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MATHEMATICS BY KAPIL KANT SHARMA, PH:9837513953, , 126 Progressions, , General term of an Arithmetic progression, Basic Level, , 1., , The sequence, , 5, 7, , ,, , 6, 7, , (a) H.P., 2., , (b) G.P., , If the 9, , th, , x 2 yz, , th, , (b) 79, , th, , (d) f (n) , , 1, b, , a n , n, , , ;n N, , [Rajasthan PET 1999], , (d) p q r, , term will be, (c) 80, , th, , [Rajasthan PET 1989], , (d) 81, , term of the series is, , (b) – 33, , (b), , The 10th term of the sequence, 243, , (c) 33, , (d) 10 a – 4, , (b), , y 2 xz, , 3,, 300, , (c), , z 2 xy, , (d) None of these, , (c), , 363, , (d), , 12 , 27 , ......is, 432, , Which term of the sequence (– 8 + 18i), (– 6+15i), (– 4 + 12i), ........is purely imaginary, (a) 5th, , 11., , (d) 3 : 1, , It x , y, z are in A.P., then its common difference is, , (a), 10., , (c) p r q, , (b) p q r, , If (a 1), 3a, (4 a 2) are in A.P. then 7, , (a), 9., , (c) f (n) (an b) kr n ; n N, , (b) f (n) kr n ; n N, , term of an A.P. is 35 and 19 is 75, then its 20, , (a) 10 a 4, 8., , (c) 1 : 3, , If the p th term of an A.P. be q and q th term be p, then its rth term will be, , (a) 78, 7., , , n, (d) 3 , p, , , Which of the following sequence is an arithmetic sequence, , (a) p q r, 6., , , n, (c) 3 , p, , , p, , (b) 3 , n, , , , (b) 2 : 1, , (a) f (n) an b; n N, , 5., , (d) None of these, , If the 9th term of an A.P. be zero, then the ratio of its 29 th and 19th term is, (a) 1 : 2, , 4., , (c) A.P., , 1 , 2 , 3, , p th term of the series 3 3 3 ..... will be, n , n , n, , p, , (a) 3 , n, , , , 3., , , 7 ........ is, , (b) 7th, , (c) 8th, , (d) 6th, , If (m +2)th term of an A.P. is (m+2)2–m2 , then its common difference is, (a) 4, , (b) – 4, , (c) 2, , (d) – 2
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Progressions 127, 12., , For an A.P., T2 T5 T3 10, T2 T9 17, then common difference is, (a) 0, , (b) 1, , (c) – 1, , (d) 13, , Advance Level, 13., , If tan n tan m , then the different values of will be in, (a) A.P., , 14., , (b) G.P., th, , If the p , q, , th, , and r, , th, , [Karnataka CET 1998], , (c) H.P., , (d) None of these, , term of an arithmetic sequence are a, b and c respectively, then the value of [a (q – r)+b(r, , – p)+ c (p – q)]=, [MP PET 1985], , (a) 1, 15., , (d), , 1, 2, , 4, 9, , (b), , 7, 16, , (c), , 3, 7, , (d), , [MP PET 1986], , 8, 15, , The 6th term of an A.P. is equal to 2, the value of the common difference of the A.P. which makes the product, a1a4 a5 least is given by, (a), , 17., , (c) 0, , If nth terms of two A.P.'s are 3n + 8 and 7n +15, then the ratio of their 12th terms will be, (a), , 16., , (b) – 1, , 8, 5, , (b), , 5, 4, , (c), , 2, 3, , (d) None of these, , If p times the p th term of an A.P. is equal to q times the q th term of an A.P., then ( p q)th term is, [MP PET 1997; Karnataka CET 2002], , (a) 0, 18., , (b) 1, , The numbers t(t 2 1) , , , (c) 2, , (d) 3, , 1 2, t and 6 are three consecutive terms of an A.P. If t be real, then the next two terms of, 2, , A.P. are, (a) –2, –10, 19., , If the pth term of the series 25, 22, (a) 11, , 20., , 1, y, 3, , (c) 13, 2, y, 3, , th, , (d) 14, , term is (x + y), then its first term is, (c) x , , 4, y, 3, , [AMU 1989], , (d) x , , 5, y, 3, , The number of common terms to the two sequences 17, 21, 25, ......417 and 16, 21, 26, ..... 466 is, (b) 19, , (c) 20, , (d) 91, , In an A.P. first term is 1. If T1 T3 T2T3 is minimum, then common difference is, (a) –5/4, , 23., , (b) x , , (d) None of these, , 1, 3, 1, , 20 , 18 ,...... is numerically the smallest, then p=, 2, 5, 4, , (b) 12, , (a) 21, 22., , (c) 14, 22, , The second term of an A.P. is (x – y) and the 5, (a) x , , 21., , (b) 14, 6, , (b) –4/5, , (c) 5/4, , (d) 4/5, , Let the sets A={2, 4, 6, 8,......} and B= {3, 6, 9, 12, .....}, and n(A) = 200, n(B) = 250. Then, (a) n(A B) = 67, , (b) n(A B) = 450, , (c) n(A B) = 66, , (d) n(A B) = 384, , Sum to n terms of an Arithmetic progression, Basic Level
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128 Progressions, , 24., , The sum of first n natural numbers is, (a) n(n – 1), , 25., , The sum of the series, (a) , , 26., , 5, 6, , 1, 2, , (c) 1, , (d) , , 3, 2, , (c) 1681, , [MP PET 1984], , (d) 1682, [MP PET 1984], , (c) n 2, , (d) (n 1)2, , If the sum of the series 2+ 5+ 8+11 ....... is 60100, then the number of terms are, (b) 200, , (c) 150, , [MNR 1991; DCE 2001], , (d) 250, , If the first term of an A.P. be 10, last term is 50 and the sum of all the terms is 300, then the number of terms, are [Rajasthan PET 1987], (b) 8, , (c) 10, , (d) 15, , The sum of the numbers between 100 and 1000 which is divisible by 9 will be, (b) 57228, , (c) 97015, , [MP PET 1982], , (d) 62140, , If the sum of three numbers of a arithmetic sequence is 15 and the sum of their squares is 83, then the, numbers are, [MP PET 1985], (a) 4, 5, 6, , 32., , [MNR 1985], , (b) (2n)2, , (a) 55350, 31., , n(n 1), 2, , The sum of 1+3+5+7+..... upto n terms is, , (a) 5, 30., , (d), , 1 1 1, ...... to 9 terms is, 2 3 6, , (b) 1683, , (a) 100, 29., , (c) n(n + 1), , The sum of all natural numbers between 1 and 100 which are multiples of 3 is, , (a) (n 1)2, 28., , n(n 1), 2, , (b) , , (a) 1680, 27., , (b), , [MP PET 1984; Rajasthan PET 1995], , (b) 3, 5, 7, , (c) 1, 5, 9, , (d) 2, 5, 8, , If the sum of three consecutive terms of an A.P. is 51 and the product of last and first term is 273, then the, numbers are, [MP PET 1986], , (a) 21, 17, 13, 33., , (c) 22, 18, 14, , (d) 24, 20, 16, , There are 15 terms in an arithmetic progression. Its first term is 5 and their sum is 390. The middle term is [MP PET 19, (a) 23, , 34., , (b) 20, 16, 12, , If S n nP , , (b) 26, , (c) 29, , (d) 32, , 1, n (n 1)Q, where S n denotes the sum of the first n terms of an A.P. then the common difference is, 2, [JEE West Bengal 1994], , (a) P + Q, 35., , (b) 136557, , (c) 161575, , [Rajasthan PET 1997], , (d) 156375, , (b) 3, , (c) 2, , (d) 1, , The number of terms of the A.P. 3, 7, 11, 15 ...... to be taken so that the sum is 406 is, (a) 5, , 38., , (d) Q, , Four numbers are in arithmetic progression. The sum of first and last term is 8 and the product of both middle, terms is 15. The least number of the series is, [MP PET 2001], (a) 4, , 37., , (c) 2Q, , The sum of numbers from 250 to 1000 which are divisible by 3 is, (a) 135657, , 36., , (b) 2P + 3Q, , (b) 10, 2, , [Kerala (Engg.) 2002], , (c) 12, , (d) 14, , (c) 43, 45, ......, 85, , (d) 43, 45, ....., 89, , 2, , The consecutive odd integers whose sum is 45 – 21 are, (a) 43, 45, ....., 75, , (b) 43, 45,...... 79
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Progressions 129, 39., , If common difference of m A.P.'s are respectively 1, 2,...... m and first term of each series is 1, then sum of their, mth terms is, (a), , 40., , 1, m (m 1), 2, , (b), , (b) 550 × 130, , (d) None of these, , (c) 552 × 128, , (d) None of these, , (c) p4, , (d) None of these, , If S n n p and S m m p, m n, in A.P., then S p is, (b) p3, , An A.P. consists of n (odd terms) and its middle term is m. Then the sum of the A.P. is, (a) 2 mn, , 43., , 1, m (m 2 1), 2, , 2, , 2, , (a) p2, 42., , (c), , The sum of all those numbers of three digits which leave remainder 5 after division by 7 is, (a) 551 × 129, , 41., , 1, m (m 2 1), 2, , (b), , 1, mn, 2, , (c) mn, , (d) mn2, , The minimum number of terms of 1 3 5 7 ..... that add up to a number exceeding 1357 is, (a) 15, , (b) 37, , (c) 35, , (d) 17, , Advance Level, , 44., , If the ratio of the sum of n terms of two A.P.'s be (7n+1) : (4n+27), then the ratio of their 11th terms will be[AMU 1996], (a) 2 : 3, , 45., , (b) 3 : 4, , (c) 4 : 3, , (d) 5 : 6, , The interior angles of a polygon are in A.P. If the smallest angle be 120° and the common difference be 5, then, the number of sides is, [IIT 1980], , (a) 8, 46., , (b) 10, , (d) 6, , The sum of integers from 1 to 100 that are divisible by 2 or 5 is, (a) 3000, , 47., , (c) 9, , (b) 3050, , (c) 4050, , [IIT 1984], , (d) None of these, , If the sum of first n terms of an A.P. be equal to the sum of its first m terms, (m n), then the sum of its first, (m + n) terms will be, [MP PET 1984], , (a) 0, 48., , (b) n, , (c) m, , If a1, a2 ,......., an are in A.P. with common difference d, then the sum of the following series is, , sin d (coses a1 . cosec a2 cosec a2 .cosec a3 ....... cosec an 1 cosec an ), (b) cot a1 cot an, , (a) sec a1 sec an, 49., , (d) m + n, [Rajasthan PET 2000], , (c) tan a1 tan an, , (d) cosec a1 cosec an, , (c) 2n, , (d) 4n 3, , The odd numbers are divided as follows, 1 3, 5 7 9 11, 13 15 17 19 21 23, , ., ., ., ., , ., ., ., ., , ., ., ., ., , ., ., ., ., , Then the sum of n, , ., ., ., .th, , (a) 2n 2 [2n 2n 1 1], , ., ., ., ., , row is, (b), , 1, (2n 1), 2
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130 Progressions, , 50., , If the sum of n terms of an A.P. is 2n 2 5 n, then the n th term will be, (a) 4 n 3, , 51., , m 1, n 1, , (d) 11, th, , th, , (b), , n 1, m 1, , (c), , a, , x log 3 a x ..... log n a x , , (b) x a a, , 2m 1, 2n 1, , (d), , 2n 1, 2m 1, , a 1, will be, 2, , (c) x a 1 / a, , (d) x a1 / a, , n2, (b) cot 1 , , n , , (c) tan 1 (n 1) tan 1 1, , (b) 6, , S 3n, , Sn, , (c) 8, , (d) All of these, [MNR 1993; UPSEAT 2001], , (d) 10, , If the sum of the first n terms of a series be 5 n 2 2n, then its second term is, (b) 17, , (c) 24, , [MP PET 1996], , (d) 42, , All the terms of an A.P. are natural numbers. The sum of its first nine terms lies between 200 and 220. If the, second term is 12, then the common difference is, (a) 2, , 59., , (c) 9, 2, , Let S n denotes the sum of n terms of an A.P. If S 2n 3S n , then ratio, , (a) 7, 58., , [MP PET 1983], , (d) 40, , Sum of first n terms in the following series cot 1 3 cot 1 7 cot 1 13 cot 1 21 ..... is given by, , (a) 4, 57., , (b) 7, , The value of x satisfying log a x log, , n , (a) tan 1 , , n2, , 56., , (c) 80, , The ratio of sum of m and n terms of an A.P. is m : n , then the ratio of m and n term will be[Roorkee 1963; MP PET 19, , (a) x a, 55., , (b) 35, , 2, , (a), 54., , (d) 4 n 7, , If the sum of two extreme numbers of an A.P. with four terms is 8 and product of remaining two middle term is, 15, then greatest number of the series will be, [Roorkee 1965], (a) 5, , 53., , (c) 4 n 6, , (b) 4 n 5, , The nth term of an A.P. is 3n 1 . Choose from the following the sum of its first five terms, (a) 14, , 52., , [Rajasthan PET 1992], , (b) 3, , (c) 4, , (d) None of these, , If S1 a2 a4 a6 .....up to 100 terms and S 2 a1 a3 a5 ...... up to 100 terms of a certain A.P. then its common, difference d is, (a) S1 S 2, , 60., , (c), , S1 S 2, 2, , (d) None of these, , In the arithmetic progression whose common difference is non-zero, the sum of first 3n terms is equal to the, sum of the next n terms. Then the ratio of the sum of the first 2n terms to the next 2n terms is, (a), , 61., , (b) S 2 S1, , 1, 5, , (b), , 2, 3, , (c), , 3, 4, , (d) None of these, , If the sum of n terms of an A.P. is nA n 2 B, where A, B are constants, then its common difference will be[MNR 1977], (a) A – B, , (b) A + B, , (c) 2A, , (d) 2B, , Arithmetic mean, Basic Level, 62., , A number is the reciprocal of the other. If the arithmetic mean of the two numbers be, are, , 13, , then the numbers, 12
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Progressions 131, (a), 63., , n 1, 2, , (c), , 2 5, ,, 5 2, , (d), , 3 2, ,, 2 3, , (b), , [Rajasthan PET 1986], , n 1, 2, , (c), , n, 2, , (d) n, [MP PET 1985], , (b) 7, 11, 15, 19, , (c) 5, 11, 15, 22, , (d) 7, 15, 19, 21, , The mean of the series a, a + nd, a + 2nd is, (a) a (n 1)d, , 66., , 3 4, ,, 4 3, , The four arithmetic means between 3 and 23 are, (a) 5, 9, 11, 13, , 65., , (b), , The arithmetic mean of first n natural number, (a), , 64., , 1 4, ,, 4 1, , [DCE 2002], , (b) a nd, , (c) a (n 1)d, , (d) None of these, , If n A.M. s are introduced between 3 and 17 such that the ratio of the last mean to the first mean is 3 : 1, then, the value of n is, (a) 6, , (b) 8, , (c) 4, , (d) None of these, , Advance Level, , 67., , The sum of n arithmetic means between a and b, is, n(a b ), (a), 2, , 68., , [Rajasthan PET 1986], , (n 1) (a b ), (c), 2, , (b) n (a b), , (d) (n 1) (a b), , Given that n A.M.'s are inserted between two sets of numbers a, 2b and 2a, b, where a, b R. Suppose further, that m th mean between these sets of numbers is same, then the ratio a : b equals, (a) n – m + 1 : m, , 69., , (b) n – m + 1 : n, , (c) n : n – m + 1, , (d) m : n – m + 1, , Given two number a and b. Let A denote the single A.M. and S denote the sum of n A.M.'s between a and b, then, S/A depends on, [Pb. CET 1992], , (a) n, a, b, 70., , 41, 35, , 19 ,, 2, 2, , (b) a nd, , [Pb. CET 1998], , (c) a (n 1)d, , (d) None of these, , (b) 20 ,, , 41 43, ,, 2 2, , (c) 20 ,, , 61 62, ,, 2 3, , [MNR 1997], , (d) 20 , 22, 24, , If f (x y, x y) xy , then the arithmetic mean of f (x , y) and f (y, x ) is, (a) x, , 73., , (d) n, , If 11 AM's are inserted between 28 and 10, then three mid terms of the series are, (a), , 72., , (c) n, a, , The A.M. of series a (a d) (a 2d) ..... (a 2nd ) is, (a) a (n 1)d, , 71., , (b) n, b, , (b) y, , (c) 0, , If A.M. of the roots of a quadratic equation is, , [AMU 2002], , (d) 1, , 8, 8, , then the quadratic, and the A.M. of their reciprocals is, 5, 7, , equation is, (a) 7 x 2 16 x 5 0, 74., , (b) 7 x 2 16 x 5 0, , (c) 5 x 2 16 x 7 0, , (d) 5 x 2 8 x 7 0, , If a1=0 and a1, a2, a3,.....an are real numbers such that | ai | |ai–1+1| for all i, then A.M. of the numbers a1, a2,, ......an has the value x where, (a) x<1, , 75., , (b) x , , 1, 2, , (c) x , , 1, 2, , If A.M. of the numbers 5 1 x and 5 1 x is 13 then the set of possible real values of x is, , (d) x , , 1, 2
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132 Progressions, 1, (a) {5 , }, 5, , (b) 1, 1, , (c) { x | x 2 1| 0, x R}, , (d) None of these, , Properties of A.P., Basic Level, , 76., , If 2x, x+ 8, 3x + 1 are in A.P., then the value of x will be, (a) 3, , 77., , (d) – 2, , 1, 2, , (b) 1,, , 1, 3, , (c) 1,, , [IIT 1990], , 3, 2, , (d) None of these, , If am denotes the m th term of an A.P., then am , (a), , 79., , (c) 5, , 7, , If log32, log3(2x –5) and log3 2 x are in A.P., then x is equal to, 2, , , (a) 1,, 78., , (b) 7, , [MP PET 1984], , am k am k, 2, , (b), , am k am k, 2, , (c), , am k, , 2, am k, , (d) None of these, , If 1, logy x, logz y, – 15 logxz are in A.P., then, (a) z 3 x, , (c) z 3 y, , (b) x y 1, , (d) x y 1 z 3, , (e) All of these, 80., , If, , 1, 1, 1, are in A.P., then, ,, ,, p q r p q r, , (a) p, q, r are in A.P., 81., , (b) p 2 , q 2 , r 2 are in A.P., , (c), , 1 1 1, , , are in A.P., p q r, , (d) None of these, , If a, b, c, are in A.P., then b 2 ac is equal to, (a), , 82., , [Rajasthan PET 1995], , 1, (a c)2, 4, , (b), , 1, (a c)2, 4, , [Roorkee 1975], , (c), , 1, (a c)2, 2, , (d), , 1, (a c)2, 2, , If a1, a2 , a3 ,..... are in A.P. then a p , aq , ar are in A.P. if p, q, r are in, (a) A.P., , (b) G.P., , (c) H.P., , (d) None of these, , Advance Level, , 83., , If the sum of the roots of the equation ax 2 bx c =0 be equal to the sum of the reciprocals of their squares,, then bc 2 , ca 2 , ab 2 will be in, (a) A.P., , 84., , If, , (c) H.P., , (d) None of these, , 1, 1, 1, ,, ,, be consecutive terms of an A.P., then (b – c)2, (c – a)2, (a – b)2 will be in, b c c a ab, , (a) G.P., 85., , (b) G.P., , [IIT 1976], , (b) A.P., , (c) H.P., , If a 2 , b 2 , c 2 are in A.P., then (b+ c)–1, (c a)1 and (a b)1 will be in, , (d) None of these, [Roorkee 1968; Rajasthan PET 1996]
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Progressions 133, (a) H.P., 86., , (b) G.P., , 88., , 89., , (b) 2, 3, 4, , (a) (1, 2 ), , (b) (1, 2), , If a, b, c are in A.P. then, , (a c)2, , (b 2 ac), , (a) 1, , (b) 2, , 93., , (d) (1, 2), [Roorkee 1975], , (c) 3, , (d) 4, , (b) 2 (f – d), , [Pb. CET 1989, 91], , (c) 2 (d – c), , (d) d – c, , If p, q, r are in A.P. and are positive, the roots of the quadratic equation px + qx + r = 0 are all real for [IIT 1995], r, 7 4 3, p, , (b), , p, 7 4 3, r, , (c) All p and r, , n 1, , (b), , a1 an, , n 1, a1 an, , (d) No p and r, 1, , If a1 , a2 , a3 , ....... an are in A.P., where ai 0 for all i, then the value of, , a1 a2, , , , 1, a2 a3, , n 1, , (c), , a1 an, , (d), , Given a d b c where a, b, c, d are real numbers, then, (a) a, b, c, d are in A.P., (c) (a b), (b c), (c d), (a d) are in A.P., , (d), , (b) abc, , an 1 an, , [IIT 1982, , n 1, a1 an, , 1, 1, 1, 1, ,, ,, ,, are in A.P., ab b c cd ad, , If a, b, c are in A.P., then (a + 2b – c) (2b+ c – a) (c + a – b) equals, 1, abc, 2, , 1, , ........ , , [Kurukshetra CEE 1998], , 1 1 1 1, (b) , , ,, are in A.P., a b c d, , (a), 94., , (c) (1, 2), , [IIT 1984], , 2, , (a), 92., , (d) 4, 5, 6, , If a, b, c, d, e, f are in A.P., then the value of e – c will be, , (a), 91., , (c) 3, 4, 5, , [Roorkee 1974], , If a, b, c are in A.P., then the straight line ax + by + c = 0 will always pass through the point, , (a) 2 (c – a), 90., , (d) None of these, , If the sides of a right angled triangle are in A.P., then the sides are proportional to, (a) 1, 2, 3, , 87., , (c) A.P., , [Pb. CET 1999], , (c) 2 abc, , (d) 4 abc, , If the roots of the equation x 3 12 x 2 39 x 28 0 are in A.P., then their common difference will be, [UPSEAT 1994, 99, 2001; Rajasthan PET 2001], , (a) 1, 95., , If 1, log 9 (3, , (b) 2, 1 x, , n(n 1) a 2 a1, ., 2, an 1, , (c) 1 log 4 3, , (d) log 4 3, , (b), , (c) 3, , [Rajasthan PET 2002], , (d) None of these, , a2 n 1 a1 a2 n a2, a an, , ......... n 2, is equal to, a2 n 1 a1 a2 n a2, an 2 an, , n(n 1), 2, , (c) (n 1)(a 2 a1 ), , (d) None of these, , If the non-zero numbers x, y, z are in A.P. and tan 1 x , tan 1 y, tan 1 z are also in A.P., then, (a) x y z, , 99., , (b) 2a, , If a1, a2 , a3 , ....... a2n 1 are in A.P. then, (a), , 98., , (b) 1 log 3 4, , [AIEEE 2002], , If a, b, c, d, e are in A.P. then the value of a+b+4c – 4d + e in terms of a, if possible is, (a) 4a, , 97., , (d) 4, , 2), log 3 (4 . 3 1) are in A.P., then x equals, , (a) log 3 4, 96., , (c) 3, , x, , (b) xy yz, , (c) x 2 yz, , (d) z 2 xy, , If three positive real numbers a, b, c are in A.P. such that abc =4, then the minimum value of b is
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134 Progressions, (a) 2 1 / 3, , (c) 2 1 / 2, , (b) 2 2 / 3, , (d) 2 3 / 2, , 100. If sin , sin 2 , 1, sin 4 and sin 5 are in A.P., where , then lies in the interval, (b) ( / 3, / 3), , (a) ( / 2, / 2), , (c) ( / 6, / 6), , (d) None of these, , 101. If the sides of a triangle are in A.P. and the greatest angle of the triangle is double the smallest angle, the ratio, of the sides of the triangle is, (a) 3 : 4 : 5, , (b) 4 : 5 : 6, , (c) 5 : 6 : 7, , (d) 7 : 8 : 9, , c, 102. If a, b, c of a ABC are in A.P., then cot , 2, , (a) 3 tan, , A, 2, , (b) 3 tan, , [T.S. Rajendra 1990], , B, 2, , (c) 3 cot, , A, 2, , (d) 3 cot, , B, 2, , 103. If a, b, c are in A.P. then the equation (a b)x 2 (c a)x (b c) 0 has two roots which are, (a) Rational and equal, , (b) Rational and distinct, , 104. The least value of 'a' for which 5 1 x 5 1 x ,, (a) 10, , (c) 12, , x , f (x )dx 4 , where f (x ) x , 0, x , , , , (a) 1, , (d) Complex conjugates, , a, , 25 x 25 x are three consecutive terms of an A.P. is, 2, , (b) 5, , 105. , , , are in A.P. and, , (c) Irrational conjugates, , 2, , (b) –1, , x, x , x , , (d) None of these, , x , x 1 , then the common difference d is, x , , (c) 2, , (d) – 2, , 106. If the sides of a right angled triangle form an A.P. then the sines of the acute angles are, (a), , 3 4, ,, 5 5, , (b), , 3,, , 1, 3, , 5 1, ,, 2, , (c), , 5 1, 2, , (d), , 3 1, ,, 2 2, , 107. If x, y, z are positive numbers in A.P., then, (a) y 2 xz, (c), , (b) y 2 xz, , x y, yz, has the minimum value 2, , 2y x 2y z, , (d), , x y, yz, , 4, 2y x 2y z, , General term of Geometric progression, Basic Level, 108. If the 4 th , 7 th and 10 th terms of a G.P. be a, b, c respectively, then the relation between a, b, c is, [MNR 1995; Karnataka CET 1999], , (a) b , , ac, 2, , 109. 7th term of the sequence, (a) 125 10, 110. If the 5th term of a G.P. is, (a), 111., , 3, 4, , (b) a 2 bc, , (c) b 2 ac, , (d) c 2 ab, , (c) 125, , (d) 125 2, , 2 , 10 , 5 2 , ....... is, (b) 25 2, , 1, 16, , then the 4th term will be, and 9th term is, 3, 243, , (b), , 1, 2, , (c), , 1, 3, , If the 10th term of a geometric progression is 9 and 4 th term is 4, then its 7th term is, , [MP PET 1982], , (d), , 2, 5, [MP PET 1996]
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Progressions 135, (a) 6, 112., , (b) 36, , 4, 9, , (d), , 9, 4, , The third term of a G.P. is the square of first term. If the second term is 8, then the 6 th term is, (a) 120, , 113., , (c), , The 6, , th, , (b) 124, term of a G.P. is 32 and its 8, , (a) – 1, , (c) 128, th, , (d) 132, , term is 128, then the common ratio of the G.P. is, , (b) 2, , [MP PET 1997], , (c) 4, , [Pb. CET 1999], , (d) – 4, , 114. The first and last terms of a G.P. are a and l respectively, r being its common ratio; then the number of term in, this G.P. is, (a), , 115., , log l log a, log r, , (b) 1 , , log l log a, log r, , (c), , If first term and common ratio of a G.P. are both, (a) 2n, , (b) 4n, , log a log l, log r, , (d) 1 , , log l log a, log r, , 3 i, . The absolute value of nth term will be, 2, , (c) 1, , (d) 4, , 116. In any G.P. the last term is 512 and common ratio is 2, then its 5 th term from last term is, (a) 8, 117., , (b) 16, , (c) 32, , (d) 64, , Given the geometric progression 3, 6, 12, 24, ...... the term 12288 would occur as the, th, , (a) 11 term, , th, , (b) 12 term, , (c) 13, , th, , term, , [SCRA 1999], , (d) 14, , th, , term, , 118. Let {tn } be a sequence of integers in GP in which t4 : t6 1 : 4 and t2 t5 216 . Then t1 is, (a) 12, , (b) 14, , (c) 16, , (d) None of these, , Advance Level, , 119., , , are the roots of the equation x 2 3 x a 0 and , are the roots of the equation x 2 12 x b 0 . If , , , , form an increasing G.P., then (a, b) =, (a) (3, 12), , (b) (12, 3), , [DCE 2000], , (c) (2, 32), , (d) (4, 16), , 120. If ( p q)th term a G.P. be m and (p – q)th term be n, then the pth term will be [Rajasthan PET 1997; MP PET 1985, 99], (a) m / n, , (b), , mn, , (c) mn, , 121. If the third term of a G.P. is 4 then the product of its first 5 terms is, (a) 4 3, , (b) 4 4, , (c) 4 5, , (d) 0, [IIT 1982; Rajasthan PET 1991], , (d) None of these, , 122. If the first term of a G.P. a1 , a2 , a3 ,......... is unity such that 4 a2 5a3 is least, then the common ratio of G.P. is, (a) , , 2, 5, , (b) , , 3, 5, , (c), , 2, 5, , 123. Fifth term of a G.P. is 2, then the product of its 9 terms is, (a) 256, , (b) 512, , 124. If the nth term of geometric progression 5,, (a) 11, , (b) 10, , (d) None of these, [Pb. CET 1990, 94; AIEEE 2002], , (c) 1024, , (d) None of these, , 5 5 5, 5, , , ,..... is, , then the value of n is, 2 4 8, 1024, , (c) 9, , [Kerala (Engg.) 2002], , (d) 4, , Sum to n terms of Geometric progression, Basic Level
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136 Progressions, , 125. The sum of 100 terms of the series .9+ .09 + .009...... will be, 100, , 106, , 106, , 1 , (a) 1 , 10 , , 1 , (c) 1 , 10 , , 1 , (b) 1 , 10 , , 100, , 1 , (d) 1 , 10 , , 126. If the sum of three terms of G.P. is 19 and product is 216, then the common ratio of the series is, (a) , , 3, 2, , 3, 2, , (b), , (c) 2, , [Roorkee 1972], , (d) 3, , 127. If the sum of first 6 terms is 9 times to the sum of first 3 terms of the same G.P., then the common ratio of the, series will be, [Rajasthan PET 1985], , (a) – 2, , (b) 2, , (c) 1, , (d), , 1, 2, , 128. If the sum of n terms of a G.P. is 255 and nth term is 128 and common ratio is 2, then first term will be[Rajasthan PET 19, (a) 1, , (b) 3, , (c) 7, , (d) None of these, , 129. The sum of 3 numbers in geometric progression is 38 and their product is 1728. The middle number is[MP PET 1994], (a) 12, , (b) 8, , (c) 18, , (d) 6, , 130. The sum of few terms of any ratio series is 728, if common ratio is 3 and last term is 486, then first term of, series will be, [UPSEAT 1999], , (a) 2, 131., , (b) 1, , (c) 3, , The sum of n terms of a G.P. is 3 , (a), , 3, 16, , 3 n 1, , then the common ratio is equal to, 4 2n, , 3, 256, , (b), , (d) 4, , (c), , 39, 256, , (d) None of these, , 132. The value of n for which the equation 1 r r 2 ..... r n (1 r) (1 r 2 )(1 r 4 ) (1 r 8 ) holds is, (a) 13, , (b) 12, 13, , 133. The value of the sum, , (i i, n, , n 1, , (c) 15, , (d) 16, , ), where i 1 , equals, , [IIT 1998], , n 1, , (a) i, , (b) i – 1, , 134. For a sequence a1, a2 ......... an given a1 2 and, , (a), , 20, [4 19 3], 2, , 1 , , (b) 3 1 20 , 3, , , , (c) – i, an 1, 1, = . Then, 3, an, , (d) 0, , 20, , a, , r, , is, , r 1, , (c) 2(1 – 3–20), , 135. The sum of (x 2)n 1 (x 2)n 2 (x 1) (x 2)n 3 (x 1)2 .....( x 1)n 1 is equal to, , (d) None of these, [IIT 1990]
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Progressions 137, (a) (x 2)n 2 (x 1)n, , (b) (x 2)n 1 (x 1)n 1, , (c) (x 2)n (x 1)n, , (d) None of these, , Advance Level, , 136. The sum of the first n terms of the series, , 1 3 7 15, , ........ is, 2 4 8 16, , [IIT 1988; MP PET 1996; Rajasthan PET 1996, 2000; Pb. CET 1994; DCE 1995, 96], , (b) 1 2, , (a) 2 n 1, n, , n, , (c) n 2 n 1, , (d) 2 n 1, , 137. If the product of three consecutive terms of G.P. is 216 and the sum of product of pair – wise is 156, then the, numbers will be, [MNR 1978], , (a) 1, 3, 9, , (b) 2, 6, 18, , (c) 3, 9, 27, , (d) 2, 4, 8, n, , 138. If f (x ) is a function satisfying f (x y) f (x ) f (y) for all x , y N such that f (1) 3 and, , f(x) 120 ., , Then the value, , x 1, , of n is, [IIT 1992], , (a) 4, , (b) 5, , (c) 6, , (d) None of these, , 139. The first term of a G.P. is 7, the last term is 448 and sum of all terms is 889, then the common ratio is [MP PET 2003], (a) 5, , (b) 4, , (c) 3, , (d) 2, , 140. The sum of a G.P. with common ratio 3 is 364, and last term is 243, then the number of terms is, (a) 6, , (b) 5, , (c) 4, , [MP PET 2003], , (d) 10, , 141. A G.P. consists of 2n terms. If the sum of the terms occupying the odd places is S 1 , and that of the terms in the, even places is S 2 , then S 2 / S 1 is, (a) Independent of a, 142. Sum of the series, (a) n , , 1 n, (3 1), 2, , (b) Independent of r, , (c) Independent of a and r (d) Dependent on r, , 2 8 26 80, , , ..... to n terms is, 3 9 27 81, , (b) n , , 1 n, (3 1), 2, , [Karnataka CET 2001], , (c) n , , 1, (1 3 n ), 2, , (d) n , , 1 n, (3 1), 2, , 143. If the sum of the n terms of G.P. is S product is P and sum of their inverse is R, then P 2 is equal to[IIT 1966; Roorkee 19, (a), , R, S, , (b), , R, (c) , S , , S, R, , n, , S , (d) , R, , n, , 144. The minimum value of n such that 1 3 3 2 ..... 3 n 1000 is, (a) 7, , (b) 8, , (c) 9, , (d) None of these, , 145. If every term of a G.P. with positive terms is the sum of its two previous terms, then the common ratio of the, series is, [Rajasthan PET 1986], , (a) 1, , (b), , 2, 5, , (c), , 5 1, 2, , (d), , 5 1, 2
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138 Progressions, 49, , 146. If (1.05 )50 11 .658 , then, , (1.05 ), , n, , equals, , [Roorkee 1991], , n 1, , (a) 208.34, , (b) 212.12, , (c) 212.16, , (d) 213.16, , aa, aa, aa, a a, 147. If a1, a2 , a3 ..... an are in G.P. with first term 'a' and common ratio 'r' then 2 1 2 2 2 2 3 2 2 3 4 2 ....... 2 n 1 n 2, a1 a2 a2 a3 a3 an, an 1 an, , is equal to, (a), , nr, 1 r2, , (b), , (n 1)r, 1 r2, , (c), , nr, 1r, , (d), , (n 1)r, 1r, , 148. The sum of the squares of three distinct real numbers which are in G.P. is S 2 . If their sum is S , then, (a) 1 2 3, , (b), , 1, 2 1, 3, , (c) 1 3, , (d), , 1, 1, 3, , Sum to infinite terms, Basic Level, 149. If the sum of the series 1 , (a) x 2, , 2, 4, 8, , , .... is a finite number, then, x x2 x3, , (b) x 2, , (c) x , , 1, 2, , 150. If y x x 2 x 3 x 4 ..... , then value of x will be, (a) y , 151., , 1, y, , (b), , y, 1y, , [UPSEAT 2002], , (d) None of these, [MNR 1975; Rajasthan PET 1988; MP PET 2002], , (c) y , , 1, y, , (d), , y, 1y, , If the sum of an infinite G.P. be 9 and the sum of first two terms be 5, then the common ratio is, (a), , 1, 3, , (b), , 3, 2, , (c), , 3, 4, , (d), , 2, 3, , . . ., , 152., , 2. 3 5 7 =, , (a), , 2355, 1001, , [IIT 1983; Rajasthan PET 1995], , (b), , 2370, 997, , (c), , 2355, 999, , (d) None of these, , 153. The first term of a G.P. whose second term is 2 and sum to infinity is 8, will be [MNR 1979; Rajasthan PET 1992, 95], (a) 6, , (b) 3, , (c) 4, , (d) 1, , 154. The sum of infinite terms of a G.P. is x and on squaring the each term of it, the sum will be y, then the common, ratio of this series is, [Rajasthan PET 1988], , (a), , x 2 y2, x 2 y2, , 155. If 3 3 3 2 ........ , (a), , 15, 23, , (b), , x 2 y2, x 2 y2, , (c), , x2 y, x2 y, , x2 y, x2 y, , 45, , then the value of will be, 8, , (b), , 7, 15, , [Pb. CET 1989], , (c), , 7, 8, , 156. The sum can be found of a infinite G.P. whose common ratio is r, (a) For all values of r, 0), , (d), , (b) For only positive value of r (c) Only for 0 < r < 1, , (d), , 15, 7, [AMU 1982], , (d) Only for – 1 < r < 1(r
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Progressions 139, 157. The sum of infinity of a geometric progression is, (a), , 7, 16, , (b), , 3, 4, and the first term is . The common ratio is [MP PET 1994], 3, 4, , 9, 16, , (c), , 1, 9, , (d), , 7, 9, , 158. The value of 4 1 / 3. 4 1 / 9. 4 1 / 27 ..... is, (a) 2, , [Rajasthan PET 2003], , (b) 3, , (c) 4, , (d) 9, , 159. 0.14189189189…. can be expressed as a rational number, (a), , 7, 3700, , (b), , 7, 50, , [AMU 2000], , (c), , 525, 111, , (d), , 21, 148, , 160. The sum of the series 5.05 1.212 0.29088 ... is, (a) 6.93378, , [AMU 2000], , (b) 6.87342, , (c) 6.74384, , (d) 6.64474, , 161. Sum of infinite number of terms in G.P. is 20 and sum of their square is 100. The common ratio of G.P. is [AIEEE 2002], (a) 5, , (b) 3/5, , (c) 8/5, , (d) 1/5, , 162. If in an infinite G.P. first term is equal to the twice of the sum of the remaining terms, then its common ratio is [Rajast, (a) 1, , (b) 2, , (c) 1/3, , 163. The sum of infinite terms of the geometric progression, (a), , (b) ( 2 1) 2, , 2 ( 2 1) 2, , x, , 164. If x > 0, then the sum of the series e, , e, , 2 x, , e, , 3 x, , 1, , 1, ..... is, 2 1 2 2 2, ,, , ,, , [Kerala (Engg.) 2002], , (c) 5 2, , (d) 3 2 5, , 1, (c), 1 e x, , 1, (d), 1 ex, , ...... is, , 1, (b) x, e 1, , 1, (a), 1 e x, , 2 1, , (d) – 1/3, , 165. The sum of the series 0.4 0.004 0.00004 ....... is, 11, 41, (a), (b), 100, 25, , [AMU 1989], , [AMU 1989], , (c), , 40, 99, , (d), , 2, 5, , 166. A ball is dropped from a height of 120 m rebounds (4/5)th of the height from which it has fallen. If it continues, to fall and rebound in this way. How far will it travel before coming to rest ?, (a) 240 m, 167. The series C , (a) – 1/2, , (b) 140 m, 2, , 3, , (c) 1080 m, , (d) , , 4, , C, C, C, , , ..... has a finite sum if C is greater than, 2, 1 C (1 C), (1 C)3, , (b) – 1, , (c) – 2/3, , (d) None of these, , Advance Level, 168. If A 1 r z r 2 z r 3 z ..... , then the value of r will be, 1/z, , (a) A(1 A)z, , A 1, (b) , , A , , 169. The sum to infinity of the following series 2 , (a) 3, 170., , (b) 4, , 1/z, , 1, , (c) 1 , A, , , (d) A(1 A)1 / z, , 1 1 1, 1, 1, 1, , , , , ....., will be, 2 3 22 3 2 23 3 3, , (c), , 7, 2, , [AMU 1984], , (d), , 9, 2, , x 1 a a 2 ....... (a 1) , y 1 b b 2 ....... (b 1) . Then the value of 1 ab a 2 b 2 ...... is[MNR 1980; MP PET 1985]
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140 Progressions, , (a), 171., , xy, x y 1, , (b), , xy, x y 1, , (c), , The value of alog b x , where a 0 .2, b 5 , x , , xy, x y 1, , (d), , xy, x y 1, , 1 1, 1, , ...... to is, 4 8 16, , (a) 1, (b) 2, (c) 1/2, (d) 4, 172. The sum of an infinite geometric series is 3. A series, which is formed by squares of its terms have the sum also, 3. First series will be, \, [Rajasthan PET 1999; Roorkee 1972; UPSEAT 1999], , (a), , 1 1 1 1, , , ,, ,....., (b), 2 4 8 16, , 3 3 3 3, , , ,, ,....., 2 4 8 16, , 1 1 1 1, , ,, ,, ,....., (c), 3 9 27 81, , (d) 1,, , 1 1, 1, , 2 , 3 ,....., 3 3, 3, , 173. If 1 cos cos 2 ....... 2 2, then , (0 ) is, (a) / 8, , [Roorkee 2000], , (b) / 6, , (c) / 4, , (d) 3 / 4, , 174. Consider an infinite G.P. with first term a and common ratio r, its sum is 4 and the second term is 3/4 , then, [IIT Screening 2000; DCE 2001], , 7, 3, (a) a , r , 4, 7, , 3, 1, (b) a , r , 2, 2, , 3, (c) a 2, r , 8, , (d) a 3, r , , 1, 4, , 175. Let n ( 1) be a positive integer, then the largest integer m such that (n m 1) divides (1 n n 2 ..... n127 ), is[IIT 1995], (a) 32, , (b) 63, , (c) 64, , (d) 127, , 176. If |a|<1 and |b|<1, then the sum of the series a(a b) a (a b ) a (a b ) ..... upto is, 2, , (a), , a, ab, , 1 a 1 ab, , (b), , a2, ab, , 2, 1, , ab, 1a, , 2, , (c), , 2, , 3, , 3, , 3, , b, a, , ab 1a, , (d), , b2, ab, , 2, 1, , ab, 1b, , 177. If S is the sum to infinity of a G.P., whose first term is a, then the sum of the first n terms is, a, , (a) S 1 , S, , , , a, (b) S 1 1 , S , , , n, , n, , , a, , (c) a 1 1 , S , , , n, , (d) None of these, , 178. If S denotes the sum to infinity and S n the sum of n terms of the series 1 , S Sn , , 1 1 1, .....,, 2 4 , , such that, , 1, , then the least value of n is, 1000, , (a) 8, 179. If, , [UPSEAT 2002], , exp., , (b) 9, 2, , 4, , 4, , {(sin x+sin x+sin x+....+), , (c) 10, loge2}, , satisfies, , the, , (d) 11, equation, , x 9 x 8 0,, 2, , then, , the, , value, , cos x, , ,0 x , is, cos x sin x, 2, , (a), , 1, ( 3 1), 2, , (b), , 1, ( 3 1), 2, , (c) 0, , (d) None of these, , Geometric mean, Basic Level, 180. If G be the geometric mean of x and y, then, (a) G 2, , (b), , 1, G2, , 1, 1, , , G2 x 2 G2 y 2, , (c), , 2, G2, , (d) 3G 2, , 181. If n geometric means be inserted between a and b, then the nth geometric mean will be, , of
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Progressions 141, n, , b, (b) a , a, , b n 1, (a) a , a, , 182. If, , n 1, n, , n, , 1, , b n 1, (c) a , a, , b n, (d) a , a, , (c) 1/2, , (d) None of these, , an b n, be the geometric mean of a and b, then n=, a b n 1, n 1, , (a) 0, , (b) 1, , 183. The G.M. of roots of the equation x 18 x 9 0 is, 2, , [Rajasthan PET 1997], , (a) 3, (b) 4, (c) 2, 184. If five G.M.'s are inserted between 486 and 2/3 then fourth G.M. will be, (a) 4, , (b) 6, , (c) 12, , (d) 1, [Rajasthan PET 1999], , (d) – 6, , 185. If 4 G.M’s be inserted between 160 and 5 them third G.M. will be, (a) 8, , (b) 118, , (c) 20, , 186. The product of three geometric means between 4 and, (a) 4, (b) 2, 187. The geometric mean between –9 and –16 is, (a) 12, , (b) – 12, , (d) 40, , 1, will be, 4, , (c) – 1, , (d) 1, , (c) – 13, , (d) None of these, , Advance Level, 188. If n geometric means between a and b be G1 , G2 , ..... Gn and a geometric mean be G, then the true relation is, (b) G1 . G2 ...... Gn G1 / n, , (a) G1 . G2 ...... Gn G, , (c) G1 . G2 ...... Gn G n, , (d) G1 . G2 ...... Gn G 2 / n, , 189. If x and y be two real numbers and n geometric means are inserted between x and y. now x is multiplied by k, 1, and y is multiplied, and then n G.M’s. are inserted. The ratio of the n tn G.M’s. in the two cases is, k, n 1, , 1, , (a) k n 1 : 1, , (b) 1 : k n 1, , (c) 1 : 1, , (d) None of these, , Properties of G.P., Basic Level, , 190. If a, b, c are in G.P., then, (b) a(b 2 c 2 ) c(a 2 b 2 ), , (a) a(b 2 a 2 ) c(b 2 c 2 ), , (c) a 2 (b c) c 2 (a b), , (d) None of these, , 191. If x is added to each of numbers 3, 9, 21 so that the resulting numbers may be in G.P., then the value of x will be[MP PE, (a) 3, , (b), , 1, 2, , 1, 3, , (c) 2, , (d), , (c) log a (log e a) log a (log e b), , (d) log a (log e b) log a (log e a), , 192. If log x a, a x / 2 and log b x are in G.P., then x =, (b) log a (log a b), , (a) log a (log b a), n, , 193. If, , , n 1, , n, , n,, , n, , n, , 10, ., n2,, 3 n 1, , n 1, , 3, , are in G.P. then the value of n is
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142 Progressions, (a) 2, , (b) 3, th, , 194. If p, q, r are in A.P., then p , q, , (c) 4, , th, , (d) Nonexistent, , th, , and r terms of any G.P. are in, , (a) AP, , (b) G.P., , (c) Reciprocals of these terms are in A.P., , (d) None of these, , 195. If a, b, c are in G.P., then, , [Rajasthan PET 1995], , (b) a 2 (b c), c 2 (a b), b 2 (a c) are in G.P., , (a) a 2 , b 2 , c 2 are in G.P., a, b, c, ,, ,, are in G.P., b c ca ab, , (c), , (d) None of these, , 196. Let a and b be roots of x 2 3 x p 0 and let c and d be the roots of x 2 12 x q 0, where a, b, c, d form an, increasing G.P. Then the ratio of (q + p) : (q – p) is equal to, (a) 8 : 7, , (b) 11 : 10, , (c) 17 : 15, , (d) None of these, , 197. If the roots of the cubic equation ax bx cx d 0 are in G.P., then, 3, , (a) c 3 a b 3 d, , 2, , (b) ca 3 bd 3, , (c) a 3 b c 3 d, , (d) ab 3 cd 3, , 198. If x1 , x 2 , x 3 as well as y1 , y 2 , y3 are in G.P. with the same common ratio, then the points (x1, y1 ), (x 2 , y2 ) and (x 3 , y3 ) [IIT 199, (a) Lie on a straight line (b) Lie on an ellipse, triangle, , (c) Lie on a circle, , (d) Are, , vertices, , of, , a, , 199. Let f (x ) 2 x 1 . Then the number of real values of x for which the three unequal numbers f (x ), f (2 x ), f (4 x ) are in, GP is, (a) 1, , (b) 2, , (c) 0, , (d) None of these, , 200. Sr denotes the sum of the first r terms of a G.P. Then S n , S 2n S n , S 3n S 2n are in, (a) A.P., b, , 1/ x, , 201. If a, , (b) G.P., 1/y, , c, , 1/z, , (a) A.P., , (c) H.P., , (d) None of these, , and a, b, c are in G.P., then x, y, z will be in, (b) G.P., , [IIT 1969; UPSEAT 2001], , (c) H.P., , (d) None of these, , 202. If x, y, z are in G.P. and a x b y c z , then, (a) log a c log b a, , (b) log b a log c b, , [IIT 1966, 1968], , (c) log c b log a c, , (d) None of these, , General term of Harmonic progression, Basic Level, , 203. Three consecutive terms of a progression are 30, 24, 20. The next term of the progression is, (a) 18, , (b) 17, , 204. The 5th term of the H.P., 2, 2, (a) 5, , 1, 5, , 1, 7, , (c) 16, , (d) None of these, , 1, 1, , 3 ,...... will be, 2, 3, , (b) 3, , 1, 5, , [MP PET 1984], , (c) 1 / 10, , (d) 10
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Progressions 143, 205. If 5th term of a H.P. is, (a), , 1, 1, and 11th term is, , then its 16th term will be, 69, 45, , 1, 89, , (b), , 206. If the 7th term of a H.P. is, (a), , 1, 37, , 207. If 6th term of a H.P. is, (a), , 1, 28, , 1, 85, , (c), , [Rajasthan PET 1987, 97], , 1, 80, , (d), , 1, 79, , 1, 1, , then the 20th term is, and the 12th term is, 10, 25, , (b), , 1, 41, , (c), , [MP PET 1997], , 1, 45, , (d), , 1, 49, , 1, 1, , then first term of that H.P. is, and its tenth term is, 105, 61, , (b), , 1, 39, , (c), , 1, 6, , [Karnataka CET 2001], , (d), , 1, 17, , Advance Level, , 208. The 9th term of the series 27+ 9 + 5, (a) 1, , 10, 17, , (b), , 2, 6, 3 ..... will be, 5, 7, , 10, 17, , [MP PET 1983], , (c), , 16, 27, , (d), , 17, 27, , 209. In a H.P., pth term is q and the qth term is p. Then pqth term is, (a) 0, , (b) 1, th, , 210. If a, b, c be respectively the p , q, (a) 1, , [Karnataka CET 2002], , (d) pq ( p q ), , (c) pq, th, , (b) 0, , th, , and r, , bc, terms of a H.P., then p, 1, , ca ab, q, r equals, 1 1, , (c) – 1, , (d) None of these, , Harmonic mean, Basic Level, , 211., , If, , an 1 b n 1, be the harmonic mean between a and b, then the value of n is, an b n, , (a) 1, , (b) – 1, , (c) 0, , [Assam PET 1986], , (d) 2, , H a H b, , 212. If the harmonic mean between a and b be H, then, H a H b, , (a) 4, , (b) 2, , (c) 1, , 213. If H is the harmonic mean between p and q, then the value of, (a) 2, , (b), , pq, pq, , (c), , [AMU 1998], , (d) a + b, H H, is, , p, q, , pq, pq, , [MNR 1990; UPSEAT 2000; 2001], , (d) None of these, , 214. H. M. between the roots of the equation x 2 10 x 11 0 is, 1, (a), 5, , 5, (b), 21, , 21, (c), 20, , [MP PET 1995], , 11, (d), 5
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144 Progressions, , 215. The harmonic mean of, , a, a, and, is, 1 ab, 1 ab, , a, , (a), , (b), , 1a b, , 2 2, , a, 1 a 2b 2, , 63, 120, , (b), , (c) a, , (d), , 1, a a 2b 2, , 6, is, 13, , 216. The sixth H.M. between 3 and, (a), , [MP PET 1996], , [Rajasthan PET 1996], , 63, 12, , (c), , 126, 105, , (d), , 120, 63, , Advance Level, 217. If there are n harmonic means between 1 and, , 1, and the ratio of 7th and (n 1)th harmonic means is 9 : 5, then, 31, , the value of n will be, [Rajasthan PET 1986], , (a) 12, , (b) 13, , (c) 14, , (d) 15, , 218. If m is a root of the given equation (1 ab)x (a b )x (1 ab) 0 and m harmonic means are inserted between a, 2, , 2, , 2, , and b, then the difference between last and the first of the means equals, (a) b – a, , (b) ab (b – a), , (c) a (b – a), , (d) ab(a – b), , Properties of Harmonic progression, Basic Level, , 219. If, , 1, 1, 1 1, , , then a, b, c are in, b a b c a c, , (a) A.P., , [MNR 1984; MP PET 1997; UPSEAT 2000], , (b) G.P., , 220. If a, b, c are in H.P., then, (a) A.P., , (c) H.P., , (d) In G.P. and H.P. both, , a, b, c, ,, ,, are in, b c ca ab, , (b) G.P., , [Roorkee 1980], , (c) H.P., , (d) None of these, , 221. If a, b, c, d are any four consecutive coefficients of any expanded binomial, then, (a) A.P., , (b) G.P., , (c) H.P., , (d) None of these, , 222. log 3 2, log 6 2, log12 2 are in, (a) A.P., , ab b c cd, ,, ,, are in, a, b, c, , [Rajasthan PET 1993, 2001], , (b) G.P., , (c) H.P., , (d) None of these, , 223. If a, b, c are in H.P., then for all n N the true statement is, (a) a c 2b, n, , n, , n, , (b) a c 2b, n, , n, , n, , [Rajasthan PET 1995], , (c) a c 2b, n, , n, , n, , (d) None of these, , 224. Which number should be added to the numbers 13, 15, 19 so that the resulting numbers be the consecutive term, of a H.P., (a) 7, , (b) 6, , (c) – 6, , Advance Level, , (d) – 7
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Progressions 145, 225. If b 2 , a 2 , c 2 are in A.P., then a c, b c, c a will be in, , [AMU 1974], , (a) A.P., (b) G.P., 226. If a, b, c, d be in H.P., then, , (c) H.P., , (d) None of these, (d) ac bd b 2 d 2, , (a) a 2 c 2 b 2 d 2, (b) a 2 d 2 b 2 c 2, (c) ac bd b 2 c 2, 227. If a1, a2 , a3 ,......, an are in H.P., then a1a2 a2a3 ........ an 1an will be equal to, (a) a1an, , [IIT 1975], , (c) (n 1)a1an, , (b) na1an, , (d) None of these, , 228. If x, y, z are in H.P., then the value of expression log( x z ) log( x 2 y z ) will be, (a) log( x z ), , (b) 2 log( x z ), , x y, yz, , y,, are in H.P., then x, y, z are in, 2, 2, (a), A.P.(b), 230. If a, b, c, d are in H.P., then, (a) a + d > b + c, (b) ad > bc, , [Rajasthan PET 1985, 2000], , (d) 4 log( x z ), , (c) 3 log( x z ), , 229. If, , [Rajasthan PET 1989; MP PET 2003], , G.P., , (c) H.P., , (d), , [Rajasthan PET 1991], , (c) Both (a) and (b), , (d) None of these, , Arithmetio-geometric progression, Basic Level, 231. If |x| <1, then the sum of the series 1 2 x 3 x 2 4 x 3 ........ will be, 1, 1, 1, (a), (b), (c), 1 x, 1x, (1 x )2, 232. The sum of 0.2+0.004 + 0.00006 + 0.0000008+...... to is, 2000, 1000, 200, (a), (b), (c), 9801, 9801, 891, , (d), , 1, (1 x )2, , (d) None of these, , 233. The n th term of the sequence 1.1, 2.3, 4.5, 8.7,...... will be, (a) 2n (2n 1), , (c) 2n 1 (2n 1), , (b) 2n 1 (2n 1), , (d) 2n (2n 1), , Advance Level, 234. The sum of infinite terms of the following series 1 , , 4 7 10, , , .....will be, 5 52 53, , [MP PET 1981; Rajasthan PET 1997; Roorkee 1992; DCE 1996, 2000], , 3, (a), 16, , 35, (b), 8, , 235. The sum of the series 1+ 3x+ 6x2+10x3+....... will be, 1, 1, (a), (b), 1x, (1 x )2, , (c), , 35, 4, , (d), , 35, 16, , (c), , 1, (1 x )2, , (d), , 1, (1 x )3, , 236. 21 / 4 . 4 1 / 8. 8 1 / 16 .16 1 / 32....... is equal to, (a) 1, , (b) 2, , 2 3, 4, , , ........ upto n terms is, 5 52 53, 25, 4n 5, 3, 2n 5, , , (a), (b), n 1, 16 16 5, 4 16 5 n 1, , [MNR 1984; MP PET 1998; AIEEE 2002], , 3, (c), 2, , (d), , 237. The sum of 1 , , 5, 2, [MP PET 1982], , (c), , 3, 3n 5, , 7 16 5 n 1, , 238. The sum of i – 2 – 3i + 4 + ....... upto 100 terms, where i 1 is, , (d), , 1, 5n 1, , 2 3 5n2
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146 Progressions, (a) 50 (1 i), , (b) 25 i, , (c) 25 (1 i), , (d) 100 (1 i)
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Method of Difference, Basic Level, 239. nth term of the series 2 4 7 11 ...... will be, (a), , n2 n 1, 2, , [Roorkee 1977], , (b) n 2 n 2, , (c), , n2 n 2, 2, , (d), , n 2 2n 2, 2, , 240. If tn denotes the nth term of the series 2 3 6 11 18 .... then t50 is, (a) 49 2 1, , (c) 50 2 1, , (b) 49 2, , (d) 49 3 2, , 241. First term of the 11th group in the following groups (1), (2, 3, 4), (5, 6, 7, 8, 9), ..... is, (a) 89, , (b) 97, , (c) 101, , (d) 123, , 242. The sum of the series 6 66 666 ...... upto n terms is, (a) (10, , n 1, , 9n 10 ) / 81, , (b) 2 (10, , n 1, , 9 n 10 ) / 27, , [IIT 1974], , (c) 2 (10 9 n 10 ) / 27, n, , (d) None of these, , 243. Sum of n terms of series 12 16 24 40 ..... will be, (a) 2 (2 n 1) 8 n, , (b) 2(2 n 1) 6n, , [UPSEAT 1999], , (c) 3(2 n 1) 8 n, , (d) 4(2 n 1) 8 n, , 244. If |a|<1 and |b|<1, then the sum of the series 1 (1 a)b (1 a a2 )b 2 (1 a a2 a3 )b 3 ...... is, (a), , 1, (1 a) (1 b ), , (b), , 1, (1 a) (1 ab), , (c), , 1, (1 b) (1 ab), , (d), , 1, (1 a)(1 b)(1 ab), , nth Term of Special series, Basic Level, , 245. n th term of the series, (a) n 2 2n 1, 246. The nth term of series, , (a), , n 1, 2, , 13 13 2 3 13 2 3 3 3, + .......will be, , , 1, 13, 135, , (b), , n 2 2n 1, 8, , (c), , [Pb. CET 2000], , n 2 2n 1, 4, , (d), , n 2 2n 1, 4, , 1 12 123, , , ......... will be, 1, 2, 3, , (b), , n 1, 2, , [AMU 1982], , (c), , n2 1, 2, , (d), , n2 1, 2, , 247. If a1 a2 2, an an 1 1(n 2), then a5 is, (a) 1, , (b) – 1, , (c) 0, , (d) – 2, , Advance Level, , 248. The number 111.......1 (91 times) is a, (a) Even number, , (b) Prime number, , (c) Not prime, , (d) None of these, , 249. The difference between an integer and its cube is divisible by, (a) 4, , (b) 6, , (c) 9, , [MP PET 1999], , (d) None of these, , 250. In the sequence 1, 2, 2, 4, 4, 4, 4, 8, 8, 8, 8, 8, 8, 8, 8,......, where n consecutive terms have the value n , the, 1025th term is, (a) 29, , (b) 210, , (c) 211, , (d) 28, , 251. Observe that 13 1, 2 3 3 5, 3 3 7 9 11, 4 3 13 15 17 19 . Then n3 as a similar series is, n(n 1), , (n 1)n, , (n 1)n, , , 1 1 2 , 1 1 ...... 2 , 1 2n 3 , (a) 2 , 2, , , 2, , 2,
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(b) (n 2 n 1) (n 2 n 3) (n 2 n 5) ..... (n 2 3n 1), (c) (n 2 n 1) (n 2 n 3) (n 2 n 5) ..... (n 2 n 1), (d) None of these, , Sum to n terms and infinite number of terms, Basic Level, , 252. The sum of the series 3 . 6 + 4 . 7 + 5 . 8 +....... upto (n – 2) terms, (a) n 3 n 2 n 2, , (b), , 1, (2n 3 12 n 2 10 n 84 ), 6, , [EAMCET 1980], , (c) n 3 n 2 n, , (d) None of these, , 253. The sum of the series 1 (1 2) (1 2 3) ...... upto n terms, will be, (a) n 2 2n 6, , (b), , n(n 1) (2n 1), 6, , [MP PET 1986], , (c) n 2 2n 6, , (d), , n(n 1) (n 2), 6, , 254. The sum to n terms of the series 2 2 4 2 6 2 ....... is, (a), , n(n 1) (2n 1), 3, , (b), , 2n(n 1) (2n 1), 3, , [MP PET 1994], , (c), , n(n 1) (2n 1), 6, , (d), , n(n 1) (2n 1), 9, , 255. 11 2 12 2 13 2 ....... 20 2 , (a) 2481, , [MP PET 1995], , (b) 2483, , (c) 2485, , (d) 2487, , 256. The sum to n terms of (2n 1) 2 (2n 3) 3 (2n 5) ..... is, (a) (n 1)(n 2)(n 3) / 6, 257., , (b) n (n 1)(n 2) / 6, , [AMU 2001], , (c) n (n 1)(2n 3), , (d) n (n 1)(2n 1) / 6, , 13 2 3 3 3 4 3 ..... 12 3, , 12 2 2 3 3 4 2 ..... 12 2, , (a), , 234, 25, , (b), , [MP PET 1998], , 243, 35, , (c), , 263, 27, , (d) None of these, , 258. Sum of the squares of first n natural numbers exceeds their sum by 330, then n=, (a) 8, 259., , (b) 10, , (c) 15, , 1, 1, 1, 1, , , ..... , equals, 1 . 2 2 .3 3 .4, n. (n 1), , (a), , 1, n (n 1), , (b), , [Karnataka CET 1998], , (d) 20, , [AMU 1995; Rajasthan PET 1996; UPSEAT 1999, 2001], , n, n 1, , (c), , 2n, n 1, , (d), , 260. The sum to n terms of the infinite series 1 . 3 2 2 .5 2 3 . 7 2 ..... is, (a), , n, (n 1) (6 n 2 14 n 7 ), 6, , (b), , n, (n 1) (2n 1) (3 n 1), 6, , 2, n (n 1), [AMU 1982], , (c) 4 n 3 4 n 2 n, , (d) None of these, , Advance Level, 261. The sum of all the products of the first n natural numbers taken two at a time is, (a), , 1, n2, n (n 1) (n 1) (3n 2) (b), (n 1) (n 2), 24, 48, , (c), , 1, n (n 1) (n 2) (n 5), 6, , (d) None of these, , 262. The sum of the series 1. 3. 5 + 2. 5. 8 +3. 7. 11+.....up to 'n' terms is, n(n 1) (9 n 23 n 13 ), n(n 1) (9 n 23 n 12 ), (b), 6, 6, 2, , 2, , (a), , [Dhanbad Engg. 1972], , (n 1) (9 n 23 n 13 ), 6, 2, , (c), , 263. The sum of first n terms of the given series 12 2 .2 2 3 2 2 .4 2 5 2 2 .6 2 ....... is, n is odd, the sum will be, (a), , n(n 1)2, 2, , (d), , n (9 n 2 23 n 13 ), 6, , n(n 1)2, , when n is even. When, 2, [IIT 1988; AIEEE 2004], , (b), , 1 2, n (n 1), 2, , (c) n(n 1)2, , (d) None of these
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ar , is, r 1 , , , n, , log b, , 264. The value of, , r 1, , a, n, log n, 2, b, , , , , , , n, , (a), , (b), , 265. The sum of the series, (a), , a n 1 , n, log 2 , 2, b , , (c), , a n 1 , n, log n 1 , 2, b , , (d), , a n 1 , n, log n 1 , 2, b , , 1, 3, 2, , , ......... to n terms is, 1 1 2 14 1 2 2 2 4, 1 32 34, , n(n 2 1), n2 n 1, , (b), , n(n 1), 2(n 2 n 1), , (c), , n(n 2 1), 2(n 2 n 1), , (d) None of these, , 266. For any odd integer n 1 , n3 (n 1)3 ...... (1)n 113 , (a), , 1, (n 1)2 (2n 1), 2, , (b), , 1, (n 1)2 (2n 1), 4, , (c), , 267. The sum of the infinite terms of the sequence, (a), , 1, 18, , (b), , [IIT 1996], , 1, (n 1)2 (2n 1), 2, , (d), , 1, (n 1)2 (2n 1), 4, , (d), , 1, 72, , 5, 9, 13, 2 2 2 2 ..... is, 2, 3 .7, 7 .11, 11 .15, 2, , 1, 36, , (c), , 1, 54, , 268. The sum of the infinite series 1 2 2 2 x 3 2 x 2 ..... is, (a) (1 x ) /(1 x )3, 269. If in a series t n , , (a), , n, , then, (n 1)!, , 20 !1, 20 !, , n, , 270., , (b), , n, , m, , m 1, , r 1, , (c) x /(1 x )3, , (b) (1 x ) /(1 x ), , (d) 1 /(1 x )3, , 20, , t, , n, , is equal to, , n 1, , 21 !1, 21 !, , (c), , 1, 2(n 1) !, , (c), , 1 , r2 , 2 r 1, , (d) None of these, , r r is equal to, 2, , r 1, , 1 , r2 , 2 r 1, n, , (a) 0, , (b), , , , n, , , r 1, , , r, , , , n, , n, , , , r , , (d) None of these, , r 1, , 271. For all positive integral values of n, the value of 3 .1 .2 3 .2 .3 3 .3 .4 ..... 3 .n.(n 1) is, (a) n (n 1)(n 2), , (b) n (n 1)(2n 1), , 272. The sum of (n 1) terms of, (a), , n, n 1, , (c) (n 1) n (n 1), , [Rajasthan PET 1999], , (d), , (n 1) n (n 1), 2, , 1, 1, 1, , , ...... is, 1 12 123, , (b), , 2n, n 1, , [Rajasthan PET 1999], , (c), , 2, n (n 1), , (d), , 2 (n 1), n2, , 273. The sum of (n 1) terms of 1 (1 3) (1 3 5) ....... is, (a), , n (n 1)(2n 1), 6, , (b), , n 2 (n 1), 4, , [Rajasthan PET 1999], , (c), , n (n 1)(2n 1), 6, , (d) n 2, , 274. The sum 1(1 !) 2(2 !) 3(3 !) ..... n(n !) equals, (a) 3(n !) n 3, , (b) (n 1) ! (n 1) !, , 275. Sum of the n terms of the series, (a), , 2n, n 1, , 276. The sum of the series 1 , (a) 1, 277., , [AMU 1999], , (b), , 3, 1, , 2, , , , (c) (n 1) ! 1, , 5, 1 2, 2, , 2, , , , 7, 1 22 32, 2, , 4n, n 1, , ... is, , (c), , 6n, n 1, , (d) 2(n !) 2n 1, [Pb. CET 1999; Rajasthan PET 2001], , (d), , 9n, n 1, , 1 .3 1 .3 .5, , .... is, 6, 6 .8, , (b) 0, , [UPSEAT 2001], , (c) , , 11 3 12 3 .... 20 3, , (d) 4, [Pb. CET 1997; Rajasthan PET 2002], , (a) Is divisible by 5, (c) Is an even integer which is not divisible by 5, , (b) Is an odd integer divisible by 5, (d) Is an odd integer which is not divisible by 5, , 278. The sum of all numbers between 100 and 10,000 which are of form n 3 (n N ) is equal to, (a) 55216, , (b) 53261, , (c) 51261, , (d) 53216, , [IIT 1989]
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279. The cubes of the natural numbers are grouped as 1 3 , (2 3 ,3 3 ), (4 3 ,5 3 ,6 3 ),..... then sum of the numbers in the nth, group is, (a), , 1 3 2, n (n 1)(n 2 3), 8, , (b), , 1 3 2, n (n 16 )(n 2 12 ), 16, , (c), , n3 2, (n 2)(n 2 4 ), 12, , (d) None of these, , 280. The value of the expression 2(1 )(1 2 ) 3(2 1)(2 2 1) 4(3 1)(3 2 1) ..... (n 1)(n 1)(n 2 1) is, n(n 1) , (a) , , 2 , , 281. If, , 2, , 2, , 2, , n(n 1) , (c) , n, 2 , , n(n 1) , (b) , n, 2 , , (d) None of these, , 1, 1, 1, 1, 1, 1, 2, ....up, to, , , , , , , then 2 2 2 ..... equals to, 2, 2, 2, 6, 1, 2, 3, 1, 3, 5, , (b) 2 / 16, , (a) 2 / 6, n, , 282. The value of, , , r 1, , 1, a rx a (r 1)x, , n, , (a), n, , 283. Let, , , , (d) None of these, , is, , a nx a, x, , (b), , a a nx, , (c) 2 / 8, , (c), , n( a nx a), x, , (d) None of these, , n, , r 4 f (n) . Then, , n 1, , (2r 1), , 4, , is equal to, , r 1, , (a) f (2n) 16 f (n) ,for all n N, , n 1, (b) f (n) 16 f , , when n is odd, 2 , , n, (c) f (n) 16 f , when n is even, 2, , (d) None of these, , 284. The sum to n terms of the series, (a), , 1, 3(n 1)(n 2)(n 3), , (b), , 1, 1, 1, , , ..... is, 1. 2 . 3 . 4 2. 3 . 4 . 5 3. 4 . 5 . 6, , 1, 6(n 2)(n 3)(n 4 ), , (c), , 15, 4 n(n 1)(n 5), , (d) None of these, , Relation between A.P., G.P. and H.P., Basic Level, 285. If a and b are two different positive real numbers, then which of the following relations is true [MP PET 1982,2002], (a) 2 ab (a b), , (b) 2 ab (a b), , (c) 2 ab (a b), , (d) None of these, , 286. If a, b, c are in A.P. as well as in G.P., then, (a) a b c, , (b) a b c, , [MNR 1981; AMU 1998], , (c) a b c, , (d) a b c, , 287. If three numbers be in G.P., then their logarithms will be in, (a) A.P., , (b) G.P., , (c) H.P., , [BIT 1992], , (d) None of these, , 288. If the arithmetic, geometric and harmonic means between two distinct positive real numbers be A, G and H, respectively, then the relation between them, [MP PET 1984; Roorkee 1995], (a) A G H, , (b) A G H, , (c) H G A, , (d) G A H, , 289. If the arithmetic, geometric and harmonic means between two positive real numbers be A, G and H, then, [AMU 1979,82; MP PET 1993], , (a) A 2 GH, 290. If a, b, c are in A.P. then, (a) A.P., , (b) H 2 AG, , (c) G AH, , a 1 2, , ,, are in, bc c b, , (b) G.P., , (d) G 2 AH, [MNR 1982; MP PET 2002], , (c) H.P., , (d) None of these, , 291. The geometric mean of two numbers is 6 and their arithmetic mean is 6.5. The numbers are, (a) (3, 12), , (b) (4, 9), , (c) (2, 18), , [MP PET 1994], , (d) (7, 6), , 292. In the four numbers first three are in G.P. and last three in A.P. whose common difference is 6. If the first and, last numbers are same, then first will be, [IIT 1974], (a) 2, , (b) 4, , (c) 6, , (d) 8
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293. If A1 , A2 are the two A.M.'s between two numbers a and b and G1 , G2 be two G.M.'s between same two, numbers, then, , A1 A 2, , G1 . G 2, [Roorkee 1983; DCE 1998], , ab, ab, , (a), , (b), , ab, 2 ab, , 2 ab, ab, , (c), , (d), , ab, ab, , 294. If the A.M. and H.M. of two numbers is 27 and 12 respectively, then G.M. of the two numbers will be [Rajasthan PET 198, (a) 9, , (b) 18, , (c) 24, , 295. The A.M., H.M. and G.M. between two numbers are, , (d) 36, , 144, , 15 and 12, but necessarily in this order. Then H.M.,, 15, , G.M. and A.M. respectively are, (a) 15 ,12 ,, , 144, 15, , (b), , 144, ,12 ,15, 15, , (c) 12 ,15 ,, , 144, 15, , (d), , 144, ,15 ,12, 15, , 296. If G.M. =18 and A.M.=27, then H.M. is, (a), , 1, 18, , (b), , [Rajasthan PET 1996], , 1, 12, , (c) 12, , (d) 9 6, , 297. If sum of A.M. and H.M. between two numbers is 25 and their G.M. is 12, then sum of numbers is, (a) 9, 298. If, , (b) 18, , (c) 32, , (d) 18 or 32, , a bx b cx c dx, , , (x 0), then a, b, c, d are in, a bx b cx c dx, , (a) A.P., , (b) G.P., , [Rajasthan PET 1986], , (c) H.P., , (d) None of these, , 299. The numbers 1,4, 16 can be three terms (not necessarily consecutive) of, (a) No A.P., , (b) Only one G.P., , (c) Infinite number of A.P’s., , (d)Infinite numbers of G.P’s., , 300. In a G.P. of alternately positive and negative terms, any terms is the A.M. of the next two terms . Then the, common ratio is, (a) – 1, , (b) – 3, , 301. If a, b, c are in A.P., then a , (a) A.P., , 1, 2, , (c) – 2, , (d) , , (c) H.P., , (d) None of these, , 1, 1, 1, ,b ,c , are in, bc, ca, ab, , (b) G.P., , 302. The A.M. of two given positive numbers is 2. If the larger number is increased by 1, the G.M. of the numbers, becomes equal to the A.M. of the given numbers. Then the H.M. of the given numbers is, (a), , 3, 2, , (b), , 2, 3, , 1, 2, , (c), , (d) None of these, , Advance Level, , qth, rth and sth terms of an A.P. be in G.P., then ( p q), (q r), (r s) will be in, , 303. If pth, (a) G.P., , (b) A.P., , (c) H.P., , (d) None of these, , 304. If a, b, c are the positive integers, then (a b)(b c)(c a) is, (a) 8 abc, , (b) 8 abc, a, , b, , 305. If a, b, c are in A.P., then 3 ,3 ,3, (a) A.P., , c, , [DCE 2000], , (c) 8 abc, , (d) None of these, , shall be in, , (b) G.P., , [Pb. CET 1990], , (c) H.P., , (d) None of these, , 306. If a, b, c, d and p are different real numbers such that (a b c )p 2(ab bc cd )p (b 2 c 2 d 2 ) 0, then a, b,, 2, , 2, , 2, , 2, , c, d are in, [IIT 1987], , (a) A.P., , (b) G.P., , (d) ab cd, , (c) H.P., , 307. If the first and (2n 1)th terms of an A.P., G.P. and H.P. are equal and their nth terms are respectively a, b and c,, then, , [IIT 1985,88], , (b) a c b, , (a) a b c, 308. If the (m 1) , (n 1), th, , th, , and (r 1), , th, , (c) ac b 0, 2, , (d) (a) and (c) both, , terms of an A.P. are in G.P. and m, n, r in H.P., then the value of the ratio of, , the common difference to the terms of the A.P. is, , [MNR 1989; Roorkee 1994]
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(a) , , 2, n, , (b), , 2, n, , (c) , , n, 2, , (d), , n, 2, , 309. Given a x b y c z d u and a, b, c, d are in G.P., then x, y, z, u are in [Dhanbad Engg. 1972; Roorkee 1984; Rajasthan PET 2, (a) A.P., , (b) G.P., , (c) H.P., , (d) None of these, , 310. If a, b, c are in G.P. and log a log 2b, log 2b log 3c and log 3 c log a are in A.P., then a, b, c are the length of the, sides of a triangle which is, (a) Acute angled, 311., , (b) Obtuse angled, , (c) Right angled, , If a, b, c are in A.P., b, c, d are in G.P. and c, d, e are in H.P., then a, c, e are in, (a) No particular order, , (b) A.P., 2, , 2, , 312. If a, b, c are in A.P. and a , b , c, (a) a b c, , 2, , (c) G.P., , (d) H.P., , are in H.P., then, , (b) 2b 3a c, , (d) Equilateral, [AMU 1988,2001; MP PET 1993], , [MNR 1986,88; IIT 1977,2003], , (c) b 2 (ac / 8), , (d) None of these, , 313. The harmonic mean of two numbers is 4 and the arithmetic and geometric means satisfy the relation, 2 A G 2 27 , the numbers are, [MNR 1987; UPSEAT 1999,2000], , (a) 6, 3, , (b) 5, 4, , (d) – 3, 1, , (c) 5, – 2.5, , 314. In a G.P. the sum of three numbers is 14, if 1 is added to first two numbers and subtracted from third numbers,, the series becomes A.P., then the greatest number is, [Roorkee 1973], (a) 8, , (b) 4, , (c) 24, , (d) 16, , 315. If a, b, c are in G.P. and x, y are the arithmetic means between a, b and b, c respectively, then, , a c, is equal to, , x y, , [Roorkee 1969], , (a) 0, , (b) 1, , (c) 2, , (d), , 1, 2, , 316. If a, b, c are in A.P. and a, b, d in G.P., then a, a – b, d – c will be in, (a) A.P., , (b) G.P., , [Ranchi BIT 1968], , (c) H.P., , (d) None of these, , 317. If x, 1, z are in A.P. and x, 2, z are in G.P., then x, 4, z will be in, (a) A.P., 318., , (b) G.P., , [IIT 1965], , (c) H.P., , (d) None of these, , 1 1 1 5, x y z 15 , if 9, x , y, z , a are in A.P.; while, if 9, x , y, z , a are in H.P., then the value of a will be [IIT 1978], , x y z 3, , (a) 1, , (b) 2, , (c) 3, , (d) 9, , 319. If 9 A.M.'s and H.M.'s are inserted between the 2 and 3 and if the harmonic mean H is corresponding to, 6, , arithmetic mean A, then A , [Dhanbad Engg. 1987], H, (a) 1, , (b) 3, th, , 320. If the p , q, , th, , th, , and r, , (c) 5, , (d) 6, , term of a G.P. and H.P. are a, b, c, then a(b c) log a b(c a) log b c(a b) log c [Dhanbad Engg. 1976], , (a) – 1, , (b) 0, , (c) 1, , (d) Does not exist, , 321. If the product of three terms of G.P. is 512. If 8 added to first and 6 added to second term, so that number may, be in A.P., then the numbers are, [Roorkee 1964], (a) 2, 4, 8,, , (b) 4, 8, 16, , (c) 3, 6, 12, , (d) None of these, , 322. If the ratio of H.M. and G.M. between two numbers a and b is 4 : 5 , then ratio of the two numbers will be [IIT 1992; MP, (a) 1 : 2, , (b) 2 : 1, , (c) 4 : 1, , (d) 1 : 4, , 323. If the A.M. and G.M. of roots of a quadratic equations are 8 and 5 respectively, then the quadratic equation will, be [Pb.CET 1990], (a) x 2 16 x 25 0, , (b) x 2 8 x 5 0, , (c) x 2 16 x 25 0, , (d) x 2 16 x 25 0, , 324. Let a1 , a2 ,..... a10 be in A.P. and h1 , h2 ,....., h10 be in H.P. If a1 h1 2 and a10 h10 3, then a4 h7 is, (a) 2, , (b) 3, , (c) 5, , 325. If ln(a c), ln(c a), ln(a 2b c) are in A.P., then, (a) a, b, c are in A.P., 326. If, , A1 , A2 ; G1 , G2, , and, , (b) a 2 , b 2 , c 2 are in A.P., , [IIT 1999], , (d) 6, [IIT Screening 1994; Rajasthan PET 1999], , (c) a, b, c are in G.P., , (d) a, b, c are in H.P., , H 1 , H 2 be two A.M’s, G.M’s and H.M’s between two numbers respectively, then, , G1G2 H1 H 2, , equals, H1 H 2 A1 A2, [Rajasthan PET 1997; AMU 2000], , (a) 1, , (b) 0, , (c) 2, , (d) 3
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1, 1, 1, are in, ,, ,, 1 ln x 1 ln y 1 ln z, , 327. If x 1, y 1, z 1 are in G.P., then, (a) A.P., , (b) H.P., , (c) G.P., , [IIT 1998; UPSEAT 2001], , (d) None of these, , 328. If p, q, r are in one geometric progression and a, b, c in another geometric progression, then cp, bq, ar are in, [Roorkee Qualifying 1998], , (a) A.P., , (b) H.P., , 329. If first three terms of sequence, , (c) G.P., , (d) None of these, , 1, 1, are in geometric series and last three terms are in harmonic series,, , a, b,, 16, 6, , then the value of a and b will be, (a) a , , 1, ,b 1, 4, , (b) a , , [UPSEAT 1999], , 1, 1, ,b , 12, 9, , (c) (a) and (b) both are true, , 330. If a x b y c z and a, b, c are in G.P., then x, y, z are in, (a) A. P., , (b) G. P., , (d), , None of these, , [Pb. CET 1993; DCE 1999; AMU 1999], , (c) H. P., , (d) None of these, , 331. If G1 and G 2 are two geometric means and A the arithmetic mean inserted between two numbers, then the, value of, , G12 G 22, is, , G2, G1, [DCE 1999], , (a), 332. If, , A, 2, , (b) A, , (c) 2 A, , (d) None of these, , a, b, c, ,, ,, are in H.P., then a, b, c are in, bc ca ab, , (a) A.P., , (b) G.P., 1, , 333. If a, b, c are in A.P., then, (a) A.P., , [Rajasthan PET 1999], , a b, , (c) H.P., ,, , 1, a c, , ,, , 1, b c, , (d) None of these, , are in, , (b) G.P., , [Roorkee 1999], , (c) H.P., , (d) None of these, , 334. The sum of three decreasing numbers in A.P. is 27. If 1, 1, 3 are added to them respectively, the resulting, series is in G.P. The numbers are, (a) 5, 9, 13, , [AMU 1999], , (b) 15, 9, 3, , (c) 13, 9, 5, , (d) 17, 9, 1, , 335. If in the equation ax 2 bx c 0, the sum of roots is equal to sum of square of their reciprocals, then, , c a b, , ,, a b c, , are in, [Rajasthan PET 2000], , (a) A.P., , (b) G.P., , 336. If a, b, c are in A.P., then 2, (a) A.P., , ax 1, , ,2, , bx 1, , (c) H.P., ,2, , , x 0 are in, , (b) G.P. only when x 0, , 337. If b c, c a, a b are in H.P., then, (a) A.P., , cx 1, , (d) None of these, [DCE 2000; Pb. CET 2000], , (c) G.P. if x 0, , (d) G.P. for all x 0, , a, b, c, ,, ,, are in, bc ca ab, , (b) G.P., , [Rajasthan PET 2000], , (c) H.P., , (d) None of these, , 338. The common difference of an A.P. whose first term is unity and whose second, tenth and thirty fourth terms, are in G.P., is, [AMU 2000], , (a), , 1, 5, , (b), , 1, 3, , (c), , 1, 6, , (d), , 1, 9, , 339. The sum of three consecutive terms in a geometric progression is 14. If 1 is added to the first and the second, terms and 1 is subtracted from the third, the resulting new terms are in arithmetic progression. Then the, lowest of the original term is, [MP PET 2001], , (a) 1, , (b) 2, , (c) 4, , (d) 8, , 340. a, g, h are arithmetic mean, geometric mean and harmonic mean between two positive numbers x and y, respectively. Then identify the correct statement among the following, (a) h is the harmonic mean between a and g, , [Karnataka CET 2001], , (b) No such relation exists between a, g and h
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(c) g is the geometric mean between a and h, , (d) a is the arithmetic mean between g and h, , 341. Let the positive numbers a, b, c, d be in A.P., then abc, abd, acd, bcd are, (a) Not in A.P./G.P./H.P. (b) In A.P., , [IIT Screening 2001], , (c) In G.P., , (d) In H.P., , 342. If (y x ), 2(y a) and (y z ) are in H.P., then x a, y a, z a are in, (a) A.P., , (b) G.P., , [Rajasthan PET 2001], , (c) H.P., , (d) None of these, , 343. If A and G are arithmetic and geometric means and x 2 Ax G 0 , then, 2, , (b) A G, , (a) A G, , 2, , [UPSEAT 2001], , (c) A G, , (d) A G, , 344. If A is the A.M. of the roots of the equation x 2 2ax b 0 and G is the G.M. of the roots of the equation, x 2 2bx a 2 0, then, [UPSEAT 2001], , (b) A G, , (a) A G, , (c) A G, , (d) None of these, , 345. If a,b, c are three unequal numbers such that a, b, c are in A.P. and b – a, c – b, a are in G.P., then a : b : c is [UPSEAT 20, (a) 1 : 2 : 3, , (b) 2: 3 : 1, , (c) 1 : 3 : 2, , (d) 3 : 2 : 1, , 346. If a, b, c are in A.P. and a 2 , b 2 , c 2 are in H.P., then, (a) a b c, , (b) a 2 b 2 , , c2, 2, , [UPSEAT 2001], , (c) a, b, c are in G.P., , (d), , a, , b , c are in G.P., 2, , 347. Let a1 , a2 , a3 be any positive real numbers, then which of the following statement is not true, (a) 3 a1a2 a3 a13 a23 a33, , (b), , 1, 1, 1, , , (c) (a1 a 2 a 3 ), a1 a 2 a 3, , , 9, , , , [Orissa JEE 2002], , a1 a 2 a 3, , , 3, a 2 a 3 a1, , 1, 1, 1, (d) (a1 a 2 a 3 ) , , a1 a 2 a 3, , 3, , , 27, , , , 348. If a1 , a 2 ,.... an are positive real numbers whose product is a fixed number c, then the minimum value of, , a1 a2 ... an1 2an is, [IIT Screening 2002], , (b) (n 1) c 1 / n, , (a) n(2c)1 / n, , (d) (n 1)(2c)1 / n, , (c) 2nc 1 / n, , 349. Suppose a, b, c are in A.P. and a 2 , b 2 , c 2 are in G.P. If a < b < c and a b c , , 3, , then the value of a is, 2, [IIT Screening 2002], , (a), , 1, , 1, , (b), , (c), , 2 3, , 2 2, , 1, 1, , 2, 3, , (d), , 1, 1, , 2, 2, , n, , 5 n 1 , 5 , 350. Two sequences {tn } and {s n } are defined by tn log n 1 , sn log , then, 3 , 3 , , [AMU 2002], , (a) {tn } is an A.P., {s n } is a G.P., , (b) {tn } and {s n } are both G.P., , (c) {tn } and {s n } are both A.P., , (d) {s n } is a G.P., {t n } is neither A.P. nor G.P., , a b a b, 351. If b c b c, 2 1, 0, , = 0 and 1 / 2, then a, b, c are in, , (a) A.P., , (b) G.P., , (c) H.P., , (d) None of these, , 352. If x, y, z are in G.P. and tan 1 x , tan 1 y, tan 1 z are in A.P., then, (a) x y z or y 1, , (b) z 1 / x, , (c) x y z, but their common value is not necessarily zero (d), , x yz0
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353. If in a progression a1 , a2 , a3 ....., etc., (ar ar 1 ) bears a constant ratio with ar .ar 1 then the terms of the, progression are in, (a) A.P., 354. If, , (b) G.P., , (c) H.P., , (d) None of these, , (c) H.P., , (d) None of these, , a a3 , a2 a3, a a3, then a1 , a2 , a3 , a4 are in, 2, 3 2, , a1 a 4, a1 a 4, a1 a 4 , , (a) A.P., , (b) G.P., , 355. If a, a1 , a2 , a3 ,..... a2n1 , b are in A.P., a, b1 , b 2 , b 3 ,..... b 2n1 , b are in G.P. and a, c1 , c 2 , c 3 ,..... c 2n1 , b are in H.P., where a, b, are positive, then the equation an x 2 b n x c n 0 has its roots, (a) Real and unequal, , (b) Real and equal, , (c) Imaginary, , (d) None of these, , 356. If a, x, b, are in A.P., a, y, b are in G.P. and a, z, b are in H.P. such that x 9 z and a 0, b 0 then, (b) x 3 | y |, , (a) | y | 3 z, , (c) 2 y x z, , (d) None of these, , 357. If a, b, c are in G.P. and a, p, q in A.P. such that 2a, b p, c q are in G.P. then the common difference of the A.P., is, (a), , (b) ( 2 1)(a b), , 2a, , (c), , (d) ( 2 1)(b a), , 2 (a b ), , Applications of Progressions, Basic Level, , 358. If x, y, z are positive then the minimum value of x log y log z y log z log x z log x log y is, (a) 3, , (b) 1, , (c) 9, , 359. a, b, c are three positive numbers and abc 2 has the greatest value, , (a) a b , , 1, 1, ,c , 2, 4, , (b) a b , , 1, 1, ,c , 4, 2, , (d) 16, 1, . Then, 64, , (c) a b c , , 1, 3, , (d) None of these, , 360. If a 0, b 0, c 0 and the minimum value of a(b 2 c 2 ) b(c 2 a2 ) c(a2 b 2 ) is abc , then the is, (a) 2, , (b) 1, , (c) 6, , 361. If x, y, z are three real numbers of the same sign then the value of, (a) [2,), , (b) [3,), , (d) 3, x y z, lies in the interval, y z x, , (c) (3,), , (d) (,3), , 362. The sum of the products of the ten numbers 1, 2, 3, 4, 5 taking two at a time is, (a) 165, , (b) – 55, , (c) 55, , (d) None of these, , 363. Let S1 , S 2 ..... be squares such that for each n 1, the length of a side of S n equals the length of a diagonal of, , S n 1. If the length of a side of, , S 1 is 10 cm, then for which of the following values of n is the area of S n less, , then 1 sq cm, (a) 7, , [IIT 1999], , (b) 8, , (c) 9, , (d) 10, , 364. Jairam purchased a house in Rs. 15000 and paid Rs. 5000 at once. Rest money he promised to pay in annual, installment of Rs. 1000 with 10% per annum interest. How much money is to be paid by Jairam, (a) Rs. 21555, , (b) Rs. 20475, , (c) Rs. 20500, , [UPSEAT 1999], , (d) Rs. 20700
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365. The sum of the integers from 1 to 100 which are not divisible by 3 or 5 is, (a) 2489, , (b) 4735, , [MP PET 2000], , (c) 2317, , (d) 2632, , (c) Divisible by n, , (d) Never less than n, , 366. The product of n positive numbers is unity. Their sum is, (a) A positive integer, , (b) Equal to n , , 1, n, , 367. If a, b, c, d are positive real numbers such that a b c d 2, then M (a b)(c d ) satisfies the relation [IIT Screening, (a) 0 M 1, , (b) 1 M 2, , (d) 3 M 4, , (c) 2 M 3, , 368. The sum of all positive divisors of 960 is, (a) 3048, , [Karnataka CET 2000], , (b) 3087, , (c) 3047, , (d) 2180, , 369. 2 sin 2 cos is greater than, , [AMU 2000], 1, , (a), , 1, 2, , (b), , , 1 , 1, , , 2 , , 2, , (c) 2, , 2, , (d) 2 , , 370. If the altitudes of a triangle are in A.P., then the sides of the triangle are in, , [EAMCET 2002], , (a) A.P., , (b) H.P., , (c) G.P., , (d) Arithmetico-geometric progression, , 371. A boy goes to school from his home at a speed of x km/hour and comes back at a speed of y km/hour, then the, average speed is given by, (a) A.M., , (b) G.M., , (c) H.M., , (d) None of these, , 372. A monkey while trying to reach the top of a pole height 12 metres takes every time a jump of 2 metres but slips, 1 metre while holding the pole. The number of jumps required to reach the top of the pole, is, (a) 6, , (b) 10, , (c) 11, , (d) 12, , 373. Balls are arranged in rows to form an equilateral triangle. The first row consists of one ball, the second row of, two balls and so on. If 669 more balls are added then all the balls can be arranged in the shape of a square and, each of the sides then contains 8 balls less than each side of the triangle did. The initial number of balls is [Roorkee 19, (a) 1600, , (b) 1500, , (c) 1540, , (d) 1690, , 374. If a, b and c are three positive real numbers, then the minimum value of the expression, (a) 1, , (b) 2, , (c) 3, , (d) 6, , 375. If x 1 0, i 1,2,....., 50 and x 1 x 2 ..... x 50 50, then the minimum value of, (a) 50, , 1, 1, 1, , ..... , equals to, x1 x 2, x 50, , (c) (50 )3, , (b) (50 )2, , bc ca ab, , , is, a, b, c, , (d) (50 )4, , 1 1 1, 376. If a, b and c are positive real numbers, then least value of (a b c) is, a b c, , (a) 9, , (b) 3, , (c) 10/3, , (d) None of these, , (c) 23, , (d) 24, , 377. In the value of 100 ! the number of zeros at the end is, (a) 11, , (b) 22, , 378. If (1 p)(1 3 x 9 x 2 27 x 3 81 x 4 243 x 5 ) 1 p 6 , p 1 then the value of, (a) 1/3, , (b) 3, , p, is, x, , (c) 1/2, , n , 1, 379. Let f (n) , where[x] denotes the integral part of x. Then the value of, 2, 100, , , , (a) 50, 380. Ar ; r 1,2,3,....., n, , (b) 51, are n points on the parabola, , (c) 1, y2 4x, , x 1 , x 2 , x 3 ,....., x n are in G.P. and x 1 1, x 2 2, then y n is equal to, , (d) 2, 100, , f (n) is, n 1, , (d) None of these, in the first quadrant. If, , Ar (x r , y r ),, , where
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(a) 2, , n 1, 2, , (b) 2 n 1, , (c) ( 2 )n 1, , n, , (d) 2 2, , 381. The lengths of three unequal edges of a rectangular solid block are in G.P. The volume of the block is 216 cm 3, and the total surface area is 252 cm 2 . The length of the longest edge is, (a) 12 cm, , (b) 6 cm, , (c) 18 cm, , (d) 3 cm, , 382. ABC is right-angled triangle in which B 90 and BC a. If n points L1, L2 ,....., Ln on AB are such that AB is, divided in n 1 equal parts and L1 M1, L2 M2 ,....., Ln Mn are line segments parallel to BC and M1, M2 ,....., Mn are on AC, then the sum of the lengths of L1 M1, L2 M2 ,....., Ln Mn is, (a), , a(n 1), 2, , (b), , (c), , an, 2, , (d) Impossible to find from the given data, , ***, , a(n 1), 2
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156 Progressions, , Assignment (Basic and Advance Level), , Progressions, , 1, c, , 2, b, , 3, b, , 4, a, , 5, b, , 6, b, , 7, a, , 8, b, , 9, b, , 10, a, , 11, a, , 12, c, , 13, a, , 14, c, , 15, a, , 16, c, , 17, a, , 18, c, , 19, b, , 20, d, , 21, c, , 22, a, , 23, c,d, , 24, d, , 25, d, , 26, b, , 27, c, , 28, b, , 29, c, , 30, a, , 31, b, , 32, a, , 33, b, , 34, d, , 35, d, , 36, d, , 37, d, , 38, d, , 39, b, , 40, a, , 41, b, , 42, c, , 43, b, , 44, c, , 45, c, , 46, b, , 47, a, , 48, b, , 49, d, , 50, a, , 51, d, , 52, b, , 53, c, , 54, d, , 55, d, , 56, b, , 57, b, , 58, c, , 59, d, , 60, a, , 61, d, , 62, d, , 63, b, , 64, b, , 65, b, , 66, b, , 67, a, , 68, d, , 69, d, , 70, b, , 71, a, , 72, c, , 73, c, , 74, c, , 75, b, , 76, c, , 77, d, , 78, a, , 79, e, , 80, b, , 81, b, , 82, a, , 83, a, , 84, b, , 85, c, , 86, c, , 87, b, , 88, d, , 89, c, , 90, a, , 91, a, , 92, b, , 93, d, , 94, c, , 95, b, , 96, d, , 97, a, , 98, a, , 99, b, , 100, d, , 101, b, , 102, a, , 103, a, , 104, c, , 105, a, , 106, a, , 107, a,d, , 108, c, , 109, d, , 110, b, , 111, a, , 112, c, , 113, b, , 114, b, , 115, c, , 116, c, , 117, c, , 118, a, , 119, c, , 120, b, , 121, c, , 122, a, , 123, b, , 124, a, , 125, a, , 126, b, , 127, b, , 128, a, , 129, a, , 130, a, , 131, a, , 132, c, , 133, b, , 134, b, , 135, c, , 136, c, , 137, b, , 138, a, , 139, d, , 140, a, , 141, d, , 142, d, , 143, d, , 144, d, , 145, d, , 146, c, , 147, b, , 148, a,b, , 149, a, , 150, d, , 151, d, , 152, c, , 153, c, , 154, c, , 155, b, , 156, d, , 157, a, , 158, a, , 159, d, , 160, d, , 161, b, , 162, c, , 163, a, , 164, d, , 165, c, , 166, c, , 167, a, , 168, b, , 169, c, , 170, a, , 171, d, , 172, a, , 173, d, , 174, d, , 175, c, , 176, b, , 177, b, , 178, c, , 179, b, , 180, b, , 181, c, , 182, c, , 183, a, , 184, b, , 185, c, , 186, d, , 187, b, , 188, c, , 189, a, , 190, b, , 191, a, , 192, c, , 193, c, , 194, b, , 195, a, , 196, c, , 197, a, , 198, a, , 199, c, , 200, b, , 201, a, , 202, b, , 203, b, , 204, d, , 205, a, , 206, d, , 207, c, , 208, a, , 209, b, , 210, b, , 211, b, , 212, b, , 213, a, , 214, d, , 215, c, , 216, a, , 217, c, , 218, b, , 219, c, , 220, c, , 221, c, , 222, c, , 223, b, , 224, d, , 225, c, , 226, c, , 227, c, , 228, b, , 229, b, , 230, c, , 231, d, , 232, b, , 233, c, , 234, d, , 235, d, , 236, b, , 237, a, , 238, a, , 239, c, , 240, d, , 241, c, , 242, b, , 243, d, , 244, c, , 245, c, , 246, a, , 247, b, , 248, c, , 249, b, , 250, b, , 251, c, , 252, b, , 253, d, , 254, b, , 255, c, , 256, d, , 257, a, , 259, b, , 260, a, , 261, a, , 262, a, , 263, b, , 264, c, , 265, b, , 266, d, , 267, d, , 268, a, , 269, b, , 270, c, , 271, a, , 272, d, , 273, c, , 274, c, , 275, c, , 276, d, , 277, b, , 278, b, , 279, a, , 280, b, , 281, c, , 282, a,b, , 283, a, , 284, d, , 285, b, , 286, d, , 287, a, , 288, a, , 289, d, , 290, d, , 291, b, , 292, d, , 293, a, , 294, b, , 295, b, , 296, c, , 297, c, , 298, b, , 299, a, , 300, c, , 301, a, , 302, a, , 303, a, , 304, b, , 305, b, , 306, b, , 307, d, , 308, a, , 309, c, , 310, b, , 311, c, , 312, a, , 313, a, , 314, a, , 315, c, , 316, b, , 317, c, , 318, a, , 319, c, , 320, b, , 321, b, , 322, c,d, , 323, c, , 324, d, , 325, d, , 326, a, , 327, b, , 328, c, , 329, c, , 330, c, , 331, c, , 332, c, , 333, a, , 334, d, , 335, a, , 336, d, , 337, a, , 338, b, , 339, b, , 340, c, , 341, d, , 342, b, , 343, b, , 344, c, , 345, a, , 346, d, , 347, d, , 348, a, , 349, d, , 350, a, , 351, b, , 352, a, , 353, c, , 354, c, , 355, c, , 356, b, , 357, b,d, , 358, a, , 359, b, , 360, c, , 361, b, , 362, b, , 363, b,c,d, , 364, c, , 365, d, , 366, d, , 367, a, , 368, a, , 369, d, , 370, b, , 371, c, , 372, c, , 373, c, , 374, d, , 375, a, , 376, a, , 377, d, , 378, b, , 379, b, , 380, c, , 381, a, , 382, c, , 258, b
