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SEQUENCES AND SERIES

Only One Correct Answer Type

1. If a, b and c be in G.P. and x, y be the arithmetic means between a, b and b, c respectively, then  is
A: 2

 

2. If Sn be the sum of 'n' terms of an A.P. and   is independent of 'n', then the common difference is
A: 2a

 

3. The maximum value of a2b3c4 subject to a + b + c = 18 is
A: 4263 84

 

4. There are 'n' A.P.'s whose common differences are 1, 2, 3, ....., n respectively the first term of each being unity. Then sum of their nth term is
A: 

 

5. If the 1st and the (2n - 1)th term of an A.P., G.P., and H.P. are equal and their nth terms are a, b and c respectively, then
A: a = b = c

 

A: 
 

7. If a1, a2, a3, ..... is an A.P. such that a1 + a5 + a10 + a15 + a20 + a24 = 225, then
     a1 + a2 + a3 + ....... + a23 + a24 is equal to
A: 900

 

8. If x, y, z are real numbers satisfying the equation
    16(25x2 + 4y2) + 25z2 - 160xy - 40yz - 100zx = 0, then x, y, z are in
A: A.P.

 

9. If a, b, c, d are in H.P., then ab + bc + cd is equal to
A: 3ad

 


 
A: 

 

A: H.P.
 

12. A G.P. consists of an even number of terms. If the sum of all the terms is five times the sum of those terms occupying the odd places, then common ratio is
A: 5

 

13. Between 1 and 31 'm' arithmetic means are inserted, so that the ratio of the 7th and (m - 1)th means is 5 : 9. Then the value of n is
A: 14

 

14. If the A.M. and G.M. between two numbers are in the ratio m : n, then the numbers are in the ratio
A:  

 

15. Sum of the series S = 1 +  (1 + 2) + (1 + 2 + 3) + (1 + 2 + 3 + 4) + ......  upto 20 terms is
A: 115

 

16. If A and G are the A.M. and G.M. respectively between two numbers, then the numbers are
A: 

 

17. Sum of the series S = 12 - 22 + 32 - 42 + ...... - 2002+ 20032 is
A: 2007006

 

18. 0.7 + 0.77 + 0.777 + ....... + n terms =
A: 

 

19. If x, y and z are positive real numbers such that x + y + z = a, then
A:  

 

A: H.P.

 

21. The greatest value of x2y3z4, (if x + y + z = 1, x, y, z > 0) is
Ans: 

 

22. The G.M. of two numbers is 6. Their A.M. 'A' and H.M. 'H' satisfy the equation 90A + 5H = 918, then
Ans: A = , A = 10

 


Ans: 6

 

24. If (1 + 3 + 5 + ....... + p) + (1 + 3 + 5 + ....... + q) = (1 + 3 + 5 + ....... + r) where each set of parenthesis contains the sum of consecutive integers, then the smallest possible value of p + q + r (p > 6) is
Ans: 21

 

25. Three distinct real numbers a, b, c are in G.P. such that a + b + c = xb, then
Ans: x < −1 or x > 3

 


 
Ans: G.P.

 

27. If a, b, c, d are non zero real numbers such that
(a2 + b2 + c2) (b2 + c2 + d2) (ab + bc + cd)2, then a, b, c, d are in
Ans: G.P.

 

28. If the ratio between the sum of first 'n' terms of two A.P.'s is 7n + 1 : 4n + 27, then the ratio of 11th term is
Ans: 148 : 111

 

29. If one geometric mean G and two arithmetic means p and q be inserted between two numbers, then G2 is equal to
Ans: (2p - q)(2q - p)

 

Ans: 0

 


 
Ans: 

 

32. The sides a, b, c of ∆ ABC are in G.P. and log a - log 2b, log 2b - log 3c, log 3c - log a are in A.P., then ∆ ABC is
Ans: obtuse angled

 

Ans: 
 

34. In a geometrical progression consisting of positive terms, each term equals the sum of the next two terms. Then the common ratio of this progression equals to
Ans: 

 

35. If a, a1, a2, a3, ......, a2n - 1, b are in A.P.; a, b1, b2, b3, ........, b2n - 1, b are in G.P. and a, c1, c2, c3, ....., c2n - 1, b are in H.P., where a, b are positive, then the equation anx2 - bnx + cn = 0 has its roots
Ans: imaginary

 


 
Ans: 

 


 Ans: 

 

 

 Ans: 

 

Ans: 2

 

Ans: n2


41. A three digit number whose consecutive numbers form a G.P. If we subtract 792 from this number, we get a number consisting of the same digits written in the reverse order. Now if we increase the second digit of the required number by 2, the resulting number will form an A.P., then the number is
Ans: 931

 

 Ans: 
 

43. An employee gets Rs. 300 per month in his 11th year of service and got Rs. 495 per month in his 24th year of service. If his monthly salary is in A.P., his initial salary and his increment are
Ans: Rs. 150, Rs. 15

 

Ans: H.P.
 

45. Sum of all odd integers between 2 and 100 that are divisible by 3 is
Ans: 867

 

46. The numbers 32 sin2x−1, 14, 34− 2 sin 2x form first three terms of an A.P., then x =
Ans: 

 

47. If the A.M. of the roots of a quadratic in x is 3 and G.M. is  , then the quadratic equation is
Ans: x2 − 6x + 8 = 0

 


 Ans: 
     
49. If the sum of an infinitely decreasing G.P. is 3 and the sum of squares of its terms is , then the sum of the cubes of the terms is
Ans: 

 

50. If three non-zero numbers x, y, z are in A.P. and tan-1 x, tan-1 y, tan-1 z are in A.P., then
Ans: x = y = z

 

One or more than one correct answer type
 

51. If b1, b2, b3 (b1 > 0) are three successive terms of a G.P. with common ratio r, the value of r for which the inequality b3 > 4b2 - 3b1 holds is given by
   A) r > 3       B) r < 1       C) r = 3.5      D) r = 5.2
Ans: A, B, C, D

 

52. If (r)n denotes the number rrr ...... (n digits), where r = 1, 2, 3, ......, 9 and a = (6)n, b = (8)n, c = (4)2n, then
      A) a2 + b + c = 0    B) a2 + b - c = 0      C) a2 + b - 2c = 0     D) a2 + b - 9c = 0
Ans: B

 

53. If a1, a2, a3, ........, an are in H.P. and 'd' be the common difference of the corresponding A.P., then the expression a1a2 + a2a3 + ....... + an-1an is equal to
     A)                  B) (n - 1)(a1 - an)
     C) n (a1 - an)              D) (n - 1) a1an
Ans: A, D

 

54. The sum of infinite terms of a decreasing G.P. is equal to the greatest value of the function
f(x) = x3 + 3x - 9 in the interval [-2, 3] and the difference between thefirst two terms is f '(0). The common ratio of the G.P. is
     A)             B) 
     C)            D) 
Ans: B

 

55. If p, q, r are positive and are in A.P., the roots of the quadratic equation px2 + qx + r = 0 are all real for
      A)        B) 
      C)          D) 
Ans: A, B, C, D

 

     A) x2 + 2x + 15 = 0       B) x2 + 2x - 15 = 0
    C) x2 - 6x - 8 = 0          D) x2 - 9x + 20 = 0
Ans: B

 

57. Let a and b be two positive real numbers. Suppose A1, A2 are two arithmetic means; G1, G2 are two geometric means and H1, H2 are two harmonic means between a and b, then
        A)      B) 
        C)            D) 
Ans: A, B, C

 

58. Given that x + y + z = 15 where a, x, y, z, b are in A.P. and  when a, x, y, z, b are in H.P., then
       A) G.M. of a and b is 3                       B) One possible value of a + 2b is 11
       C) A.M. of a and b is 6                       D) Greatest value of a - b is 8
Ans: A, B, D

 

59. Let Sn = (1)(5) + 2(52) + (3)(53) + ....... + (n) (5n) =   [(4n - 1) 5a + b], then
      A) a = n + 1          B) a = n        C) b = 5         D) b = 25
Ans: A, C

 


     A) x, y and z are in H.P.          B)  are in A.P. 
     C) x, y, z are in G.P.                 D) , are in G.P.
Ans: A, B

 

61. Let S1, S2, ....... be squares such that for each n 1, the length of a side of Sn equals to the length of a diagonal of Sn+1. If the length of a side of S1 is 10 cm, then for which of the following value(s) of 'n' is the area of Sn less than 1 sq. cm?
       A) 7           B) 8            C) 9          D) 10
Ans:  B, C, D

 


       A) a + c = b + d                          B) e = 0
       C)       D) c/a is an integer
Ans: A, B, C, D

 

63. If roots of x3 + bx2 + cx + d = 0 are
       A) in A.P., then 2b3 - 9bc + 27d = 0          B) in G.P., then b3d = c3
       C) in G.P., then 27d3 = 9bcd2 - 4c3d         D) equal, then c3 = b3 + 3bc
Ans: A, B

 

 then
       A)      B)      

C)              D) 
Ans: B, C

 

65. The three sides of a right-angled triangle are in G.P. The tangents of the two acute angles may be
    A)       B) 
    C)          D) 
Ans: B, D

 

Comprehension Passage Type
Passage - 1

Let ABCD is a unit square and 0 < a < 1. Each side of the square is divided in the ratio a : 1 - a, as shown in figure. These points are connected to obtain another square. The sides of new square are divided in the ratio a : 1 - a and points are joined to obtain another square. The process is continued indefinitely. Let an denote the length of side and An the area of the nth square.                

 

Ans: 
 

67. The value of 'a' for which side of nth square equals the diagonals of (n + 1)th square is
Ans: 

 

Ans: 
 

Passage - 2 
Consider the sequence in the form of groups (1), (2, 2), (3, 3, 3,), (4, 4, 4, 4), (5, 5, 5, 5, 5), ...

 

69. The 2000th term of the sequence is
Ans: 63

 

70. The sum of first 2000 terms is
Ans: 84336

 

71. The sum of the remaining terms in the group after 2000th term in which 2000th term lies is
Ans: 1008

 

Matrix Matching Type
72.
Match the conditions for the equation ax3 + bx2 + cx + d = 0 having roots in


Ans: A-P; B-Q; C-R; D-S

 

73.

Ans: A-S; B-P; C-Q; D-R.
 

Integer Type

Ans: 4

 

Ans: 9

 

Ans: 9

 

Ans: 2

 

Ans: 2
 

79. If 9 A.M.s and again 9 H.M.s are inserted between 2 and 3 and if A is any A.M. and H, the corresponding H.M., then 
Ans: 5

 

Ans:

 


 Ans: 1

 

Ans: 7

 

83. If a, b, c are positive and [(1 + a)(1 + b)(1 + c)]7 > 7ka4b4c4, then k is
Ans: 7

 

Ans: 7

Posted Date : 19-02-2021

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