Preparred by A.Immanuvel Maduram Thangiah, St. John s HSS, Palayamkottai Key for March 2015 Maths Questions Pl.visit 12th-maths-key.weebly.

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1 Prprrd A.Immnuvl Mdurm Thngh, S. John s HSS, Plmo K for Mrh 5 Mhs Qusons Pl.vs h-mhs-.wl.om Mrh 5 Hghr Sondr Mhms A I Answr ll h Qusons. =. Answr :. Infnl mn soluon. Answr : d. ll h ov. Answr : 8. Answr :.,, 5. Answr : d. p q q p. Answr : Answr : d. oh h s 8. Answr : d. nowhr 9. Answr :.. Answr : d. :. Answr :. 8. Answr :.. Answr :. 8. Answr :. os nθ 5. Answr :.. Answr :. u 7. Answr :. s. Answr : d.<θ<. Answr : d. 9,. Answr :.*=. Answr :.. Answr : d. 5. Answr :.. Answr : d. sn n n 7. Answr : d. 8. Answr :. n 9. Answr :. Answr : d. lm=n prpndulr o 8. Answr : onssn 9. Answr :.Aos+Bsn. Answr :.. Answr : d.,. Answr :. d+d=. Answr :.,. Answr : Answr :. I. F. I. F. Q d. Answr : d. = 7. Answr :. B = 8. Answr : d Answr :. nd Pg

2 Prprrd A.Immnuvl Mdurm Thngh, S. John s HSS, Plmo K for Mrh 5 Mhs Qusons Pl.vs h-mhs-.wl.om. If h quon l m suh h l m n, hn h ssm hs n. non ro unqu soluon. rvl soluon. Infnl mn soluon d. No Soluon.. If X s onnuous rndom vrl hn P<X<=.P X.P<X.P X< d. ll h ov. If p+q= + hn q s 8 d8. Th pon of nrson of h lns r nd r 5 s s.,,.,,.,, d.,, 5. p q s quvln o.p q.q p.p q q p d. p q q p. If rds r drwn from wll shuffld p of 5 rds, h pro-l h h r of h sm olours whou rplmn, s. 5 d Th urv 9 = s smmrl ou. -s. -s. = d. oh h s 8. Th pon of nflon of h urv = s. =. =. = d. nowhr 9. Gvn EX+ = 8 & EX = hn vlu of s... d.. Volum of sold ond rvolvng h r of h llps ou mor nd mnor s r n h ro. :. :. : d. :. If ompound smn s md up of hr smpl smn, hn h numr of rows n h ruh l s d.. In h group Z 5 {[]}, 5, [] s.5.. d.. Th lngh of h r of h urv / + / = s. 8.. d.9. If h vlu of s. os nθ. sn nθ. sn nθ d. os nθ 5. Th vlu of [,, ] s qul o... d.. If u, hn. u s un vor. 7. If hn u. u d. u. s prlll o. s prpndulr o. d. nd r un vors 8. If ρa=ρ*a B+ hn h ssm s onssn nd hs nfnl mn soluon onssn nd hs unqu soluon onssn d nonssn 9. Th omplmnr funon of D += s.a+b.aos+bsn.a+b d.a+b. If f s n odd funon hn f d s. f d. f d. d. f d. Th lngh of h sm-mor nd h lngh of sm mnor s of h llps r 9 nd Pg

3 Prprrd A.Immnuvl Mdurm Thngh, S. John s HSS, Plmo K for Mrh 5 Mhs Qusons Pl.vs h-mhs-.wl.om.,.,., d.,. Th dffrnl quon of ll rls wh nr h orgn s. d+d=. d d=. d+d= d. d d=. Th funon f = s drsng n.,.,., d.,. For Posson dsruon wh prmr =.5 h vlu of h nd momn ou h orgn s d Th soluon of lnr dffrnl quon P Q whr P nd Q r funons of, s. I. F. I. F. Q d. I. F. I. F. Q d. I. F. I. F. Q d d. I. F. I. F. Q d. Th urv = - us h -s.=, =. =,=. =, = d. = 7. If A nd B r n wo mrs suh h AB = nd A s non sngulr, hn. B =. B s sngulr. B s non sngulr d. B = A 8. Th nr of h hprol whos lus rum s qul o hlf of s onug s s.. 5. d Th vlu of d s... d.. In h lw of mn, h vlu of ssfs h ondon.θ>.θ<.θ< d.<θ<. Th modulus nd mplud of h ompl numr r rspvl. 9,. 9,., d. 9,. Whh of h followng s no nr opron on R.*=.*=.*= d.*= d d. Th slop of h norml o h urv = h pon whos oordn s s... d.. Th ln s ngn o h prol hn s... d. 5. If h mr. 5 hs n nvrs hn h vlus of. s n rl numr. d.. Th ngl wn h ln r nd h pln r n q s onnd h rlon. n os. n os q. sn n d. sn n n n 7. Dgr of h dffrnl quon / d d d d... d. 8. If s h nh roo of un hn n n d. n 9. Th wor don h for F n movng h pon of pplon from,, o,, long srgh ln s gvn o 5 uns. Th vlu of s 8 d 8. Th ondon h h ln l+m+n= m ngn o h rngulr hprol = s nd Pg

4 Prprrd A.Immnuvl Mdurm Thngh, S. John s HSS, Plmo K for Mrh 5 Mhs Qusons Pl.vs h-mhs-.wl.om nd Pg. l + m =n.m =ln. l - m =n d. lm=n II. Answr An NINE AND 55 TH =. Fnd h rn of h mr 7 L A = 7 A~ 7 C C A~ 5 5 R R R A~ R R R Th ls quvln mr s n hlon form. And hs wo non-ro rows. ρ A =. Solv usng d. mhod =7, += Th mr qn. s 7 L 7 7 B Crmmr s rul nd Th soluon s s {=, = }. Vrf. d, f,,, d d. d. If A,, s on nd of dmr AB of h sphr =, hn fnd h oordns of B. Fnd h ngl wn h ln nd h pln +++5= Th qn. of sphr s = Hr u = Co-ff.of = u = / v = Co-ff.of = v = w= Co-ff.of = + w = + Cnr s u, v, w = /,, If A,, s on nd of dmr AB L h oordns of B,, Th md pon of AB s h nr /,,

5 Prprrd A.Immnuvl Mdurm Thngh, S. John s HSS, Plmo K for Mrh 5 Mhs Qusons Pl.vs h-mhs-.wl.om..,, ;, ; ;,, ; ; ; Th oordns of B r,, =,, L Ɵ h ngl wn h ln nd h pln., n n = 9 = n sn n n sn ngl sn P rprsns h vrl ompl numr Z. Fnd h lous of P, f 5 5. L = Whh s h lous of P.. Fnd h squr roo of 7+ L 7 On squrng oh sds w g, 7+= + Equng h rl nd mgnr prs = = Also + = = = Solvng nd w g =9 nd = =± nd =± Bu s posv nd hv h sm sgn. =, = or =, = 7 or 7. Evlu Lm Drmn h domn h onv of h urv = Lm whh s n ndrmn form Applng L Hopl s Rul w g Lm whh s n ndrmn form Agn pplng L Hopl s Rul w g Lm = = =, Hr s lws v Th urv s onv downwrds, 8. A wh ngls θ do h urvs= nd = nrs? nd Pg 5

6 Prprrd A.Immnuvl Mdurm Thngh, S. John s HSS, Plmo K for Mrh 5 Mhs Qusons Pl.vs h-mhs-.wl.om 9. If u=log + u u, Fnd. u Dffrn u prll w.r.. u Dffrn u prll w.r.. u u 5. Evlu : Sol n :- L I = I / d / n sn I d os sn sn os + I= sn os d d n d 5. Solv: d d d d d d d Ingrng w g os d os sn os d os sn sn sn os os d os sn sn os sn os d d I I d n nd Pg

7 Prprrd A.Immnuvl Mdurm Thngh, S. John s HSS, Plmo K for Mrh 5 Mhs Qusons Pl.vs h-mhs-.wl.om log s h soluon. 5. Prpr ruh l for p q p q Thr r TWO smns p,q Th no of rows n h l r = p q p q p q p q p q T T T F T T F F T T F T F T T F F F T T 5. Fnd h mn nd vrn of h dsruon : Mn E f d d E f d d 9! f,, lswhr! n n d Gmm d n 9 9 Vr E [ E X ] Mn ; Vrn 9 5. Mrs n n pud s gvn o 8 sudns of shool ws found o normll dsrud. % of h sudns sord low mrs nd % of h sudns sord ov 9 mrs. Fnd h numr of sudns sord wn & 9. L X h mrs of sudn. 9! P<=% nd P>9=% P<X<9=P <X< {P<+P>9} = % % % = 8% Ou of 8 sudns numr of sudn sord wn & 9 = 55. Fnd h quon of h hprol whos nr s, h dsn wn h drs s / nd h dsn wn h fo s nd h rnsvrs s s prlll o -s. 8 8 = sudns. Show h h s of four mrs, undr mulplon of mrs. [OR],,, form n ln group nd Pg 7

8 Prprrd A.Immnuvl Mdurm Thngh, S. John s HSS, Plmo K for Mrh 5 Mhs Qusons Pl.vs h-mhs-.wl.om III. Answr An NINE AND 7 TH = nd Pg 8

9 Prprrd A.Immnuvl Mdurm Thngh, S. John s HSS, Plmo K for Mrh 5 Mhs Qusons Pl.vs h-mhs-.wl.om 5. Solv mr drmnn mhod + =, + =, + = Th gvn ssm s onssn nd hs nfnl Th mr quon s mn soluons. T = Now h ssm hs n rdud o wo L quons Th r + = + nd + = 8 L L 8 L 8 L L L 8 B Crmmr s rul,, =, = nd = Sn Δ= & Δₓ= Δ= Δ= nd on mnor Soluon {,, } 57. Aluds of rngl r onurrn-prov vor mhod Now BE CA OB CA In ABC, l AD, BE h wo luds nrsng O. In ordr o prov h h luds r onurrn, s suffn o prov h CO s lr o AB. Tng O s h orgn. l h poson vrors of A. B. C,, rspvl. Thn OA, OB, OC Now AD BC OA BC OA BC 58. Fnd ll h vlus of L ---- = ros θ + sn θ r= r OB CA Addng & w g BA OC ---- OC AB Hn h hr luds r ongrun. nd hn prov h h produ of h vlus s. Equng h rl & mgnr prs, w g os θ = nd sn θ = +v nd +v θ ls n h I Qudrn. θ = α θ = [os sn ] [os sn ] [os sn ] Add π wh h rgumn, w g, [os sn ] Appl D Movr s Thorm, w g, os sn =,,, nd Pg 9

10 Prprrd A.Immnuvl Mdurm Thngh, S. John s HSS, Plmo K for Mrh 5 Mhs Qusons Pl.vs h-mhs-.wl.om Th vlus r s, s, s 5, s 7 Now hr Produ s= s 5 7 = s s = os sn = + = Hn provd 59. Fnd h s, vr, fous, drr, lngh nd quon of lusrum nd drw hr grph for h prols 8 +7= 8 +7= + = = = 8 = 8 Ths s of h form Y = X Whr X=, Y= nd =8 Ths prol opn rghwrd Rffrrd o X,Y s = X, =Y As of h prol s X-s,., Y= Vr s, X=, Y= Fous s,=, X=, Y= Eqn. of Drr X= X= Eqn. Lusrum X= X= Lngh of Lusrum =8 of Rffrrd o, s =X+, =Y+ Y= =+= = X= =+= Y= =+= Vr s, X= =+= Y= =+= Fous s, X= = += = X= = += = =8. A ho-ho plr n pr ssson whl runnng rlss h h sum of h dsns from h wo ho-ho pols from hm s lws 8m. Fnd h quon of h ph rd hm f h dsn wn h pols s m. From h gvn d, h wo ho-ho pols n s h fo F nd F. L P, h poson of h plr. F P + F P = 8 = 8 = & F F = = = nd Pg

11 Prprrd A.Immnuvl Mdurm Thngh, S. John s HSS, Plmo K for Mrh 5 Mhs Qusons Pl.vs h-mhs-.wl.om Also = 9 7 = 7 Th quon of h ph s.., 7. Fnd h quon of h hprol f s smpos r prlll o + = nd +8=,, s h nr of h hprol nd psss hrough,. Th smpos r prlll o h lns + = & +8= Th quons of smpos r of h form ++m= nd +n= Bu hs r pssng hrough h nr, +8+m= nd 8+n= m= nd n= Th quons of smpos r + = nd += Th omnd quon of smpos s + += Th quon of h hprol s of h form + ++= Ths hprols psss hrough, + ++= = Th quon of h hprol s + ++=. Fnd h lol mnmum nd mmum vlus of f= + Sol n :- L f = + f = 9 + f = 8+ = + = A h rl pon f = 9 + = = =, nd / A =, f = = Thus h II drvv s gvs no nformon ou h rmum nur of f = = s no h m. or mn. pon. f = + f =, s no h rmum pon. A = /, f / = + v > = / s h mnmum pon. f = + f / = / / + / / = / 5 / 9 + / / = 7 / 5 s h lol mnmum vlu h mnmum pon s /, 7 / 5. Us dffrnls o fnd nd Pg

12 Prprrd A.Immnuvl Mdurm Thngh, S. John s HSS, Plmo K for Mrh 5 Mhs Qusons Pl.vs h-mhs-.wl.om. Fnd h r wn h ln = nd h urv =. From h fgur, us -s o fnd h r. To fnd h lms of, solv h wo quons. = ½ + = + 8= + = = nd = Th rqurd r Rgh lf d d 8 8 = 9 Sq. Uns. 5. Fnd h surf r of h sold gnrd rvolvng h lod sn, os ou s s X-s To g h nrsng pon wh -s,pu =.., os os, os os Gvn h = + sn & = + os os os d d = + os & = sn d d d d Surf r d d d d d os sn d d os os d os os sn os os sn os os d nd Pg

13 Prprrd A.Immnuvl Mdurm Thngh, S. John s HSS, Plmo K for Mrh 5 Mhs Qusons Pl.vs h-mhs-.wl.om 8 os d T os d d d. Solv: d d Ths qn.n wrn s Pu = v.--- Dff. Su.& n w g dv v v d v dv v v d v d d d d dv v v dv v d v d v v d dv Tng ngron on oh sds v v d dv pu v v 8v dv d d d 8 v dv d 8 v dv d I Sq. Uns log log log v 8 log v log v log 8log log log 8 log 8 v log v Th sum of Rs. s ompoundd onnuousl h nomnl r of nrs % pr nnum. In how mn rs wll h moun w h orgnl prnpl? Log =.9 L A h Prnpl moun nd vrs prulr m rs. R of nrs % W.. da da da A A. A d d d da. d Tng ngron on oh sds A log A. log A A. log.. A Whn =, h moun A= Now w hv o fnd, whn A=.. log. log.9 7.rs.. Afr 7 rs h moun wll dould. 8. Show h h s G of ll mrs of h form whr R {} s group undr mr mulplon. L G / R {} w shll show L A= G, B= G h G s group undr mr mulplon., Closur om: AB G nd Pg

14 Prprrd A.Immnuvl Mdurm Thngh, S. John s HSS, Plmo K for Mrh 5 Mhs Qusons Pl.vs h-mhs-.wl.om nd Pg Closur om s ru. Assov om: Mr mulplon s lws ssov. Assov om s ru. Idn om: L E= B h df n. of, B h df n. of M, AE= = Thus E = G suh h AE = A for ll AG Idn om s ru. Invrs om: L A = G, AG B h df n. of nvrs, AA =E B h df n. of Mr Mulplon, AA = = Thus A = G suh h AA =E AG Invrs om s ru. G s group undr mr mulplon. Hn provd. 7. Drv h quon of pln n h nrp form. L, & h X,Y & Z nrps of h pln rspvl. Th pln psss hrough h pons,,,,, &,, Th v.qn. s s s r.. s s,. Equng h Co-ffns of, nd vors = s, =s nd = nd s s,, Addng hs hr w g s s Crsn form: Hr,, =,,,,, =,, &,, =,, Th Cr. qn. s.., + + = + + = G A G, A AE

15 Prprrd A.Immnuvl Mdurm Thngh, S. John s HSS, Plmo K for Mrh 5 Mhs Qusons Pl.vs h-mhs-.wl.om [OR] 9. A rndom vrl X hs h followng prol mss funon X 5 PX= Fnd, Evlu p X<, p X5 & p < X, Fnd h smlls vlu of for whh p X > ½. Sn s prol mss funon, Th smlls vlu of s nd Pg 5

16 Prprrd A.Immnuvl Mdurm Thngh, S. John s HSS, Plmo K for Mrh 5 Mhs Qusons Pl.vs h-mhs-.wl.om A noon, shp A s m Ws of shp B. Shp A s slng Es 5 m/hr nd shp B s slng norh 5 m/hr. How fs s h dsn wn h shps hngng. p.m. L P & Q h nl poson of shps A & B Afr hours, from h gvn d, shps A & B rvlld ms nd ms. From h dgrm = + Dffrnng w g, d d d d d d Gvn h =, =. Thn 9 d d d And d d d d d Km / hr. Th dsn wn h shps s hngng 95 h r Km / hr. 9 nd Pg

17 Prprrd A.Immnuvl Mdurm Thngh, S. John s HSS, Plmo K for Mrh 5 Mhs Qusons Pl.vs h-mhs-.wl.om nd Pg 7