CBSE ANNUAL EXAMINATION ALL INDIA SET Wih Complee Eplnions M Mrks : SECTION A Q If R = {(, y) : + y = 8} is relion on N, wrie he rnge of R Sol Since + y = 8 h implies, y = (8 ) R = {(, ), (, ), (6, )} So, rnge of R = {,, } Time Allowed : Hours Q If n n y, y <, hen wrie he vlue of + y + y y Sol We hve n n y n y y n y y y y+ y Q If A is squre mri such h A = A, hen wrie he vlue of 7A (I + A), where I is n ideniy mri Sol We hve, 7A (I + A) = 7A (I + A)[ (I + A) (I + A)] 7A (I A)[II IA + AI+ AA] 7A (I A)[I A+ A] { AA A A, AI A IA 7A (I A)(I A) 7A (I A A ) I [Given h A = A y z Q If y w 5, find he vlue of + y y z Sol We hve y w 5 By equliy of mrices, we hve : y, y On solving hese eqs, we ge : =, y = So, + y = Q5 If 7 8 7, find he vlue of 6 Sol Given 7 8 7 6 Q6 If Sol f () sin d, hen wrie he vlue of f () We hve f () sin d Differeniing wr boh sides, Q7 Evlue : Sol We hve [log ] f () [ sin ] f () [ sin ] [sin ] sin 7 [log7 log 5] log 5 Q8 Find he vlue of p for which he vecors i ˆ j ˆ 9kˆ nd ˆ i pj ˆ k ˆ re prllel Sol As he vecors re prllel, so heir dr s mus be proporionl Therefore, 9 p p Q9 Find (b c), if ˆ ˆ ˆ ˆ ˆ ˆ = i j k, b i j k nd c = i ˆ ˆj kˆ For vrious suffs on Mhs, plese visi : wwwheopgupcom Pge
CBSE Annul Em Pper (All Indi) Compiled By OP Gup (+9-9788) Sol As (b c) = [ b c] = So, (b c) = ( ) ( ) ( 6) Q If he cresin equion of line re y z 6, wrie he vecor equion for he line 5 7 Sol We hve y z 6 y ( ) z 5 7 5 7 y y z z On compring o we hve he direcion rios s 5, 7, nd poin on he b c line s (,, ) Therefore he required vecor equion of line is r = iˆ j ˆ+ k ˆ (kˆ 5i ˆ+ 7j) ˆ Q If he funcion f : R R be given by Sol SECTION B find fog nd gof nd hence find fog () nd gof ( ) We hve And, fog () f [ g ()] f fog() 6 f () nd g : R R be given by Also, gof () g[ f ()] g( ) ( ) And, gof ( ) ( ) Q Prove h : n cos ; OR If n n, find he vlue of Sol LHS : Le Y = n Pu cos θ θ cos (i) cosθ cosθ Le Y = n cosθ sin θ cosθ cosθ = n cosθ sin θ n n θ cosθ sin θ n θ = n = n = n cosθ sin θ n θ n n θ = n n θ θ cos [By (i) = RHS OR The given equion is n n g(),, For vrious suffs on Mhs, plese visi : wwwheopgupcom Pge
CBSE Annul Em Pper (All Indi) Compiled By OP Gup (+9-96558) n n () n n n n ( ) 6 y Q Using properies of deerminns, prove h : 5 y Sol LHS : Le y 5 y 8y 8 y 5 8y 8 [Applying R R R nd R R 8R [Epnding long C [( 5) ()( )] [ 5 ] RHS Q Find he vlue of dy θ θ θ, if e (sin θ cos θ) nd y e (sin θ cos θ) Sol On differeniing nd y wr θ boh he sides, we ge : θ θ θ θ e (sin θ cos θ) e (sin θ cos θ) e (cos θ sin θ) e sin θ dθ θ dy θ θ θ And, y e (sin θ cos θ) e (sin θ cos θ) e (cos θ sin θ) e cos θ dθ dy dy dθ θ e cosθ co θ θ dθ e sin θ dy So, co(/) θ/ b d y dy Q5 If y Pe Qe, hen prove h ( b) by b Sol We hve y Pe Qe [On dividing boh sides by b (b) e y Pe Q [On diff wr boh sides b b e y bye ( b) P( b) e b (b) e (y by) P( b) e b On muliplying boh he sides by e, we ge : e (y by) P( b) [Agin diff wr boh sides e (y by ) e (y by) e [y by y by] d y dy ie, y ( b)y by or, ( b) by HP Q6 Find he vlues of for which y [( )] is n incresing funcion OR Find he equions of ngen nd norml o he curve b e y he poin (, b) b For vrious suffs on Mhs, plese visi : wwwheopgupcom Pge
CBSE Annul Em Pper (All Indi) Compiled By OP Gup (+9-9788) Sol dy We hve y [( )] y [ ] dy [ ][ ] For dy [ ][ ] =,, Inervl Sign of dy y is (, ) Negive Decresing (, ) Posiive Incresing (, ) Negive Decresing (, ) Posiive Incresing Since dy in (,) (, ) so, y is incresing in (,) (, ) y y dy OR Given b b dy b y [ ][ ] b ( ) b Slope of ngen (,b) b nd, Slope of norml (,b) b So, eq of ngen is : y b ( ) b y = b nd, eq of norml is : y b ( ) by = ( + b ) b sin Q7 Evlue : OR Evlue : cos 5 6 Sol Le I Using sin (i) cos f () f ( ), Adding (i) nd (ii), we ge Pu cos ( ) sin ( ) ( ) sin I cos ( ) cos (ii) sin I cos sin d Also when, = d I I d I sin I cos nd when = b I [n ] [n () n ( )] I OR Le I I 5 6 ( 5) I 5 6 5 6 I [ 5 6] 5 + 5 5 I 5 6 log + + C ( 5) 5 6 For vrious suffs on Mhs, plese visi : wwwheopgupcom Pge
CBSE Annul Em Pper (All Indi) Compiled By OP Gup (+9-96558) 5 or, I 5 6 log + + 5 6 C Q8 Find he priculr soluion of he differenil equion dy y y, given h y = when = Sol We hve dy y y dy ( )( y) dy ( ) ( y) log y C Given h y = when = so, log () C C = required soluion is : log y dy n Q9 Solve he differenil equion : ( ) y e n dy n dy e Sol Given ( ) y e y This is liner differenil equion of he form dy P()y Q() So, n e P(),Q() n Now, IF e e n n n e required soluion is : y( e ) e C n n y( e n e ) d C [Pu e d n y( e n n ) C y( e ) e C n n y C e e is he required soluion Q Show h he four poins A, B, C nd D wih posiion vecors i ˆ 5j ˆ+ kˆ, ˆj kˆ, i ˆ 9j ˆ+ kˆ nd ( ˆi ˆj + k) ˆ respecively re coplnr OR The sclr produc of he vecor = ˆi ˆj + kˆ wih uni vecor long he sum of vecors b = i ˆ j ˆ 5kˆ nd c ˆi j ˆ+ kˆ is equl o one Find he vlue of nd hence find he uni vecor long b c Sol The poins A, B, C nd D re coplnr if ABAC AD Now, AB i ˆ 6ˆj k, ˆ AC ˆi ˆj k, ˆ AD 8i ˆ ˆj kˆ 6 ABAC AD ( ) 6( ) ( ) 8 So, A, B, C nd D re coplnr b c OR Given b c b c b c (i ˆ ˆj + k)(i ˆ ˆ j ˆ 5k) ˆ (iˆ ˆj + k)( ˆ iˆ j ˆ+ k) ˆ i ˆ j ˆ 5kˆ ˆi j ˆ+ k ˆ 5 + ( ) 6 ( ) For vrious suffs on Mhs, plese visi : wwwheopgupcom Pge 5
CBSE Annul Em Pper (All Indi) Compiled By OP Gup (+9-9788) Also, uni vecor long b c is given s : b c ( )i ˆ 6j ˆ k ˆ ( )iˆ 6j ˆ kˆ i ˆ 6ˆj kˆ b c ( ) 6 ( ) ( ) 6 ( ) 7 Q A line psses hrough (,, ) nd is perpendiculr o he lines r = (i ˆ ˆj k) ˆ (i ˆ j ˆ+ k) ˆ nd r = (i ˆ ˆj k) ˆ μ(i ˆ+ ˆj + k) ˆ Obin is equion in vecor nd Cresin form Sol Given lines re r = (i ˆ ˆj k) ˆ (i ˆ j ˆ+ k) ˆ nd r = (i ˆ ˆj k) ˆ μ(i ˆ+ ˆj + k) ˆ A line perpendiculr o he given lines will be in he direcion of ˆi ˆj kˆ (i ˆ j ˆ+ k) ˆ (i ˆ j ˆ+ k) ˆ = 6i ˆ j ˆ+ 6k ˆ or,b = i ˆ ˆj kˆ Posiion vecor of given poin (,, ) is, i ˆ ˆj kˆ Using r = b, he required vecor equion of line is : r = i ˆ ˆj k ˆ (i ˆ ˆj k) ˆ And, Cresin form of he line is : y z Q An eperimen succeeds hrice s ofen s i fils Find he probbiliy h in he ne five rils, here will be les successes Sol Le he probbiliy of success be p nd h of filure be q Therefore, p = q Since p + q = so, q + q = q, p P( les successes) P(r ) P() + P() + P(5) n r n r Using P(r) = C p q, where n = 5 r 5 5 5 55 5 5 5 5 ie, P(r ) C C C 7 8 5 98 59 So, required probbiliy 5 SECTION C Q Two schools A nd B wn o wrd heir seleced sudens on he vlues of sinceriy, ruhfulness nd helpfulness The school A wns o wrd ` ech, `y ech nd `z ech for he hree respecive vlues o is, nd sudens wih ol wrd money of `6 School B wns o spend ` o wrd is, nd sudens on he respecive vlues (by giving he sme wrd money for he hree vlues s before) If he ol moun of wrds for one prize on ech vlue is `9, using mrices, find he wrd money for ech vlue Apr from hese hree vlues, sugges one more vlue which should be considered for wrd Sol Le he wrd money spen on he vlues of sinceriy, ruhfulness nd helpfulness be ` ech, `y ech nd `z ech respecively + y + z = 6, + y + z =, + y + z = 9 6 The given siuion cn be epressed s : y z 9 6 where A, X y, B AX B X A B (i) z 9 Now, A = ( ) ( ) + ( ) = 5, so A eiss Consider A ij s he cofcors of he elemen ij of mri A For vrious suffs on Mhs, plese visi : wwwheopgupcom Pge 6
CBSE Annul Em Pper (All Indi) Compiled By OP Gup (+9-96558) A, A, A, A, A, A, A 5, A 5, A 5 5 5 So, A 5 dj A dj A 5 A 5 5 5 5 6 By (i), X 5 y 5 5 9 z By equliy of mrices, we ge :, y, z Hence he wrd money for he vlues of sinceriy, ruhfulness nd helpfulness is `, ` nd ` respecively Also, he vlue of Obedience cn be included for he wrds Q Prove h he liude of he righ circulr cone of mimum volume h cn be inscribed in sphere of rdius r is r Also show h he mimum volume of he cone is 8 of he 7 volume of he sphere Sol Le VAB be cone of mimum volume inscribed in sphere of rdius r Le OC = Then AC r = Rdius of cone, VC = OC + VO = + r = Heigh of cone Then volume of cone, V = (AC) (VC) V (r )(r ) r dv r (r ) r r A d V And, 6 r dv For poins of locl mim & locl minim, we hve ie, r r (r )(r ) r =, r d V r We shll rejec r So, r r r = So, V is mimum r r r Now, heigh of cone VC = + r r = Hence Proved r r 8 Now, volume of he cone = r r r r 7 7 8 mimum volume of cone volume of sphere Hence Proved 7 Q5 Evlue : cos sin Sol Le I = cos sin Dividing Nr nd Dr by cos, we ge : For vrious suffs on Mhs, plese visi : wwwheopgupcom Pge 7
CBSE Annul Em Pper (All Indi) Compiled By OP Gup (+9-9788) sec ( n ) sec I = n n ( ) I d I d Now, pu v d dv Also v I dv n C v ( ) Pu n sec d Dividing Nr & Dr by v v n So, I n C I n C n Q6 Using inegrion, find he re of he region bounded by he ringle whose verices re (, ), (, 5) nd (, ) Sol Le A(, ), B(, 5) nd C(, ) form ringle ABC Equion of side AB : y = ( 7), Equion of side BC : y = ( ), Equion of side CA : y = ( 5) So, Required re of ABC = y y y AB BC CA = ( 7) ( ) ( 5) + + = ( 7) ( ) ( 5) + + = 7 9 SqUnis Q7 Find he equion of he plne hrough he line of inersecion of he plnes + y + z = nd + y + z = 5 which is perpendiculr o he plne y + z = Also find he disnce of he plne obined bove, from he origin OR Find he disnce of he poin (,, 5) from he poin of inersecion of he line r = i ˆ j ˆ+ k ˆ (i ˆ+ j ˆ+ k) ˆ nd he plne r(i ˆ j ˆ+ k) ˆ = Sol The equion of plne hrough he line of inersecion of he plnes + y + z = nd + y + z = 5 is given s : y z + λ( + y+ z 5) = ie, (+ ) (+ )y (+ )z 5λ = (i) Since (i) is perpendiculr o he plne y + z =, So (+ )() (+ )( ) (+ )() = = Using bb cc Subsiuing he vlue of = in (i), we hve : + + y + z 5 = ie, z = For vrious suffs on Mhs, plese visi : wwwheopgupcom Pge 8
CBSE Annul Em Pper (All Indi) Compiled By OP Gup (+9-96558) + Also, he disnce of he plne z = from origin (, ) is Unis ( ) OR Given line is r = i ˆ j ˆ+ k ˆ (i ˆ+ j ˆ+ k) ˆ so, he posiion vecor of coordines of ny rndom poin on his line is OP ( )i ˆ+ ( ) ˆj + ( )kˆ For he given line o inersec he plne r(i ˆ j ˆ+ k) ˆ =, he posiion vecor of coordines of ny rndom poin OP mus sisfy he equion of plne for some vlue of ie, ( )i ˆ+ ( ) ˆj + ( )k ˆ (iˆ j ˆ+ k) ˆ = ( ) + ( )( ) + ( ) = he poin of inersecion is OP iˆ + j ˆ kˆ ie, P(,, ) So, he required disnce of P from (,, 5) is ( ) ( ) ( 5) Unis Q8 A mnufcuring compny mkes wo ypes of eching ids A nd B of Mhemics for clss XII Ech ype of A requires 9 lbour hours of fbricing nd hour for finishing Ech ype of B requires lbour hours for fbricing nd lbour hours for finishing For fbricing nd finishing, he mimum lbour hours vilble per week re 8 nd respecively The compny mkes profi of `8 on ech piece of ype A nd ` on ech piece of ype B How mny pieces of ype A nd ype B should be mnufcured per week o ge mimum profi? Mke i s n LPP nd solve grphiclly Wh is he mimum profi per week? Sol Le he number of pieces of ype A nd ype B mnufcured per week be nd y respecively To mimize : Z ` (8 + y) Subjec o consrins :, y, 9 y 8 y 6, y Corner poins of fesible region Vlue of Z (in `) O(, ) A(, ) B(, 6) 68 C(, ) 6 Scle : smll boes on boh he es = unis So, mimum profi of `68 is obined when pieces of ype A nd 6 pieces of ype B re mnufcured by he compny per week Q9 There re hree coins One is wo-heded coin (hving hed on boh fces), noher is bised coin h comes up heds 75% of he imes nd hird is lso bised coin h comes up ils % of he imes One of he hree coins is chosen rndom nd ossed, nd i shows heds Wh is he probbiliy h i ws he wo-heded coin? OR Two numbers re seleced rndom (wihou replcemen) from he firs si posiive inegers Le X denoe he lrger of he wo numbers obined Find he probbiliy disribuion of he rndom vrible X, nd hence find he men of he disribuion Sol Le E : choosing firs (wo heded) coin, E : choosing second (bised) coin, E : choosing hird coin Also, le A : he coin showing heds 75 6 P(E ) P(E ) P(E ), P(A E ), P(A E ), P(A E ) P(A E )P(E ) By Byes' Theorem, P(E A) P(A E )P(E ) P(A E )P(E ) P(A E )P(E ) For vrious suffs on Mhs, plese visi : wwwheopgupcom Pge 9
CBSE Annul Em Pper (All Indi) Compiled By OP Gup (+9-9788) 7 5 OR 65 Tol number of wys of selecing wo numbers form si posiive inegers Le X denoes he lrger of he Two numbers seleced So, vlues of X :,,, 5, 6 5 So, P(X ), P(X ), P(X ), P(X 5), P(X 6) 5 5 5 5 5 disribuion cn be wrien s : 6 C 5 X 5 6 P(X) /5 /5 /5 /5 5/5 5 7 Therefore, men of he disribuion = X P(X) = 5 6 5 5 5 5 5 5 NOTE: Only hose Quesions from Se nd re given here which re no in common wih Se SET Wih Complee Eplnions Q9 Evlue : e e log Sol Le I log e e Pu log d Also when d I log log log log e nd, when = e Q Find vecor of mgniude 5, mking n ngle of wih -is, wih y-is nd n cue ngle wih θ wih z-is Sol Le â be he uni vecor in he direcion of vecor Since vecor mkes n ngle of wih -is, wih y-is nd n cue ngle wih θ wih z- is herefore, cos cos cos θ [Using cos α cos β cos γ ie, cos θ cos θ θ Therefore, ˆ 5 5 cos ˆi cos ˆj cos kˆ = = 5i ˆ 5kˆ b+ c c+ + b b c Q9 Using properies of deerminns, prove h : q+ r r + p p+ q p q r y+ z z+ + y y z Sol LHS : Le b+ c c+ + b q+ r r+ p p+ q [Apply C C (C C ) y+ z z+ + y For vrious suffs on Mhs, plese visi : wwwheopgupcom Pge
CBSE Annul Em Pper (All Indi) Compiled By OP Gup (+9-96558) c+ + b c+ + b p r + p p+ q p r + p p+ q z+ + y z+ + y c b [Tking common from C p r q [By C C C, C C C z y b c ( ) p q r [By C C y z b c p q r RHS y z dy b Q If sin ( cos ) nd y = bcos( cos), show h, Sol We hve sin ( cos ) cos ( cos ) sin d dy And, y = bcos ( cos ) bcos sin bsin ( cos ) d dy dy d bcos sin bsin ( cos ) b[cos sin sin ( cos)] d cos ( cos ) sin [ cos ( cos ) sin ] Since we know h, sin nd cos, dy b[( )] b So, Hence Proved [( ) ] Q Find he priculr soluion of he differenil equion ( y ) y(+ )dy, given h y = when = Sol We hve ( y ) y(+ )dy y dy (+ ) ( y ) ie, y dy (+ ) ( y ) log + log + y log k log + log + y log C, where log C log k + C (i) + y Given h y = when = so by (i), + C C = + Therefore, he required priculr soluion is (+ ) + y or, y + Q Find he vecor nd Cresin equions of he line pssing hrough he poin (,, ) nd perpendiculr o he lines y z nd y z 5 Sol The dr s of given lines y z nd y z 5 re respecively,, ;,, 5 Le he dr s of required line be, b, c er So by using bb cc for lines, we hve : b c nd b 5c b c 8 he dr's re, 7, for he required line For vrious suffs on Mhs, plese visi : wwwheopgupcom Pge
CBSE Annul Em Pper (All Indi) Compiled By OP Gup (+9-9788) So, he eqion of line in Cresin form is : y z nd, he vecor form of he line is 7 given s : r = i ˆ ˆj + k ˆ (i ˆ 7ˆj + k) ˆ n co sin cos I n co sin cos cos sin sin cos d Q8 Evlue : Sol Le Pu Also cos sin sin cos I d I I sin (sin cos ) C d sin C Q8 Prove h he heigh of he cylinder of mimum volume h cn be inscribed in sphere of rdius R is R Also find he mimum volume Sol Le r nd h be he rdius nd heigh of cylinder inscribed in sphere of rdius R In BDA, BD AD AB [By using Pyhgors heorem h (r) (R) R h r (i) Now volume of he cylinder, V r h R h V h [By using (i) V R h h Differeniing wr h boh he sides : dv R h dh dv h Agin differeniing wr h boh he sides : 6h dh dv For poins of locl mim & minim, dh R h h R R h Now, R d V R dh R h = So, V is mimum R h Hence, heigh of he cylinder of mimum volume h cn be inscribed in sphere of rdius R is R For vrious suffs on Mhs, plese visi : wwwheopgupcom Pge
CBSE Annul Em Pper (All Indi) Compiled By OP Gup (+9-96558) Also volume, Hence mimum volume is R R R V R Cubic Unis R Cubic Unis Q9 If Sol, find he vlue of 8 We hve 8 SET Wih Complee Eplnions n 8 n n 8 n n () = Q If nd b re perpendiculr vecors, b = nd = 5, find he vlue of b Sol We hve b = b = ( b)( b) = 69 b b = 69 5 b () = 69 [As b b 5 b = 69 b = b = Q9 Using properies of deerminns, prove h : Sol LHS : Le b c b c bc b b b Apply R R,R R,R R b bc bc c b c [By R R R R c c c b c b c b c bc b b b c c c [Tking common from R b c For vrious suffs on Mhs, plese visi : wwwheopgupcom Pge
CBSE Annul Em Pper (All Indi) Compiled By OP Gup (+9-9788) Q If Sol bc b c b b b c c c bc b c b c bc b c bc bc c b RHS cos ( cos ) nd We hve And, [By C C C, C C C [Epnding long R y sin ( sin ), find he vlue of dy cos ( cos ) cos cos 6cos sin sin = sin (cos ) dy d y sin ( sin ) sin sin cos 6 sin cos = cos ( sin ) dy dy d cos ( sin ) co d sin (cos ) dy So, co (/) = / Q Find he priculr soluion of he differenil equion y when For vrious suffs on Mhs, plese visi : wwwheopgupcom Pge d dy log + y, given h dy dy Sol We hve log + y y y + y e e e e dy e y y e e C e + e + C 7 Given h y when so, e + e + C C y Therefore, he required priculr soluion is : e + e 7 7y z 7 7 y 5 6 z Q Find he vlue of p so h he lines l : nd l : re p p 5 perpendiculr o ech oher Also find he equions of line pssing hrough poin (,, ) nd prllel o he line l y z y 5 z 6 Sol Given lines re l : nd l p : p 5 7 7 p p So, he dr s of hese lines re,,;,, 5 7 7 Since l l so, by using bb cc we hve : p p ( ) () ()( 5) p 7 7 7
CBSE Annul Em Pper (All Indi) Compiled By OP Gup (+9-96558) Also, he equion of line pssing hrough (,, ) nd prllel o l is : y z ie, y z [Since prllel lines hve proporionl dr s 7 7 Q8 If he sum of he lenghs of he hypoenuse nd side of righ ringle is given, show h he re o of he ringle is mimum, when he ngle beween hem is 6 Sol Le he lengh of he side AB of righ ringle be nd h of hypoenuse AC be y Given h + y = k (i) Are of ringle, A BC AB A y Le S A (y ) S [(k ) ] S [k k ] ds [k 6k ] d S [k k] For he locl poins of mim nd/or minim, ds k ie, [k 6k ] d S k [k k ] k/ k Th implies, S is mimum nd, hence A is lso mimum k Now, cos A y k k k cos A k cos A cos A cos ie, A 6 Q9 Evlue : sin sin cos cos Sol Le I sin sin cos cos o Hence Proved For vrious suffs on Mhs, plese visi : wwwheopgupcom Pge 5
CBSE Annul Em Pper (All Indi) Compiled By OP Gup (+9-9788) sec [Dividing Nr & Dr by n n ( n )sec n n ( ) cos Pu n sec d I d [Divide Nr & Dr by I d Pu u d du Also u u I du u ( ) I u n C I n C I n C So, n I n C n Der Suden/Techer, I would urge you for lile fvour Plese noify me bou ny error(s) you noice in his (or oher Mhs) work I would be beneficil for ll he fuure lerners of Mhs like us Any consrucive criicism will be well cknowledged Plese find below my conc info when you decide o offer me your vluble suggesions I m looking forwrd for response Also I would wish if you inform your friends/sudens bou my effors for Mhs so h hey my lso benefi Le s lern Mhs wih smile :-) For ny clrificion(s), plese conc : MhsGuru OP Gup [Mhs (Hons), E & C Engg, Indir Awrd Winner] Conc Nos : +9-965 5 8 +9-978 8 Mil me : heopgup@gmilcom Officil Web-pge : wwwheopgupcom @heopgup For vrious suffs on Mhs, plese visi : wwwheopgupcom Pge 6