LEVEL I Fid the sum of first terms of the AP, if it is kow tht + 5 + 0 + 5 + 0 + = 5 The iterior gles of polygo re i rithmetic progressio The smllest gle is 0 d the commo differece is 5 Fid the umber of sides of the polygo The rtio betwee the sum of terms of two AP s is 7 + : + 7 Fid the rtio betwee their th terms The r th, s th d t th terms of certi GP re R, S d T respectively Prove tht R s t S t r T r s = 5 The sum of three umbers i GP is If the first two umbers re icresed by d third is decresed by, the resultig umbers form AP Fid the umbers of GP 6 If oe GM G d two rithmetic mes p d q be iserted betwee y give umbers, the show tht G = (p q) (q p) 7 The sum of ifiite geometric series is 6 d the sum of its first terms is 60 If the reciprocl of its commo rtio is iteger, fid ll possible vlues of the commo rtio, d the first term of the series 8 Evlute: + + + + + 00 99 9 If pth, q th, r th terms of AP be, b, c respectively, the prove tht p(b c) + q (c ) + r ( b) = 0 0 Fid S of the GP whose first term is 8 d the fourth term is 9 If H be the HM betwee & b, the show tht (H ) (H b) = H Fid the sum of terms of the series, the rth term of which is (r + ) r Let x = + + 6 + 0 +, < ; y = + b + 0b + 0b +, b < Fid S = + (b) + 5 (b) i terms of x d y After strikig the floor certi bll rebouds /5 th of the height from which it hs flle Fid the totl distce tht it trvels before comig to rest if it getly dropped from height of 0 meter 5 Show tht the umber is composite umber 9 digits
LEVEL II Let S deote the sum of first terms of AP If S = S, the show tht the rtio S /S is equl to 6 If the roots of the equtio x x + 9x 8 = 0 re i AP, the fid the commo differece If x = + + + + to ( < ) d y = + b + b + b + to ( b < ), the prove tht + b + b + b + to = xy x y (i) Evlute: 6 terms 5 (ii) Sum to terms the series + + 5 6 + 5 If the AM of d b is twice s gret s their GM, the show tht : b = ( ):( ) 6, b, c re the first three terms of geometric series, If the hrmoic me of d b is d tht of b d c is 6, fid the first five terms of the series 7 A AP d HP hve the sme first term, the sme lst term d the sme umber of terms ; prove tht the product of the r th term from the begiig i oe series d the r th term from the ed i the other is idepedet of r 8 The sum of first te terms of AP is equl to 55, d the sum of the first two terms of GP is 9, fid these progressios, if the first term of AP is equl to commo rtio of GP d the first term of GP is equl to commo differece of AP 9 The series of turl umbers is divided ito groups (); (,, ); (5, 6, 7, 8, 9); d so o Show tht the sum of the umbers i the th group is ( ) + 0 Suppose x d y re two rel umbers such tht the rth me betwee x d y is equl to the rth me betwee x d y whe rithmetic mes re iserted betwee them i both the cses Show y tht r x A AP d GP with positive terms hve the sme umber of terms d their first terms s well s lst terms re equl Show tht the sum of the AP is greter th or equl to the sum of the GP
Solve the followig equtios for x d y, R S / / log0 x log0 x log 0 x y 5 ( y ) 0 7 0 ( y ) 7 log T 0 x Give tht x = b y = c z = d u d, b, c, d re i GP, show tht x, y, z, u i HP The first d lst terms of AP re d b There re ltogether ( + ) terms A ew series is formed by multiplyig ech of the first terms by the ext term Show tht the sum of ew series is ( )( ) ( b ) b 6 5 Sum the followig series to terms d to ifiity : (i) (ii) 5 5 7 5 7 9 (iii) r( r )( r )( r ) r (v) If A = d B = tht A = B RS T (iv) 7 70 7 0 r r ( ) ( ) ( ) ( )( ) UV W, the show
INEQUALITIES () If x i > 0, (i =,, ), the prove tht (x + x + + x ) x x x (b) If, re o-zero rel umbers, prove tht (i) If,, re positive rel umbers, show tht b c c b (ii) If, b, c re three distict positive rel umbers Prove tht 6 or, b c bc (b + c) + c (c + ) + b ( + b) > 6bc (iii) (iv) (v) (vi) If, b, c re three distict positive rel umbers, prove tht ( + b ) + b ( + c ) + c ( + ) > 6bc If, b, c, d re distict positive rel umber, prove tht 8 ( + b 8 ) + b 8 ( + c 8 ) + c 8 ( + d 8 ) + d 8 ( + 8 ) > 8 b c d Show tht, if, b, c, d be four positive uequl qutities d s = + b + c + d, the (s - ) (s - b) (s - c) (s - d) > 8 bcd If, b, c, d re distict positive rel umbers such tht s = + b + c + d, the prove tht bcd > 8(s - ) (s - b) (s - c) (s - d) If i < 0 for ll i =,,, prove tht ( - + ) ( - + )( - + ) > ( ) (where is eve) (i) (ii) Prove tht b b b b b 5 Prove tht xyz xyz x y z x y z x y z x y z x y z 6 If oe of b, b,,b is zero, prove tht b 7 Show tht ( ) (b b ) b / ( )
8 By cosiderig the sequece,,,, where 0 < <, prove tht (i) - > - ( - ) (ii) - < ( - ) 9 If x, y, z re postive d x + y + z =, prove tht 8 x y z 0 If 5 < 5 for fixed positive iteger 6, show tht ( + ) 5 < 5 + (i) If, b, c re the sides of trigle, the prove tht + b + c > b + bc + c (ii) I trigle ABC prove tht b c b c c b (iii) (iv) If, b, c be the legth of the sides of sclee trigle, prove tht ( + b + c) > 7 ( + b - c) (b + c - ) (c + - b) If, b, c re positive rel umbers represetig the sides of sclee trigle, prove tht b c b + bc + c < + b + c < (b + bc + c) or, d hece prove b bc c ( b c) tht (b + bc + c ) < ( + b + c ) < (b + bc + c) or b bc c If A, B d C re the gles of trigle, prove tht : (i) A B C si si si 8 (ii) A B C cos cos cos 8 (iii) cos A + cos B + cos C (iv) A B C t t t (i) If is positive iteger, prove tht {( )!} (ii) /() (!) If is positive iteger, show tht (iii) For every positive rel umber d for every positive iteger prove tht / By ssigig weights d to the umbers d + (x/) respectively, prove tht if x x x >, the 5 Prove tht 5( ) 6 I
SET I If,re i AP the p, q, r re i AP if p, q, r re i (A) AP (B) GP (C) HP (D) oe of these The product of positive umbers is uity The their sum is (A) positive iteger (B) divisible by (C) equl to + (D) ever less th If p, q, r, s N d they re four cosecutive terms of AP the the p th, q th, r th, s th terms of GP i (A) AP (B) GP (C) HP (D) oe of these If i progressio,, etc, ( r r + ) bers costt rtio with r r + the the terms of the progressio re i (A) AP (B) GP (C) HP (D) oe of these 5 If the, re i (A) AP (B) GP (C) HP (D) oe of these 6 Let x, y, z be three positive prime umbers The progressio i which x, y, z c be three terms ( ot ecessrly cosecutive ) is (A) AP (B) GP (C) HP (D) oe of these 7 Let f(x) = x + The the umber of rel umber of rel vlues of x for which the three uequl umbers f(x), f(x), f(x) re i GP is (A) (B) (C) 0 (D) oe of these 8 If 0, r N,, re i AP the r is equl to (A) ( - ) (B) ( ) (C) (D) oe of these 9 If, + re i AP the (A) ( ) ( ) (B) is equl to (C) ( + ) ( - ) (D) oe of these 0 Let, be i AP d p, q, r be i GP The q : p is equl to r p (A) (B) q p q p (C) r q r q (D) oe of these q p If i AP, t = log 0, t + = log 0 b d t + = log 0 c the, b, c re i
(A) AP (B) GP (C) HP (D) oe of these If!,! d ( + )! re i GP the!, 5! d ( + )! re i (A) AP (B) GP (C) HP (D) oe of these I AP, the pth term is q d the (p + q)th term is 0 The the qth term is (A) - p (B) p (C) p + q (D) p - q I sequece of ( + ) terms the first ( + ) terms s i AP whose commo differece is, d the lst ( + ) terms re i GP whose commo rtio is 05 If the middle terms of the AP d GP re equl the the middle term of the sequece is (A) (B) (C) (D) oe of these 5 5 5 If x 9y 5z xyz the x, y, z i x y z (A) AP (B) GP (C) HP (D) oe of these 6 If, b, c, d d p re distict rel umbers such tht ( + b + c )p - (b + bc + cd)p + (b + c + d ) 0 the, b, c, d re i (A) AP (B) GP (C) HP (D) oe of these 7 The lrgest term commo to the sequeces,,,,to 00 terms d, 6,, 6,to 00 terms is (A) 8 (B) 7 (C) 8 (D) oe of these 8 The iterior gles of covex polygo re i AP, the commo differece beig 5 0 If the smllest gle is the the umber of sides is (A) 9 (B) 6 (C) 7 (D) oe of these 9 The miimum umber of terms of + + 5 + 7 + tht dd up to umber exceedig 57 is (A) 5 (B) 7 (C) 5 (D) 7 0 I the vlue of 00! the umber of zeros t the ed is (A) (B) (C) (D) If ( r)r, N, r N is expressed s the sum of k cosecutive odd turl umbers the k is equl to (A) r (B) (C) r + (D) + The sum of ll odd proper divisiors of 60 is (A) 77 (B) 78 (C) 8 (D) oe of these I the squece,,,,,,,,,,, where cosecutive terms hve the vlue, the 50th term is
(A) 7 (B) 6 (C) 8 (D) oe of these I the sequece,,,,,,, 8, 8, 8, 8, 8, 8, 8, 8,, where k cosecutive terms hve the vlue k (k =,,, 8,), the 05 th term is (A) 9 (B) 0 (C) (D) 8 5 Let {t } be sequece of itegers i GP i which t : t 6 = : d t + t 5 = 6 The t is (A) (B) (C) 6 (D) oe of these 6 If 5c b log, log d log re i AP, where, b, c re i GP, the, b, c re the 5c b legths of sides of (A) isosceles trigle (B) equilterl trigle (C) sclee trigle (D) oe of these 7 If x, y, z re i AP, where the distict umbers x, y, z re i GP, the the commo rtio of the GP is (A) (B) (C) (D) 8 If x > 0 d is kow positive umber the the lest vlues of x + x is (A) (B) (C) (D) oe of these 9 If,, -, b re i AP,, b, b, b,, b -, b re i GP d, c, c, c,, c -, b re i HP, where, b re positive, the the equtio x - b x + c = 0 hs its roots (A) rel d uequl (B) rel d equl (C) imgiry (D) oe of these 0 If, x, b re i AP,, y, b re i GP d, z, b re i HP such tht x = 9z d > 0, b > 0 the (A) y = z (B) x = y (C) y = x + z (D) oe of these
SET II If three umbers re i HP the the umbers obtied by subtrctig hlf of the middle umber from ech of them re i (A) AP (B) GP (C) HP (D) oe of these, b, c, d, e re five umbers i which the first three re i AP d the lst three re i HP If the three umbers i the middle re i GP the the umbers i the odd plces re i (A) AP (B) GP (C) HP (D) oe of these If, b, c re i AP the, b, c re i bc c b (A) AP (B) GP (C) HP (D) oe of these The AM of two give positive umbers is If the lrger umber is icresed by, the GM of the umbers becomes equl to the AM of the give umbers The the HM of the give umbers is (A) (B) (C) (D) oe of these 5 Let, b be two positive umbers, where > b d GM = 5 HM for the umbers The is (A) b (B) b (C) b (D) b 6 If,,, b re i AP d, g, g, g,, g, b re i GP d h is the HM of d b the (A) h g g g g g g is equl to (B) h (C) h (D) oe of these 7 Let = 0 d,, be rel umbers such tht i = i - + for ll i the the AM of the umbers,, hs the vlue A where (A) A (B) A < - (C) A (D) A 8 Let there be GP whose first term is d the commo rtio is r If A d H re the rithmetic me d the hrmoic me respectively for the first terms of the GP, A H is equl to (A) r - (B) r (C) r (D) oe of these 9 If the first d the ( - ) th terms of AP, GP d HP re equl d their th terms re, b d c respectively the (A) = b = c (B) b c (C) + c = b (D) c - b = 0 0 b b is the HM betwee d b if is (A) 0 (B) (C) - (D)
If the hrmoic me betwee P d Q be H the H is equl to P Q (A) (B) PQ P Q (C) P Q PQ (D) If p, q, r be three positive rel umbers, the the vlue of (p + q) (q + r) (r + p) is (A) > 8 pqr (B) < 8 pqr (C) 8 pqr (D) oe of these 5 Let t r r(r!) The tr is equl to r (A) 5! - (B) 5! + (C) 6! - (D) oe of these Four umbers re i rithmetic progressio The sum of first d lst terms is 8 d the product of both middle terms is 5 The lest umber of the series is (A) (B) (C) (D) 5 If rithmtic me re iserted betwee d 8, the the sum of the resultig series is obtied s 00, the the vlue of is (A) 6 (B) 8 (C)9 (D)0 6 The sum of three cosecutive terms i geometric progressio is If is dded to the first d the secod term d is subtrcted from the third term The resultig ew terms re i rithmetic progressio The the lowest of the origil terms is (A) (B) (C) (D) 8 7 The gles of trigle re i A P d the rtio of the gretest to the smllest gle is : The the smllest gle is (A) 6 (B) (C) (D) oe of these 8 Let S be the sum, P be the product d R be the sum of the reciprocls of terms of GP The P R : S is equl to (A) : (B) (commo rtio) : (C) (first term) : (commo rtio) (D) oe of these b c 9 If, b, c re i AP, the,, b c c b (A) AP (B) GP (C) HP (D) oe of these b 0 If is the AM betwee d b, the is equl to b (A) - (B) - (C) 0 (D)
The legth of the side of squre is meter A secod squre is formed by joiig the middle poits of the sides of the squre The third squre is formed by joiig the middle poits of the sides of the secod squre d so othe the sum of the re of squres which crried up to ifiity is (A) (B) (C) (D) If S P ( ) Q, where S deotes the sum of the first terms of AP, the commo differece is (A) P + Q (B) P + Q (C) Q (D) Q The three sides of right gled trigle re i G P The tgets of the two cute gles re (A) (C) 5 5 d 5 d 5 5 5 (B) d (D) oe of the foregoig pirs of umbers log ylogz logzlog x log xlog y If x, y, z re positive the the miimum vlue of x y z is (A) (B) (C) 9 (D) 6 5, b, c re three positive umbers d bc hs the gretest vlue The 6 (A) b, c (C) b c (B) b, (D) oe of the c 6 The sum of ll the umbers betwee 00 d 00 which re divisible by 7 is (A) 987 (B) 789 (C) 879 (D) 879 7 Observe tht =, = + 5, = 7 + 9 +, = + 5 + 7 + 9The s similr series is ( ) ( ) ( ) (A) (B) ( + + ) + ( + + ) + ( + + 5) + + ( + ) (C) ( + ) + ( + ) + ( + 5) + + ( + ) (D) oe of these / 8 Let r / t r The t r is equl to r 0 r
(A) 0 0 (B) 9 0 (C) 0 (D) oe of these 9 The sum of the series : + + 5 6 + 00 is (A) 000 (B) 5050 (C) 55 (D) 5500 F() 0 Suppose tht F( + ) = for =,,, d F() = The F(0) equls (A) 50 (B) 5 (C) 5 (D) oe of these
SET III If x, x, x re o-zero rel umbers such tht (x + x, + x - ) (x + x + + x ) (x x + x x + + x - x ) the x, x,,x re i (A) AP (B) GP (C) H P (D) oe of these If, b d c re three positive rel umbers, which oe of the followig hold? (A) b c bc c b (C) (b c) (c ) ( b) 8 bc If N d >, which oe of the followig holds? (B) b c bc (D) oe of these (A) > 5 ( - ) (B) (C) (D) ll of these If x, y d z re positive rel umbers, such tht x + y + z =, the (A) 9 (B) ( - x) ( - y) ( - z) 8xyz x y z (C) ( - x) ( - y) ( - z) 8 7 (D) ll of these 5 If, b d c re three positive rel umbers, the the miimum vlue of the expressio b c c b b c is (A) (B) (C) (D) 6 W I Fidig the umber of shot rrged i complete pyrmid the bse of which is equilterl trigle re s follows : Suppose tht ech side of the bse cotis shot, the the umber of shot i the lowest lyer is + ( ) + ( ) + + ; ( ) ie or ( ) If we write,, for d obti the shot i the d, rd,lyer If the bse of pyrmid is rectgle, the umber of shot rrged i complete pyrmid c be fid out s follow Let m d be the umber of shot i the log d short side respectively of the bse The top lyer cosist of sigle row of m ( ) or m + shot ; i the ext lyer the umber is (m + ) i the ext lyer the umber is (m + ) d so o i the lowest lyer the umber is (m + ) S = (m + ) + (m + ) + (m + ) + + (m + ) = (m ) ( + + + + ) ( + + + )
= ( m ) ( ) ( )( ) 6 ( ) = {(m ) } 6 ( ) (m ) = 6 6 The umber of shot rrged i complete pyrmid squre bse of side is (A) (C) ( ( ) ( ) 6 ) ( ) 6 (B) ( )( ) 6 (D) oe of these 7 The umber of shot rrged i icomplete pyrmid, the bse of which is rectgle, where, b deotes the umber of shots i the two side of the top lyer, d the umber of lyers (A) [6b ( b) ( ) ( ) ( )] (B) [6b ( b)( ) ( ) ( )] 6 6 (C) [6b ( b) ( ) ( )( )] (D) oe of these 6 8 The umber of shot i icomplete squre pile of 7 courses, hvig 0 shot i ech side of the bse is (A) 000 (B) (C) 800 (D) oe of these 9 The umber of shot i complete rectgulr pile of 5 courses, hvig 0 shot i the loger side of the its bse is (A) 800 (B) 850 (C) 80 (D) 80 0 The umber of shot required to complete rectgulr pile hvig 5 d 6 shot i the loger d shorter side, respectively, of its upper course is (A) 80 (B) 85 (C) 8 (D) 90 W II The sum of terms of series ech term of which is composed of r fctors i rithmeticl progressio, the first fctors of the severl terms beig i the sme rithmeticl progressio Let the series be deoted by u + u + u + + u, where u = ( b) ( b) ( b)( r b ) Replcig by, we hve u = ( b) ( b)( b)( r b) ( b) u r b u v, sy Replcig by + we hve ( rb) u v Therefore by subtrctio ; (r + ) b u = v + v
Similrly, (r + ) b u = v v, (r + ) b u = v v, (r +) b u = v v By dditio, (r + ) b S = v + v ; v v ( rb)u tht is, S = = C, sy ; (r )b (r )b where C is qutity idepedet of, which my be fouded by scribig to some prticulr vlue The bove result gives us the followig coveiet rule : Write dow the th term, ffic the ext fctor t the ed divide by the umber of fctors thus icresed d by the commo differece The sum of terms of series 5 + 5 7 + 5 7 9 + is (A) ( + 8 7 ) (B) ( + 8 + 7 ) (C) ( 8 + 7 ) (D) oe of these The sum of terms of series 5 9 + 6 0 + 7 + (A) ( ) ( 8) ( 9) (B) ( )( 8)( 9) (C) ( ) ( 8)( 9) (D) oe of these The sum of terms of series + 5 + 5 6 + (A) ( )( ) ( )( ) (B) ( ) ( )( ) ( ) 5 5 (C) ( )( ) ( )( ) 5 (D) oe of these The sum of terms of series 7 + 7 0 + 7 0 + (A) (7 90 5 50) (B) (7 90 5 50) (C) (7 90 5 50) (D) oe of these 5 The sum of terms of series 7 + 5 8 + 6 9 + (A) ( ) ( 6)( 7) (B) ( )( 6) ( 7) (C) ( )( 6)( 7) (D) oe of these W III To fid the gretest vlue of m b c p whe + b + c + is costt ; m,, p,beig positive itegers Sice m,, p,re costts, the expressio m b c p will be gretest m p whe b c is gretest But this lst expressio is the product of m + + p + m p
b c fctors whose sum is m, or + b + c +, d therefore costt m p b c Hece m b c p will be gretest whe the fctors,,, re ll equl, tht is, whe m p m b c b c p m p Thus the gretest vlue is m p b c m p m p mp 6 The gretest vlue of ( + x) ( x) for y rel vlue of x umericlly less th is (A) 7 7 7 (B) 5 8 7 7 7 (C) 6 0 7 7 7 (D) oe of these 7 A odd iteger is divided ito two itegrl prts whose product is mximum, the these two prts re (A), (B), + (C), + (D) oe of these 8 The miimum vlue of ( x) (b x), ( x > c, > c, b > c) is c x (A) ( c) b c (B) c b c (C) b c (D) oe of these 9 The mximum vlue of (7 x) ( + x) 5 whe x lies betwee 7 d (A) 5 5 (B) 5 5 (C) 5 (D) oe of these (5 x)( x) 0 The miimum vlue of, (x > ) is x (A) 7 (B) 8 (C) 9 (D) oe of these True & Flse The p th term of AP is d q th term is b The the sum of its (p + q) terms is p q b b p q The gles of trigle re i AP d tget of smllest gle is, the the gles re 5 0, 60 0, 75 0 The sums of terms of three rithmeticl progressios re S, S d S The first term of ech
is uity d the commo differeces re, d respectively The S + S = S If, b, c re i AP, the,, bc c b re i AP 5 If x, y, z re i AP, the x + z 8y = zyz Fill i the blks 6 The sum of three umbers i AP is 5 wheres sum of their squres is 8 The umbers re 7 Four umbers i AP whose sum is 0 d sum of their squres is 0, re 8 The sum of ll two digit umbers which whe divided by, yield uity s remider is 9 Let x, x + d x re the I st three terms of AP, its sum of I st upto 0 terms is 0 If, b, c re the positive rels umbers d ( c) = (b c) the, b, c re i
LEVEL I ANSWER 900 9 6 8 5 6,, OR,, 6 7 r, or =, or d = 08, or 60 9 8 8 99 00 + 0 7/6 b + + + S ( b) where = x / & b = y / 080 m LEVEL II d (i) 6 (ii) ( ), whe be eve & ( ), whe be odd 6 8,, 7, 6, 68 8 AP is + 5 + 8 + + & GP is + 6 + + + or AP is & GP is 5 65 6 5 79 8 6 6 x = 0 5, y = 0 5 (i) S = (/) - [/{( + ) ( + )}] ; S = / (ii) S = (/) - [/{6( + ) ( + ) }] ; S = / (iii) (/5) ( + ) ( + ) ( + ) ( + ) (iv) /( + )
SET I A D B C 5 C 6 D 7 C 8 B 9 A 0 C B A B A 5 C 6 B 7 D 8 A 9 B 0 D A A A B 5 A 6 D 7 B 8 C 9 C 0 B SET II B B A A 5 A 6 A 7 C 8 A 9 D 0 A A A C D 5 B 6 B 7 A 8 A 9 A 0 D B D B A 5 B 6 C 7 C 8 B 9 B 0 B SET III B A D B 5 D 6 C 7 B 8 B 9 C 0 D B A C A 5 B 6 A 7 B 8 A 9 B 0 C T T F F 5 F 6 (, 5, 7) 7 (,, 6, 8) 8 0 9 50 or 80 0 AP