1. QUESTION BANK ( ) Class - XII Subject - MATHEMATICS (ONE MARK QUESTIONS)

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1 . QUESTION BANK (-) Clss - XII Sujet - MATHEMATICS (ONE MARK QUESTIONS) os sin. If A = find < < sin os when A A' = I.. If B is skew smmeti mti wite whethe the mti (ABA') is smmeti o skew smmeti.. If A = show tht A-A' is skew smmeti whee A T denotes tnspose of A. 5. If A is skew smmeti mti of ode s.t. = = & = 5 find mti A. 5. If A is skew smmeti mti of ode find A If is singul mti find Constut the mti A = [ij] whee ij = i - j 8. If A is sque mti of ode s.t. AdjA = 6 find A & A If A = find A -.. Find if [ =. If = 5 find &. 6. Using deteminnts find the e of the tingle with veties ( ) ( -) & (- 6).. Using deteminnts find the eqution of the stight line joining the points () & ( 6).. Find the ofto of in 5

2 5. Give n emple of sl mti. 6. If A ij is the ofto of ij of mti then find the vlue of A A A. 7. Find the point on the uve = - whee the tngent is to the -is. 8. If f() = f l () = find the vlue of the deivtive of log f (e ) w..t t the point =. 9. Find fo whih f() = ( sin) is inesing.. If noml to the uve t point Pon = f () is pllel to the -is wht is the d vlue of t P. d. Disuss the ppliilit of Rolle's theoem fo i. f () = on [- ] ii. f () = ( ) on [ ] iii. f () = ( - ) / on [ ]. Give n emple of funtion whih is ontinuous t ll points ut not diffeentile t thee points.. Diffeentite tn - sin os w..t. os sin. If = t d = t find t t = -. d 5. If A = [- - -] find AA' whee A' is tnspose of A. 6. Evlute : If A = find k if A = k A 8. Find the lest vlue of.s.t. f() = is stitl inesing on ( ) 9. Find the points on the uve = t whih slope of tngent is equl to the - oodinte.

3 . Evlute d. Evlute d /. Evlute / sin d. Evlute d 5 os. Evlute d log 5 sin.5 5. Evlute whee [ ] d whee [] epesents the getest intege funtion. log 6. Evlute e d 7. Wite the ode nd degee of the D.E. d d i) = d d d d d ii) sin = d d d 8. Solve the D.E. d d = 9. Find unit veto long if = i ˆ ˆ j k ˆ & = iˆ ˆj kˆ. Find the veto in the dietion of the veto 5 iˆ ˆj kˆ whih hs mgnitude 8 units.. Find the pojetion of the veto = iˆ ˆj kˆ on the veto = iˆ ˆj kˆ. Find if vetos & e s.t. = = nd. =

4 . If & e two unit vetos nd θ is the ngle etween them show tht sin θ / = ˆ ˆ. If = find the ngle etween nd 5. Conside two points P & Q with P.V. OP = & OQ =. Find the P.V. of R whih divides the line joining P & Q in the tio : etenll. 7. Evlute : i ˆ ( ˆj kˆ) ˆj ( kˆ iˆ) kˆ ( ˆj iˆ) 8. If iˆ ˆj ˆ v = = j kˆ = kˆ iˆ find unit veto in the dietion of 9. Find the ngle etween the vetos iˆ j kˆ & ˆ i ˆj kˆ 5. If is unit veto nd ( ) ( ) = 8 5. Find the dietion osines of the veto iˆ ˆj kˆ 5. Find s.t. ( i ˆ ˆ j k ˆ ) is unit veto 5. Find λ s.t. iˆ j k & ˆj 6 ˆj λkˆ e (i) Pllel (ii) Pependiul find 5. If f l () is n even funtion wht tpe of funtion is f()? 55. Mention the points of disontinuit of the following funtions : (i) (ii) f() = [] f ( ) = 56. Find the solute mimum nd solute minimum vlue of : f ( ) = os ; [ ]

5 5 57. Find the slope of the tngent to the uve 7 = t the point whee it ( ) ( ) uts the -is. 58. At wht point on the uve = does the tngent mke n ngle of 5 o with the - is? 59. Find the slope of noml to the uve = t = t t t =. 6. Evlute : i) e d ii) sin os iii) sin os v) d 6. If ( k) d = find the vlue of k. e ( ) 6. Evlute d os ( e ) 6. Find the genel solution of the diffeentil eqution : d iv) vi) 7 sin log d d d d d = 6. Find the ode nd degee of the following diffeentil eqution : d d i) sin = d d ii) d d = d d 65. Wht is the no. of it onstnts in the ptiul solution of diffeentil eqution of ode? 66. Find the integtig fto of the diffeentil eqution :

6 6 d log = log d 67. Find unit veto pllel to the sum of the vetos : iˆ ˆj kˆ & iˆ ˆj kˆ 68. Find nd if = = 5 nd = If is n veto in spe show tht : iˆ)ˆ v i j ˆ v = ( ( ˆ) j ( k ˆ) kˆ 7. If iˆ λ ˆj kˆ & 5ˆ i ˆj µ kˆ e olline find λ µ 7. If veto mkes ngles : α β γ with the z - is espetivel find the vlue of : sin α sin β sin γ 7. Find the ngle etween the following pi of lines : = ( iˆ 5 ˆj kˆ) λ (ˆ i ˆj 6kˆ) nd = ( ˆ i ˆj kˆ) µ (ˆ i 5kˆ ) 7. Find the eqution of the plne pssing though the point ( - ) nd pllel to the plne z =. 7. Find the inteepts ut off the plne 5 - z = on the z es. 75. Find the veto eqution of the line pssing though the point ( ) nd pllel to the line = ( iˆ 7 ˆj kˆ) λ (8ˆ i j kˆ ) 76. Find the distne of the point ( ) fom the plne = ( ˆ i 6 ˆj kˆ) = 77. The Ctesin eqution of line AB is: z = =. Find the dietion tios of line pllel to AB. 78. Find the veto eqution of the plne whih is t distne of 5 units fom the oigin nd is pepediul to the veto iˆ ˆj 6kˆ 79. z Find the ngle etween the line = = 5 7 & the plne - z = 8. A line in the plne mkes ngle with the -is. Find the dietion tios nd 6 dietio osines of the line.

7 7 8. Redue the euqtion. (ˆ i 6 ˆj kˆ) = 5 to the noml fom nd hene find the length of pependiul fom the oigin to the plne. 8. Find the eqution of the plne whih psses though the point ( - ) & is pependiul to the line though the points ( -) & ( - -5). 8. If kz = is the eqution of the plne though the oigin tht ontins the line = = z find the vlue of K. 8. If * = find the vlue of ( * ) * 85. Let * e in opetion on N given : * = L.C.M. of &. Find the vlue of * If * defined on Z s : * = in opetion? Justif ou nswe. 87. Chek whethe the in opetion * defined on Q s follows is ssoitive : * = Fou Mks Questions 88. Epess s the sum of smmeti & skew smmeti mti. 89. If A = using piniple of mthemtil indution show tht : ( A) n = n I n n- A n N 9. If A = using piniple of M.I. show tht : A n n = n n n N n 9. If f () = 5 7 find f (A) if A = 9. Show tht A = stisfies the eqution A A I =. Hene find A -.

8 8 9. If A & B e sque mties of the sme ode s.t. AB = BA then pove using piniple of mthemtil indution tht : AB n = B n A. Futhe P.T. (AB) n = A n B n N n 9. If e in A.P. evlute Using popeties of deteminnts P.T. : i. z z z z z = z ii. z q p z z q p p q = iii. = ( ). iv. = v. = z z z vi. 6 6 = q p p q p p q p p vii. If z e diffeent & z z z = pove tht z = -/ 96. If e positive nd e the p th q th th tems of G.P. using popeties of deteminnts pove

9 9 tht : log log log = q p 97. Using popeties of deteminnts P.T. i. ) ( ) ( ) ( = () ii. = ( ) 98. Solve using popeties of deteminnts i. 6 = ii = iii = 99. Find if A = & AA = whee A is tnspose of mti A.. If A = = & I find k s.t. A = ka I. Hene find A -.. Find the mti = u z s t u z. If A = 5 show tht

10 i. A A is smmeti ii. A-A is skew smmeti Whee A denotes tnspose of mti A. Disuss ontinuit of f () = - - t =.. Show tht the funtion defined s : F () = > < < 5 Is ontinuous t = ut not diffeentile t =. 5. Find if : i. f () = > = < ) ( ) sin ( os sin is ontinuous t = ii. f () = > is evewhee diffeentile iii. f () = < < sin os ot sin is ontinuous iv. f () = < < is ontinuous 8 v. f () = > = < Find the points of disontinuit (if n) fo :

11 i. f () = < < 6 ii. f () = - iii. f () = 5 / 7 5 / < < 7. Find if f () = = >. is diffeentile t 8. Find p if f () = < is ontinuous p p 9. Find d d if : i. = (sin - ) sin - ii. = tn - < < < < fo & sin sin sin sin iii. = sin- < < iv. = os - if < < & - < < v. = tn - sin os vi. = vii. = viii. = i. = e -

12 . = tn () i. ii. iii. iv. = sin = tn ( ) = = v. = sin os sin os... vi. = vii. = sin - 5 viii. = tn -. If d = povetht = d ( ). d If = log find e d. If = sin (log ) P.T. =.. If = tn log P.T. ( ) ( ) =. If = sin P.T. ( ) - =. m d d d d 5. If = { } P. T.( ) m =

13 6. If = d d P.T. = d d 7. If = os θ = sin θ find d tθ = d 8. If = n- log find the vlue of ( n) 9. If = n -n Pove tht : = n (n). If = sint & = sinpt pove tht : ( ) p =. If = log pove tht : = ( ). Diffeentite i. tn - w..t. tn - ; ii. sin-...ot w t < <. Veif Rolles theoem fo the following funtions : i. f() = ( ) (-) on [- ] ii. f () = e - sin on [ ] iii. f() = sin os on [ /] iv. f() = ()e -/ on [- ]. If the tngent to the uve = t P ( -6) is pllel to the line = 5 find &. 5. At wht points will the tngent to the uve = 5 6 e pllel to the -is? Also find the equtions of the tngents to the uve t those points. 6. If the uves = & = ut t ight ngles pove tht =. 7. Show tht the stight line = touhes the uve = e -/ t the point whee the uve osses the - is.

14 8. The pessue P & the volume V of gs e onneted the eltion PV / = whee is is onstnt. Find the % inese in the pessue oesponding to ½% deese in volume. 9. Find the eqution(s) of tngent (s) to the uve = 6 whih e pependiul to the line =. Show tht the uves = 8 & = 7 touh eh othe. Also find the eqution of the ommon tngent.. A mn of height 8 m is moving w fom lmp post t. m/s. If the height of the lmp post is.5 m find the te t whih his shdow is lengthening.. A mn is moving w fom towe 85 m high t speed of m/s. Find the te t whih his ngle of elevtion of the top of the towe is hnging when he is t distne of 6m fom the foot of the towe.. The volume of spheil lloon is inesing t the te of m /s. Find the te of hnge of its sufe e when its dius is 6 m.. A ldde m long lens ginst wll. The foot of the ldde is pulled long the gound w fom the wll t the te of.5 m/s. How fst is the ngle θ etween the ldde & the gound hnging when the foot of the ldde is m w fom the wll? 5. Wte is unning into n inveted one t the te of m /min. The height of the one is m & the dius of its se is 5 m. How fst is the wte level ising when the wte stnds 7.5 m ove the se? 6. A ptile moves long the uve = 5. Find the points on the uve t whih the o-odinte hnges five times s fst s the o-odinte. 7. Let I e the intevl disjoint fom (- ). Pove tht f () = / is stitl inesing on I. 8. Solve the following diffeentil equtions i. d log d d = ii. ot = sin ; = when = iii. d d = ( ) iv. v. ( ) d = d d = (sin - ); () =

15 5 vi. vii. viii. d = e - d ( ) d d = ot d ( ) (d d) = d d i. e / = (e) =. d = ( ) d i. ( ) d d = ii. d = tn d e d iii. = ;( #) d iv. d ( ) = e d v. e / d = (e / ) d; 9. Fom the diffeentil eqution of the fmil of iles touhing the -is t the oigin.. Fom the diffeentil eqution epesenting the fmil of uves given (-) = whee is n it onstnt.. Veif tht = os (log ) sin (log ) is solution of the diffeentil eqution : d d d = d. Evlute : i. d ( ) ( ) ii. d 6 5 iii. log ( tn ) d

16 6 / os iv. os sin d ( ) v. d ( ) vi. f () d whee f () = sin vii. os sin / d viii. ot d / i. ( tn ot ). d d i. / sin os d 9 6sin / ii. sin d iii. iv. / / os log sin d d v. d os sin / / vi. e sin d vii. d ( )

17 7 viii. d ( )( ) i. sin os d sin. d sin sin ( α ) d i. ( sin os ) d ii. sin d os iii. d os( )os( ) iv. d tn v. os os os6 d d vi. sin ( os ) vii. 6 7 d ( 5)( ) viii. ot - ( ) d i. / sin os d sin os tn. d se tn d i. os ii. / / (sin - os ) d

18 8 iii. sin d f ( ) d =. Pove tht :. Evlute s limit of sum : f ( ) d If f ( ) = f ( ) if f ( ) = f ( ) ( )d / 5. Pove tht (logsin logsin ) d = log 6. d If f ( ) = d suh tht f () = Find f () 7. Evlute : d i. ( ) d ii. / / d iii. ( 5 ) / iv. / ( ) / d d v. sin os vi. d 8. Fom the D.E. of the fmil of uves = sin ( ) if : i. e pmetes ii. e pmetes

19 9 9. Find veto of mgnitude whih is pependiul to eh of the vetos î - ˆj kˆ & ˆ i ˆj kˆ 5. If α = iˆ ˆj & β = i ˆj kˆ epess β s the sum of two vetos β & β whee β α β α & 5. Thee vetos stisf the ondition of = Find the vlue of... if = = & = 5. If iszeoveto & = = 5& = 7 find ngle etween & 5. If = iˆ ˆj kˆ & = ˆj kˆ v find veto s. t. = nd. = 5. Let iˆ j ˆj kˆ = ˆ = & = 7ˆ i kˆ Find veto d whih is pependiul to oth s &. d = 55. If thee vetos e v is zeo veto povetht = = suh tht 56. Fo n two vetos & show tht : = ) ( )( ) (. ) ( 57. If e vetos s. t.. =. nd = nd then pove tht = 58. If epesent the vetos BC CA AB of Λ ABC sin A sin B sin C Pove tht : = = 59. If e position vetos of the veties of tingle pove tht veto e of the tingle is given : ( ) 6. Show tht ( ) ( ) = ( ) nd intepet the esult geometill. Find the e of gm whose digonls e epesented the vetos iˆ ˆj kˆ & ˆ i ˆj kˆ 6. Find if iˆ j ˆ kˆ & ˆ i ˆj kˆ e pependiul vetos of equl mgnitudes. 6. Let 5 ˆ ˆ ˆ 5 ˆ ˆ & ˆ ˆ ˆ = i j k = i j k = i j k Find veto d s. t. d d & d. =

20 6. If e thee mutull pependiul vetos of equl mgnitudes show tht is equll inlined to the vetos & 6. Find the o-odintes of the foot of pependiul dwn fom the point ( ) to the line joining the points ( 6) & (5 ). Also find the length of pependiul. 65. Find the point on the line ) z = = t distne fom the point ( z 66. Find the eqution of the plne pllel to the line = = ontins the point (5 -) nd psses though the oigin. whih 67. Find the distne of the point ( -) fom the plne z = 5 mesued z pllel to the line = = Find the shotest distne etween the lines; = ( 6i j kˆ) λ ( i ˆj kˆ) = i kˆ µ (ˆ i ˆj kˆ) 69. Find the veto eqution of the plne pssing though the intesetion of the plnes z =. z = nd pependiul to the plne z =. Also find the inlintion of this plne with the - plne. 7. Find the distne of the point (- -5 ) fom the point of intesetion of the line : ˆ ˆ ˆ ˆ = i j k λ (ˆ i j kˆ) & the plne.(ˆ i ˆj kˆ) = 5 7. Find the Ctesin & veto eqution of the plne pssing though the intesetion of the plnes.(ˆ i 6 ˆ) j = &.(ˆ i ˆj kˆ) = whih e t unit distne fom the oigin. 7. Find the o-odintes of the foot of pependiul dwn fom oigin to the plne z 6 = 7. Find the eqution of the plne though the points ( - ) & (5 -) nd pependiul to the plne 5z = 7. Pove tht the ngle etween n two digonls of ue is os Show tht the lines

21 d z d = = α δ α α δ nd z = = β γ β β γ e opln Also find the eqution of the plne ontining these lines. 76. Find the veto eqution of the line pssing though the point ( ) & pllel to the plnes ˆ.(ˆ i j kˆ) = 5.(i ˆj kˆ) = Find the veto eqution of the plne ontining the lines : = iˆ ˆj kˆ t (ˆ i kˆ) & = iˆ ˆj kˆ s ( ˆj kˆ ) 78. Find the points on the line point P ( ) z = = t distne of 5 units fom the 79. Find the distne of the point (6 5 9) fom the plne detemined the points A ( - ) B (5 ) & C (- - 6). 8. Let N e the set of ntul numes & R e eltion on N N defined ( ) R ( d) d = fo ll ( ) ( d) EN N. Show tht R is n equivlene eltion on N N. 8. Let N e the set of ll ntul numes & R e the eltion on N N defined ( ) R ( d) iff d ( ) = ( d) Emine whethe R is n equivlene eltion on N N. 8. Show tht the eltion R on the set A = { Z : } E given R = {( ) : is multiple of } is n equivlene eltion. Find the set of ll elements elted to. 8. Show tht the eltion R on the set A = { 5} given : R { ) is even} is n equivlene eltion. Show tht ll the elements of { 5} e elted to eh othe nd ll the elements of { } e elted to eh othe ut no element of { 5} is elted to n element of { } 8. Show tht the eltion R in the set of el numes defined s : R = {( ) : < } is neithe efleive no smmeti no tnsitive. 85. Find if the eltion R defined in the set A = {. } defined s : R = {( ) : = } is efleive smmeti & tnsitive.

22 86. Let A = { }. Find the no. of eltions ontining ( ) & ( ) whih e efleive & tnsitive ut not smmeti. 87. Let A = { }. Find the no. of equivlene eltions in A ontining ( ) & ( ) 88. Let R e the eltion defined on the set A = { 5 6 7} R = {( ) : oth & e eithe even o odd}. Show tht R is n equivlene eltion. Futhe show tht ll elements of the suset { 5 7} e elted to eh othe & ll elements of the suset { 6} e lso elted to eh othe ut no element of the suset { 5 7} is elted to n element of the suset { 6}. 89. Let A = { } & R e the eltion defined on A s follows : R = {( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )} Find whethe R is efleive smmeti & tnsitive. 9. Let L e the set of ll lines in XY plne nd R e the eltion in L defined s : R = {L L ) : : L is pllel to L }. Show tht R is n equivlene eltion. Find the set of ll lines elted to the line = Show tht the opetion* on Q {} defined : * = Q {} is : i. ommuttive ii. ssoitive Also find the identit element nd invese fo the opetion* on Q {}. 9. Let A = N N & * e in opetion on A defined s : ( ) * ( d) = ( d). Show tht * is ommuttive & ssoitive. Find the identit element of * on A (if n). 9. Conside the in opetion * on the set { 5} defined s : * = h..f. of &. Wite the opetion tle & find if * is i. ommuttive ii. ssoitive Also find if identit element eists fo * on the given set. 9. Let A = R {} & B = R {}. If f : A B is define f () = is ijetion. Also find f. show tht f

23 95. Find if f : Q {} Q defined s f () = i. One-one is : ii. onto 96. Show tht f : [- ] R given f () = of the funtion f : [- ) Rnge (f). is one-one onto. Also find invese 97. Conside F : R [ ) given : f () = whee R is the set of ll nonnegtive nd numes. Show tht f is invetile & find f. 98. Show tht f : R {-} R {} given f () = 99 Let f : R - R e funtion defined s : is invetile. Also find f. f() = of f?. Is f invetile? Find f - : Rnge f R -. Wht is the nge. Let R e eltion defined on the set R of el numes s : R = {( ) R R; < }. Find if the eltion R is efleive smmeti tnsitive.. Let * e in opetion on the set of integes given s : * = Z Is * (i) ommuttive (ii) ssoitive Find the identit element fo * on I. Also find if * ( * ) =.. Pove tht : tn - tn = tn 7 7. i. Simplif : ot ( ) - ii. Solve : tn - (os ) = tn - ( ose ) 5. Evlute : os sin sin 5

24 5. Pove tht : tn - = os 6. An opetion * defined on the set of positive tionl numes Q is given : * = Q Pove tht * is in opetion on Q. Find the identit element fo * in Q. Also the invese of Q. 7. Pove tht : 9 sin Simplif : 9 = sin { } i. tn - { } ii. sin - 9. Pove tht : Sin - 6 os tn 5 6 =. Solve fo : i. ot - ot - ( ) = > o ii. tn (os - ) = sin ot - iii. sin - = os tn iv. tn - tn- =. Evlute : i. tn - 5 se tn 5 7 8

25 5 ii. sin tn. Pove tht : sin sin Cot - = sin sin. Two ds e dwn suessivel fom well shuffled pk of 5 ds. Find the poilit distiution of the no. of es. Also find men & stndd devition.. In inomil distiution the sum of men & vine of 5 tils is.75. Find the distiution. Also find P (X > ) if X is inomil vile. 5. In shool thee e students out of whih e gils. It is known tht out of % of the gils stud in lss XII. Wht is the poilit tht student hosen ndoml studies in lss XII given tht the hosen student is gil? 6. A g X ontins white & ed lls nd g Y ontins white & 5 ed lls. One ll is dwn t ndom fom one of the gs & is found to e ed. Find the poilit tht it ws dwn fom g Y. 7. ds numeed to e pled in o mied up thooughl nd then d is dwn t ndom fom the o. If it is known tht the nume on the dwn d is moe thn find the poilit tht it is n even nume. 8. In multiple hoie emintion with thee possile nswes (out of whih onl one is oet) fo eh of the five questions wht is the poilit tht ndidte would get fou o moe oet nswe just guessing? 9. A g ontins tikets numeed to. Two tikets e dwn without eplement. Wht is the poilit tht the seond tiket hs n even nume given tht the fist hs n odd nume?. In hudle e ple hs to oss hudles. The poilit tht he will le eh hudle is /. Wht is the poilit tht he will knok down fewe thn hudles.. A d fom pk of 5 ds is lost. Fom the emining ds of the pk two ds e dwn & e found to e spdes. Find the poilit of the lost d eing Spde.. A speks the tuth 8 times out of times. A die is tossed. He epots getting 5. Wht is the poilit tht it ws tull 5?. The poilit of shoote hitting tget is ¾. How mn minimum no. of times must he/she fie so tht the poilit of hitting the tget t lest one is moe thn.99?

26 6. A etin tem wins with poilit.7 loses with poilit. & ties with poilit.. The tem pls thee gmes. Find the poilit tht the tem wins t lest two gmes ut does not lose. 5. Bg ontins ed & lk lls. Bg II ontins ed & 5 lk lls. One ll is tnsfeed fom g I to g II & then ll is dwn fom g II. The ll so dwn is found to e lk in olou. Find the poilit tht the tnsfeed ll is ed. 6. A mn tkes step fowd with poilit. & kwd with poilit.8. Find the poilit tht t the end of eleven steps he is one step w fom the stting point. 7. A n hit tget times out of 5 times B n hit tget times out of 5 times & C n hit times out of times. Find the poilit tht two out of A B nd C will hit the tget. 8. A student tkes his emintion in sujets A B C & D. To qulif he must pss in A nd t lest othe sujets. His hnes of pssing in A B C & D e /5 ¾ 5/6 / espetivel. Find the hnes of his qulifing. 9. Thee e gs A & B ontining lk & white; white & lk lls espetivel. A die is thown. If o 5 tuns up ll is dwn fom g A. Othewise ll is dwn fom g B. Find the poilit of getting lk ll.. A is known to tell the tuth in 5 ses out of 6 nd he sttes tht white ll ws dwn fom g ontining 9 ed & white ll. Find the poilit tht white ll ws dwn.. In shool 8% of the gils & % of the os hve n intelligent quotient of moe thn. In the shool 6% of the students e gils. A student with intelligent quotient moe thn is seleted. Find the poilit tht the student seleted is gil.. A fmil hs hilden. Find the poilit tht oth e os if it is known tht :. t lese one of the hilden is o.. the elde hild is o.. A g ontins lls. Two lls e dwn t ndom nd e found to e white. Wht is the poilit tht ll lls e white? SIX MARK QUESTIONS. Using mti method solve the following sstem of equtions : 5 z = 7 z = 5

27 7 7 z = 7 5. If A = find A -. Hene solve the following sstem of equtions: z = z = 9 z = 6. Use the podut 6 9 to solve the sstem of equtions : z = z = z = 7. Using element ow tnsfomtions find the invese of : 8. Using element olumn tnsfomtions find the invese of : 9. Find the o-odintes of the points on the uve = the tngents t whih pss though the oigin. Also find the equtions of the tngents.. The uve = 5 touhes the -is t P (- ) & uts the -is t the point Q whee its gdient is. Find the eqution of the uve ompletel.. If the uves = & = k ut t ight ngles show tht k = 5.. Evlute : i. d sin os

28 8 d ii. sin se iii. sin d iv. d os sin v. log ( ) d. /. Pove tht : log sin d = log. Evlute s limit of sum : ( 5) d nd ( ) d 5. A window is in the fom of etngle suounded semiile. If the peimete of the window is P m show tht the window will llow the mimum possile P light onl when the dius of the semiile is. m 6. A o of onstnt volume C is to e twie s long s it is wide. The mteil on the top & fou sides osts thee times s muh pe sque m s tht in the ottom. Wht e the most eonomil dimensions? 7. A sque tnk of pit 5m hs to e dug out. The ost of lnd is Rs. 5 pe sq. m. The ost of digging ineses with the depth & fo the whole tnk is (depth) upees. Find the dimensions of the tnk fo the lest totl ost. 8. The peimete of etngle is m. Find the length of its sides when the e is mimum. 9. A o is onstuted fom etngul metl sheet m 6 m utting sques of sides m fom the ones of the sheet & then tuning up the pojeted potions. Fo wht vlues of will the volume of the o e mimum. 5. How ould wie m long e divided into pts if one pt is to ent into ile the othe pt is to e ent into sque nd the plne figues e to hve es the sum of whih is minimum?

29 9 5. The setion of window onsists of etngle sumounted n equiltel tingle. If the peimete e given s 6 m find the dimensions of the window in ode tht the mimum light m e dmitted. 5. If the sum of lengths of the hpotenuse nd side of ight ngled tingle is given show tht the e of the tingle is mimum when the ngle etween them is. 5. Find the mimum e of the isoseles tingle insied in the ellipse = with its vete t one end of the mjo is. 5. A point on the hpotenuse of tingle is t distne & fom the sides. Show tht the minimum length of the hpotenuse is ( / / ) /. 55. Show tht the ight iul one of lest uved sufe e & given volume hs n ltitude equl to times the dius of the se. 56. Show tht the semi-vetil ngle of ight iul one of given sufe e & mimum volume is Sin - (/). 57. Show tht the volume of the getest linde whih n e insied in one of height h & semi-vetil ngle is h An open o with sque se is to e mde out of given quntit of sheet of e. Show tht the mimum volume of the o is The totl e of pge is m. The omined width of the mgin t the top & ottom is 6 m & the side is m. Wht must e the dimensions of the pge in ode tht the e of the pinted mtte is mimum? 6. Using integtion find the e of the egion : {( ) : < < 6} 6. Find the e ounded the uve = -is nd the lines = - = using integtion. 6. Sketh the egion ounded the uves = using method of integtion. 5 & =. Find its e 6. Find the e of the egion in the st qudnt enlosed the -is the line = & the ile =.

30 6. Using integtion find the e ounded the uves = sin = os & the is s.t. < < 65. Using integtion find the e enlosed etween the iles = & ( ) =. 66. Find integtion find the e enlosed the given uves : i. = & = - ii. = & = iii. = 5 & = 5 iv. = 6 & = v. = & = Using integtion find the e ounded the lines : i. = = & = 7 ii. = = 6 & 5 = 68. Using integtion find the e of the egion : i. { ) : < < /} ii. { ) : < < < < < < } iii. {( ) : > 6 < 6} iv. {( ) : 9 6 > 6} 69. Using integtion find the e of Λ ABC with veties A (- ) B ( ) & C ( ) 7. Using integtion find the e of the egion ounded the uve = 7. Using integtion find the e ounded the uves : i. = & = - ii. = = - & the -is 7. Find the e of the smlle pt of the ile = ut off the line =. 7. Find the e enlosed etween the ile = 6 the pol = 6 & the -is.

31 7. It is given tht the te t whih some tei multipl is popotionl to the instntneous nume pesent. If the oiginl no. of tei doules in two hous in how mn hous will it e five times? 75. Find the imge of the point ( ) in the plne z = Find the veto eqution of the line of shotest distne etween the lines : 6 z z = = & = = Also find the S.D. etween the lines. 77. Show tht the points ( - -) ( 5 ) ( 9 ) & (- ) e opln. Wite the veto eqution & Ctesin eqution of the ommon plne. 78. Mon wnts to invest t the most Rs. in svings etifite nd Ntionl Svings Bonds. She hs to invest t lest Rs. in svings etifites nd t lest Rs. in Ntionl Sving Bonds. If the te of inteest on svings etifite is 8% p.. nd the te of inteest on ntionl svings onds % p.. how muh mone should she invest to en mimum el inome? 79. A ik mnuftue hs two depots A & B with stoks of 5 & 5 iks espetivel. He eeives odes fom thee uildes P Q & R fo 5 & iks espetivel. The ost (in Rs.) of tnspoting iks to the uildes fom the depots is given elow : To P Q R Fom A 6 5 B How should the mnuftue fulfil the odes so s to keep the ost of tnspottion minimum? 8. An eoplne n mimum of pssenges. A pofit of Rs. is mde on eh eeutive lss tiket nd pofit of Rs. 6 is mde on eh eonom lss tiket. The iline eseves t lest sets fo eeutive lss. Howeve t lest times s mn pssenges pefe to tvel eonom lss s the eeutive lss. Detemine how mn tikets of eh tpe must e sold in ode to mimize the pofit fo the iline. Wht is the mimum pofit? 8. A mn owns field e of sq. m. He wnts to plnt fuit tees in it. He hs sum of Rs. to puhse oung tees. He hs the hoie of two tpes of tees. Tpe A equies sq. m. of gound pe tee & osts Rs. pe tee nd tpe B equies sq. m. of gound pe tee & osts Rs. 5 pe tee. When full gown tpe A podue n vege of kg of fuit whih n e sold t pofit of Rs.

32 pe kg. nd tpe B podues n vege of kg. fuits whih n e sold t pofit of Rs..5 pe kg. How mn tees of eh tpe should e plnted to hieve mimum pofit when the tees e full gown? Wht is tht pofit? 8. A dietiin hs to develop speil diet using two foods P & Q. Eh pket (ontining g) of food P ontins units of lium units of ion 6 units of holesteol & 6 units of vitmin A. Eh pket of the sme quntit of food Q ontins units of lium units of ion units of holesteol nd units of vitmin A. The diet equies t lest units of Clium t lest 6 units of ion nd t the most units of holesteol. How mn pkets of eh food should e used to minimize the mount of vitmin A in the diet? Wht is the minimum mount of vitmin A? 8. A to ompn mnuftues two tpes of dolls A & B. Mket tests nd ville esoues hve indited tht the omined podution level should not eeed dolls pe week nd the demnd fo dolls of tpe B is t most hlf of tht fo dolls of tpe A. Futhe the podution level of dolls of tpe A n eeed thee times the podution of dolls of othe tpe t most 6 units. If the ompn mkes pofit of Rs. & Rs. 6 pe doll espetivel on dolls A & B how mn of eh tpe should e podued weekl in ode to mimize the pofit? 8. A smll fim mnuftues items A & B. The totl no. of items A & B tht it n mnuftue in d is t the most. Item A equies one hou to mke while item B tkes onl hlf n hou. The mimum time ville pe d is 6 hous. If the pofit on one unit of item A is Rs. nd on one unit of item B is Rs. 6 how mn of eh tpe should e podued to mimize the pofit? Solve the polem gphill. 85. A home deoto mnuftues two tpes of lmps A & B. Lmp A equies hous of the utte s time nd hou of the finishe. Lmp B equies hou of the utte s nd hous of the finishe s time. The utte hs hous nd the finishe hs 76 hous of time ville eh month. Pofit on eh lmp of tpe A & B is Rs. 6 Rs. espetivel. Assuming tht he n sell ll tht he podues how mn lmps of eh tpe should e mnuftued to otin mimum pofit? Solve gphill. 86. Given thee identil oes I II & III eh ontining two oins. In o I oth oins e gold oins in o II oth e silve oins & in o III thee is one gold & one silve oin. A peson hooses o t ndom & tkes out oin. If the oin is of gold wht is the poilit tht the othe oin in the o is lso gold oin? 87. Conside the epeiment of tossing oin. If the oin shows hed toss it gin ut if it shows til then thow die. Find the onditionl poilit of the event tht the die shows nume gete thn given tht thee is t lest one til. 88. Fou defetive tiles e mied with ten good ones. A smple of is dwn t ndom fom the lot. Find the poilit distiution of no. of defetive tiles dwn. Hene find men & vine of the poilit distiution.

33 89. B emining the hest X- the poilit tht T.B. is deteted when peson is tull suffeing is.99. The poilit of inoet dignosis is.. In etin it in pesons suffes fom T.B. A peson is seleted t ndom & is dignosed to hve T.B. Wht is the hne tht he tull hs T.B.? 9. A lss hs 5 students whose ges (in es) e &. One student is seleted in suh mnne tht eh hs the sme hne of eing hosen nd the ge X of the seleted student is eoded. Wht is the poilit distiution of the ndom vile X? Find men vine & stndd devition of X. 9. A fto hs mhines X Y Z poduing & olts pe d espetivel. The mhine X podues % defetive olts Y podues.5% & Z podues % defetive olts. At the end of d olt is dwn t ndom & is found defetive. Wht is the poilit tht this defetive olt hs een podued mhine X? 9. A & B e pling gme. A thows die & B tosses oin tun tun. A wins the gme if he gets nume moe thn on the die & B wins if he gets hed. Find thei espetive hnes of winning if A stts the gme. 9. In gme mn wins upee fo si nd loses upee fo n othe nume when fi die is thown. If he thows the die fo mimum of thee times nd he quits the gme on getting si find the epeted vlue of the mount he wins/loses. 9. Assume tht the hnes of ptient hving het ttk is %. It is lso ssumed tht medittion & og ouse edues the isk of het ttk % & pesiption of etin dug edues its hne 5%. At time ptient n hoose n one of the two options with equl poilities. It is given tht fte going though one of the two options the ptient seleted t ndom suffes het ttk. Find the poilit tht the ptient followed ouse of medittion & og. 95. If fi oin is tossed times find the poilit of getting : i. etl si heds ii. iii. t lest si heds t most si heds