ) (B) (, ) (C) (, ) (D) (, + ) [JEE 000 (Scr.) ++ out of 5] Q.5() If f () = e ( ), then f() is (A) incresing on, (B) decresing on R (C) incresing on R (D) decresing on, (b) Let < p <. Show tht the eqution 4 p = 0 hs unique root in the intervl, nd identify it. [ JEE 00, + 5 ] Q.6 The length of longest intervl in which the function sin 4sin is incresing, is (A) (B) (C) (D) [JEE 00 (Screening), ] Q.7() Using the reltion ( cos) <, 0 or otherwise, prove tht sin (tn) >, (b) Let f : [0, 4] R be differentible function. (i) Show tht there eist, b [0, 4], (f (4)) (f (0)) = 8 f () f (b) (ii) Show tht there eist, with 0 < < < such tht 0,. 4 4 0 f(t) dt = ( f ( ) + f ( ) ) [JEE 00 (Mins), 4 + 4 out of 60] http://kj59.wordpress.com
ln, 0 Q.8() Let f () =. Rolle s theorem is pplicble to f for [0, ], if = 0, 0 (A) (B) (C) 0 (D) f ( ) f () (b) If f is strictly incresing function, then Lim is equl to 0 f () f (0) (A) 0 (B) (C) (D) [JEE 004 (Scr)] Q.9 If p () = 5 0 00 45 + 05, using Rolle's theorem, prove tht t lest one root of p() lies between (45 /00, 46). [JEE 004, out of 60] Q.0() If f () is twice differentible function nd given tht f() =, f() = 4, f() = 9, then (A) f '' () =, for (, ) (B) f '' () = f ' () =, for some (, ) (C) f '' () =, for (, ) (D) f '' () =, for some (, ) [JEE 005 (Scr), ] (b) f () is differentible function nd g () is double differentible function such tht f () nd f '() = g (). If f (0) +g (0) = 9. Prove tht there eists some c (, ) such tht g (c) g"(c)<0. [JEE 005 (Mins), 6] http://kj59.wordpress.com
MAXIMA - MINIMA EXERCISE I Q. A cubic f() vnishes t = & hs reltive minimum/mimum t = nd =. 4 If f ()d =, find the cubic f (). Q. Investigte for mim & minim for the function, f () = [ (t ) (t ) + (t ) (t ) ] dt Q. Find the mimum & minimum vlue for the function ; () y = + sin, 0 (b) y = cos cos 4, 0 Q.4 Suppose f() is rel vlued polynomil function of degree 6 stisfying the following conditions ; () f hs minimum vlue t = 0 nd (b) f hs mimum vlue t = (c) for ll, Limit 0 f() 0 ln 0 0 =. Determine f (). Q.5 Find the mimum perimeter of tringle on given bse nd hving the given verticl ngle. Q.6 The length of three sides of trpezium re equl, ech being 0 cms. Find the mimum re of such trpezium. Q.7 The pln view of swimming pool consists of semicircle of rdius r ttched to rectngle of length 'r' nd width 's'. If the surfce re A of the pool is fied, for wht vlue of 'r' nd 's' the perimeter 'P' of the pool is minimum. Q.8 For given curved surfce of right circulr cone when the volume is mimum, prove tht the semi verticl ngle is sin. Q.9 Of ll the lines tngent to the grph of the curve y = minimum nd mimum slope. 6, find the equtions of the tngent lines of Q.0 A sttue 4 metres high sits on column 5.6 metres high. How fr from the column must mn, whose eye level is.6 metres from the ground, stnd in order to hve the most fvourble view of sttue. Q. By the post office regultions, the combined length & girth of prcel must not eceed metre. Find the volume of the biggest cylindricl (right circulr) pcket tht cn be sent by the prcel post. Q. A running trck of 440 ft. is to be lid out enclosing footbll field, the shpe of which is rectngle with semi circle t ech end. If the re of the rectngulr portion is to be mimum, find the length of its sides.. http://kj59.wordpress.com
Q. A window of fied perimeter (including the bse of the rch) is in the form of rectngle surmounted by semicircle. The semicirculr portion is fitted with coloured glss while the rectngulr prt is fitted with clen glss. The cler glss trnsmits three times s much light per squre meter s the coloured glss does. Wht is the rtio of the sides of the rectngle so tht the window trnsmits the mimum light? Q.4 A closed rectngulr bo with squre bse is to be mde to contin 000 cubic feet. The cost of the mteril per squre foot for the bottom is 5 pise, for the top 5 pise nd for the sides 0 pise. The lbour chrges for mking the bo re Rs. /-. Find the dimensions of the bo when the cost is minimum. Q.5 Find the re of the lrgest rectngle with lower bse on the -is & upper vertices on the curve y =. Q.6 A trpezium ABCD is inscribed into semicircle of rdius l so tht the bse AD of the trpezium is dimeter nd the vertices B & C lie on the circumference. Find the bse ngle of the trpezium ABCD which hs the gretest perimeter. b Q.7 If y = ( ) ( 4) mimum. hs turning vlue t (, ) find & b nd show tht the turning vlue is Q.8 Prove tht mong ll tringles with given perimeter, the equilterl tringle hs the mimum re. Q.9 A sheet of poster hs its re 8 m². The mrgin t the top & bottom re 75 cms nd t the sides 50 cms. Wht re the dimensions of the poster if the re of the printed spce is mimum? Q.0 A perpendiculr is drwn from the centre to tngent to n ellipse + y =. Find the gretest vlue b of the intercept between the point of contct nd the foot of the perpendiculr. Q. Consider the function, F () = ( t t)dt, R. () (b) (c) (d) (e) Find the nd y intercept of F if they eist. Derivtives F ' () nd F '' (). The intervls on which F is n incresing nd the invervls on which F is decresing. Reltive mimum nd minimum points. Any inflection point. Q. A bem of rectngulr cross section must be swn from round log of dimeter d. Wht should the width nd height y of the cross section be for the bem to offer the gretest resistnce () to compression; (b) to bending. Assume tht the compressive strength of bem is proportionl to the re of the cross section nd the bending strength is proportionl to the product of the width of section by the squre of its height. Q. Wht re the dimensions of the rectngulr plot of the gretest re which cn be lid out within tringle of bse 6 ft. & ltitude ft? Assume tht one side of the rectngle lies on the bse of the tringle. Q.4 The flower bed is to be in the shpe of circulr sector of rdius r & centrl ngle. If the re is fied & perimeter is minimum, find r nd. Q.5 The circle + y = cuts the -is t P & Q. Another circle with centre t Q nd vrble rdius intersects the first circle t R bove the -is & the line segment PQ t S. Find the mimum re of the tringle QSR. http://kj59.wordpress.com
EXERCISE II Q. The mss of cell culture t time t is given by, M (t) = t 4e () Find Lim M(t) nd Lim M(t) t t (b) (c) Show tht dm = M( M ) dt Find the mimum rte of growth of M nd lso the vlue of t t which occurs. Q. Find the cosine of the ngle t the verte of n isosceles tringle hving the gretest re for the given constnt length l of the medin drwn to its lterl side. Q. From fied point A on the circumference of circle of rdius '', let the perpendiculr AY fll on the tngent t point P on the circle, prove tht the gretest re which the APY cn hve is sq. units. 8 Q.4 Given two points A (, 0) & B (0, 4) nd line y =. Find the co-ordintes of point M on this line so tht the perimeter of the AMB is lest. Q.5 A given quntity of metl is to be csted into hlf cylinder i.e. with rectngulr bse nd semicirculr ends. Show tht in order tht totl surfce re my be minimum, the rtio of the height of the cylinder to the dimeter of the semi circulr ends is /( + ). Q.6 Depending on the vlues of p R, find the vlue of '' for which the eqution + p + p = hs three distinct rel roots. Q.7 Show tht for ech > 0 the function e. ² hs mimum vlue sy F (), nd tht F () hs minimum vlue, e e/. Q.8 For > 0, find the minimum vlue of the integrl ( 4 )e d. Q.9 Find the mimum vlue of the integrl e d where. Q.0 Consider the function f () = () (b) http://kj59.wordpress.com ln 0 Find whether f is continuous t = 0 or not. Find the minim nd mim if they eist. (c) Does f ' (0)? Find Lim f '(). (d) 0 for Find the inflection points of the grph of y = f ().. 0 when 0 0 Q. Consider the function y = f () = ln ( + sin ) with. Find () the zeroes of f () (b) inflection points if ny on the grph (c) locl mim nd minim of f () (d) symptotes of the grph (e) sketch the grph of f () nd compute the vlue of the definite integrl f () d. 5
Q. A right circulr cone is to be circumscribed bout sphere of given rdius. Find the rtio of the ltitude of the cone to the rdius of the sphere, if the cone is of lest possible volume. Q. Find the point on the curve 4 ² + ²y² = 4 ², 4 < ² < 8 tht is frthest from the point (0, ). 5 Q.4 Find the set of vlue of m for the cubic + = log ) 4 ( m hs distinct solutions. Q.5 Let A (p, p), B (q, q), C (r, r) be the vertices of the tringle ABC. A prllelogrm AFDE is drwn with vertices D, E & F on the line segments BC, CA & AB respectively. Using clculus, show tht mimum re of such prllelogrm is : 4 (p + q) (q + r) (p r). Q.6 A cylinder is obtined by revolving rectngle bout the is, the bse of the rectngle lying on the is nd the entire rectngle lying in the region between the curve y = & the is. Find the mimum possible volume of the cylinder. Q.7 For wht vlues of does the function f () = + ( 7) + ( 9) hve positive point of mimum. Q.8 Among ll regulr tringulr prism with volume V, find the prism with the lest sum of lengths of ll edges. How long is the side of the bse of tht prism? Q.9 A segment of line with its etremities on AB nd AC bisects tringle ABC with sides, b, c into two equl res. Find the length of the shortest segment. Q.0 Wht is the rdius of the smllest circulr disk lrge enough to cover every cute isosceles tringle of given perimeter L? Q. Find the mgnitude of the verte ngle of n isosceles tringle of the given re A such tht the rdius r of the circle inscribed into the tringle is the mimum. Q. Prove tht the lest perimeter of n isosceles tringle in which circle of rdius r cn be inscribed is 6 r. Q. The function f () defined for ll rel numbers hs the following properties (i) f (0) = 0, f () = nd f ' () = k( )e for some constnt k > 0. Find () the intervls on which f is incresing nd decresing nd ny locl mimum or minimum vlues. (b) the intervls on which the grph f is concve down nd concve up. (c) the function f () nd plot its grph. Q.4 Find the minimum vlue of sin + cos + tn + cot + sec + cosec for ll rel. Q.5 Use clculus to prove the inequlity, sin in 0. You my use the inequlity to prove tht, cos in 0 http://kj59.wordpress.com
EXERCISE III Q. A conicl vessel is to be prepred out of circulr sheet of gold of unit rdius. How much sectoril re is to be removed from the sheet so tht the vessel hs mimum volume. [ REE '97, 6 ] Q.() The number of vlues of where the function f() = cos + cos ttins its mimum is : (A) 0 (B) (C) (D) infinite (b) Suppose f() is function stisfying the following conditions : (i) f(0) =, f() = (ii) f hs minimum vlue t = 5 nd b (iii) for ll f () = b b ( b) b b Where, b re some constnts. Determine the constnts, b & the function f(). [JEE '98, + 8] Q. Find the points on the curve + by + y = c ; c > b > > 0, whose distnce from the origin is minimum. [ REE '98, 6] Q.4 The function f() = t (e t ) (t ) (t ) (t ) 5 dt hs locl minimum t = (A) 0 (B) (C) (D) [ JEE '99 (Screening), ] Q.5 Find the co-ordintes of ll the points P on the ellipse ( / ) + (y /b ) = for which the re of the tringle PON is mimum, where O denotes the origin nd N the foot of the perpendiculr from O to the tngent t P. [JEE '99, 0 out of 00] Q.6 Find the normls to the ellipse ( /9) + (y /4) = which re frthest from its centre. [REE '99, 6] Q.7 Find the point on the stright line, y = + which is nerest to the circle, 6 ( + y ) + 8 y 50 = 0. [REE 000 Mins, out of 00] Q.8 Let f () = [ for 0. Then t = 0, ' f ' hs : for 0 (A) locl mimum (C) locl minimum (B) no locl mimum (D) no etremum. [ JEE 000 Screening, out of 5 ] Q.9 Find the re of the right ngled tringle of lest re tht cn be drwn so s to circumscribe rectngle of sides '' nd 'b', the right ngles of the tringle coinciding with one of the ngles of the rectngle. [ REE 00 Mins, 5 out of 00 ] Q.0() Let f() = ( + b ) + b + nd let m(b) be the minimum vlue of f(). As b vries, the rnge of m (b) is (A) [0, ] (B) 0, (C), (D) (0, ] (b) The mimum vlue of (cos ) (cos )... (cos n ), under the restrictions O <,,..., n < nd cot cot... cot n = is (A) (B) (C) (D) n/ n n [ JEE 00 Screening, + out of 5 ] http://kj59.wordpress.com
Q.() If,,..., n re positive rel numbers whose product is fied number e, the minimum vlue of + + +... + n + n is (A) n(e) /n (B) (n+)e /n (C) ne /n (D) (n+)(e) /n [ JEE 00 Screening] (b) A stright line L with negtive slope psses through the point (8,) nd cuts the positive coordintes es t points P nd Q. Find the bsolute minimum vlue of OP + OQ, s L vries, where O is the origin. [ JEE 00 Mins, 5 out of 60] Q.() Find point on the curve + y = 6 whose distnce from the line + y = 7, is minimum. [JEE-0, Mins- out of 60] (b) For circle + y = r, find the vlue of r for which the re enclosed by the tngents drwn from the point P(6, 8) to the circle nd the chord of contct is mimum. [JEE-0, Mins- out of 60] Q.() Let f () = + b + c + d, 0 < b < c. Then f (A) is bounded (B) hs locl mim (C) hs locl minim (D) is strictly incresing [JEE 004 (Scr.)] ( ) (b) Prove tht sin 0,. (Justify the inequlity, if ny used). [JEE 004, 4 out of 60] Q.4 If P() be polynomil of degree stisfying P( ) = 0, P() = 6 nd P() hs mimum t = nd P'() hs minim t =. Find the distnce between the locl mimum nd locl minimum of the curve. [JEE 005 (Mins), 4 out of 60] Q.5() If f () is cubic polynomil which hs locl mimum t =. If f () = 8, f () = nd f '() hs locl mim t = 0, then (A) the distnce between (, ) nd (, f ()), where = is the point of locl minim is 5. (B) f () is incresing for [, 5 ] (C) f () hs locl minim t = (D) the vlue of f(0) = 5 e (b) f () = e e 0 nd g () = 0 t (A) locl mim t = + ln nd locl minim t = e (B) locl mim t = nd locl minim t = (C) no locl mim (D) no locl minim f dt, [, ] then g() hs [JEE 006, 5mrks ech] (c) If f () is twice differentible function such tht f () = 0, f (b) =, f (c) =, f (d) =, f (e) = 0, where < b < c < d < e, then find the minimum number of zeros of g() f '() f ().f"() intervl [, e]. [JEE 006, 6] in the http://kj59.wordpress.com
ANSWER KEY TANGENT & NORMAL EXERCISE I Q. y = or + y = Q. = when t =, m ; 5 4y = if t, m = /] Q. (0, ) Q.7 T : y = 0 ; N : + y = 0 Q.8 + y = / & + y = / Q.9 () n = Q. = Q.4 ; 4y = Q.6 = / ; b = /4 ; c = Q.0 e / Q. (b) b = b Q. = tn C EXERCISE II Q.5 Q. /9 m/min Q. (i) 6 km/h (ii) km/hr Q. (4, ) & ( 4, /) Q.4 /8 cm/min Q.5 + 6 cu. cm/sec Q.6 /48 cm/s Q.7 0.05 cm/sec Q.8 4 cm/s Q.9 00 r / (r + 5)² km² / h Q.0 66 7 5 Q. () m/min., (b) 4 88 Q. = tn 4 7 m m Q. 4 cm/sec. m/min. Q.4 () r = ( + t) /4, (b) t = 80 Q.5 () 5.0, (b) 80 7 EXERCISE III Q. + y = 0 or y = 0 Q. D Q.4 D Q.5 D * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * MONOTONOCITY EXERCISE I Q. () I in (, ) & D in (, ) (b) I in (, ) & D in (, 0) (0, ) (c) I in (0, ) & D in (, ) (, ) (d) I for > or < < 0 & D for < or 0 < < Q. (, 0) (, ) Q. () I in [0, /4) (7/4, ] & D in (/4, 7 /4) (b) I in [0, /6) (/, 5/6) (/, ] & D in (/6, /) (5/6, /)] Q.5 continuous but not diff. t = Q.6 < 5 or > 5 Q.7 () (/6)+(/)ln, (/) (/)ln, (b) lest vlue is equl to (/e) /e, no gretest vlue, (c) & 0 Q.8 [, ) Q.0 (, ] [, ) Q. [ 7, ) [, ] Q. incresing in (/, /) & decresing in [0, /) (/, ] Q. 0 Q.4 in (, ) nd in (, ) Q.5 (6, ) Q.6 0 http://kj59.wordpress.com
Q.7 () (, 0] ; (b) in 5, nd in 5 (,), { } 5 ; (c) = ; (d) removble discont. t = (missing point) nd non removble discont. t = (infinite type) (e) Q.4 (, 0) (, ) Q.5 (b ) /4 EXERCISE II mb n Q. c = which lies between & b Q.6 =, b = 4 nd m = m n Q.7 y = 5 9 nd y = 5 + Q.8 0 EXERCISE III Q. C Q. () A, C ; (b) D Q.4 () B ; (b) D ; (c) C Q.5 () A, (b) cos cos p Q.6 A Q.8 () D ; (b) C Q.0 () D * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * MAXIMA - MINIMA EXERCISE I Q. f () = + + Q. m. t = ; f() = 0, min. t = 7/5 ; f(7/5) = 08/5 Q. () M t =, M vlue =, Min. t = 0, Min vlue = 0 (b) M t = /6 & lso t = 5 /6 nd M vlue = /, Min t = /, Min vlue = Q.4 f () = 6 5 5 + 4 Q.5 P m = cosec Q.6 75 sq. units Q.7 r = A 4 s = A 4 Q.9 + 4y 9 = 0 ; 4y + 9 = 0 Q.0 4 m Q. / cu m Q. 0 ', 70 ' Q. 6/(6 + ) Q.4 side 0', height 0' Q.5 sq. units Q.6 = 60 0 Q.7 =, b = 0 Q.9 width m, length m Q.0 b Q. () (, 0), (0, 5/6) ; (b) F ' () = ( ), F '' () =, (c) incresing (, 0) (, ), decresing (0, ) ; (d) (0, 5/6) ; (, /) ; (e) = / Q. () = y = d, (b) = d, y = d Q. 6' 8' Q.4 r = A, = rdins Q.5 4 http://kj59.wordpress.com
EXERCISE II Q. () 0,, (c) 4, t = ln 4 Q. cos A = 0.8 Q.4 (0, 0) Q.6 p < < p p + p if p > 0 ; + p < < p if p < 0 Q.8 4 when = 7 7 Q.9 Mimum vlue is (e + e ) when = Q.0 () f is continuous t = 0 ; (b) e ; (c) does not eist, does not eist ; (d) pt. of inflection = Q. () =,, 0,,, (b) no inflection point, (c) mim t = nd nd no minim, (d) = nd =, (e) ln Q. 4 Q. (0, ) & m. distnce = 4 Q.4 m, 6 Q.6 4 4V ( c b)( b c) Q.7 (, ) (, 9/7) Q.8 H = = Q.9 Q.0 L/4 Q. Q. () incresing in (0, ) nd decresing in (, 0) (, ), locl min. vlue = 0 nd locl m. vlue = (b) concve up for (, ) ( +, ) nd concve down in ( ), ( + ) (c) f () = e Q.4 / EXERCISE III Q. sq. units Q. () B, (b) = 4 ; b = 5 4 ; f() = 4 ( 5 + 8) Q. c c ( b), ( b) & c c ( b), ( b) Q.4 () B, D, Q.5 ± b, ± b b Q.6 ± ± y = 5 Q.7 ( 9/, ) Q.8 A Q.9 b Q.0 () D ; (b) A Q. () A ; (b) 8 Q. () (, ) ; (b) 5 Q. () D Q.4 65 4 Q.5 () B, C; (b) A, B, (c) 6 solutions * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * http://kj59.wordpress.com