Objective Mathematics

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Lucas McCormick
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. If sum of '' terms of a sequece is give by S Tr ( )( ), the 4 5 67 r (d) 4 9 r is equal to : T.

1 . If sum of '' terms of a sequece is give by S Tr ( )( ), the r (d) 4 9 r is equal to : T. Let a, b, c be distict o-zero real umbers such that a, b, c are i harmoic progressio ad a, b, c are i arithmetic progressio, the : b + ac = 0 4b + ac = 0 b ac = 0 (d) 4b ac = 0. Let a, b, c are i A.P. ad a, b, c are i G.P., if a < b < c ad a + b + c = /, the value of 'a' is : r (d) 4. If a, b, c R, the maximum value of bc ac ab is b c a c a b ( a b c ) abc ( a b c ) (d) abc 5. If the sum of first terms of a A.P. is c, the the sum of squares of these terms is : (4 ) c 6 (4 ) c (d) (4 ) c (4 ) c 6 p q r s 6. Let R {} ad ( l), ( l), ( l), ( l ) be i G.P., the pqr, pqs, prs, qrs are i : A.P. H.P. G.P. (d) A.G.P. 7. Let T, Tr Tr Tr r N ad S... T T T T, the S00 4 S00 S00 8. Let S (d) 0 S r, the (r ) is give by : r r S 8S S4 4 S S 6 S (d) S4 6 S 9. Let {x } represets G.P. with commo ratio 'r' such that xk xk k k 0, the umber of possible values for 'r' is/are : (d) 4 0. Let x, y be o-zero real umbers ad the expressio x + y 48x 4 y 4 is ot less tha 'k', the value of 'k' is equal to : 8 (d) 8. Let 0 A.M.'s ad 0 H.M.'s be iserted i betwee ad. If 'A' be ay A.M. ad 'H' be the correspodig H.M., the H(5 A) is equal to : 6 0 (d) 8 [ 9 ] Mathematics for JEE-0

2 Sequeces ad Series. Let a, b, c ad the iequality R bx ( ( a c) 4 b ) x ( a c) 0 holds true for a b c all real value of 'x', the e, e, e are i : A.P. G.P. H.P. (d) oe of these.. Let 'A ' deotes the sum of terms of a A.P. ad A A A, the A is equal to : (d) 0 0. I a sequece of (4 + ) terms, the first ( + ) terms are i A.P. whose commo differece is, ad the last ( + ) terms are i G.P. whose commo ratio is /. If the middle terms of the A.P. ad G.P. are equal, the the middle term of sequece is :. ( )..,. (d) ( ).. Let a, a, a,..., a 50 be 50 distict umbers i. 4. If a 0, roots of equatio are i G. P., the : ac db a b c d ax bx cx d a c d b (d) ab 0 cd 5. Let a, b, c be o-zero real umbers ad 4a + 9b + 6c = (ab + 6bc + 4ac), the a, b, c are i : A.P. H.P. G.P. (d) A.G.P. 6. I a set of four umbers, if first three terms are i G.P. ad the last three terms are i A.P. with commo differece 6, the sum of the four umbers, whe the first ad the last terms are equal, is give by : (d) 8 7. Let the real umbers,, be i A.P. ad satisfy the equatio p, p, x ( x ) px q 0, the : q, 7 (d) q, 7 / 50 r A.P., ad ar a a50 r 5 ( ) ( ), 7 where N, the value of is equal to : 4 8 (d) 0. Let three umbers be removed from the geometric sequece {a } ad the geometric mea of the remaiig terms is 5 7. If a..., 4 8 the value of '' ca be : (d). Let x, y R ad x y 6, the the least value of x + 4y is equal to : 0 8 (d) 0 4. Let S... terms ad S lim( S ), 4 8 if S S, the least value of '' is : I ABC, if all the sides are i A.P., the the correspodig ex-radii are i : A.P. G.P. H.P. (d) oe of these. 8r 9. Let S, the lim ( S) is equal to : 4 r 4r 4 (d) 0 [ 0 ] 0 (d) 6 5. Let the sides of a triagle be i arithmetic progressio. If the greatest agle of triagle is double the smallest agle, the the cosie value of the smallest agle is equal to : (d) 4 Mathematics for JEE-0

3 6. If a, b R, where a, A, A, b are i arithmetic progressio, a, G, G, b are i geometric progressio ad a, H, H, b are i harmoic progressio, the which of the followig relatios are correct? (d) G G G G H H H H G G A A A A H H GG ( a b)( b a) H H 9ab A A ( a b)( b a) H H 9ab 7. Let four cosecutive itegers form a icreasig arithmetic progressio ad oe of these umbers is equal to the sum of the squares of the other three umbers, the : the smallest umber is 0. the largest umber is. sum of all the four umbers is. (d) product of all the four umbers is For two distict positive umbers, let A, G, H deote the AM, GM ad HM respectively. For, N, if A ad H has arithmetic, geometric ad harmoic meas as A, G, H respectively, the : A A A A4... G G G G4... H > H > H > H 4 >... (d) G = G = G = G 4 = Let {a } represets the arithmetic sequece for which 0. Let a = x, a = x ad a = x +, the : a a a = 0 a 5 (d) a a a 4... ( ) ad b + a =. If b > a for all > 0, where N, the, possible values of atural umber ' 0 ' ca be : (d) Followig questios are assertio ad reasoig type questios. Each of these questios cotais two statemets, Statemet (Assertio) ad Statemet (Reaso). Each of these questios has four alterative aswers, oly oe of them is the correct aswer. Select the correct aswer from the give optios : Both Statemet ad Statemet are true ad Statemet is the correct explaatio of Statemet. Both Statemet ad Statemet are true but Statemet is ot the correct explaatio of Statemet. Statemet is true but Statemet is false. (d) Statemet is false but Statemet is true.. Statemet : Let three positive umbers i geometric progressio represet the sides of a triagle, the the commo ratio of the G.P. ca be si 5 Statemet : the commo ratio of the G.P. i cosideratio lies i betwee si. 0 si 0 ad. Statemet : I a triagle ABC, if cot A, cot B, cot C forms a A.P., the,, also form a b a c b a c A.P. Statemet :,, a b c form a H.P.. Statemet : If [.] ad {.} deote the greatest iteger fuctio ad the fractioal part, the x, [x], {x} ca ever form a geometric progressio for ay positive ratioal value of x Statemet : x, [ x], { x } ca form a G.P. for x R 7, oly if x si. 0 [ ] Mathematics for JEE-0

4 Sequeces ad Series 4. Statemet : If a, b, c R, the the miimum value of a( b c ) b( c a ) c( a b ) is equal to 6abc Statemet : for a, a, a, a4,... a R, ( AM )( HM ) ( GM ) N {} 5. Statemet : Let S N, the S l( )..., 4 5 Statemet : l ( + ) > l () N [ ] Mathematics for JEE-0

5 Comprehesio passage () ( Questios No. - ) Let V r deote the sum of the first r terms of a arithmetic progressio (A.P.) whose first term is r ad the commo differece is (r ). Let Tr Vr Vr ad Qr Tr Tr for r =,,.... The sum V + V V is : 5. Let Q { a, b, c}, where a < b < c, the the roots of the quadratic equatio ax + bx + c = 0 are : real real ad equal real ad uequal (d) o-real 6. Sum of all the elemets of set P Q is equal to : 56 9 (d) 5 ( )( ) ( )( ) ( ) (d) ( ). T r is always : a odd umber a prime umber a eve umber (d) a composite umber. Which oe of the followig is a correct statemet? Q, Q, Q,... are i A.P. with commo differece 5 Q, Q, Q,... are i A.P. with commo differece 6 Q, Q, Q,... are i A.P. with commo differece (d) Q = Q = Q =... Comprehesio passage () ( Questios No. 4-6 ) Let P ad Q be two sets each of which cosistig of three umbers i A.P. ad G.P. respectively. Sum of the elemets of set P is ad product of the elemets of set Q is 8, where the commo differece ad the commo ratio of A.P. ad G.P. are represeted by 'd' ad 'r' respectively. If sum of the squares of the terms of A.P. is 8 times the sum of the terms of G.P., where d = r, ad d, r questios. I, the aswer the followig 4. Total umber of terms i the set of P Q is/are : 0 (d) 7. Let x ad y be two real umbers such that the k th mea betwee x ad y is equal to the k th mea betwee x ad y whe arithmetic meas are placed betwee them i both the situatios. The value of y expressio is equal to... k x 8. Let S ad r r ( ) S '..., ( ) ( )( ) 6 S ' the value of is equal to... S 9. Let a A.P. ad a G.P. each has as the first term ad as the secod term, where 0. If sum of ifiite terms of G.P. is 4 ad the sum of first terms of ( ) A.P. ca be writte as, the value of k 'k' is equal to Let sum of the squares of three distict real umber i geometric progressio be S ad their sum is p ( S ). If p R, the total umber of possible itegral values of 'p' is/are.... Let a, b, c, d, e R ad s = a + b + c + d + e, if ( s a)( s b)( s c)( s d)( s e) miimum value of abcde is 4, the value of is... [ ] Mathematics for JEE-0

6 Sequeces ad Series. Match the followig colums (I) ad (II) Colum (I) Colum (II) Let 009, r r ( r ) the sum of all (p) the digits of the umber ' ' is The largest positive term of the harmoic progressio (q) 4 whose first two terms are 5 ad, If I / 4 0 is equal to ta x dx, where N, ad (r),,... form a A.P., the (s) I I I I I I commo differece of this A.P. is (d) Value of log 9 7. Match the followig colums (I) ad (II). Colum (I) is equal to (t) 6 Colum (II) If p is prime umber ad x N, where (p) i arithmetic progressio p log x x p, the first three smallest possible values of x are If a, a, a, a 4, a 5 are five o-zero distict umbers (q) i geometric progressio such that a, a, a are i A.P., a, a, a 4 are i G.P. ad a, a 4, a 5 are i H.P., the a, a, a 5 are (r) i harmoic progressio ta 70º, ta 50º + ta 0º ad ta 0º are (d) If a, b are positive distict real umber ad,, are (s) ot is arithmetic progressio three roots of x a x b b a such that b a x a x b ad c, the a, b, c are (t) ot i geometric progressio [ 4 ] Mathematics for JEE-0

7 4. Match the followig colums (I) ad (II). Colum (I) If sum of first positive itegers is 5 times the sum of (p) Colum (II) their squares, the is If, 0, are i G..P., the the value (q) 7 of is x x 7 If log, log ( 5) ad log are i A.P., the (r) 4 value of x is (d) Let S, S, S,... be squares such that for each, (s) 6 legth of side of S equals the legth of diagoal of S. If legth of S is.5 cm, the for which values of is the area of S less tha sq. cm. (t) 5. Match the followig colums (I) ad (II). Colum (I) If altitudes of a triagle are i A.P., the sides of triagle are i Colum (II) (p) A.P. a b a b If b c b c 0 ad, the a, b, c are i (q) G.P. 0 If a a a a a a a, the a4 a a4 a a4 a, a, a, a 4 are i (r) H.P. (d) If (y x), (y a) ad (y z) are i H.P., the (x a), (y a), (z a) are i (s) A.G.P. [ 5 ] Mathematics for JEE-0

8 Sequeces ad Series... (d) Ex (d) (b, c) 7. (b, c,d) 8. (a, d) 9. (a, c) 0. (b, c).. (d) (d) (d) Ex ( ) 8. ( ) 9. ( 8 ) 0. ( 9 ). ( 5 ). r. s, t 4. q 5. r t q, s r q p p, t p r (d) q (d) r, s, t (d) p, q, r, s q [ 6 ] Mathematics for JEE-0

Q.11 If S be the sum, P the product & R the sum of the reciprocals of a GP, find the value of

Q.11 If S be the sum, P the product & R the sum of the reciprocals of a GP, find the value of Brai Teasures Progressio ad Series By Abhijit kumar Jha EXERCISE I Q If the 0th term of a HP is & st term of the same HP is 0, the fid the 0 th term Q ( ) Show that l (4 36 08 up to terms) = l + l 3 Q3