KENDRIYA VIDYALAY SANGATHAN: CHENNAI REGION CLASS XII PRE-BOARD EXAMINATION Q.No. Value points Marks 1 0 ={0,2,4} 1.

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1 KENDRIYA VIDYALAY SANGATHAN: CHENNAI REGION CLASS XII PRE-BOARD EXAMINATION 7-8 Answer ke SET A Q.No. Vlue points Mrks ={,,4} tn os.5 For orret proof 5 LHS M,RHS M 4 du dv os + / os. e d d 7 8 d Writing I =, 4 ( ) Corret Ans. I = sin 4 + C Sum= sin sin sin + Epressing into vrile nd seprle C Given E nd F re independent eventsp E F = P E P F Now P E F = P(EUF) = P E P F + P E P F = P E P F =P(E)P(F) Hene E nd F re independent.

2 4 f π 6 = lim f π 6 ½ k = lim π 6 sin + os + π 6 = lim π 6 sin + os + π 6 ½ = lim π 6 = os π 6 sin + sin π 6 os + π 6 = lim π 6 sin + π 6 + π 6 ½ f ½ For LHD nd RHD t -, LHD nd RHD t for eh For onlusion 5 Writing ; Differentiting, d d Agin differentite nd getting orret proof

3 6 7 8

4 9 d d tn.5 I. F e tn tn e tn.5

5 Let E represent the event of orret set up nd E represent the event of inorret setup. Let A e the event tht the mhine produes eptle items. P(E) =.8, P ( E) =., P ( A/E) =.9.9, P ( A/E) =.4.4 Writing the formul for P ( E/A 8 Sustituting the vlues nd getting the orret nswer = 85 4 f is - f is onto f f f 6 ( ) (4) (6) 4

6 5

7 6 AB: + =, BC : = ; AC: + = 7 B solving nd getting A( 4, ), B (, ) nd C (, ) Corret figure nd shding : ½ Are of tringle ABC = d ( ) d 7 ( ) d For orret integrtion : ½ Getting orret nswer s 6 sq.units 7 ½ Put t = sin os Writing I = = 5 6 t 4 dt 9 6( t dt = ) = = log 5 log t t

8 8 5 4 An point on the line s B (,, ) Dr`s of AB = 9 5 8,, dr`s of norml to plne = 4,, AB is prllel to the plne Getting 5 nd B4,, Distne AB = 7 9 Let kes of one kind nd kes of nother kind e mde LPP: Mimie: Z = + Sujet to the onstrints: , 5 + 6,, Corret grph with shding Vlue of Z = t (, ), m. no of kes = t (, ) nd 5 t ( 5, ) nd

9 KENDRIYA VIDYALAY SANGATHAN: CHENNAI REGION CLASS XII PRE-BOARD EXAMINATION 7-8 Answer ke SET - B Q.No. Vlue points Mrks ={,,4} tn os For orret proof os d os sin Put t = os sin LHS M,RHS M We get I = t dt = log t = log ( os sin ) + C du d / dv d os os. e + 9 ot ot 6 Sum= sin sin sin + Given End F re independent eventsp E F = P E P F Now P E F = P(EUF) = P E P F + P E P F = P E P F =P(E)P(F) Hene E nd F re independent..5 Epressing into vrile nd seprle Soln. C

10 4 k = lim π 6 sin + os + π 6 f π 6 = lim f π 6 = lim π 6 sin + os + π 6 ½ ½ 5 = lim π 6 os π 6 sin + sin π 6 os = lim π 6 sin + π 6 + π + π 6 6 = f For LHD nd RHD t -, LHD nd RHD t for eh For onlusion d m d d Squring on oth sides, differentiting nd dividing to get the nswer d ½ ½ 6 `

11 7 l + = where is onstnt Are A = l l da l dl d A l dl Proving l = = / 8 On dividing, I = d = d ( ) / B prtil frtions, getting seond integrl = d Corret nswer s I = log log C 4 9 d.5

12 d d I. F e tn tn tn e tn.5.5 An point on the I line = ƛ +, = ƛ, = - An point on II line = µ+ 4, =, = µ- Let the line interset t (,, ) B equting the oordintes, getting ƛ= nd µ= Getting point of intersetion ( 4,, - ) Let E person seleted mle nd E seleted femle A The seleted person is gre hired P(E) = P(E) = ½ P(A/E) = 5/, P(A/E) = 5/ Finding P(E/A) = / f is - f is onto f f f ( ) (4) (6) 4 6

13 5 C C C C C C, = Appl R R R nd R R R nd then epnd Proving = ( + + ) ( + + ).5.5

14 6 ½ Put t = sin os Writing I = = dt 9 6( t dt = 5 6 t 4 = 5 = t 4. log t 4 = log 7 Getting the point of intersetion (, ), (,5 ) nd ( 6, ) Corret figure nd shding ).5

15 .5 + The required re is d d d 5 Integrting nd getting the orret nswer= 6.5 sq units An point on the line s B (,, ) Dr`s of AB =,, dr`s of norml to plne = 4,, AB is prllel to the plne 5 Getting nd B4,, 7 Distne AB = 9 Let kes of one kind nd kes of nother kind e mde LPP: Mimie: Z = + Sujet to the onstrints: , 5 + 6,, Corret grph with shding Vlue of Z = t (, ), t (, ) nd 5 t ( 5, ) nd m. no of kes =

16 KENDRIYA VIDYALAY SANGATHAN: CHENNAI REGION CLASS XII PRE-BOARD EXAMINATION 7-8 Answer ke SET - C Q.No. Vlue points Mrks ={,,4} 4 5 du dv os / os. e d d 5 4 tn os For orret proof e I = e d dt = t putting t = e tn ( e ) C = LHS M,RHS M 9 Given End F re independent eventsp E F = P E P F Now P E F = P(EUF) = P E P F + P E P F = P E P F =P(E)P(F) Hene E nd F re independent. Sum= sin sin sin + Epressing into vrile nd seprle tn C os tn os Solving nd getting os e 4.5.5

17 .5.5 d d I. F e tn tn tn e tn 4 k = lim π 6 sin + os + π 6 f π 6 = lim f π 6 = lim π 6 sin + os + π 6 ½ ½ = lim π 6 os π 6 sin + sin π 6 os + π 6 = lim π 6 sin + π 6 + π 6 ½ 5 = f For LHD nd RHD t -, LHD nd RHD t for eh For onlusion d d ( ) Differentiting gin nd proving the result ½

18 6 ` 7 Let r e the rdius nd h e the height of the linder For orret figure 8 Writing volume of the linder V = r h 4R h h dv 4R h dh 4 dv R h dh d V R dh d I= log(log ) d (log ) Integrte prts the first integrl, we get 4.5

19 9 d I = log(log ) d log (log ) Integrte gin d prts, we get log I = log(log ) log Let E e the event tht the lost rd is spde E e the event tht the lost rd is non-spde A e the event tht two rds drwn from 5 re spde

20 P(E ) = P(E ) = ½ C C P(A/E ) = nd P ( E ) = 5C 5C B ppling the formul nd getting P ( E /A ) = 5 Let the diretion rtios of the required line e,, 4 The eqution of required line e This line is perpendiulr to given lines = nd = Solving nd getting =, =, = 6 4 Eqution of the required line is 6 4 Let kes of one kind nd kes of nother kind e mde LPP: Mimie: Z = + Sujet to the onstrints: , 5 + 6,, Corret grph with shding Vlue of Z = t (, ), t (, ) nd 5 t ( 5, ) nd m. no of kes = 5 p p p

21 B properties, Δ = p, R R R R R R, we get ) )( ( ) )( ( ) ( p p ) ( ) )( )( ( B epnding, we get ) )( )( )( ( p 6 Getting point of intersetion, Corret figure with shding Required re = 4 4 d = 4 sin 4 Appl the limits nd getting the re = 8 sq. units

22 7 ½ 8 Put t = sin os Writing I = = dt 9 6( t dt 6 t 4 5 = ) = 5 t = 4. log t 4 = log 5 4 An point on the line s B (,, ) Dr`s of AB =,, dr`s of norml to plne = 4,, AB is prllel to the plne 5 Getting nd B4,, 7 Distne AB = 9 f is -

23 f is onto f f f 6 ( ) (4) (6) 4

EXPECTED ANSWERS/VALUE POINTS SECTION - A

EXPECTED ANSWERS/VALUE POINTS SECTION - A 6 QUESTION PPE ODE 65// EXPETED NSWES/VLUE POINTS SETION - -.... 6. / 5. 5 6. 5 7. 5. ( ) { } ( ) kˆ ĵ î kˆ ĵ î r 9. or ( ) kˆ ĵ î r. kˆ ĵ î m SETION - B.,, m,,, m O Mrks m 9 5 os θ 9, θ eing ngle etween