Einstein Classes, Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road New Delhi , Ph. : ,

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1 MSS SEQUENCE AND SERIES CA SEQUENCE A sequece is fuctio of tul ubes with codoi is the set of el ubes (Coplex ubes. If Rge is subset of el ubes (Coplex ubes the it is clled el sequece (Coplex sequece. Exple (i (ii (Hee ech te of the sequece c be obtied by ddig to the pecedig te 5...(Hee ech te of the sequece c be obtied by subtctig fo the pecedig te (iii (Hee ech te c be obtied by ultiplyig the pecedig te by. CB SERIES By ddig o subtctig the tes of sequece we get expessio which is clled seies. Thus ( +...(i 5 + ( +... to...(ii + ( ( to...(iii Above expessios e the exples of seies. CC TYPES OF SEQUENCES...(iv CA CB (i Fiite Sequeces A sequece is sid to be fiite if it hs fiite ube of tes. (ii Ifiite Sequeces A sequece is sid to be ifiite if it hs ifiite ube of tes. Aithetic Sequece o Aithetic Pogessio A Aithetic sequece is sequece i which the diffeece betwee y te d its just pecedig te is costt thoughout. This costt is clled the coo diffeece. th te of A.P. T = + ( d CC Su of tes of A.P. S [ ( d] If l is lst te (o th te the S ( d S [ l] l Eistei Clsses Uit No. 0 0 Vdh Rig Rod Plz Viks Pui Ext. Oute Rig Rod New Delhi 0 08 Ph

2 CD Popeties of A.P.. If...e i A.P. The ± k ± k ± k... e i A.P.. If...e i A.P. The k k k...e i A.P. MSS CE & k k k... e i A.P.. If...e i A.P. & b b b e i A.P. The ± b ± b ± b...e i A.P. 4. If...e i A.P. & b b b e i A.P. The b b b...e ot i A.P. 5. If... e i A.P. The su of the tes equidistt fo the begiig d ed is costt i.e. k + (k = + (fist te + lst te 6. If th te of y sequece is lie expessio i the sequece is A.P. If th te of fo A + B the coo diffeece of A.P. is A. 7. If su of tes of y sequece is qudtic i the sequece is A.P. If the su of te is i the fo A + B + C the coo diffeece of A.P. is A. Selectio of Tes i A.P. No. of Choose tes Coo No. of Choose Coo tes is Diffeeces tes is tes Diffeece odd eve d + d d 4 d d d + d + d 5. d d + d d 6 5d d d + d d + d + d + 5d. The lest vlue of fo which 5 + x + 5 x 5 x + 5 x e thee cosecutive tes of A.P. is ( 5 0 (c oe of these. The fist secod d iddle tes of A.P. e b c espectively. Thei su is (c c(c c(b b(c ( c (c b b c b. If S S...S q e the sus of tes of q A.Ps whose fist tes e...q d coo diffeeces e 5...(q espectively. Show tht S + S + S S q = q(q. 4. The fist d lst te of A.P. e & l espectively. If s be the su of ll the tes of the A.P. show tht the coo diffeece is l s ( l Eistei Clsses Uit No. 0 0 Vdh Rig Rod Plz Viks Pui Ext. Oute Rig Rod New Delhi 0 08 Ph

3 MSS 5. Show tht if...e i A.P. d S = k k the S k. k 6. The su of tes of itheticl pogessio e S S S...S. the fist te d coo diffeeces e... espectively. Pove tht S + S +... S = ( + ( The pth te of A.P. is d qth te is b. Pove tht su of its (p + q tes is p q b b p q [Aswes ( c ( b] CA Geoetic Sequece o Geoetic Pogessio CB A Geoetic pogessio (G.P. is sequece i which the tio of y te d its just pecedig te is costt thoughout. This costt is clled the coo tio. Exple (i th Te of G.P. (ii If is fist te d is the coo tio of geoetic pogessio the th te t = ( CC Su of Fist Tes S = If < the S (su of ifiite tes = ( CD Popeties of G.P.. If...e i G.P. The k k k...e i G.P. &. If...e i G.P. k k k...e i G.P. The...e i G.P.. If...& b b b...e i G.P. The b b b...e i G.P. & b b b...e i G.P. 4. If... & b b b...e i G.P hvig diffeet coo tio the ± b ± b ± b...e ot i G.P. 5. If... e i G.P. The log log log...e i A.P. & Viceves 6. If... e i G.P. The = =. =... Eistei Clsses Uit No. 0 0 Vdh Rig Rod Plz Viks Pui Ext. Oute Rig Rod New Delhi 0 08 Ph

4 MSS 4 7. If...e i G.P. The = = 4 ; 4 = 5 ;... Selectio of tes i G.P. No. of Choose tes Coo No. of Choose Coo tes is tio tes is tes tio odd eve 4 5. Pctice Pobles If b c e i G.P. d log c log b c log b e i A.P. the the coo diffeece o the A.P. is ( / (c / / log.5... to. The vlue of ( 0.6 is ( (c 4 oe of these. If x y z e i GP d x = b y = c z the ( log b = log c log c b = log c (c log b = log c b oe of these 4. The cosecutive digits of thee digit ube e i G.P. If the iddle digit be icesed by the they fo o A.P. If 79 is substcted fo this ube the we get the ube cosistig of the se thee digits but i evese ode. Fid the ube. 5. Does thee exist G.P. cotiig 7 8 d s thee of its tes? If it exists how y such pogessios e possible? 6. Fid the su of the ifiitely decesig G.P. whose thid te the thee ties the poduct of the fist d fouth tes d the secod te fo i the idicted ode A.P. with the coo diffeece equl to / Pove tht the su of fist tes of the seies is [Aswes ( b ( c ( c (4 9(6 ] C4A Aithetic - Geoetic Seies A seies is sid to be ithetico-geoetic seies if its ech te is foed by ultiplyig the coespodig te of A.P. d G.P. e.g. x + 4x + 6x + 8x 4... Hee e i A.P. d x x x x 4...e i G.P. C4B th Te of Aithetico-Geoetic Seies th te of ithetico-geoetic seies c be obtied by ultiplyig the th te of A.P & th te of G.P. e.g. The th tes of the seies x + 4x + 6x + 8x is { + ( }(x. x =. x C4C Su of tes of ithetico- geoetic seies Let S = + ( + d + ( + d [ + ( d] Multiplyig both sides of (i by coo tio of (G.P. d wite s follows S = + ( + d +... [ + ( d]...(i...(ii Eistei Clsses Uit No. 0 0 Vdh Rig Rod Plz Viks Pui Ext. Oute Rig Rod New Delhi 0 08 Ph

5 Fo (i (ii S ( = + [d + d d ] [ + ( d] MSS 5 d( S ( [ ( d] S d( ( [ ( d] ( d ( Su of Ifiite Tes S Pctice Pobles. The su if i i upto 00 tes whee i = is ( 50( i 5i (c 5 ( + i 00 ( i [Aswes ( ] C5 Hoic Pogessio A sequece is sid to be Hoic Pogessio (H.P. if the ecipocls of its tes e i Aithetic Pogessio (A.P. If... e i H.P. the. th te of this H.P. fo stt... e i A.P. T (. th te of this H.P. fo ed ( ( T ( ( (. th te of H.P.fo stt th te of H.P.fo lst i.e. T T Pctice Pobles fist te lst te. Show tht the sequece... is H.P. Also fid its 0th te Show tht is H.P. Fid its 8th te d th te.. The 8th d 4th te of H.P. e d espectively. Fid its 0th te. Also fid its geel te. Eistei Clsses Uit No. 0 0 Vdh Rig Rod Plz Viks Pui Ext. Oute Rig Rod New Delhi 0 08 Ph

6 MSS 6 4. Which te of the H.P. 7...is 0? 5. If 0th te of H.P. is d 0th te is 7. Fid its lgest te. 6. If th te of H.P. is d th te is the pove tht its th te is. 7. The th te of H.P. is d the th te is. Pove tht its th te is. 8. If ( + th te of H.P. is twice the ( + th te pove tht its ( + + th te is twice the ( + th te. q p p q 9. If pth qth d th te of H.P. e b c espectively pove tht 0. b c 0. If > b > c > e i G.P. the show tht log log b log c e e e e i H.P. b c d c d d b b c. If b c d e i H.P. the pove tht e i H.P. b c d. Thee ubes fo H.P. The su of the ubes is d the su of thei ecipocls is. Fid the ubes.. If... e i HP. Pove tht = ( 4. Fid the lgest positive te of the H.P. whose fist two tes e d 5 x y z 5. If d p q be i A.P. the pove tht x y z e i H.P. px qy z 6. Let b c be thee distict el ube i A.P. such tht < b < d c <. If 0 0 x y b z c the pove tht x y z e i H.P. 0. [Aswes ( /8 ( 4 0 ( 6 (4 st te (5 ( 6 o 6 (4 6] 4 4 Eistei Clsses Uit No. 0 0 Vdh Rig Rod Plz Viks Pui Ext. Oute Rig Rod New Delhi 0 08 Ph

7 MSS 7 C6A Aithetic Me Geoetic Me d Hoic Me Aithtic Me Geoetic Me Hoic Me Defiitio If thee tes i A.P. the the iddle te is clled the Aithetic e (A.M. betwee the othe two. e.g. If b c e i A.P. the b is ithetic e of d c If thee tes e i G.P. the iddle te is clled the Geoetic e (G.M. betwee the othe two. e.g. If b c e i G.P. the b is geoetic e of d c If thee tes e i H.P. the iddle te is clled the Hoic e (H.M. betwee the othe two. e.g. If b c e i H.P. the b is hoic e of d c Sigle e of positive ubes A (A = Aithetic e G = (..... / (G = Geoetic e Specil Cse If d b e two give ubes the H i (H = Hoic e i b A G = / b H b e betwee two ubes d b. A A A A A...;A b (b (b (sy G G G G G G b b b...g / / (sy H H H H H...H b ( b (sy ( b ( b ( b ( b Su of ithetic es betwee two give ubes i ties the sigle A.M. betwee the. i.e. su of A.M. = (A.M. Poduct of Geoetic e betwee two give ubes is th powe of the sigle G.M. betwee the. i.e. poduct of G.M. = (oe G.M. The su of ecipocls of Hoic es betwee two give ubes is ties the ecipocl of sigle H.M. betwee the. Eistei Clsses Uit No. 0 0 Vdh Rig Rod Plz Viks Pui Ext. Oute Rig Rod New Delhi 0 08 Ph

8 C6B Popeties of A.M. G.M. d H.M.. If A G d H be ithetic d geoetic es of two ubes d b. The G = AH. The equtio hvig d b s its oot is witte i the fo x Ax + G = 0.. Applictio to the questios of Iequlities I geel A.M. G.M. H.M. 4. Applictio to questios of Mxi d Mii MSS 8 The ithetic e of positive ubes which e ot ll equl to oe othe is gete th thei geoetic e. I the theoe the iequlities becoe equlities whe ll the ubes e equl. We dw the followig coclustios Suppose tht x y z...w e positive vibles d tht c is costt the. If x + y + z w = c the vlue of xyz...w is getest whe x = y =...= w = c/ so tht the getest vlue of xyz...w is (c/.. If xyz...w = c the vlue of x + y +... w is tlest whe x = y =... = w so thtthe lst vlue of x + y w is c /. Applictios. Fid the getest vlue of xyz fo positive vlues of x y z subject to the coditio yz + zx + xy =. Sice yz + zx + xy = the vlue of (yz (zx (xy is getest whe yz = zx = xy tht is whe x = y = z =. Hece the getest vlue of (yz (zx (xy is 6 d the getest vlue xyz is 8.. If the su of the sides of tigle is give pove tht the e is getest whe the tigle is equiltel. Let b c be the sides of the tigle d let + b + c = s. If is the e the Now = s(s (s b (s c. (s + (s b + (s c = s = costt Hece the vlue of (s (s b (s c is getest whe s = s b = s c Theefoe the vlue of is getest whe = b = c.. Fid the lest vlue of x + 4y fo positive vlues of x d y subject to the coditio x y = 6. Sice x y = 6 if µ e y costts we hve (x (x (µy (µy (µy = 6 µ. Theefoe x + x + µy + µy + µy is lest whe x = µy = (6 µ /5. Hece the lest vlue of x + µy is 5(6 µ /5. Puttig = d µ = 4 it follows tht the lest vlue of x + 4y is Pctice Pobles. If A A ; G G d H H e two ithetic geoetic d hoic es espectively betwee two qutities d b pove tht 4 5 (i GG H H A A (ii A H H H = A H = G G = b. If 9 hoic es be iseted betwee d pove tht A + H 6 = 5 whee A is y of the A.M s d H the coespodig H.M.. If A be the A.M. d H the H.M. betwee two ubes d b the show tht A b A H b H A. H Eistei Clsses Uit No. 0 0 Vdh Rig Rod Plz Viks Pui Ext. Oute Rig Rod New Delhi 0 08 Ph

9 MSS 9 4. ithetic es hve bee iseted betwee d i such wy tht the tio of the 7th d the ( th es is 5 9. Fid the vlue of. 5. If G be the geoetic e betwee two give ubes d A A be the two ithetic e betwee the pove tht G = (A A (A A. [Aswes (4 = 4] C7 Miscelleous ppoch of sutio. Method Wokig Rule fo Sutio of Seies Fid the th te of the seies. Siplify the th tes. Now evlute S t with the help of. (. Method of diffeeces ( ( 6 d ( If the diffeece of the successive tes of sequece is i A.P. o G.P. we fid the th te of this sequece by ethod of diffeece which is give s follows S = t + t + t t...( S = t + t t + t...( Fo ( (; 0 = t + (t t + (t t (t t t t = t + (t t + (t t (t t So. V Method S t To fid the su of the seies of the fos (i (ii whee......e i A.P. Solutio of fo (i Let d be the coo diffeece of A.P. the = + ( d Let su of the seies d th te e deoted by S d T espectively. The S T...(... Let V {levig fist fcto fo D to T }...(... Eistei Clsses Uit No. 0 0 Vdh Rig Rod Plz Viks Pui Ext. Oute Rig Rod New Delhi 0 08 Ph

10 MSS 0 So V V V V V V V = T ( + {fo (} = T {[ + ( d] [ + ( + d]} = d( T (V V d( T d( o T {V V } Puttig = 4... we get (V d( T 0 (V d( V T (V d( V T T Addig the bove equtios we get (V d( V V ( d T + T + T T = (V V 0 S ( d... Hece the su of tes is... ( ( Usig the bove pove the followig S Eistei Clsses Uit No. 0 0 Vdh Rig Rod Plz Viks Pui Ext. Oute Rig Rod New Delhi 0 08 Ph

11 MSS (i ( (ii ( ( 4 ( ( Solutio of Fo (ii Let S be the su d T be the th te of the seies the S = T = ( Let V = ( (Tkig oe ext fcto i T fo V V = V V = ( + = T ([ + ( + d] [ + ( d] = ( + d T ( d T (V V Puttig =... we get (V ( d T 0 (V ( d V T V T (V V ( d T (V V ( d Addig the bove equtios we get T T T...T (V V 0 ( d S ( ( [(... (... ] {Hee = d} Usig the bove pove the followig (i ( + = ( Pctice Pobles. Fid the su upto tes of the seies (. The seies of tul ubes is divided ito goups s follows {} ; { 4} ; { }... so o. Show tht the su of the ubes i the th goup is ( +.. Su up the seies to tes. Eistei Clsses Uit No. 0 0 Vdh Rig Rod Plz Viks Pui Ext. Oute Rig Rod New Delhi 0 08 Ph

12 4. Fid the su of the followig seies to tes MSS ( totes (c (x + y + (x + xy + y + (x + x y + xy + y Fid the su of tes of the seies [Aswes (. S = ( (4 (. ( 9. 4 ( 5 x ( x (c. (x y ( x y ( y y 6 (5. ] Eistei Clsses Uit No. 0 0 Vdh Rig Rod Plz Viks Pui Ext. Oute Rig Rod New Delhi 0 08 Ph

13 MSS SINGLE CORRECT CHOICE TYPE. If x + bx + cx + d is divisible by x + c the b c d e i ( A.P. G.P. (c H.P. oe of these. If ( + th ( + th d ( + th tes of AP e i GP; d e i HP the the tio of the fist te d coo diffeece of this AP is ( / / (c / /. If b c d d p e distict o-zeo el ubes such tht ( + b + c p (b + bc + cdp + (b + c + d 0. The b c d e i ( A.P. G.P. (c H.P. oe 4. If b c d e f e i A.P. the e c is equl to ( (c (d c (c (f (d c 5. Let S S... be sques such tht fo ech the legth of side of S equls the legth of digol of S +. If the legth of side of S is 0 c the fo which of the followig vlues of is the e of S less th sq. c ( 8 9 (c 0 ll 6. Choose the coect stteet fo the followig ( (c thee cot be A.P. whose tes e distict pie ubes. 0 cot be the tes of G.P. 5 cot be tes of sigle A.P. ll e coect 7. If x y d z e positive el ubes diffeet fo d x 8 = y = z 8 the log y x log z y 7log x z e i ( A.P. G.P. (c H.P. oe 8. If the sides of ight gled tigle e i G.P. the cosie of the gete cute gle is ( The thee seccessive tes of G.P. will fo the sides of tigle if the coo tio stisfies the iequlity ( (c oe 0. A sque of side legth is give A secod sque is de by joiig the iddle poits of the sides of the st sque d the d sque is de by joiig the iddle poits of the sides of the d sque. This pocess is epeted idefiitely. The e of the su of the es of ll the sques is ( (c 4. If b c be distict positive ubes i G.P. d log c log b c log b be i A.P. the coo diffeece of the pogessio is ( (c 5 oe. If + b + c = d > 0 b > 0 d c > 0 the getest vlue of b c is 0. ( (c oe. If log x y log z x log y z e i G.P. xyz = 64 d x y z e i A.P. the the vlue of y is ( 4 (c (c 5 5 Eistei Clsses Uit No. 0 0 Vdh Rig Rod Plz Viks Pui Ext. Oute Rig Rod New Delhi 0 08 Ph

14 4. The su of tes of the seies 5 ( (c 5 5. If... upto... 7 of... to is 4 ( (c the the vlue 6. If is oot of the equtio x 0x + 64 = 0 d 0 4 ( cos cos cos cos...to e log e 4 the ube of solutio of betwee 0 to is ( (c If 0 the c b c b (i (ii ( (c b c e i H.P. b c e i A.P. both e coect oly (i is coect oly (ii is coect both e icoect 8. The coefficiet of x 98 i the cotiued poduct (x + (x + (x +... (x + 00 is ( (c oe 9. If b c e i HP the the vlue of MSS 4 ( (c 4 b b c is b b c 0. If oe ithetic e A d two geoetic es p d q be iseted betwee two give qutities the the vlue of ( (c A A 4 p q is q p A A. If log x + log y 6 the the sllest possibel vlue of x + y is ( 8 (c 6 0. If b c e positive fctios d + b + c = the the xiu vlue of is. b c ( 6 (c A A A... e poits i the fist qudt o the pbol y = 4x. If the x-coodites of the poits be i HP the fist two beig y-coodite of the poit A is is ( (c 4 4. If ( k k the the. The vlue of = 4 + b + c + d + e the d 6 (c e = 0 ll the bove 5. The oots of equtio x + ( x + 9 = 0 lie betwee 6 d d h h...h 0 [] e i H.P. whee [] deotes the itegl pt of the h h 8 = ( 6 (c 9 Eistei Clsses Uit No. 0 0 Vdh Rig Rod Plz Viks Pui Ext. Oute Rig Rod New Delhi 0 08 Ph

15 6. If b c be the pth qth d th tes espectively of A.P. d G.P. both the the poduct of oots of equtio b b c c x bcx + c b c b = 0 is equl to ( (c bc 7. Let =... (55 digits b = c = the ( = b + c = bc (c b = c c = b 8. If pth qth d th te of G.P. be 7 8 d espectively the the equtio px + qx = 0 hs ( oly oe oot i (0 o oot i (0 (c both oots i (0 igiy oots 9. ABC is ight-gled tigle i which B = 90 0 d BC =. If poits L L...L o AB e such tht AB is divided i + equl pts d L M L M...L M e lie seget pllel to BC d M M...M e o AC the the vlue of of L M + L M L M is ( (c ( ( ipossible to fid fo the give dt 0. If b c be i G.P. whee s b c c b e i b c H.P. the the vlue of is ½ (c ( (c. If b c e i H.P.; b c d e i G.P. d c d e e i A.P. the the vlue of e i tes of d b is MSS 5. If S deotes the su of ifiity d S the su of thes of the seies.... such 4 8 tht S S < 000 the the lest vlue of ( 0 (c. If b c e i A.P; e i H.P. d b c e i G.P. (coo tio is ot equl to the b c is ( (c 4. If ithetic es... e iseted betwee 50 d 00 d hoic es h h..h e iseted betwee the se two ubes the h is equl to ( 500 (c Suppose b c e thee positive el ubes i A.P. such tht bc = 4. The iiu vlue of b is ( ( 4 ( (c ( 6 oe b ( ( b b (c ( b b ( b b ( b Eistei Clsses Uit No. 0 0 Vdh Rig Rod Plz Viks Pui Ext. Oute Rig Rod New Delhi 0 08 Ph

16 MSS 6 EXCERCISE BASED ON NEW PATTERN COMPREHENSION TYPE Copehesio- Let x & x be the oots of the equtio x x + A = 0 d let x & x 4 be the oots of the equtio x x + B = 0. It is kow tht the ubes x x x x 4 (i the se ode fo icesig G.P.. The coo tio of G.P. is ( ± ± (c ± ±. The vlue of A is ( 8 (c 6. The vlue of B is ( ( 8 (c 6 Copehesio- If A G H e the ithetic geoetic d hoic es of two positive el ubes d b d if A = k H 4. Which of the followig eltio is coect ( A = kg A = kg (c G = k A G = k A 5. If the tio of to b will exists the choose the possible vlue of k is ( (c 6. Fo k = the vlue of b is ( (c 4 Copehesio- Weighted Mes Let... be positive el ubes d... be positive tiol ubes. The we defie weighted Aithetic Me (A * weighted Geoetic Me (G* d weighted Hoic e (H + s A G * * (... (.. d H * Aithetic Me of th powe Let... be positive el ubes (ot ll equl d let be el ube. The... if R [0 ]. Howeve if (0 the... Obviously if {0 } the Which of the followig is coect? ( A* G* H* A* H* G* (c H* G* A* G* A* H* 8. If b c e positive el ubes such tht + b + c = 8 the the xiu vlue of b c 4 is ( 7.. (c If b c e positive el ubes tht + b + c = the the iiu vlue of ( (c MATRIX-MATCH TYPE Mtchig- Colu - A b c b c c b Colu - B (A If b c e i G.P. d (P / b c b c e i H.P. the the vlue of + 4b + c (B If the pth qth d th (Q 0 tes of H.P. e b d c espectively the the vlue of b(p q + bc(q + c( p is is Eistei Clsses Uit No. 0 0 Vdh Rig Rod Plz Viks Pui Ext. Oute Rig Rod New Delhi 0 08 Ph

17 (C If b c be distict (R 5/ positive ube i G.P. d log c log b c log b be i A.P. the coo diffeece of the pogessio is (D A sque is dw by (S joiig the id-poit of the sides of give sque. A thid sque is dw iside the secod sque i the se wy d this pocess cotiues idefiitely. If side of the fist sque is 4 c deteie the su of the es of ll the sques Mtchig- (T 64 Let A A A...A be ithetic es betwee d 07 d G G G...G be geoetic es betwee d 04. Poduct of geoetic es is 45 d su of ithetic es is 05 7 Colu - A Colu-B (A The vlue of is (P 6 (B The vlue of is (Q 0 (C The vlue of G i is (R 4 i (D The coo diffeece (S 9 of the pogessio A A A 5...A is (T oe MULTIPLE CORRECT CHOICE TYPE. If log y x log z y 5log x z e i A.P. the ( z = x x = y (c z = y x = y = z. If b c e i H.P. the the vlue of b c c ( bc b 4 c is b c MSS 7. If b b b (b > 0 e thee successive tes of G.P. with coo tio the vlue of fo which the iequlity b > 4b b holds is give by ( > < (c =.5 = If log x x/ d log b x e i G.P. the x is equl to ( log (log b log (log e log (log e b (c log (log b log (log e b log (log e 5. Let x b be i AP; y b be i GP d z b be i HP. If x = y + d = 5z the ( y = xz x > y > z (c = 9 b = = /4 b = 9/4 6. Thee positive ubes fo GP. If the iddle ube is icesed by 8 the thee ubes fo AP. If the lst ube is lso icesed by 64 log with the pevious icese i the iddle ube the esultig ube fo GP gi. The ( coo tio = fist ube = 4/9 (c coo tio = 5 fist ube = 4 7. If x y z e positive ubes i AP the ( (c y xz y xz x y y z y x y z vlue x y y z 4 y x y z hs the iiu 8. Fo the A.P. give by......the equtios stisfied e ( + + = 0 + = 0 (c + 4 = = 0 9. If the fist & the ( + th tes of A.P. G.P. & H.P. of positive tes e equl d thei ( + th tes e b & c espectively the ( = b = c b c (c + c = b c = b (c b b oe of these Eistei Clsses Uit No. 0 0 Vdh Rig Rod Plz Viks Pui Ext. Oute Rig Rod New Delhi 0 08 Ph

18 0. Betwee two uequl ubes if e two AMs; g g e two GMs d h h e two HMs the g. g is equl to ( h h (c h h Assetio-Reso Type Ech questio cotis STATEMENT- (Assetio d STATEMENT- (Reso. Ech questio hs 4 choices (A (B (C d (D out of which ONLY ONE is coect. (A (B (C (D Stteet- is Tue Stteet- is Tue; Stteet- is coect expltio fo Stteet- Stteet- is Tue Stteet- is Tue; Stteet- is NOT coect expltio fo Stteet-.... e i A.P. Stteet- is Tue Stteet- is Flse Stteet- is Flse Stteet- is Tue STATEMENT- MSS 8. b c e thee uequl positive ubes. STATEMENT- The poduct of thei su d the su of thei ecipocl exceeds 9. STATEMENT- AM of positive ubes exceeds thei HM. 4. STATEMENT- If thee positive ubes i G.P. epeset sides of tigle the the coo tio of the G.P. ust lie betwee 5. 5 d STATEMENT- Thee positive el ubes c fo tigle if su of y two is gete th the thid. 5. STATEMENT- If the su of tes of A.P. is give by S = + b + c whee b c e idepedet of the = 0. STATEMENT- The coo diffeece of A.P. ust be b STATEMENT- + = + + fo.. STATEMENT ( > N. STATEMENT- The su of the fist tul ubes is equl to. (Aswes EXCERCISE BASED ON NEW PATTERN COMPREHENSION TYPE... d d d 9. c MATRIX-MATCH TYPE. [A-Q; B-Q; C-P; D-S]. [A-S; B-R C-Q D-P] MULTIPLE CORRECT CHOICE TYPE. b c d. b c. b c d 4. b 5. c 6. d 7. d 8. b d 9. b d 0. b d ASSERTION-REASON TYPE. A. D. A 4. A 5. C Eistei Clsses Uit No. 0 0 Vdh Rig Rod Plz Viks Pui Ext. Oute Rig Rod New Delhi 0 08 Ph

19 MSS 9 INITIAL STEP EXERCISE (SUBJECTIVE. The su of sques of thee distict el ubes which e i G.P. is S. If thei su is S show tht (.. If exp {(si x + si 4 x + si 6 x log e } stisfy the equtio x 9x + 8 = 0 fid the vlue of. Show tht cos x fo 0 x cos x si x x 4 x x x is equl to x x. 4. Fid the su of tes of the seies log log log b b 5 log b A A.P. d G.P. ech hs s the fist te d b s the secod te (0 < b <. If S deotes the su of ifiity of the G.P. pove tht the su of fist tes of the A.P. c be witte s ( s. 6. If P is fist of ( > ithetic es betwee two positive ubes d q the fist of hoic es betwee the se two ubes show tht q cot lie betwee P d P. 7. If... e i A.P. ofc o o diffeece d the pove tht sec sec + sec sec +... tes t t si d 8. Pove tht the su to tes of the seies...is. 9. Suppose x & y e two el ubes such tht the th e betwee x & y is equl to the th e betwee x & y whe ithetic es e iseted betwee the i both the cses. Show tht y x 0. If b c e thee distict el ubes i G.P. d + b + c = xb the pove tht eithe x < o x >.. Fid ll the ubes x d y such tht x x + y x + y fo A.P. while the ubes (y + xy + 5 (x + fo G.P.. Fid out the lgest te of the sequece A A.P. G.P. d H.P. hve the se fist d the se ( th tes. If thei th tes e b c espectively the show tht b c fo G.P The vlue of xyz is 55 o ccodig s the 55 seies x y z b is A.P. o H.P. Fid the vlues of d b give tht they e positive iteges. 5. If < & b < the pove tht su upto of the seies ( + b + ( + b + ( + b +...is b. b 6. Fid the tul ube fo which k f ( k 6( whee the fuctio f stisfies the eltio f(x + y = f(x. f(y fo ll tul ubes x y d futhe f( =. 7. The vlue of the expessio.( ( + ( ( ( ( ( =. Fid give tht = d + + = If is oot of equtio x ( c x( + c ( + c = 0 d if H.M. s e iseted betwee d c show tht the diffeece betwee the fist d the lst e is equl to c( c. Eistei Clsses Uit No. 0 0 Vdh Rig Rod Plz Viks Pui Ext. Oute Rig Rod New Delhi 0 08 Ph

20 MSS 0 FINAL STEP EXERCISE (SUBJECTIVE. Pove tht the su of ll poducts of fist positive ubes ie iteges tke two t tie is ( ( ( + ( + i.e. pove tht 4 i i j ( ( ( + ( +. ji 4. Let S deote the su of fist tes of the seies & deotes su of fist tes of the seies Thepove tht 8 S S + = 0.. If... e i ithetic pogessio with coo diffeece d the pove tht s ( s ( 7 d. ( d 4. If S S d S deote the sus upto (> tes of thee sequeces i A.P. whose fist tes e uity d coo diffeeces e i H.P. pove tht SS SS SS S S S 4 5. Show tht ( ( (...(.( (. 6. If A A A... A e ithetic es betwee x & y d H H...H e hoic es betwee x & y show tht A H + = xy fo. 7. If H H H...H e hoic es betwee d b the show tht H H H H b b 8. Let d be two distict positive oots of x x + b = 0 d b b S the show tht S b If b c e i H.P. the pove tht b c b 4. b c b 0. The fist thee tes of geoetic pogessio e give such tht if fou substcted fo the thid te the these ubes becoes the fist thee tes of ithetic pogessio. If we subtct uity fo the secod te d five fo the thid te of the give geoetic pogessio we get the ubes x y z such tht log z ;log z d (x (y fo hooic pogessio. Obti expessio fo the su of fist tes of the G.P. d the A.P. Hece show tht the su of fist two te of G.P. is equl to tht of the A.P.. If > b > 0 show tht fo evey positive itege. Pove tht b b... b b b... b (666...to digits + (888...to digits = ( to digits.. > 0 b > 0 e the fist te of the two G.P. s with coo tio x d y espectively whee x > y > 0. Pove tht the tio of the su of ( + tes to tht of tes of the fist G.P. is gete th the tio of the su of ( + tes to tht of tes of the secod G.P. 4. If b c e positive el ubes pove tht b c b c c c b b b c. 5. If b c d e f e i G.P. the pove tht x 5 + bx 4 + cx + dx + cx + f is divisible by x 4 + cx + e. Eistei Clsses Uit No. 0 0 Vdh Rig Rod Plz Viks Pui Ext. Oute Rig Rod New Delhi 0 08 Ph

21 MSS ANSWERS (SINGLE CORRECT CHOICE TYPE. b. c. c.. b. c. c. b. b. b. b. b 4. b 4. d 4. d 4. c 5. d 5. b 5. b d b b 8. b b 9. c 0. b 0. d 0. d ANSWERS SUBJECTIVE (INITIAL STEP EXERCISE.. ( ( 4. log logb. y = x =. 4. = b = 7; = 7 b = 6. = 7. ( ( 8. ifiite ANSWERS SUBJECTIVE (FINAL STEP EXERCISE 0. su of fist tes of G.P. = ( (7 o 54 su of fist tes of A.P. = ( o 9 Eistei Clsses Uit No. 0 0 Vdh Rig Rod Plz Viks Pui Ext. Oute Rig Rod New Delhi 0 08 Ph