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1 Wnglsh Tuton Centre Puduvl + Mths Q & A Mthl Pulshers Krud PROVE BY FACTOR METHOD OF DETERMINANTS. ). ). ). ). ) 6. ) ) ) ). ) ) 8. ) 9 ) Solve ) PROPERTIES OF DETERMINANTS. 0 / / /. 0. ). ) ) ) z z z z z z. 0 z z z Show tht z= PRODUCT OF DETERMINANTS

2 Wnglsh Tuton Centre Puduvl + Mths Q & A Mthl Pulshers Krud VECTORS. ) Show tht gven vetors re Coplnr ) Show tht gven vetors re Coplnr ) Show tht gven vetors re Coplnr d) S.T. ponts gven vetors re oplnr 9. ) Show tht ponts whose p.v gven re ollner ) Show tht ponts whose p.v gven re ollner 0 ) Show tht ponts whose p.v gven re ollner 6 d) Show tht ponts whose p.v gven re ollner e) Show tht ponts whose p.v gven re ollner 8 f) Show tht ponts whose p.v gven re ollner g) Fnd m f gven vetors re ollner m 6 nd. ) Show tht Vertes of trngle whose p.v s gven form equlterl trngle ) Show tht Vertes of trngle whose p.v s gven form equlterl trngle ) Fnd entrod of trngle whose p.v of vertes d) Fnd length of sdes f vertes p.v of trngle 6. ST gven vetors form rght ngled trngle. ) Fnd Mgntude nd dretonl osnes of ) Fnd Mgntude nd d of sum of vetors ) Fnd Mgntude nd d of sum of vetors 6. ) Fnd Unt vetors n the dreton of ) Fnd Unt vetors n the dreton of. ) Fnd Unt vetors prllel to ) Fnd Unt vetors prllel to sum of 8 ) Fnd Unt vetors prllel to where d) Fnd Unt vetors prllel to whose mgntude unts BINOMIAL THEOREM. ) Fnd the oeffent of ) Fnd the oeffent of. ) Fnd the onstnt term 0 ) Fnd the term ndependent of ) Fnd term ndependent of 9 d) Fnd the term ndependent of ) Fnd mddle term n 8 ) Fnd mddle term n 6 ) Fnd mddle term n 6 d) Fnd mddle term n )

3 Wnglsh Tuton Centre Puduvl + Mths Q & A e) Fnd mddle term n BINOMIAL SERIES. Fnd frst four terms n the epnson of ) ) ) ) ) ) d) 6. ) Fnd oeffent of 8 n ) ) Fnd th term n the epnson ) ) Fnd r+) th term n the epnson ) 6. Evlute orret to deml ples ) 6 ) 00 ) 8. ) If s lrge show tht. ) If tna = tn B = show tht A+B = ) If s lrge show tht 9 ) If tn α = tn β = show tht α+β =. If A + B = show tht ) If s smll show tht ) + tna+tnb) = ) ota ) otb ) = 8. ) Show tht ) Hene dedue the vlue of tn ½ n ) n n n n..!. If tnα = nd tn β = show tht α+β = ) Show tht 6. If os = + prove tht os = ) n n n n..... ) Show tht sn 0 sn0 sn80 = 8 MATHEMATICAL INDUCTION 9. Prove mthemtl nduton tht ) Prove tht sn0 sn0 sn60 sn80 = 6 ) n +n s even ) n+n - ) s odd ) Show tht os0 os0 os80 = 8 0. Prove mthemtl nduton tht d) Prove tht os0 os0 os60 os80 = n n ) )... n 6 8. If A + B + C = π prove tht n n n ) )... n 6 ) sna + snb + snc = sna snb snc n n ) )... n ) osa + osb osc = sna snb osc ) sna snb + snc = osa snb osc d) 6... n n n ) e)... n ) n 9. If A + B + C = π prove tht nn ) ) os A+ os B os C= f)... n ) sn Asn B os C ) sn A + sn B + sn C = + osa osb osc g) 8... n n n ) ) sn A sn B sn C A B C + + = -sn sn sn h)... n n 0. If A + B + C = 90 show tht. Prove mthemtl nduton tht ) n - s dvsle for ll nturl numers n sna + snb + snc ot A ot B sna + snb - snc ) n - s dvsle for ll nturl numers n Mthl Pulshers Krud ) 0 n - + s dvsle d) nn + ) n + ) s dvsle 6 e) S = n + n + n n s dvsle for ll f) n 6n - s dvsle 6 g) n ns dvsle - TRIGNOMETRY. If A B re ute ngles ) If sna = ; os B= fnd osa + B) ) If sna = snb = fnd sn A + B). If α + β) nd α - β) re ute ) If osα+β) = ; sn α β) = fnd tn α

4 Wnglsh Tuton Centre Puduvl + Mths Q & A TRIGNOMETRIC EQUATION I. Solve :. os + sn + os = 0. os θ + snθ = 0 I. In trngle ABC prove tht. sna snb = sna B). sn B C) = 0. snb C) + snc A) + sna B) = 0. sn θ osθ + ¼ = 0 sn A - B).. sn + sn = sn A +B). tnθ + ) tnθ = 0 B C A. os sn 6. tn = tn. tnθ otθ = 6. B C) sn C A) sna B) 0 sn A sn B sn C II. Solve :. If osa = osb show tht the trngle s ether. sn = sn n soseles trngle or rght ngled trngle?. sn + sn = 0 II. In trngle ABC prove tht. sn + sn6 + sn = 0 8. osc osb) =. os + os + os = 0 A A. sn + sn = sn 9. = + ) sn + ) os III. Solve : os A os B os C 0.. sn + os =. snθ + osθ = III. In trngle ABC prove tht. + ) osa =. snθ osθ =. + ) osa = + +. seθ + tnθ =. snb snc) = 0. oseθ otθ = IV. In trngle ABC prove tht INVERSE TRIGONOMETRIC FUNCTION tn A. I. Prove tht : tn B. tn tn tn Msellenous 9. If osθ + snθ = osθ show tht. os tn tn osθ snθ = snθ. tn m tn m n. Prove tht +tna + sea) +ota osea)= n m n. If tnθ + snθ = p tnθ snθ = q nd p > q then. tn tn show tht p q = pq. If tnθ + seθ = show tht. tn ot tnθ = seθ = sn tn tn 6. tn. z z tn tn tn z tn z z π 8. Solve: tn + tn = II. Prove tht :. sn sn sn [. sn sn sn[. os os os [. os os os[ Mthl Pulshers Krud If A + B + C = π prove tht ) tna + tnb + tnc = tna tnb tnc ) tna + tnb + tnc = tna tnb tnc + sn os 6. Prove tht tn + sn os. If tnθ = fnd tnθ 8. If sna = fnd sna 9. Prove tht: sna + sn0 +A) + sn0 +A) = 0 0. Prove tht osa + os0 +A)+os0 A) = 0

5 Wnglsh Tuton Centre Puduvl + Mths Q & A STRAIGHT LINES I. Perpendulr Dstne. Fnd length of the perpendulr from ) to the lne + 9 = 0. Fnd o-ordntes of the ponts on the strght lne = + whh re t dstne of unts from the strght lne + 0 = 0. Fnd ponts on -s whose perpendulr dstne from = 0 s. Fnd the length of the perpendulr from ) to the strght lne + + =0.. Fnd the dstne of the lne = 0 from ) long the strght lne mng wth the postve dreton of the -s. VI. Conurren. Show tht the strght lnes re onurrent + = ; + = 0 nd =.. Show tht the strght lnes re onurrent + =; + = 0 nd =. Fnd for whh strght lnes re onurrent + + = 0 + = 0 nd + = 0. Fnd for whh strght lnes re onurrent 6+ = 0 ++ = 0 nd + + = 0. Fnd for whh strght lnes re onurrent + = 0 + = 0 nd + = 0 6. If the gven equtons re onurrent + + = 0 ++ = 0 nd + + = 0 II. Fnd dstne etween prllel lnes. + 9 = 0 nd + + = 0 VII. Show tht + + = Co-ordntes of orthoentre of the trngle. + 6=0 nd + + = 0. III. Fnd the equtons of. Medns of the trngle formed the ponts ) 6) nd 6 0).. Dgonls of qudrlterl whose vertes re. Fnd the o-ordntes of the orthoentre of the trngle whose vertes re the ponts ) 6 ) nd ). Fnd the o-ordntes of orthoentre of the trngle formed the strght lnes ) ) 6) nd 6 8) = 0 8 = 0 nd 9 = 0 IV. Equton of Strght Lne. Fnd the o-ordntes of the orthoentre of the. pssng through pont ) nd mng trngle formed the strght lnes nterepts on the o-ordnte es whh re n + = 0 + = 0 nd + 9 = 0 the rto :.. through the pont ) nd hvng nterepts whose sum s 9.. Fnd the o-ordntes of the orthoentre of the trngle formed the strght lnes + = 0 + = nd + = 0.. psses through the pont ) whose PAIR OF STRAIGHT LINES nterept on the -s s tmes ts nterept on the -s. whh ut off nterepts on the es whose I. Sum &produt of slopes of pr of strght lnes. Slope of one of the strght lnes of +h + = 0 s thre tht of other sum nd produt re nd 6 respetvel. show tht h = V. Angle etween two strght lnes. Fnd the ngle etween the strght lnes. Slope of one of the strght lnes + h + = 0 s twe tht of the other + = nd + = show tht 8h = 9.. Show tht the ngle etween + = 0 nd = 0 s equl to ngle etween + = 0 nd 9 + =. Show tht the trngle whose sdes re = + 6 = 0 nd + = 8 s rght ngled. Fnd ts other ngles.. Show tht strght lnes form soseles trngle 8 = 0 +6 = 0 + = 0. Show tht strght lnes form soseles trngle II. Angles etween pr of strght lnes. Fnd the ngle etween the strght lnes + + = 0. Fnd he ngle etween pr of strght lnes ) ) = 0. If ngles etween pr of strght lnes + h + =0 s 60 Show tht + ) + ) = h III. Condton for Pr of Strght Lnes Mthl Pulshers Krud = 0 + = 0 = 0.

6 Wnglsh Tuton Centre Puduvl 6 + Mths Q & A. If the gven equton represents pr of perpendulr strght lnes fnd nd = 0. Show tht gven equton represents pr of strght lnes. Fnd ngle etween them + = 0. Show tht the gven equton represents pr of strght lnes = 0. entre s ) rumferene s 8π unts.. entre ) nd rdus. Show tht t psses through 0) Show tht ngle etween them s tn. entre ) pssng through the pont ) IV. Seprte equton of the strght lnes.. S.T the eqn. represents pr of strght lnes = 0 Fnd seprte equton of strght lnes.. S.T the eqn. represents pr of prllel lnes 6. entre t ). pssng through the pont ). Two dmeters of wth rdus unts re + = + = 8 8. Etremtes of dmeter re = 0 ) nd ) Fnd dstne etween them.. S.T the eqn. represents pr of prllel lnes = 0 Fnd the dstne etween them. 9. Desred on the lne onng the ponts ) nd ) s ts dmeter. II. Generl equton of the rle. Fnd the entre nd rdus of the rle For wht vlue of does the eqn represents = 0 pr of strght lnes?. Fnd the entre nd rdus of the rle = 0 ) ) + ) ) = 0 Also wrte the seprte equtons.. Fnd the vlues of nd f the equton. Fnd suh tht the equton represents pr of strght lnes = 0 represents rle ) + + ) + + = 0. Fnd the vlues of nd f the equton Fnd ) seprte equtons of strght lnes ) ngle etween them. represents rle ) + + ) + + = 0 6. If the equton represents pr of strght lnes fnd the vlue of = 0 Wrte down resultng equton of the rle. Fnd rumferene nd re of the rle = 0 Fnd ) seprte equtons of strght lnes ) ngle etween them. III. Prmetr Equtons 6. Fnd the prmetr equtons of the rle V. Comned equton of the strght lnes.. Fnd the omned equton of the strght lnes whose seprte equtons re + = 0 nd + = 0. Fnd the omned equton of the strght lnes whose seprte equtons re + = 6. Fnd the prmetr equton of the rle + = 9 8. Fnd the rtesn equton of the rle = os θ = sn θ 9. Fnd the rtesn equton of the rle + = 0 nd + + = 0 = ¼ osθ = ¼ sn θ nd 0 θ π. Fnd omned equton of strght lnes through orgn one of whh s prllel to nd the other s perpendulr to + + = 0 IV. Equton of rle pssng through ponts. Fnd the equton the rle pssng through the ponts 0) ) nd ). Mthl Pulshers Krud CIRCLE I. Fnd the equton of the rle f. entre nd rdus re ) nd.. entre of the rle s ) nd re of rle s 6π squre unts.

7 Wnglsh Tuton Centre Puduvl + Mths Q & A. Fnd the equton of the rle pssng through = 0 the ponts 0) 0 )nd 0 ) = 0. Fnd the equton of the rle pssng through. Show tht the rles touh eh other. the ponts ) )nd ) = 0 V. Equton of rle hvng entre on St.Lne = 0.. Fnd equton of the rle pssng through the. Show tht eh of rles touhes other two ponts 0 ) ) nd hvng the entre on + + = 0 the lne + = = 0. Fnd the equton of the rle tht psses + = 0 through the ponts ) nd 6 ) nd hs ts II. Conentr rles: entre on the lne + = 6.. Fnd equton of rle onentr wth rle. Fnd equton of the rle whose entre s on = 0 the lne = nd whh psses through the nd pssng through the pont ). ponts ) nd ).. Fnd equton of rle onentr wth rle TANGENTS = 0. Fnd the length of the tngent nd psses through the pont ) from ) to the rle + += 0.. Fnd equton of rle onentr wth rle. Fnd the length of the tngent = 0 from ) to the rle = 0 nd hvng rdus.. Fnd the equton of the tngent to the rle III. Orthogonl rles: + = t ). Prove tht the gven rles re orthogonl. Fnd the equton of tngent to = = 0 t ) = 0. Fnd the equton of tngent to. Prove tht the gven rles re orthogonl = 0 t ) = 0 6. Fnd length of hord nterepted the rle + = = 0 nd the lne = III. Equton of Orthogonll touhng rles:. Fnd length of hord nterepted the rle. Fnd rle whh uts orthogonll eh of = 0 nd lne += = 0 8. Fnd the vlue of p f the lne + + = 0 + p = 0 s tngent to rle + = = 0 9. Fnd the vlue of p f the lne. Fnd rle whh uts orthogonll eh of + p = 0 s tngent to + 6 = = 0 0. Determne whether the pont le outsde/nsde = 0 ) the rle = = 0. Determne whether the pont le outsde/nsde. Fnd rle whh uts orthogonll eh of ) the rle + + = = 0 0 = 0? whh psses through the pont ). Determne whether the pont le outsde/nsde. Fnd rle whh uts orthogonll eh of ) 0 0) ) = 0 rle + + = = 0. Fnd the equton of the rle whh hs ts whh psses through ) entre t ) nd touhes the -s.. Fnd rle whh uts orthogonll eh of FAMILY OF CIRCLES = 0 I. Crles touhng eh other: + = 0. Show tht the rles touh eh other. whh psses through ) Mthl Pulshers Krud

8 Wnglsh Tuton Centre Puduvl 8 + Mths Q & A SPECIAL TYPES OF SEQUENCES AND SERIES. Fnd the nth prtl sum of the seres PARTIAL FRACTIONS I. Lner ftors none of whh s repeted ) n ) n n ) n + n.. + ) +) n. Fnd the sngle A.M etween ) + ) ) nd ) nd ) p + q) nd p q) + +. ) Fnd n rthmet mens etween nd nd. ) ) ) fnd ther sum II. Lner ftors some of whh re repeted ) Insert four A.Ms etween nd ) Fnd fve rthmet mens etween nd 9 ) + ) ) + ) v) Fnd s rthmet mens etween nd ) ) ) + ). ) Fnd n geometr mens etween nd nd fnd ther produt ) ) ) Fnd geometr mens etween 6 nd 9. III. Not ftorle nto lner ftors. ) th nd th terms of H.Pre nd 9 Fnd th term ) + ) ) ) st nd nd terms of H.P re nd 6.. ) ) Fnd 9 th term. IV. mproper frton. ) If p th nd q th terms of H.P re q nd p Show tht pq) th term s Three numers form H.P. Sum of numers s COMPOSITION OF FUNCTIONS sum of the reprols s one. Fnd the numers.. Let A = { } B = { } nd C = { 6} nd f:a B nd g : B C suh tht. If re n H.P. prove tht f) = f) = g) = g) = If re n A.P. nd m re n G.P then prove tht m re n H.P Fnd gof.. f : R R g : R R re defned 9. If s G.M of nd nd s A.M of nd nd f) = + g) =. s A.M of nd prove tht Show tht fog gof.. f g : R R e defned 0. Dfferene etween two postve numers s 8 nd tmes ther G.M s equl to tmes ther H.M. Fnd the numers.. If the A.M etween two numers s prove tht f) = + g) = Show tht fog) = gof).. f g : R R defned ther H.M s the squre of ther G.M. f) = + g) =. If s A.M of nd nd ) s G.M of Fnd : ) fog) ) ) gof) ) nd show tht : : = : : ) fof) ) v) gog) ). If re two dfferent postve numers then v) fog) ) v) gof) ) prove tht ) A.M. G.M. H.M. re n G.P. ) A.M > G.M > H.M. f : R R e funton defned f) = +. Mthl Pulshers Krud

9 Wnglsh Tuton Centre Puduvl 9 + Mths Q & A Fnd f 6. Let f : R R e defned f) = +. Show tht fof = f of = I. f g : R R re defned f) = + g)=. f Fnd f+g f g fg f g. g 8. f g : R R defned f) = + ; g) = Defne f+g) ) fg) ) v) gf) ) ) g f ) vf g) QUADRATIC INEQUATIONS I. Solve the neqult ) 0 ) < 0 ) 9 II. Solve the neqult ) + 6 > 0 ) + > 0 III. Evlute ) 8 > 0 ) + < 0 lm e lm e e.. ) III. Solve the qudrt nequton lm etn lm e sn.. ) < 0 ) 0 tn 0 + < 0 ) IV. Evlute lm log ) lm log IV. Solve the qudrt nequton.. 0 ) 0 ) ) V. Evlute lm lm.. ) ) 0 0 ) lm V. If s rel R) prove tht. 6 lm. 0 0 ) +. Rnge of f) = s + + lm 6. Compute 0 +. Rnge of f) = s + + lm Hene evlute 6 0. n't hve n vlue etween VI. Evlute n nd 9 lm. lm. n. les etween nd. 9 lm. os ) se LIMIT OF A FUNCTION I. Evlute Mthl Pulshers Krud lm lm m n lm h). 0 h 6. Fnd postve nteger n suh tht lm n n 08. Fnd postve nteger n suh tht lm n n II. Evlute lm sn. 0.. lm sn 0 lm sn 0 sn.. lm ) 0 lm lm os. 0 lm. sn 0 lm sn ) sn )

10 Wnglsh Tuton Centre Puduvl 0 + Mths Q & A DIFFERENTIATION TECHNIQUES d. Fnd f ) ) ) ) ) 6) ) log d.. Fnd f = If f) = 8+0 fnd f ) nd f ) f 0).. If for f) = ++f )= f ) = Fnd nd.. Produt rule for dfferentton ) e ) e os tn ) 6)os e ot 9 se - os e ) sn + os ) 0) e sn n ) ) e -) + ) + ) 6 sn ) os) + + ) e 8se - ose ) sn + os ) log log log ) ) e log ot = Dfferentte usng quotent rule. log log. = sn. + = ) ) ) ) ) sn e. + + tn + sn = 0 sn os log e sn + 6) ) 8) sn os log os log 6.tn + ) + tn ) = sn os 9) sn os tn 0) tn 6. Dervtve of omposte funton Chn rule) )sn log ) ) e ) ot 9)sn +) n sn 6)tn log ) )os 0) sn )sn ) ) log +) ) e. Dervtves of nverse funtons.sn.tn.tn e + ).ot - - ). os ot ) ot e n ) tn ) snlog ) ) logsn ) 8)os ) e sn ) sn +).tn 6. sn 8. Logrthm Dfferentton ) ) sn ) 6tn ) ) log tn ) sn log ) log ) sn sn 8 sn os e) 0) e log ) ) Mthl Pulshers Krud ) 9) sn sn ) ) ) 9. Dfferentton of prmetr funtons ) = os = sn ) = os t = sn t ) = se = tn ) = + sn ) = os ). ) = os os = sn sn 6) = os + log tn ) = sn ) = t = t 8) = t = t t 9) t t t 0. Dfferentton of mplt funtons. + ) se ot + = 8. tn - 9. = tn ).. e. m n + e = = + ) = e + os + = 0 m + n 0. + e -. = 00 + ) + e. If ++ g+ f+ h+ = 0 d + h + g Prove tht 0 h + + f =. Hgher order Dervtves.. Fnd the seond order dervtve of: ) log log ) ) ) sn v) ot. v) + tn.. Fnd the thrd order dervtves of ) log os) ) = ) + ot v) e m + v) os.

11 Wnglsh Tuton Centre Puduvl + Mths Q & A. If = A os + B sn. Integrte the followng wth respet to. Show tht +6 = 0 sn os. If = 00 e + 600e d os os show tht 9. Integrte the followng wth respet to.. If = e tn prove tht + ) + ) sn os = 0 6. If = log ) 6. Integrte the followng wth respet to. prove tht = sn mos n snos ) ) os pos q os os. If = sn +) prove tht = os sn sn0 sn sn++ ). Integrte the followng 8. If = sn t; = sn pt + + show tht d d ) p 0 8. Integrte the followng 9. If = os θ + θ sn θ) = sn θ θ os θ) e e e d show tht se e e 9. Integrte the followng 0. If = ) prove tht + = 0. If = os m sn ) 0. Integrte the followng prove tht ) + m ) = 0. If = e sn 6 d d prove tht ) 0. Integrte the followng INTEGRAL CALCULUS. Integrte the followng wth respet to.. Integrte the followng ) ) ) 0) 6 0 ) ) 8) ) ) 6) 9) ) e. Integrte the followng wth respet to. ) ) sn os sn ) ) ot sn os e. Integrte the followng wth respet to. ) ). Integrte the followng ) 9) 9 ) METHOD OF SUBSTITUTION. Integrte the followng l m. ) ) l m n os ) sn e d) e Mthl Pulshers Krud

12 Wnglsh Tuton Centre Puduvl + Mths Q & A. ) tn ) ot os tn e) f) os ) tn ) se os e log tn g) h) sn tn se tn ot. ) ) logse logsn ) INTEGRATION BY PARTS e e ) e e d) e e ee I. Integrte the followng e e. log. log e) f) II. Integrte the followng log. sn e. e - g) os sn. e e. Integrte the followng. e. ) etn e 6. e ) os. e 8. e esn em tn ) ) esn - 9. sn - ) ) e log e e III. Integrte the followng ) e. - sn. - tn. Integrte the followng.. ) 6 ) - sn- tn. ) ) 0 IV. Integrte the followng ) + ) log ) d). sn.. e) sn os f) sn os. se tn log ) g) h) log. sn- ) os. os sn os. ) ) 6 ) 6 9. os os 0. os ) sn d) tn se. os e. sn tn sn ) e) f). sn os se. tn IV. Integrte the followng ) ) m ) ). e os. e os ) ) d) ). e os. e os sn v) ) ) ) ). e 6. e sn ) ) d) ) ). e sn 8. e sn. Integrte the followng Msellenous) IV. Integrte the followng e e. ) ) se e e ) sn. sn os e d) sn ) V. Integrte the followng Msellenous) Mthl Pulshers Krud

13 Wnglsh Tuton Centre Puduvl + Mths Q & A. - - tn. - tn. + ) 6. VII. Integrte the followng. STANDARD INTEGRALS 9. I. Integrte the followng. + ).. 9 VIII. Integrte the followng ) IX. Integrte the followng ) +. + ) +. + ) + 9 II. Integrte the followng SQUARE COMPLETION FORMULA.. 9 I. Integrte the followng ) ) II. Integrte the followng III. Integrte the followng ) ) ) ) III. Integrte the followng IV. Integrte the followng ) V. Integrte the followng ) + VI. Integrte the followng ) ) ) Mthl Pulshers Krud IV. Integrte the followng V. Integrte the followng. + ) - ) VI. Integrte the followng )

14 Wnglsh Tuton Centre Puduvl + Mths Q & A ). 6. mngoes nd pples re n o. If two fruts re hosen t rndom the prolt tht. 6. +) ) one s mngo nd the other s n pple PARTIAL FRACTION FORMULA I. Integrte the followng ) oth re of the sme vret.. Wht s the hne tht ) non-lep er ) lep er + + should hve fft three Sunds? An nteger s hosen t rndom from the frst fft postve ntegers. Wht s the prolt tht the nteger hosen s prme or multple of Three letters re wrtten to three dfferent persons + + nd ddresses on three envelopes re lso wrtten. II. Integrte the followng Wthout loong t the ddresses wht s the. + + prolt tht ) ll letters go nto rght envelopes ) none goes nto rght envelopes A ret lu hs memers of whom onl n owl. Wht s the prolt tht n tem of + + memers tlest owlers re seleted? ). Out of 0 outstndng students n shool there - re 6 grls nd os. A tem of students s PROBABILITY seleted t rndom for quz progrmme. Fnd the prolt tht there re tlest grls.. In sngle throw of two de fnd the prolt of otnng SOME BASIC THEOREMS ON PROBABILITY ) sum of less thn. ) PA)=0.6PA or B) = 0.90PA nd B)= 0. ) sum greter thn 0 Fnd ) PB) ) P A B ) ) sum of 9 or. ) PA) = 0.8 PB) = 0.. Three ons re tossed one. Fnd the prolt Fnd ) P A ) ) PA B) of gettng ) PA B ) v) P A B ) ) etl two heds ) PA) = 0. PB) = 0.6 nd PA B) = 0.. ) tlest two heds Fnd ) PA B) ) P A B) ) tmost two heds.. A sngle rd s drwn from p of rds. Wht s the prolt tht the rd s ) PA B ) v) P A B ) v) P A B ) ) or ng d) PA) = 0. PB) = 0. nd PA B) = 0. ) or smller ) ether queen or.. A g ontns whte nd l lls. lls re drwn t rndom. Fnd the prolt tht Fnd )PAB) ) PA B ) v) P A B ) ) P A B) v) P A B ) ) ll re whte. A rd s drwn t rndom from well-shuffled de of rds. Fnd the prolt of drwng ) one whte nd l. ) ng or queen. In o ontnng 0 uls re defetve. Wht s the prolt tht mong uls hosen ) ng or spde t rndom none s defetve. ) ng or l rd Mthl Pulshers Krud

15 Wnglsh Tuton Centre Puduvl + Mths Q & A. The prolt tht grl wll get n dmsson n IIT s 0.6 the prolt tht she wll get n dmsson n Government Medl College s 0. nd the prolt tht she wll get oth s 0.. Fnd the prolt tht ) She wll get tlest one of the two sets ) She wll get onl one of the two sets. A de s thrown twe. Let A e the event. Frst de shows nd B e the event seond de shows. Fnd PA B).. The prolt of n event A ourrng s 0. nd B ourrng s 0.. If A nd B re mutull elusve events then fnd the prolt of nether A nor B ourrng 6. A rd s drwn t rndom from de of rds. Wht s the prolt tht drwn rd s ) queen or lu rd ) queen or l rd ) A spes truth n 80% ses nd B n %. The prolt tht new shp wll get n wrd ses. In wht perentge of ses re the lel for ts desgn s 0. the prolt tht t wll get to ontrdt eh other n sttng the sme ft? n wrd for the effent use of mterls s 0.. Two rds re drwn from p of rds n nd tht t wll get oth wrds s 0.. Wht s the suesson. prolt tht ) Fnd the prolt tht oth re ngs when ) t wll get tlest one of the two wrds ) The frst drwn rd s repled ) t wll get onl one of the wrds ) The rd s not repled CONDITIONAL PROBABILITY: ) Wht s the prolt of gettng two s f. A on s tossed twe. Event E = Hed on frst toss F = hed on seond toss. Fnd ) PE F) ) PE F) ) PE/F) v) P E /F) v) Are E nd F ndependent?. A nd B re two ndependent ) PA) = 0. nd PA B) = 0.8. Fnd PB). ) If PA) = 0. PB) = 0. nd PB / A) = 0. Fnd PA / B) nd PA B). ) PA) = / PB) =/ A B = smple spe) Fnd the ondtonl prolt PA / B). d) PA B) = 0.6 PA) = 0. fnd PB) e) PA B) = /6 PA B) = / P B ) = / show tht A nd B re ndependent. f) PA) = 0. PB) = 0.8 Mthl Pulshers Krud Fnd ) PA B) )PB / A) )P A B ) g) PA) = 0.0 PB) = 0.0 nd PA B) = 0.0. Verf ) PA / B) = PA) ) PA/ B ) = PA) ) PB / A) = PB) v) PB / A ) = PB) h) PA) = 0. PB) = 0.6 nd PA B) = 0. Fnd ) PAB) ) PA/B) ) PB/ A ) v) P A / B) v) P A / B ) ) PA) = 0. nd PA B) = 0.. Fnd PB) f ) A nd B re mutull elusve ) A nd B re ndependent events ) PA / B) = 0. v) PB / A) = 0.. ) X spes truth n 9 perent of ses nd Y n 90 perent of ses. In wht perentge of ses re the lel to ontrdt eh other n sttng the sme ft. ) frst rd s repled efore the seond s drwn ) not repled efore seond rd s drwn. ) Wht s the prolt of drwng ) red ng ) red e or l queen.. One g ontns whte nd l lls. Another g ontns whte nd 6 l lls. If one ll s drwn from eh g fnd the prolt tht ) oth re whte ) oth re l ) one whte nd one l. 6. A husnd nd wfe pper n n ntervew for two vnes n the sme post. The prolt of husnds seleton s /6 nd tht of wfe s seleton s /. Wht s the prolt tht ) oth of them wll e seleted ) onl one of them wll e seleted ) none of them wll e seleted

16 Wnglsh Tuton Centre Puduvl 6 + Mths Q & A. A prolem s gven to students ) Whose hnes of solvng t re nd Wht s the prolt tht ) prolem s solved ) Whose hnes of solvng t re / / nd / Wht s the prolt tht ) the prolem s solved ) etl one of them wll solve t. 8. A er s seleted t rndom. Wht s the prolt tht ) t ontns Sunds ) t s lep er ontns Sunds 9. For student the prolt of gettng dmsson n IIT s 60% nd prolt of gettng dmsson n Ann Unverst s %. Fnd prolt tht ) gettng dmsson n onl one of these ) gettng dmsson n tlest one of these. 0. A n ht trget tmes n shots B tmes n. A onsultng frm rents r from three genes shots C tmes n shots the fre volle. suh tht 0% from gen X 0% from gen Wht s the hne tht the trget s dmged Y nd 0% from gen Z. If 90% of the rs etl hts? from X 80% of rs from Y nd 9% of the rs. Two thrds of students n lss re os nd rest from Z re n good ondtons wht s the grls. It s nown tht the prolt of grl prolt tht the frm wll get r n good gettng frst lss s 0. nd tht of os s 0.0. Fnd the prolt tht student hosen t rndom wll get frst lss mrs. TOTAL PROBABILITY OF AN EVENT. An urn ontns 0 whte nd l lls. Whle nother urn ontns whte nd l lls. One urn s hosen t rndom nd two lls re drwn from t. Fnd the prolt tht oth lls re whte.. ) A ftor hs two mhnes I nd II. Mhne I produes 0% of tems of the output nd Mhne II produes 0% of the tems. Further % of tems produed Mhne I re defetve nd % produed Mhne II re defetve. If n tem s drwn t rndom ) Fnd the prolt tht t s defetve tem ) If t s defetve wht s the prolt tht t ws produed Mhne-II. ) In ftor Mhne-I produes % of the output nd Mhne-II produes % of the output. On the verge 0% tems produed I nd % of the tems produed II re defetve. An tem s drwn t rndom from d s output. ) Fnd the prolt tht t s defetve tem ) If t s defetve wht s the prolt tht t ws produed Mhne-II. ) A ftor hs two Mhnes-I nd II. Mhne-I produes % of tems nd Mhne-II produes % of the tems of the totl output. Further % of the tems produed Mhne-I re defetve wheres % produed Mhne- II re defetve. If n tem s drwn t rndom ) wht s the prolt tht t s defetve?. The hnes of X Y nd Z eomng mngers of ertn ompn re : :. The proltes tht onus sheme wll e ntrodued f X Y nd Z eome mngers re nd 0. respetvel. If the onus sheme hs een ntrodued wht s the prolt tht Z s pponted s the mnger. ondton? Also If r s n good ondton wht s prolt tht t hs me from gen Y?. Bg A ontns whte 6 l lls nd g B ontns whte l lls. One g s seleted t rndom nd one ll s drwn from t. Fnd the prolt tht t s whte. 6. There re two dentl oes ontnng respetvel whte nd red lls whte nd 6 red lls. A o s hosen t rndom nd ll s drwn from t ) fnd the prolt tht the ll s whte ) f whte prolt tht t s from frst o?. Three urns eh ontnng red nd whte hps s gven elow. Urn I : 6 red whte Urn II : red whte Urn III : red 6 whte An urn s hosen t rndom nd hp s drwn from urn. ) Fnd prolt tht t s whte. ) If hp s whte fnd prolt tht t s from urn II Mthl Pulshers Krud