IMPORTANT QUESTIONS FOR INTERMEDIATE PUBLIC EXAMINATIONS IN MATHS-IB

Transcription

1 ` K UKATP ALLY CE NTRE IMPORTANT QUESTIONS FOR INTERMEDIATE PUBLIC EXAMINATIONS IN MATHS-IB 7-8 FIITJEE KUKATPALLY CENTRE: # -97, Plot No, Opp Ptel Kunt Hud Prk, Vijngr Colon, Hderbd Ph: -66 Regd Off: 9A, ICES House, Klu Sri, Srvpri Vihr, New Delhi - 6 Ph: , , F: 65 9

2 Totl No of Questions - INTERMEDIATE PUBLIC EXAMINATION, MARCH 7 Totl No of Printed Pges - Reg No Prt - III MATHEMATICS, Pper-I (B) (English Version) Time : Hours] [M Mrks : 75 I Ver Short Answer Tpe questions: Find the vlue of the points, Find the vlue of concurrent, ' ', if the line joining the points,6 n l m SECTION - A = M, if the stright lines, p nd,,7 is prllel to the line joining nd 5 re Find the fourth verte of the prllelogrm whose consecutive vertices re,,,,6, nd,5, Find the ngle between the plnes z 5 nd z 8 5 lim 6 sin 8 lim 7 f 7, then find f ' d 8 tn e, then show tht d 9 Find d nd of t when Verif Rolle s theorem for the function f :,8 be defined b e f 5 6 II Short Answer Tpe questions: (i) Attempt n five questions (ii) Ech question crries four mrks SECTION B 5 = M A 5, nd B, re two fied points Find the eqution of locus of P, so tht the re of PAB is 9 sq units When the es re rotted through n ngle 9, find the trnsformed eqution of 5 is the perpendiculr bisector of the line segment joining the points AB, A,, find the coordintes of ' B ' cos cosb if Show tht f if b where nd b re rel constnts, is continuous t FIITJEE KUKATPALLY CENTRE: # -97, Plot No, Opp Ptel Kunt Hud Prk, Vijngr Colon, Hderbd Ph: -66 Regd Off: 9A, ICES House, Klu Sri, Srvpri Vihr, New Delhi - 6 Ph: , , F: 65 9

3 5 b then 5 5 " ' 6 Find the lengths of subtngent, subnorml t point '' t on the curve cost tsint, sint t cost 7 The volume of cube is incresing t rte of 9 cubic centimeters per second How fst is the surfce re incresing when the length of the edge is centimetres? SECTION C 5 7 = 5 M III Long Answer Tpe questions: (i) Attempt n five questions (ii) Ech question crries seven mrks 8 Find the orthocentre of the tringle whose vertices re 5, 9 Show tht the re of the tringle formed b the lines,, nd, h b nd the line l m n is n h b m h m bl The condition for the line joining the origin to the point of intersection of the circle nd the line l m to coincide Find the direction consines of two lines which re connected the reltion l m n nd mn nl lm then prove tht At point, on the curve d d, show tht the tngent is A window is in the shpe of rectngle surmounted b semicircle the perimeter of the window is ft find the mimum re BLUE PRINT (MATHS-IB) SNo Nme of the chpter Weightge Mrks CO-ORDINATE GEOMETRY Locus () Trnsformtion of es () Stright line 5 ( ) Pir of stright lines (7 + 7) D GEOMETRY 5 D-coordintes () 6 Direction Cosines & Direction Rtions 7 (7) 7 The plne () CALCULUS 8 Limits & Continuit 8 ( + + ) 9 Differentition 5 ( ) Errors Approimtions () Tngent & Norml (7 + ) Rte mesure () Rolle s & Lgrnge s Theorems () Mim & Minim 7 (7) FIITJEE KUKATPALLY CENTRE: # -97, Plot No, Opp Ptel Kunt Hud Prk, Vijngr Colon, Hderbd Ph: -66 Regd Off: 9A, ICES House, Klu Sri, Srvpri Vihr, New Delhi - 6 Ph: , , F: 65 9

4 VERY SHORT ANSWER QUESTIONS C Find the eqution of the stright line pssing through whose sum is zero, Find the eqution of the stright line pssing through Y -intercepts which re in the rtio : the re of the tringle formed b the stright lines Find the vlue of D Trnsform the eqution into norml form, nd mking non zero intercepts, nd mking X nd nd is 6 E Find the rtio in which the line divides the join of the points (, ) nd (, ) F G Prove tht the points (-5, ), (5, 5), (, 7) re colliner nd find the eqution of the line contining these points the portion of stright line intercepted between the es of coordintes is bisected t Find the eqution of the stright line H Find the equtions of the stright line pssing through the following points : p,q ) (, 5), (, 8) b) (, -), (7, -) c) (, -), (-, ) d),, t t t t I Find the equtions of the stright lines pssing through the point (, ) nd (i) prllel (ii) perpendiculr to the line pssing through the points (, ) nd (, ) J Find the eqution of stright line pssing through origin nd mking equl ngles with coordinte es K Find the eqution of the stright line mking n ngle of -is nd hs -intercept L A stright line pssing through A, mkes n ngle of Tn with the positive with OX direction Find the points on the stright line whose distnce from A is units in the positive M Find the points on the line which re t distnce of 5 units from the point (, ) N Find the eqution of line which mkes n ngle of 5 (-, -) with positive -is nd pssing through O Stte whether (, ) nd (-, -) re on the sme side or on opposite side of the stright line Find the re of the tringle formed b the coordinte es nd the line Find the set of vlues of ' ' if the points (, ) nd (, ) lie on the sme side of the stright line 5 C Find the distnce between the stright lines 5 nd 6 9 FIITJEE KUKATPALLY CENTRE: # -97, Plot No, Opp Ptel Kunt Hud Prk, Vijngr Colon, Hderbd Ph: -66 Regd Off: 9A, ICES House, Klu Sri, Srvpri Vihr, New Delhi - 6 Ph: , , F: 65 9

5 D Find the vlue of perpendiculr ' k' if the stright lines k & k 8k 6 re E Find the orthocenter of the tringle whose sides re given b, nd 7 F, b, c re in AP, show tht b c represents fmil of concurrent lines nd find the point of concurrenc G Find the eqution of the line perpendiculr to the line 6 nd mking n intercept on the -is H Find the point of concurrenc of the lines represented b 5k k k I Find the eqution of the stright line pssing through the point of intersection of the lines nd 5 nd contining the point (5, -) J 5 is the perpendiculr bisector of the line segment joining (, ) nd Find K Find the length of the perpendiculr drwn from (, ) to the line, L Find the circumcentre of the tringle formed b the lines, nd M Find the vlue of 5 k, if the ngle between the stright lines 7 nd k 5 9 is N (-, 6) is the imge of the point (, ) wrt the line L, then find the eqution of L O Find the ngle which the stright line mkes with -is Find if the distnce between 5,,7 nd,5, is 9 units Show tht the points,,, 7,, nd,, re colliner C Show tht the points,,5,,5, nd,, form right ngled isosceles tringle D Show tht the points,,,,, nd,, form n equilterl tringle E P is vrible point which moves such tht PA = PB A =,, nd B,,, prove tht P stisfies the eqution z 8 z 7 F Show tht ABCD is squre where A, B, C, D re the points,,,,,,,5, nd,6, respectivel Find the eqution of the plne if the foot of the perpendiculr from origin to the plne is,, 5 Reduce the eqution z 6 of the plne to the norml form C Find the ngle between the plnes z 5 nd z 8 FIITJEE KUKATPALLY CENTRE: # -97, Plot No, Opp Ptel Kunt Hud Prk, Vijngr Colon, Hderbd Ph: -66 Regd Off: 9A, ICES House, Klu Sri, Srvpri Vihr, New Delhi - 6 Ph: , , F: 65 9

6 D Find the eqution to the plne prllel to the ZX -plne nd pssing through,, E Find the eqution of the plne pssing through the point,, nd hving, 5, s dr s of its norml F Find the eqution of the plne pssing through the point,, nd prllel to the plne z 7 G Find the eqution of the plne through z nd z 8,, nd perpendiculr to the plnes H Find the eqution of the plne pssing through,, nd,, nd perpendiculr to z 6 I Find the eqution of the plne through the points,,,,,, 7,,6 J A plne meets the coordinte es in A,B,C centroid of the ABC is, b, c Show tht the z eqution to the plne is b c 5 C D E Find / e cos / sin b sin b nd F G H Evlute,, b, b b sin, b, b sinb cos I J e sin K f,, then find f nd f FIITJEE KUKATPALLY CENTRE: # -97, Plot No, Opp Ptel Kunt Hud Prk, Vijngr Colon, Hderbd Ph: -66 Regd Off: 9A, ICES House, Klu Sri, Srvpri Vihr, New Delhi - 6 Ph: , , F: 65 9

7 L M Find f e where f if if if N Evlute O P Q log e sin tn e e 6 e cos 8 C cos 7 D Is f defined b f sin, if, if continuous t E Show tht f cos cosb if b if where nd b re rel constnts, is continuous t F cosm cosn n G H sin I J 6 5 K cos sin FIITJEE KUKATPALLY CENTRE: # -97, Plot No, Opp Ptel Kunt Hud Prk, Vijngr Colon, Hderbd Ph: -66 Regd Off: 9A, ICES House, Klu Sri, Srvpri Vihr, New Delhi - 6 Ph: , , F: 65 9

8 7 f e sin, then find f ' f then find f ' C f log sec tn find f ' D f 7 then find f ' E f sin log, find f ' F f 6 find f ' G f log7 log, find d d H f 5 then prove tht f' f ' 8 5 then find '' n n e b e then prove tht '' n C log sin log, find d d D Find the derivtive of log E Find the second order derivtive of f log 9 F sin, find d d G log cosh, find d d 9 Find d nd of f t when C the rdius of sphere is incresed from 7 cm to 7 cm then find the pproimte increse in the volume of the sphere the increse in the side of squre is % then find the pproimte percentge of increse in its re D Show tht the length of the subnorml t n point on the curve is constnt E Show tht the length of the subtngent t n point on the curve is constnt F Find d nd if, 8 nd FIITJEE KUKATPALLY CENTRE: # -97, Plot No, Opp Ptel Kunt Hud Prk, Vijngr Colon, Hderbd Ph: -66 Regd Off: 9A, ICES House, Klu Sri, Srvpri Vihr, New Delhi - 6 Ph: , , F: 65 9

9 G Find the pproimte vlue of 999 H The side of squre is incresed from cm to cm Find the pproimte increse in the re of the squre I The dimeter of sphere is mesured to be cm the error of cm is mde in it, then find the pproimte errors of cm is mde in it, then find the pproimte errors in volume nd surfce re of the sphere J Find the slope of the tngent to the curve t Find the vlue of k, so tht the length of the subnorml t n point on the curve k k is constnt Show tht the length of the subnorml t n point on the curve vries s the cube of the ordinte of the point C Let f Prove tht there is more thn one ' c' in, such tht f ' c D Verif Rolle s theorem for the function f in, E Verif Rolle s theorem for the function / f e in, F Find the intervls on which the function f 5 8 is strictl incresing function G Find the rte of chnge of re of circle wrt rdius when r 5cm H The distnce time formul for the motion of prticle long stright line is S t 9t t 8 Find when nd where the velocit is zero I Find the intervls on which f 8 is incresing or decresing J Find the verge rte of the chnge of S f t t between t nd t SHORT ANSWER QUESTIONS Find the eqution of the locus of P, if the rtio of the distnces from P : to A (5, -) nd B (7, 6) is Find the eqution of locus of point P such tht the distnce of P from origin is twice the distnce of P from A (, ) C Find the eqution of locus of P, if the line segment joining (, ) nd (-, 5) subtends right ngle t P D Find the eqution of locus of point, the difference of whose distnces from (-5, ) nd (5, ) is 8 E Find the eqution of the locus of point, the sum of whose distnces from (, ) nd (, ) is 6 FIITJEE KUKATPALLY CENTRE: # -97, Plot No, Opp Ptel Kunt Hud Prk, Vijngr Colon, Hderbd Ph: -66 Regd Off: 9A, ICES House, Klu Sri, Srvpri Vihr, New Delhi - 6 Ph: , , F: 65 9

10 F G H The ends of the hpotenuse of right ngled tringle re (, 6) nd (6, ) Find the eqution of the locus of its third verte A (5, ) nd B (, ) re two fied points Find the eqution of the locus of P, so tht the re of tringle PAB is 9 A (, ), B (, -) nd C (-, ) re three points A point P moves such tht Show tht the eqution to the locus of P is 7 7 PA PB PC I Find the eqution of the locus of P, if A = (, ), B = (, ) nd PA PB J A (, ), B (-, ) be two given points Find the eqution of locus of P so tht the re of the tringle PAB is 85 K Find the locus of the third verte of right ngled tringle, the ends of whose hpotenuse re (, ) nd (, ) L Find the eqution of locus of P, if A (, ), B (, -) nd PA + PB = 8 Show tht the es re to be rotted through n ngle of h Tn b so s to remove the C term from the eqution h b if b, nd through n ngle if b When the origin is shifted to the point (, ), the trnsformed eqution of curve is 7 7 Find the originl eqution of the curve When the es re rotted through n ngle Find the originl eqution of the curve, the trnsformed eqution of curve is D When the es re rotted through n ngle 6, find the trnsformed eqution of E When the es re rotted through n ngle 9, find the trnsformed eqution of F When the es re rotted through n ngle, find the trnsformed eqution of cos sin p G Find the point to which the origin is to be shifted b the trnsltion of es so s to remove the first degree terms from the eq h b g f c, where h b H When the origin is shifted to (, ) b the trnsltion of es, find the trnsformed equtions of the following (i) (ii) FIITJEE KUKATPALLY CENTRE: # -97, Plot No, Opp Ptel Kunt Hud Prk, Vijngr Colon, Hderbd Ph: -66 Regd Off: 9A, ICES House, Klu Sri, Srvpri Vihr, New Delhi - 6 Ph: , , F: 65 9

11 Trnsform the eqution into (i) slope-intercept form (ii) intercept form (iii) norml form Trnsform the eqution b distnce of the stright line from the origin is p, deduce tht into the norml form when, b the perpendiculr p b C Find the vlue of p, if the following lines re concurrent (i) 5,, p 6 (ii) 7, p, 6 5 D Find the vlue of k, if ngle between the stright lines 7 nd k 5 9 is 5 E A stright line Q (, ) mkes n ngle with the negtive direction of the stright line intersects the line 7 t P, find the distnce PQ X -is the F A stright line with slope psses through Q,5 meets the line 6 t P Find the distnce PQ G Find the eqution of stright line prllel to line 7 nd pssing through the point of intersection of lines nd 6 H I P nd Q re the lengths of the perpendiculrs from the origin to the stright lines sec cosec nd cos sin cos Prove tht P Q Find the equtions of the stright lines pssing through the point (-, ) nd mking n ngle of 5 with the stright line J A vrible stright line drwn through the point of intersection of the stright lines b nd b is b b meets the coordinte es t A nd B Show tht the locus of the midpoint of AB K A stright line L with negtive slope psses through the point (8, ) nd cuts positive coordinte es t the points P & Q Find the minimum vlue of OP + OQ s L vries, where O is the origin L Ech side of squre is of length units The centre of the squre is (, 7) nd one of its digonls is prllel to Find the coordintes of its vertices M Find the re of the rhombus enclosed b the four stright lines b c N the stright lines b c, b c nd c b re concurrent, then prove tht b c bc O Find the point on the stright line, which is equidistnt from (-5, 6) nd (, ) P Find the points on the line which re t distnce of 5 units from the point, FIITJEE KUKATPALLY CENTRE: # -97, Plot No, Opp Ptel Kunt Hud Prk, Vijngr Colon, Hderbd Ph: -66 Regd Off: 9A, ICES House, Klu Sri, Srvpri Vihr, New Delhi - 6 Ph: , , F: 65 9

12 sin sin cos cosb C Check the continuit of the following function t '' if f if 8 if D Check the continuit of f given b f 9 if 5 nd 5 if t the point E Show tht f cos cosb if b if where nd b re rel constnts, is continuous t F G Find rel constnts /8 /8 b, so tht the function f given b f sin if if b if if is continuous on H I Check the continuit of the function if f if if f given below t nd J f, given b f k k, if, if is continuous function on, then find the vlue of k K Check the continuit of f given b f if 5 if 9 if if FIITJEE KUKATPALLY CENTRE: # -97, Plot No, Opp Ptel Kunt Hud Prk, Vijngr Colon, Hderbd Ph: -66 Regd Off: 9A, ICES House, Klu Sri, Srvpri Vihr, New Delhi - 6 Ph: , , F: 65 9

13 5 Find the derivtive of the following functions from first principles ( mrks ech) (i) (ii) b c (iii) sin (iv) cos (v) sin (vi) tn (vii) sec (viii) cos e then show tht d d log log C Find d d if cost tsint, sint t cost D / / / then prove tht d d E tn find d d F tn find d d G sin sin, then show tht d d sin sin n H Find the derivtive of tn I Find d d if t t, bt t J Differentite f with respect to g for f sec, g K Show tht the function f, is differentible for ll rel numbers ecept for & L Show tht tn stisfies d cos d M n n b then prove tht '' n n N 5 b then prove tht 5 '' ' 6 the slope of the tngent to the curve equtions of the tngent nd norml t tht point t point on it is /, then find the Show tht the tngent t n point on the curve c sec, c tn is given b sin c cos C Find the equtions of tngent nd norml to the curve t,5 FIITJEE KUKATPALLY CENTRE: # -97, Plot No, Opp Ptel Kunt Hud Prk, Vijngr Colon, Hderbd Ph: -66 Regd Off: 9A, ICES House, Klu Sri, Srvpri Vihr, New Delhi - 6 Ph: , , F: 65 9

14 D Show tht t n point, on the curve b e, the length of the subtngent is constnt nd the length of the subnorml is E Show tht the eqution of the tngent to the curve, b b is b n n t the point b, F Find the vlue of k so tht the length of the subnorml t n point on the curve k k is constnt G Find the ngle between the curve e / nd is H Find the equtions of the tngent nd the norml to the curve 5 t the point,5 I Find the lengths of subtngent nd subnorml t point on the curve J Find the equtions of the tngent nd norml to the curve t, bsin K Show tht the curves 6 5 nd 8 touch ech other t, 7 The volume of cube is incresing t rte of 9 cubic centimeters per second How fst is the surfce re incresing when the length of the edge is centimeters A point P is moving on the curve The coordinte of P is incresing t the rte of units per second Find the rte t which the coordinte is incresing when the point is t,8 C A prticle is moving in stright line so tht fter t seconds its distnce is fied point on the line is given b initil velocit (iii) ccelertion t s f t 8t t Find (i) the velocit t time t sec s (in cms) from t sec (ii) the D The volume of cube is incresing t the rte of 8cm /sec How fst is the surfce re incresing when the length of n edge is cm? E A continer in the shpe of n inverted cone hs height cm nd rdius 6 cm t the top it is filled with wter t the rte of cm /sec, wht is the rte of chnge in the height of wter level when the tnk is filled 8 cm F A stone is dropped intro quiet lke nd ripples move in circles t the speed of 5cm/sec At the instnt when the rdius of circulr ripple is 8cm, how fst is the enclosed rc increses G A continer is in the shpe of n inverted cone hs height 8m nd rdius 6m t the top it is filled with wter t the rte of m /minute, how fst is the height of wter chnging when the level is m? FIITJEE KUKATPALLY CENTRE: # -97, Plot No, Opp Ptel Kunt Hud Prk, Vijngr Colon, Hderbd Ph: -66 Regd Off: 9A, ICES House, Klu Sri, Srvpri Vihr, New Delhi - 6 Ph: , , F: 65 9

15 H The totl cost C in rupees ssocited with production of units of n item is given b C 5 5 Find the mrginl cost when units re produced (mrginl cost is the rte of chnge of totl cost) I A prticle is moving long line ccording to in meters nd time the ccelertion is zero t s f t t t 5t where is mesured in seconds Find the velocit nd ccelertion t time s is mesured t At wht 8 LONG ANSWER QUESTIONS Q h, k is the foot of the perpendiculr from P, on the stright line b c, then show tht h k b c b b And hence find the foot of the perpendiculr from (, ) on the stright line 5 8 Q h, k is the imge of the point P, wrt the stright line b c then show tht h k b c b b 5 And hence find the imge of (, ) wrt the stright line C Find the circumcentre of the tringle whose vertices re (, ), (, 5) nd (5, ) D Find the circumcentre of the tringle whose sides re 5, nd 5 E Find the orthocenter of the tringle whose vertices re ( 5, 7), (, ) nd ( 5, 6) F the equtions of the sides of tringle re 7, 5 nd, find the orthocenter of the tringle G Two djcent sides of prllelogrm re given b 5 nd 7 nd one digonl is 7 9 Find the equtions of the remining sides nd the other digonl H The bse of n equilterl tringle is nd the opposite verte is, Find the equtions of the remining sides I p nd q re the lengths of the perpendiculrs from the origin to the stright lines sec cosec nd cos sin cos, prove tht p q J Find the equtions of the stright lines pssing through, nd which re t distnce of units from, K Find the circumcentre of the tringle formed b the stright lines, 5 nd L Find the orthocenter of the tringle with the vertices (-, -) (6, -) nd (, 5) FIITJEE KUKATPALLY CENTRE: # -97, Plot No, Opp Ptel Kunt Hud Prk, Vijngr Colon, Hderbd Ph: -66 Regd Off: 9A, ICES House, Klu Sri, Srvpri Vihr, New Delhi - 6 Ph: , , F: 65 9

16 M Find the orthocenter of the tringle formed b the lines, 5 nd N Find the incentre of the tringle whose sides re 7, nd 5 O Find the circumcenter of the tringle whose vertices re (, ), (, -) nd (-, ) P the four stright lines b p, b q, c d r nd c d s form prllelogrm, show tht the re of the prllelogrm so formed is p q r s bc d 9 Show tht the product of perpendiculr distnces from point, to the pir of stright lines h b is h b b h the eqution S h b g f c represents pir of prllel stright lines then prove tht (i) h b (ii) f bg nd (iii) the distnce between the prllel lines g c f bc b b b C the second degree eqution nd represents pir of stright lines, then (i) bc fgh f bg ch nd (ii) S h b g f c in the two vribles h b, g c nd f bc D the eqution h b represents pir of intersecting lines, then the combined eqution of the pir of bisectors of the ngles between these lines is h b E Show tht the re of the tringle formed b the lines h b nd l m n is n h b m hlm bl F Show tht the stright lines represented b 8 nd form n equilterl tringles of re sq units G the pirs of lines represented b h b nd h b g f c form rhombus, prove tht b fg h f g H Show tht the product of the perpendiculr distnces from the origin to the pir of stright lines represented b h b g f c is c b h FIITJEE KUKATPALLY CENTRE: # -97, Plot No, Opp Ptel Kunt Hud Prk, Vijngr Colon, Hderbd Ph: -66 Regd Off: 9A, ICES House, Klu Sri, Srvpri Vihr, New Delhi - 6 Ph: , , F: 65 9

17 I Show tht the lines represented b l m m l nd l m n form n equilterl tringle with re n l m J Two equl sides of n isosceles tringle re 7 nd nd its third side psses through the point, Find the eqution of the third side K, is the centroid of the tringle formed b the lines h b nd l m Prove tht bl hm m hl bl hlm m L two of the sides of prllelogrm re represented b one of its digonls Prove tht other digonl is bp hq q hp M the eqution h b nd p q is h b g f c represents pir of intersecting lines then show tht the squre of the distnce of their point of intersection from the origin is c b f g b h perpendiculr Also show tht the squre of this distnce is f h g b if the given lines re N Let the eqution h b represents pir of stright lines Then the ngle between the lines is given b cos b b h C Show tht the lines joining the origin to the points of intersection of the curve nd the stright line re mutull perpendiculr Find the vlues of k, if the lines joining the origin to the point of intersection of the curve nd the line k re mutull perpendiculr Find the ngle between the lines joining the origin to the points of intersection of the curve 5 nd the line D Find the condition for the chord l m of the circle the origin to subtend right ngle t E Find the lines joining the origin to the points of intersection of the curve with the stright line nd lso the ngle between them F Find the condition for the lines joining the origin to the points of intersection of the circle nd the line l m to coincide FIITJEE KUKATPALLY CENTRE: # -97, Plot No, Opp Ptel Kunt Hud Prk, Vijngr Colon, Hderbd Ph: -66 Regd Off: 9A, ICES House, Klu Sri, Srvpri Vihr, New Delhi - 6 Ph: , , F: 65 9

18 G Write down the eqution of the pir of stright lines joining the origin to the points of intersection of the line 6 8 with the pir of stright lines 6 Show tht the lines so obtined mke equl ngles with coordinte es H Show tht the eqution 7 6 represents pir of stright lines Also find the ngles between them nd the coordintes of the point of intersection of the lines Show tht the lines whose dc s re given b l m n, mn nl 5lm re perpendiculr to ech other Find the ngle between the lines whose direction cosines stisf the equtions l m n, l m n C r mkes ngles,,, with the four digonls of cube find cos cos cos cos D Find the direction cosines of two lines which re connected b the reltions l 5m n nd 7l 5m n E Find the direction cosines of two lines which re connected b the reltions l m n nd F mn nl lm Find the ngle between the lines whose direction cosines re given b the equtions l m 5n nd 6mn nl 5lm G vrible line in two djcent positions hs direction cosines l, m, n l l, m m, n n, show tht the smll ngle between the two positions is given b l m n H Find the ngle between two digonls of cube I The vertices of tringle re A,,, B,,, C,, Find A, B, C nd tn for, find d d then prove tht d d C log then prove tht d d D tn cos sin, find d d E Find the derivtive of the function log sin sin FIITJEE KUKATPALLY CENTRE: # -97, Plot No, Opp Ptel Kunt Hud Prk, Vijngr Colon, Hderbd Ph: -66 Regd Off: 9A, ICES House, Klu Sri, Srvpri Vihr, New Delhi - 6 Ph: , , F: 65 9

19 F b then prove tht d log d log G f sin nd g tn then Prove f ' g ' H b nd, f b / cos cos b bcos then f ' b cos I Find d d if / / 5/6 6/7 6 7 J t sin t, cost find d d k K e cosn bsin n then prove tht k '' k' n L Differentite f with respect to g where f Tn, g Tn M cos sin bsin sin then prove '' tn ' cos the tngent t n point on the curve nd B, then show tht the length AB is constnt / / / intersects the coordintes es in A m n m n the tngent t n point P on the curve mn meets the coordinte es in A,B then show tht AP : BP is constnt C Show tht the curves nd 6 9 intersect orthogonll D Find the ngle between the curves nd E Find the ngle between the curves nd 5 F Show tht the condition for orthogonlit of the curves b b G Find the lengths of subtngent, subnorml t point t on the curve cost tsint ; sint t cost b nd b is H At n point t on the curve t sin t, cost Find the lengths of tngent, norml, subngent nd sub norml I Find the equtions of the tngents to the curve where it meets the is J Show tht the tngent t P, on the curve is FIITJEE KUKATPALLY CENTRE: # -97, Plot No, Opp Ptel Kunt Hud Prk, Vijngr Colon, Hderbd Ph: -66 Regd Off: 9A, ICES House, Klu Sri, Srvpri Vihr, New Delhi - 6 Ph: , , F: 65 9

20 K Show tht the squre of the length of subtngent t n point on the curve vries with the length of the subnorml t tht point b b L Find the ngle between the curves 8 nd M Find the ngle between the curves 9 nd (in the th qudrnt) Find two positive integers nd such tht 6 nd is mimum From rectngulr sheet of dimensions cm 8cm, four equl squres of side C D removed t the corners nd the sides re then turned up so s to form n open rectngulr bo Find the vlue of, so tht the volume of the bo is the gretest A window is in the shpe of rectngle surmounted b semi circle the perimeter of the window is ft, find the mimum re the curved surfce of right circulr clinder inscribed in sphere of rdius show tht the height of the clinder is E A wire of length L r ' r' cm re is mimum, is cut into two prts which re bent respectivel in the form of squre nd circle Wht re the lengths of the pieces of the wire respectivel so tht the sum of the res is the lest F Find two positive numbers whose sum is 5 so tht the sum of their squres is minimum G Find the mimum re of the rectngle tht cn be formed with fied perimeter H Prove tht the rdius of the right circulr clinder of gretest curved surfce re which cn be inscribed in given cone is hlf of tht of the cone I Find the bsolute mimum nd bsolute minimum of f on 8, J The profit function p of compn selling ' ' items per d is given b P 5 Find the number of items tht the compn should mnufcture to get mimum profit Also find the mimum profit wish ou ll the best FIITJEE KUKATPALLY CENTRE: # -97, Plot No, Opp Ptel Kunt Hud Prk, Vijngr Colon, Hderbd Ph: -66 Regd Off: 9A, ICES House, Klu Sri, Srvpri Vihr, New Delhi - 6 Ph: , , F: 65 9