EXERCISE a a a 5. + a 15 NEETIIT.COM

- Dowlod our droid App. Sigle choice Type Questios EXERCISE -. The first term of A.P. of cosecutive iteger is p +. The sum of (p + ) terms of this series c be expressed s () (p + ) () (p + ) (p + ) () (p + ) () p + (p + ). If,,,... re i A.P. such tht + 5 + 0 + 5 + 0 + = 5, the + + +... + + is equl to () 909 () 75 () 750 () 900. The sum of itegers from to 00 tht re divisible by or 5 is () 550 () 050 () 050 () oe of these. The iterior gles of polygo re i A.P. If the smllest gle is 0º & the commo differece is 5º, the the umber of sides i the polygo is: () 7 () 9 () 6 () oe 5. Cosider A.P. with first term '' d the commo differece 'd'. Let S k deote the sum of its first K terms. If Skx is idepedet of x, the Sx () = d/ () = d () = d () oe 6. If x R, the umbers 5 +x + 5 x, /, 5 x + 5 x form A.P. the '' must lie i the itervl: () [, 5] () [, 5] () [5, ] () [, ) 7. There re A.M's betwee d 5, such tht the 8th me: ( ) th me:: : 5. The vlue of is. () () 6 () 8 () 0 8. The third term of G.P is. The product of the first five terms is () () 5 () () oe of these 9. Suppose, b, c re i A.P. &, b, c <. If x = + + +... to ; y = + b + b +... to & z = + c + c +... to the x, y, z re i: () A.P. () G.P. () H.P. () oe 0. If S is the sum to ifiity of G.P. whose first term is, the the sum of the first terms is () S S. The sum of the series () S S () log + log + log8 S () oe of these +... + log is () ( + ) () ( + ) ( + ) () ( ) () ( + ) - Free NEET & IIT Study Meteril & Ppers - Dowlod our droid App. Pge -

- Dowlod our droid App.. For sequece { }, = d 0 =. The r r is () 0 [ + 9 ] () 0 () ( 0 ) () oe of these., be the roots of the equtio x x + = 0 d, the roots of x x + b = 0 d umbers,,, (i this order) form icresig G.P., the () =, b = () =, b = () =, b = () =, b = 6. The rtiol umber, which equls the umber. 57 with recurrig deciml is 55 () 00 79 () 997 55 () 999 () oe of these 5. If the sum of the roots of the qudrtic equtio, x + bx + c = 0 is equl to sum of the squres of their reciprocls, the c, b, c re i b () A.P. () G.P. () H.P. () oe 6. If + ( + d) + ( + d) +... + upto = 8, the the vlue of d is: () 9 () 5 () () oe of these 7. The sum of the first -terms of the series +. + +. + 5 +.6 +... is Whe is odd, the sum is () ( ) () ( ) () ( ) ( ) () ( ), whe is eve. 8. If A, G & H re respectively the A.M., G.M. & H.M. of three positive umbers, b, & c, the the equtio whose roots re, b, & c is give by: () x Ax + G x G = 0 () x Ax + (G /H)x G = 0 () x + Ax + (G /H) x G = 0 () x Ax (G /H) x + G = 0 9. If x = b y = c z = d t d, b, c, d re i G.P., the x, y, z, t re i () A.P. () G.P. () H.P. () oe of these 0. The sum is equl to: r r () () / () / () oe - Free NEET & IIT Study Meteril & Ppers - Dowlod our droid App. Pge -

- Dowlod our droid App. Multiple choice Type Questios. For the A.P. give by,,...,,..., the equtios stisfied re () + + = 0 () + = 0 () + = 0 () + 6 + 5 = 0 9. If sum of the ifiite G.P., p,,, p,,... is, the vlue of p is p p () () (). If positive umbers, b, c re i A.P. d, b, c re i H.P., the () = b = c () b = + c () b =. Idicte the correct ltertive(s), for 0 < < /, if: x = 0 cos, y = 0 si, z = 0 c 8 cos si the: () () oe of these () xyz = xz + y () xyz = xy + z () xyz = x + y + z () xyz = yz + x 5. If the rithmetic me of two positive umbers & b ( > b) is twice their geometric me, the : b is: () + : () 7 + : () : 7 () : 6. If r (r ) (r + ) = + b + c + d + e, the r () + c = b + d () e = 0 (), b /, c re i A.P. () c/ is iteger SUBJECTIVE TYPE QUESTIONS. I A.P. the third term is four times the first term, d the sixth term is 7 ; fid the series. As., 5, 8,.... The third term of A.P. is 8, d the seveth term is 0 ; fid the sum of 7 terms. As. 6. How my terms of the series 9, 6,,... must be tke tht the sum my be 66? As.. Fid the umber of itegers betwee 00 & 000 tht re (i) divisible by 7 (ii) ot divisible by 7 As. 8, 77 5. Fid the sum of ll those itegers betwee 00 d 800 ech of which o divisio by 6 leves the remider 7. As. 9668 6. Fid the sum of 5 terms of the series whose p th term is 7 p +. As. 60 - Free NEET & IIT Study Meteril & Ppers - Dowlod our droid App. Pge -

- Dowlod our droid App. 7. The sum of three umbers i A.P. is 7, d their product is 50, fid them. As., 9, 8. If, b, c re i A.P., the show tht: (i) (b + c), b (c + ), c ( + b) re lso i A.P. (ii) b + c, c + b, + b c re i A.P. 9. The third term of G.P. is the squre of the first term. If the secod term is 8, fid its sixth term. As. 8 0. The cotiued product of three umbers i G.P. is 6, d the sum of the products of them i pirs is 56; fid the umbers As., 6, 8. If the p th, q th, r th terms of G.P. be, b, c respectively, prove tht q r b r p c p q =.. The sum of three umbers which re cosecutive terms of A.P. is. If the secod umber is reduced by & the third is icresed by, we obti three cosecutive terms of G.P., fid the umbers. As., 7, or, 7,. If the p th, q th & r th terms of AP re i GP. Fid the commo rtio of the GP. As. q r p q. The sum of ifiite umber of terms of G.P. is d the sum of their cubes is 9. Fid the series. As. 6,, /,... 5. If, b, c, d re i G.P., prove tht : (i) (ii) ( b ), (b c ), (c d ) re i G.P.,, re i G.P. b b c c d 6. Sum the followig series (i) + + + +... to terms. 7 5 (ii) + + + + +... to ifiity. 6 6 56 As. (i) (ii) 8 7. Fid the sum of terms of the series the r th term of which is (r + ) r. As.. + + +. - Free NEET & IIT Study Meteril & Ppers - Dowlod our droid App. Pge -

- Dowlod our droid App. 8. Fid the th term of H.P. whose 7 th term is 0 d th term is 8. As. 9. The rithmetic me of two umbers is 6 d their geometric me G d hrmoic me H stisfy the reltio G + H = 8. Fid the two umbers. As. =, b = 8 0. If betwee y two qutities there be iserted two rithmetic mes A, A ; two geometric mes G, G ; d two hrmoic mes H, H the prove tht G G : H H = A + A : H + H.. Usig the reltio A.M. G.M. prove tht (i) (ii) t + cot ; if 0 < < (x y + y z + z x) (xy + yz + zx ) 9x y z. (x, y, z re positive rel umber) (iii) ( + b). (b + c). (c + ) bc ; if, b, c re positive rel umbers. Fid the sum of the terms of the series whose th term is (i) ( + ) (ii) As. (i) ( + ) ( + 7) (ii) ( + + ) + 6. Fid the sum to -terms of the sequece. (i) + 5 + + 9 + 6 +... to -terms. (ii) + + + +... to terms. As. (i) + (ii) 7 (0 + 9 0) 5. Fid the sum to -terms of the sequece. (i) + + +.....5.5.7 5.7.9 (ii).. +.. +. 5. +... As. (i) ( )( ) (ii) ( + ) ( + ) ( + ) ( + ) 0 ANSWER KEY Q. 5 6 7 8 9 0 A. Q. 5 6 7 8 9 0 A.,,,, Q. 5 6 A.,, ll - Free NEET & IIT Study Meteril & Ppers - Dowlod our droid App. Pge - 5

- Dowlod our droid App. EXERCISE - Sigle choice Type Questios. If x i > 0, i =,,..., 50 d x + x +.. + x 50 = 50, the the miimum vlue of x + x +...+ x equl to 50 () 50 () (50) () (50) () (50). If,,,,...,, b re i A.P. d, g, g, g,...g, b re i G.P. d h is the hrmoic me of d b, the () h g g + g g +... + g g is equl to () h () h () h. Oe side of equilterl trigle is cm. The midpoits of its sides re joied to form other trigle whose mid poits re i tur joied to form still other trigle. This process cotiues idefiitely. The the sum of the perimeters of ll the trigles is () cm () cm () 88 cm () oe of these. If the sum of terms of G.P. (with commo rtio r) begiig with the p th term is k times the sum of equl umber of terms of the sme series begiig with the q th term, the the vlue of k is: () r p/q () r q/p () r p q () r p + q 5. If A, A be two A.M.s d G, G be two G.M.s betwee d b, the () b b () b b () b b A A G G () b b is equl to 6. If P, Q be the A.M., G.M. respectively betwee y two rtiol umbers d b, the P Q is equl to () b () b () b b 7. I G.P. of positive terms, y term is equl to the sum of the ext two terms. The commo rtio of the G.P. is () cos 8 () si 8 () cos 8 () si 8 () 8. If + + +... upto = 6, the + + 5 +... = () / () / () /8 () oe of these b 9. If,,... re i A.P. with commo differece d 0, the the sum of the series (si d) [cosec cosec + cosec cosec +... + cosec cosec ] () sec sec () cosec cosec () cot cot () t t - Free NEET & IIT Study Meteril & Ppers - Dowlod our droid App. Pge - 6

- Dowlod our droid App. 0. Sum of the series S = + +... 00 + 00 is () 007006 () 00500 () 000506 () oe of these 5. If H = + + +...+, the vlue of + + +... + is () H () + H () H () H + S( 8S). If S, S, S re the sums of first turl umbers, their squres, their cubes respectively, the is S equl to () () () 9 () 0 Multiple choice Type Questios. The sides of right trigle form G.P. The tget of the smllest gle is () 5 () 5 (). If b, b, b (b i > 0) re three successive terms of G.P. with commo rtio r, the vlue of r for which the iequlity b > b b holds is give by () r > () 0 < r < () r =.5 () r = 5. 5 SUBJECTIVE TYPE QUESTIONS. The sum of the first te terms of AP is 55 & the sum of first two terms of GP is 9. The first term of the AP is equl to the commo rtio of the GP & the first term of the GP is equl to the commo differece of the AP. Fid the two progressios. As. ( + 6 + +...); (/ + 5/ + 65/6 +...). Fid the sum i the th group of sequece, (i), (, ); (, 5, 6, 7); (8, 9,..., 5);... (ii) (), (,, ), (5, 6, 7, 8, 9),... As. (i) ( + ) (ii) ( ) +. Fid the sum of the series As. 5 55 + () 65 6 555 5555 + () () +... up to + () 5. If 0 < x < d the expressio exp {( + cos x + cos x + cos x + cos x +... upto ) log e } stisfies the qudrtic equtio y 0y + 6 = 0 the fid the vlue of x. As.,, - Free NEET & IIT Study Meteril & Ppers - Dowlod our droid App. Pge - 7

- Dowlod our droid App. 5. Show tht,, 5 cot be the terms of sigle A.P. 6. I circle of rdius R squre is iscribed, the circle is iscribed i the squre, ew squre i the circle d so o for times. Fid the limit of the sum of res of ll the circles d the limit of the sum of res of ll the squres s. As. R ; R 7. Give tht re roots of the equtio, A x x + = 0 d, the roots of the equtio, B x 6 x + = 0, fid vlues of A d B, such tht,, & re i H.P. As. A = ; B = 8 8. If, b, c re positive rel umbers the prove tht b c + c + b bc ( + b + c). 9. Sum the followig series to terms. (i) (ii) r r (r + ) (r + ) (r + )........ ( ) ( ) As. (i) (/5) ( + ) ( + ) ( + ) ( + ) (ii) ( ) ( ) 0. Sum of the followig series (i) 5 (ii) As. (i) 5 5 5 6 +... 5 5 5 5 5 + (ii) + ( ) ; s = ( ) +.... If, b, c re positive rel umbers d sides of the trigle the prove tht ( + b + c) 7 ( + b c) (c + b) (b + c ). If the sum of m terms of A.P. is equl to the sum of either the ext terms or the ext p terms, the prove tht (m + ) = (m + p). m p m ANSWER KEY Q. 5 6 7 8 9 0 A. Q. A., ll - Free NEET & IIT Study Meteril & Ppers - Dowlod our droid App. Pge - 8

- Dowlod our droid App. MATCH THE COLUMN TYPE QUESTIONS EXERCISE -. Colum Colum As. (A) If log 5, log 5 ( x 5) d log 5 ( x 7/) re i A.P., (p) 6 the vlue of x is equl to (B) Let S deote sum of first terms of A.P. If S = S, (q) 9 the S S is 8 6 (C) Sum of ifiite series + + + + - - - is (r) (D) The legth,breth, height of rectgulr box re i G.P. The (s) volume is 7, the totl surfce re is 78. The the legth is (A) (p), (B) (p), (C) (q), (D) (q). Colum Colum As. (A) If log x y, log z x, log y z re i G.P., xyz = 6 d x,y,z (p) x re i A.P., the = y (B) The vlue of. 8. 86 - - - = (q) (C) If x, y, z re i A.P., the (r) (x + y z) (y + z x) (z + x y) = kxyz, where k N, the k = (D) There re m A.M s betwee d. If the rtio of the (s) 7 th d (m ) th mes is 5 : 9 the 7 m = (A) (r), (B) (p), (C) (s), (D) (p) ASSERTION/REASON TYPE QUESTIONS. Sttemet : The series for which sum to terms, S, is give by S = 5 + 6 is A.P. Sttemet : The sum to terms of A.P. hvig o zero commo differece is qudrtic i, i.e., + b. () Sttemet is true, Sttemet - is true ; Sttemet - is correct expltio for sttemet. () Sttemet is true, Sttemet- is true;sttemet is ot correct expltio for sttemet-. () Sttemet- is true, Sttemet- is flse. () Sttemet is flse, Sttemet is true. - Free NEET & IIT Study Meteril & Ppers - Dowlod our droid App. Pge - 9

- Dowlod our droid App.. STATEMENT- :,,, 8,... is G.P.,, 8, 6, is G.P. d +, + 8, + 6, 8 +,... is lso G.P. STATEMENT- : Let geerl term of G.P. with commo rtio r be T k + d geerl term of other G.P. with commo rtio r be T k +, the the series whose geerl term T k + = T k + + T k + is lso G.P. with commo rtio r. () Sttemet- is True, Sttemet- is True; Sttemet- is correct expltio for Sttemet-. () Sttemet- is True, Sttemet- is True; Sttemet- is NOT correct expltio for Sttemet- () Sttemet- is True, Sttemet- is Flse () Sttemet- is Flse, Sttemet- is True 5. Sttemet : The sum of the first 0 terms of the sequece,,,7,,6,- - - is 50. Sttemet : If the successive differeces of the terms of sequece form A.P. the geerl term of sequece is of the form + b + c. () Sttemet- is true, sttemet- is true; sttemet- is correct expltio for sttemet-. () Sttemet- is true, sttemet- is true; sttemet- is ot correct expltio for sttemet-. () Sttemet- is true, sttemet- is flse. () Sttemet-, is flse, sttemet - is true. 6. Sttemet :,6, re i G.P., the 9,,8 re i H.P. Sttemet : If middle term is dded i three cosecutive terms of G.P, resultt will be i H.P. () Sttemet- is true, sttemet- is true; sttemet- is correct expltio for sttemet. () Sttemet- is true, sttemet- is true; sttemet- is ot correct expltio for sttemet-. () Sttemet- is true,sttemet- is flse. () Sttemet- is flse, sttemet- is true. PASSAGE BASED QUESTIONS Pssge - 0 ( ) We kow tht + + +... = = f(), ( )( ) + + +... + = = g(), 6 + + +... + = ( ) = h() 7. g() g( ) must be equl to () () ( ) () () 8. Gretest eve turl umber which divides g() f(), for every, is () () () 6 () oe of these 9. f() + g() + h() is divisible by + + +... + () oly if = () oly if is odd () oly if is eve () for ll N - Free NEET & IIT Study Meteril & Ppers - Dowlod our droid App. Pge - 0

- Dowlod our droid App. Pssge - 0 I sequece of ( + ) terms the first ( + ) terms re i AP whose commo differece is, d the lst ( + ) terms re i GP whose commo rtio 0.5. If the middle terms of the AP d GP re equl, the 0. Middle term of the sequece is (). (). First term of the sequece is ().. Middle term of the GP is () TRUE/FALSE TYPE QUESTIONS () ()... (). () Noe of these () ().. Equl umbers re lwys i A.P., G.P. d H.P. [Flse]. If x > d,, x x As. True b c x 5. If, b, c be i H.P., the b, b, c b As. be 6. If y be 7. If re i G.P., the, b, c re i AP As. As. b Flse (GP) y b ce = y b ce Flse (G.P.) c, True c y c de y = y, c de b will be i AP the,b,c,d re i H.P. () (). re i A.P. the 9 x+, 9 bx+,9 cx+, x 0 re i G.P. - Free NEET & IIT Study Meteril & Ppers - Dowlod our droid App. Pge -

- Dowlod our droid App. FILL IN THE BLANKS TYPE QUESTIONS 8. The sum to terms of the series, + + 7 8 + 5 6 As. + +... is equl to 9. If t deotes the th term of the series + + 6 + + 8 +... the t 50 is As. 9 + 0. If S = + ( )d is the sum of first terms of A.P., the commo differece is As. + d x. If x > 0 the the expressio 00 x x x... x As. 0 00 is lwys less th or equl to. If G d G re two geometric mes d A is the rithmetic me iserted betwee two positive umbers the the vlue of As. A G G is. G G ANSWER KEY Q. 5 6 7 8 9 0 A. - Free NEET & IIT Study Meteril & Ppers - Dowlod our droid App. Pge -

- Dowlod our droid App. EXERCISE - SECTION - A PREVIOUS YEAR JEE QUESTIONS. Let ( > ) be positive iteger. The the lrgest iteger m such tht ( m + ) divides ( + + +... 7 ) is [IIT- 995] () 7 () 6 () 6 (). For y odd iteger, ( ) +... + ( ) =. [IIT- 996] As. ( ) ( + ). Sum of the series + + 5 +... to 6 terms is [IIT- 996] () 6 () 6 () 56 () Noe of these y. If cos (x y), cos x d cos (x + y) re i H.P., the cosx sec =. [IIT- 997] As. ± 5. Let p & q be roots of the equtio x x + A = 0, d let r & s be the roots of the equtio [IIT- 997] x 8x + B = 0. If p < q < r < s re i rit hmtic progressi o, the A = d B =. As. A =, B = 77 6. Let x be the rithmetic me d y, z be the two geometric mes betwee y two positive umbers. The y z xyz =. As. [IIT- 997] 7. If oe A.M. A d two G.M. s p d q be iserted betwee y two umbers, the the vlue of p + q is [IIT- 997] () pq A () A pq () A p q () Noe of these 8. If x =, y >, z > re i GP, the x, y, re i: [IIT- 998] z () AP () HP () GP () oe of the bove 9. Let T r deotes r th term of A.P., for r =,,,... If for some positive itegers m, we hve T m = d T = m, the Tm equls [IIT- 998] () m () () () 0 m - Free NEET & IIT Study Meteril & Ppers - Dowlod our droid App. Pge -

- Dowlod our droid App. 0. If x, x, x s well s y, y, y re i G.P. with sme commo rtio, the the poits (x, y ), (x, y ) d (x, y ) [IIT- 999] () lie o stright lie () lie o ellipse () lie o circle () re vertices of trigle. If,,... 0 be i A.P. d h, h,... h 0 be i H.P. If = h = d 0 = h 0 =, the h 7 is equl to [IIT- 999] () () () 5 () 6. For positive iteger, let () = + + + +... (. The [IIT- 999] ) () (00) 00 () (00) > 00 () (00) 00 () (00) > 00. Let S,S,... be squres such tht for ech, the legth of side of S equls the legth of digol of S +. If the legth of side of S is 0 cm, the for which of the followig vlues of is the re of S less th sq. cm? [IIT- 999] () 7 () 8 () 9 () 0. The hrmoic me of the roots of the equtio 5 x 5 () () () 6 () 8 x + 8 + 5 = 0 is [IIT- 999] 5. If, b, c, d re positive rel umbers such tht + b + c + d =, the M = ( + b) (c + d) stisfies the reltio: () 0 M () M () M () M [IIT- 000] 6. Cosider ifiite geometric series with first term '' d commo rtio r. If the sum is d the secod term is /, the: [IIT- 000] () = 7, r = 7 () =, r = 8 () =, r = () =, r = 7. The fourth power of the commo differece of rithmetic progressio with iteger etries dded to the product of y four cosecutive terms of it. Prove tht the resultig sum is the squre of iteger. [IIT- 000] 8. Let, be the roots of x x + p = 0 d, be the roots of x x + q = 0. If re i G.P., the the itegrl vlues of p d q respectively, re [IIT- 00] (), (), () 6, () 6, 9. If the sum of the first terms of the A.P., 5, 8,..., is equl to the sum of the first terms of the A.P. 57, 59, 6,..., the equls [IIT- 00] () 0 () () () 0. Let the positive umbers, b, c, d be i A.P. The bc, bd, cd, bcd re: [IIT- 00] () ot i A.P./G.P./H.P. () i A.P. () i G.P. () i H.P. - Free NEET & IIT Study Meteril & Ppers - Dowlod our droid App. Pge -

- Dowlod our droid App.. Let,,..., be positive rel umber i geometric progressio. For ech, let A, G, H be respectively the rithmetic me, geometric me & hrmoic me of,,...,. Fid expressio for the geometric me of G, G,..., G i terms of A, A,..., A, H, H,..., H. [IIT- 00] As. G =? K? (A k Hk ). If,,,..., re positive rel umbers whose product is fixed umber c, the the miimum vlue of + + +... + + is [IIT- 00] () (c) / () ( + ) c / () c / () ( + )(c) /. Suppose, b, c re i A.P. d, b, c re i G.P. if < b < c d + b + c =, the the vlue of is () () () () [IIT- 00]. Let, b be positive rel umbers. If, A, A, b re i rithmetic progressio,, G, G, b re i geometric progressio d, H, H, b re i hrmoic progressio, show tht [IIT- 00] G G H H = A A H H 5. If 0, the t x x + x x = ( b ) ( b ). 9 b is lwys greter th or equl to: [IIT- 00] () t () () () sec 6. If, b & c re i rithmetic progressio d, b & c re i hrmoic progressio, the prove tht either = b = c or, b & c re i geometric progressio. [IIT- 00] 7. A ifiite G.P. hs first term s x d sum upto ifiity s 5. The the rge of vlues of x is: [IIT- 00] () x 0 () x 0 () 0 < x < 0 () 0 x 0 8. I the qudrtic equtio x + bx + c = 0, = b c d +, +, +, re i G.P. where re the root of x + bx + c = 0, the [IIT- 005] () 0 () b 0 () c 0 () 0 9. If = + +...( ) d b =, the fid the miimum turl umber 0 such tht b > 0 As. miimum turl umber 0 = 6 [IIT- 006] - Free NEET & IIT Study Meteril & Ppers - Dowlod our droid App. Pge - 5

- Dowlod our droid App. Pssge Let V r deote the sum of the first r terms of rithmetic progressio (A.P.) whose first term is r d the commo differece is (r ). Let T r = V r + V r d Q r = T r + T r for r =,,... 0. The sum V + V +... + V is [IIT- 007] () ( + ) ( + ) () ( + ) ( + + ) () ( + ) () ( + ). T r is lwys [IIT- 007] () odd umber () eve umber () prime umber () composite umber. Which oe of the followig is correct sttemet? [IIT- 007] () Q, Q, Q,... re i A.P. with commo differece 5 () Q, Q, Q,... re i A.P. with commo differece 6 () Q, Q, Q,... re i A.P. with commo differece () Q = Q = Q =... Pssge Let A, G, H deote the rithmetic, geometric d hrmoic mes, respectively, of two distict positive umbers. For, let A d H hve rithmetic, geometric d hrmoic mes s A, G, H respectively.. Which oe of the followig sttemets is correct? [IIT- 007] () G > G > G >... () G < G < G <... () G = G = G =... () G < G < G 5 <... d G > G > G 6 >.... Which oe of the followig sttemets is correct? [IIT- 007] () A > A > A >... () A < A < A <... () A > A > A 5 >... d A < A < A 6 <... () A < A < A 5 <... d A > A > A 6 >... 5. Which oe of the followig sttemets is correct? [IIT- 007] () H > H > H >... () H < H < H <... () H > H > H 5 >... d H < H < H 6 <... () H < H < H 5 <... d H > H > H 6 >... 6. Suppose four positive umbers,,, re i GP. Let b =, b = b +, b = b + d b = b + Sttemet- : Numbers b, b, b, b re either i AP or i GP. Sttemet- : Numbers b, b, b, b re i HP. Correct swer is [] Sttemet- is true, Sttemet- is true; Sttemet- is correct expltio for sttemet-. [] Sttemet- is true, Sttemet- is true; Sttemet- is ot correct expltio for sttemet-. [] Sttemet- is true, Sttemet- is flse. [] Sttemet- is flse, Sttemet- is true. [IIT- 008] - Free NEET & IIT Study Meteril & Ppers - Dowlod our droid App. Pge - 6

- Dowlod our droid App. 7. If the sum of first terms of AP is c, the the sum of squres of these terms is [IIT- 009] [] c 6 [] c [] c [] c k 8. Let s k, k =,,...00 deote the sum of the ifiite geometric series whose first term is d the commo k! 00 00 rtio is. The the vlue of k ( k k ) S k is. 00! k As. 9. Let,,... be rel umbers stisfyig = 5, 7 > 0 d k = k k for k =,,...,.... If As. 0... = 90, the the vlue of is equl to. 0. Let,,... 00 be rithmetic progessio wiht = d S p i, p 00. For y iteger with 0, let m = 5. If S S m does ot deped o, the is... As.,9. The miimum vlue of the sum of rel umbers 5,,, 8 d 0 with > 0 is... As. 8. Let,,,... be i hrmoic progressio with = 5 d 0 = 5. The lest positive iteger for which < 0 [IIT- 0] [] [] [] [] 5 k k( k). Let S ( ) k. The S c tke vlue(s) [IIT- 0] [] 056 [] 088 [] 0 []. A pck cotis crds umbered from to. Two cosecutive umbered crds re removed from the pck d the sum of the umbers o the remiig crds is. If the smller of the umbers o the removed crds is k, the k 0 = [IIT- 0] As. 5 5. Let, b, c be positive itegers such tht b is iteger. If, b, c re i geometric progressio d the rithmetic me of, b, c is b +, the the vlue of p i is [JEE(Adv.)-0] 6 - Free NEET & IIT Study Meteril & Ppers - Dowlod our droid App. Pge - 7

- Dowlod our droid App. 6. Suppose tht ll the therms of rithmetic progressio (A.P.) re turl umbers. If the rtio fo the sum of the first seve terms to the sum of the first eleve terms is 6 : d the seveth term lies i betwee 0 d 0, the the commo differece of this A.P. is [JEE(Adv.)-05] As. 9 7. Let the hrmoic me of two positive rel umbers d b be. If q is positive rel umber such tht, 5, q, b is rithmetic progressio, the the vlue(s) of q is (re) - As.,5 SECTION - B PREVIOUS YEAR AIEEE QUESTIONS. If, log 9 ( x + ), log (. x ) re i A.P. the x equls. [] log [] log [] log [] log. Sum of ifiite umber of therms i G.P. is 0 d sum of their squre is 00. The commo rtio of G.P. is [] 5 [] /5 [] 8/5 [] /5. Fifth term of G.P. is, the the product of its 9 terms is [] 56 [] 5 [] 0 [] Noe of these [AIEEE -00] [AIEEE -00] [AIEEE -00]. Let T r be the rth term of A.P. whose first term is d commo differece is d. If for some positive itegers m,, m?, T m = d T = m, the d equls [] 0 [] [] m []? m [AIEEE -00] 5. If AM d GM of two roots of qudrtic equtio re 9 d respectively, the this qudrtic equtio is [AIEEE -00] [] x 8x + 6 = 0 [] x + 8x 6 = 0 [] x + 8x + 6 = 0 [] x 8x 6 = 0 6. If,,,...,... re i G.P. the the vlue of the determit 7. If [] 0 [] [] [] where, b, c re i A.P. d, b, c x, y b, z C 0 0 0 log log log?? log log log??? 5 log log log? 6? 7? 8 the x, y, z re i, is [AIEEE -005] - Free NEET & IIT Study Meteril & Ppers - Dowlod our droid App. Pge - 8

- Dowlod our droid App. [] HP [] Arithmetic - Geometric Progressio [] AP [] GP [AIEEE -005]... p p 8. Let,,,... be terms of. A.P. If... q q p q the 6 equls- [AIEEE -006] 7 [] [] [] [] 7 9. If,,..., re i H.P., the the expressio + +...+ is equl to - [AIEEE -006] [] ( ) ( ) [] [] ( ) [] ( ) 0. The first two terms of geometric progressio dd upto. The sum of the third d the fourth terms is 8. If the terms of the geometric progressio re ltertely positive d egtive, the the first term is [] [] [] []. Sttemet- : For every turl umber >,.... Sttemet- : For every turl umber >, ( ). For bove sttemets [] Sttemet- is true, Sttemet- is true; Sttemet- is correct expltio for sttemet-. [] Sttemet- is true, Sttemet- is true; Sttemet- is ot correct expltio for sttemet-. [] Sttemet- is true, Sttemet- is flse. [] Sttemet- is flse, Sttemet- is true. 6 0. The sum of ifiity of the series... is [AIEEE -008] [AIEEE-008] [AIEEE-009] [] [] [] 6 []. A perso is to cout 500 currey otes. Let deote the umber of otes he couts i the th miute. If = =... = 0 = 50 d 0,,... re i AP with commo differece, the the time tke by him to cout ll otes is [AIEEE-00] [] miutes [] 5 miutes [] 5 miutes [] miutes. A m sves Rs. 00 i ech of the first three moths of his service. I ech of the subsequet moths his svig icreses by Rs. 0 more th the svig of immeditely previous moth. His totl svig from the strt of service will be Rs. 00 fter [AIEEE-0] [] 9 moths [] 0 moths [] moths [] 8 moths 5. If 00 times the 00 th term of AP with o-zero commo differece equls the 50 times its 50 th term, the the 50 th term of this AP is - Free NEET & IIT Study Meteril & Ppers - Dowlod our droid App. Pge - 9

- Dowlod our droid App. [] 50 [] 50 times its 50 th term[] 50 [] zero 6. The sum of first 0 terms of the squece 0.7, 0.77, 0.777,..., is [] 7 0 99 0 9 d i [] 7 0 79 0 8 d i [] 7 0 99 0 9 d i [] 7 d 8 ANSWER KEY [SECTION - A] Q. 5 6 7 8 9 0 A., Q. 5 6 7 8 9 0 A., Q. 5 6 7 8 9 0 5 6 A. Q. 7 8 9 0 5 6 7 A., [SECTION - B] 0 79 0 Q. 5 6 7 8 9 0 A. Q. 5 6 7 8 9 0 A. [AIEEE-0] [JEE(Mi)-0] 7. Let d be the roots of equtio px + qx + r = 0, p 0. If p, q, r re i AP d =, the the vlue of is [JEE(Mi)-0] 6 7 [] [] [] [] 9 9 9 9 8. Three positive umbers form icresig GP. If the middle term i this GP is doubled, the ew umbers re i AP. The the commo rtio of the GP is [JEE(Mi)-0] [] + [] [] [] 9. If m is the A.M. of two distict rel umbers l d (l, > ) d G, G d G re three geometric mes betwee l d, the G G G equls - [JEE(Mi)-05] () l m () lm () lm () l m 0. The sum of first 9 terms of the series...is [JEE(Mi)-05] 5 () 7 () 96 () () 9 i - Free NEET & IIT Study Meteril & Ppers - Dowlod our droid App. Pge - 0