PERMUTATION AND COMBINATION
1) The value of 2ⁿ[1.3.5....(2n -3)(2n -1)] is
a) (2n)!/n! b) (2n)!/2ⁿ c) n!/(2n)! d) none
2) The value of r for which ¹⁵Cᵣ₊₃ = ¹⁵C₂ᵣ₋₆ is:
a) 2 b) 4 c) 6 d) -9
3) If ⁿCᵣ = 84, ⁿCᵣ₋₁ = 36 and ⁿCᵣ₊₁ = 126, then n equals
a) 8 b) 9 c) 10 d) 5
4) If ⁿCᵣ₋₁ = 36, ⁿCᵣ = 84 and ⁿCᵣ₊₁ = 126, then r equal
a) 1 b) 2 c) 3 d) none
5) If ⁿ⁻¹C₆ + ⁿ⁻¹C₇ > ⁿC₆, then
a) n>4 b) n>12 c) n≥13 d) n>13
6) The value of the expression: ⁴⁷C₄ + ⁵ⱼ₌₁∑⁵²⁻ʲC₃ is equal to
a) ⁴⁷C₅ b) ⁵²C₅ c) ⁵²C₄ d) ⁵²C₃
7) If aₙ = ⁿᵣ₌₀∑ 1/ⁿCᵣ, then ⁿᵣ₌₀∑ r/ⁿCᵣ equal
a) (n -1)aₙ b) naₙ c) (1/2) naₙ d) none
8) The smallest value of x satisfying the inequality ¹⁰Cₓ₋₁ > 2. ¹⁰Cₓ is
a) 7 b) 10 c) 9 d) 8
9) If ²ⁿC₃ : ⁿC₂ = 44:3, then for which of the following values of r, the value of ⁿCᵣ will be 15?
a) 3 b) 4 c) 6 d) 5
10) If x, y and r are positive integers, then
ˣCᵣ + ˣCᵣ₋₁ ʸC₁ + ˣCᵣ₋₂ ʸC₂ + .....+ ʸCᵣ =
a) x!y!/r! b) (x + y)!/r! c) ˣ⁺ʸCᵣ d) ˣʸCᵣ
11) For 2≤ r ≤ n(n/r)+ 2{n/(r -1)} + {n/(r -2)}=
a) (n +1)/(r -1) b) 2 (n +1)/(r -2) c) 2(n +2)/(r) d) (n +2)/(r)
12) There are 4 letters and 4 directed envelopes. The number of ways in which every letter is put into a wrong envelope is:
a) 8 b) 16 c) 15 d) 9
13) 66 games were played in a tournament where each player is placed against the rest. The number of players is
a) 33 b) 12 c) 13 d) 11
14) There are 10 true-false questions. The number of ways, in which they can be answered is:
a) 10! b) 10 c) 2¹⁰ d) 10²
15) A lady gives dinner party to 6 guests . The number of ways in which they may be selected from among ten friends if two of the friends, will not attend the party together is :
a) 112 b) 140 c) 164 d) none
16) The number of arrangements which can be made using all the letters of the word LAUGH if the vowels are adjascent, is:
a) 10 b) 24 c) 120 d) 48
17) All the letters of the word AGAIN are permitted in all possible ways and the words so formed (with or without meaning) are written as in dictionary, than the 50th word is:
a) NAAGI b) NAAIG c) IAANG d) INAGA
18) How many words can be formed by taking four different letters of the word MATHEMATICS ?
a) 756 b) 1680 c) 2454 d) 18
19) The number of different arrangements, which can be made from the letters of the word SERIES, taken all together is :
a) 6!/(2!2!) b) 6!/4! c) 6! d) none
20) The number of straight lines that can be drawn out of 10 points of which 7 are collinear is:
a) 22 b) 23 c) 24 d) 25
21) There are 15 points in a plane, no 3 of which are in a straight line, except 6, all of which are in a straight line. The number of straight lines , which can be drawn by joining them, is:
a) ¹⁵C₂ - 6 b) ¹⁵C₂ - ⁶C₂ c) ¹⁵C₂ - ⁶C₂ -1 d) ¹⁵C₂ - ⁶C₂ +1
22) There are n(>2) points in each of two parallel lines. Every point on one line is joined to every point on the other line by a line segment drawn within the lines. The number of points (between the lines) in which these segments intersect is:
a) ²ⁿC₂ - 2. ⁿC₂ +2 b) ²ⁿC₂ - 2. ⁿC₂ c) ⁿC₂ x ⁿC₂ d) none
23) There are 16 points in a plane, no 3 of which are in a straight line expect 8 which are all in straight line. The number triangles that can be formed by joining them equals:
a) 504 b) 552 c) 560 d) 1120
24) Number of ways in which 6 persons can be seated around a circular table so that 2 particular persons are never seated together, is equal to :
a) 480 b) 72 c) 120 d) 240
25) A polygon has 44 diagonals. The number of its side is
a) 11 b) 10 c) 22 d) 13
26) Number of diagonals in a polygon of m sides is:
a) (1/2)! m(m - 5) b) (m -2)2! c) m(m -3)/2! d) m(m -2)/2!
27) Number of divisors of the form 4n +2(n ≥ 0) of the integer 240 is:
a) 4 b) 8 c) 10 d) 3
28) The number of divisors of the number 56 excluding 1 and 56 is:
a) 8 b) 7 c) 6 d) none
29) Sum of proper divisors of 72(1 and 72 excluded) is equal to
a) 195 b) 122 c) 194 d) none
30) The exponent of 3 in (100)! is:
a) 33 b) 44 c) 48 d) 52
31) The sum of the digits in the unit place of all numbers formed with the help of 3,4,5,6, taken all at a time, is
a) 18 b) 108 c) 432 d) 144
32) In an examination there are three multiple choice questions and each question has 4 choices. Number of sequences in which a student can fail to get all answers correct is :
a) 11 b) 15 c) 80 d) 63
33) The number of 7-digit numbers which can be formed using the digits 1,2,3,2,3,3,4 is:
a) 420 b) 840 c) 2520 d) 5040
34) The number of different four digit numbers that can be formed with the digits 2,3,4,5,7 using each digit only once, is
a) 4! b) 4(4!) c) 5! d) 5(7!)
35) Six identical coins are arranged in a row. The number of ways in which the number of tails is equal to the number of heads is:
a) 20 b) 120 c) 9 d) 40
36) A five digits number divisible by 3 is to be formed using the numerals 0,1,2,3,4 and 5 without repetition. The total number of ways in which this can be done is:
a) 216 b) 600 c) 240 d) 3125
37) An n-digit number is a positive number with exactly n digits. Nine hundred distinct n-digit numbers are to be formed using only the three digits 2,5 and 7. The smallest value of n for which this is possible is
a) 6 b) 7 c) 8 d) 9
38) Number of triangles formed by joining 12 points, 7 of which are in the same straight line, is:
a) 185 b) 220 c) 92 d) none
39) The number of ways in which 10 persons can go in two boats, so that there may be 5 on each boat, supposing that two particular persons will not go in the same boat, is
a) (1/2) ¹⁰C₅ b) (⁸C₅)/2 c) 2 x ⁸C₄ d) ⁸C₄
40) Every body in a room shakes hand with every body else. The total number of hand shake ps is 66. The total number of persons in the room is:
a) 11 b) 12 c) 13 d) 14
41) How many different nine-digits numbers can be formed from the number 223355888 by rearranging its digits so that the odd digits occupy even positions?
a) 16 b) 36 c) 60 d) 180
42) A parallelogram is cut by two sets of m parallel lines to the sides. The number of parallelograms thus formed is:
a) m²/4 b) (m +1)²/4 c) (m + 2)²/4 d) {(m +1)²(m+2)²}/4
43) Number of different arrangements which can be made out of the letters in the expansion of A²B³C⁴, when written in full, is
a) 9/(2!3!4!) b) 2!3!4!(2!3!4!) c) 2!3! - 4 d) 9!(2!+3!+4!)
44) The number of ways of arranging the letter AAAAABBBCCCDEEF in a row when no two C's are together is:
a) 15!/(5!3!3!2!) - 3! b) 15!/(5!3!3!2!) - 13!/(5!3!2!) c) 12!/(5!3!2!) x ¹³P₃/3! d) 12!/(5!3!2!) x ¹³P₃.
45) The maximum number of points of intersection of 8 circles is :
a) 16 b) 24 c) 28 d) 56
46) The maximum number of points into which 4 circles and 4 straight lines intersect is:
a) 26 b) 56 c) 50 d) 72
47) A box contains two white balls, three black balla and four red balls. The number of ways in which three balls can be drawn from the box so that atleast one of balls is black balls is:
a) 74 b) 84 c) 64 d) 20
48) In a cricket championship there are 36 matches. The number of teams if each plays one match with other, is:
a) 8 b) 9 c) 10 d) none of these
49) Number of five digit telephone numbers having atleast one of their digits repeated, is:
a) 90000 b) 100000 c) 30240 d) 69760
50) 12 persons are to be arranged to a round table. If two particular persons among them are not to be side by side, the total number of arrangements is:
a) 9(10!) b) 2(10!) c) 45(8!) d) 10!
51) P,Q,R,S are to give lectures to an audience. The organiser can arrange the order of their presentation in :
a) 4 ways b) 12 ways c) 256 ways d) 24 ways
52) The number of ways in which 6 boys and 6 girls can sit alternatively is:
a) 518400 b) 1036800 c) 508400 d) none
53) There are n books having p copies of each. The number of ways in which a selection can be made from them is:
a) nᵖ b) pⁿ c) (p+1)ⁿ -1 d) (n + 1)ᵖ -1
53) If x is an integer between 0 and 21, then the minimum value of x! (21- x)! is ?
a) 9!12! b) 10!11! c) 20! d) 21!
54) On the occasion of Deewali, each student of a class sends greeting cards to each others. If there are 20 students in the class, then the total number of greeting cards exchanged by the students is:
a) ²⁰C₂ b) 2. ²⁰C₂ c) 2. ²⁰P₂ d) none of these
55) Ten different letters of an alphabet are given. Words with five letters are formed from these given letters. Then the number of words which have atleast one letter repeated is:
a) 69760 b) 30240 c) 99748 d) none of these
56) The number of ways in which 8 different flowers can be strung to form a garland so that 4 particular flowers are never separated is:
a) 4!4! B) 8!/4! c) 288 d) none of these
57) The total number of different combinations of letters which can be made from the letters of the word MISSISSIPPI is
a) 150 b) 148 c) 149 d) none of these
58) The number of ways in which ten candidates C₁, C₂, ...., C₁₀ can be ranked such that C₁ is always above C₁₀ is
a) 5! b) 2(5!) c) 10! d) (10!)/2
59) m parallel lines in a plane are intersected by a family of n parallel lines. The total number of parallelograms so formed is
a) (m -1)(n -1)/4 b) mn/4 c) m(m-1)(n-1)n/2 d) m(m-1)(n-1)n/4
60) A student is allowed to select atmost n books from a collection of (2n +1) books. If the total number of ways in which he can select the books is 63, then n is
a) 6 b) 3 c) 4 d) none of these
61) A father with 8 children takes 3 at a time to a Zoological Garden as often as he can without taking same three children together more than once. The number of times:
i) Each child will go to the garden is
a) 56 b) 21 c) 112 d) none of these
ii) He will go to the garden is:
a) 336 b) 112 c) 56 d) none of these
63) The sides AB, BC, CA of a triangle ABC have 3,4,5 interior points respectively on them. The total number of triangles that can be constructed by using these points as vertices is:
a) 220 b) 204 c) 205 d) 195
64) There are 10 lamps in a hall. Each one of them can be switched on independently. The number of ways in which the hall can be illuminated is
a) 10² b) 1023 c) 2¹⁰ d) 10!
65) Seven women and seven men are to sit round a circular table such that there is a man on either side of every woman, the number of seating arrangements is
a) (7!)² b) (6!)² c) 6!7! d) 7!
66) How many numbers between 5000 and 10000 can be formed using the digits 1,2,3,4,5,6,7,8,9, each digit appearing not more than once in each number?
a) 5. ⁸P₃ b) 5x ⁸P₈ c) 5!x ⁸P₃ d) 5! x ⁸C₃
67) The number of different words ending and beginning with a constant which can be made out of the letters of the word EQUATION is
a) 5200 b) 4320 c) 1295 d) 3000
68) Let A be a set containing 10 distinct elements. Then the total number of distinct functions from A to A is
a) 10! b) 10¹⁰ c) 2¹⁰ d) 2¹⁰ -1
69) How many 10 digit numbers can be written by using the digits 1 and 2 ?
a) ¹⁰C₁ + ⁹C₂ b) 2¹⁰ c) ¹⁰C₂ d) 10!
70) From 4 officers and 8 jawans, a committee of 6 is to be chosen to include exactly one officer. The number of such comittee is
a) 160 b) 200 c) 224 d) 300
71) Six X's have to be placed in the squares of the figure given below such that each row contains atleast one X.
The number of ways in which this can be done is
a) 26 b) 27 c) 22 d) none of these
72) If the letters of the word RACHIT are arranged in all possible ways and these words are written out as in a dictionary, then the rank of the word RACHIT is
a) 365 b) 481 c) 702 d) none of these
73) Let A={x: x is a prime number and x < 30}. The number of different rational numbers whose numerators and denominator belong to A is
a) 90 b) 180 c) 91 d) none of these
74) Let A be the set of 4-digit numbers abcd, where a> b > c > d, then n(A) equal
a) 126 b) 84 c) 210 d) none of these
75) The number of words that can be made by writing down the letters of the word CALCULATE such that each word starts and ends with a constant is:
a) 5(7!)/2 b) 3(7!)/2 c) 2(7!) d) none of these
76) ABCD is a convex quadrilateral.3,4,5 and 6 points are marked on the sides AB, BC, CD, DA respectively. The number of triangles with vertices on different sides is
a) 270 b) 220 c) 282 d) none of these
77) There are three coplanar lines. If any p points are taken on each of the lines, the maximum number of triangle with vertices at these points is:
a) 3p²(p -1) +1 b) 3p²(p -1) c) p²(4p -1) d) none of these
78) Four couples (husband and wife) decide to form a committee of four members. The number of different committees that can be formed in which no couple finds a place is:
a) 10 b) 12 c) 14 d) 16
79) Number of odd proper divisors of 3ᵖ.6ᵐ.21ⁿ is
a) (p+1)(m+1)(n+1) -2 b) (p+m+ n+1)(n+1)- 1 c) (p+1)(m+1)(n+1)- 1 d) none
80) Sum of divisors 2⁵.3⁷.5³.7² is
a) 2⁶.3⁸.5⁴.7⁸ b) 2⁶.3⁸.5⁴.7³ - 2.3.5.7 c) 2⁶.3⁸.5⁴.7³ -1 d) none
81) Eight chairs are numbered 1 to 8. Two women and three men wish to occupy one chair each. First the women choose the chairs from amongst the chairs 1 to 4 and then the men select from the remaining chairs. The number of possible arrangements is
a) ⁶C₃ x ⁴C₂ b) ⁴C₂x⁴P₃ c) ⁴P₂x ⁶P₃ d) none of these
82) There are four balls of different colours and four boxes of colours same as those of the balls. The number of ways in which the balls, one in each box, could be placed such that a ball does not go to box of its own colour, is
a) 8 b) 7 c) 9 d) none of these
83) The number of ways in which the letters of the word ARRANGE can be arranged such that both R do not come together is
a) 360 b) 900 c) 1260 d) 1620
84) In a circus there are 10 cages for accomodating 10 animals. Out of these 4 cages are so small that 5 out of 10 animals cannot enter into them. In how many ways will it be possible to accommodate 10 animals in these cages ?
a) 66400 b) 86400 c) 96400 d) none of these
85) How many words can be made out from the letters of the word INDEPENDENCE in which vowels always come together?
a) 16800 b) 16630 c) 1663200 d) none of these
86) In a certain test, aᵢ students gave wrong answers to atleast i questions, where i= 1,2, 3,....k,. No student gave more than k wrong answers. The total number of wrong answers, is:
a) a₁ + 2a₂ + 3a₃ +....+kaₖ b) a₁+ a₂+ a₃+.....+aₖ c) 0 d) none of these
87) A committee of 12 persons is to be formed from 9 women and 8 men in which atleast 5 women have to be included in a committee. Then the number of committee in which the women are in majority and men are in majority are respectively:
a) 4784,1008 b) 2702,3360 c) 6062,2702 d) 2702,1008
88) If the letters of the word KRISNA are arranged in all possible ways and these words are written out as in a dictionary, then the rank of the word KRISNA is
a) 324 b) 341 c) 359 d) none of these
89) In a city no two persons have identical set of teeth and there is no person without a tooth. Also no person has more than 32 teeth. If we disregard the shape and size of tooth and consider only the positioning of the teeth, then the maximum population of the city is
a) 2³² b) (32)² -1 c) 2³² -1 d) 2³² -1
90) The number of ways in which four letters of the word MATHEMATICS can be arranged is given by:
a) 136 b) 192 c) 1680 d) 2454
91) In a plane there are 37 straight lines of which 13 pass through the point P and 11 pass through the point Q. Besides, no three lines passing through one point, no line passes through both points P and Q and no two are parallel. Then the number of intersection points the lines have equals
a) 535 b) 601 c) 728 d) none
92) Number of positive integeral solutions of: x₁x₂x₃ = 30 is:
a) 25 b) 26 c) 27 d) 28
93) Number of non-negative integral solutions of a+ b+ c + d, n∈ N is
a) ⁿ⁺³P₂ b) (n+1)(n+2)(n+3)/6 c) ⁿ⁻¹Cₙ₋₄ d) none
94) If x, y, z are integers and x≥ 0, y≥ 1, z≥ 2, x+ y+ z = 15, then the number of values of the ordered triplet (x,y,z) is
a) 91 b) 455 c) ¹⁷C₁₅ d) none
95) Sum of non-negative integral solutions of: x₁ + x₂+...+xₙ≤ x (where n is +ve integer) is
a) ⁿ⁺³C₃ b) ⁿ⁺⁴C₄ c) ⁿ⁺⁵C₅ d) ⁿ⁺⁴C₆
96) If a denotes the number of permutations of (x +2) things, taken all at a time, b is the number of permutations of x things, taken 11 at a time and c the number of permutations of (x -11) things, taken all at a time such that a= 182bc, then the value of x is
a) 15 b) 12 c) 10 d) 18
97) A person goes in for an examination in which there are four papers with a maximum of m marks from each paper. The number of ways in which one can get 2m marks is:
a) ²ᵐ⁺³C₃ b) (m+1)(2m²+ 4m+1)/3 c) (m+1)(2m² + 4m +3)/3 d) none
98) There are n straight lines in a plane, no two of which are parallel, and no three pass through the same point. Their points of intersection are joined. Then the number of fresh lines thus obtained is
a) n(n-1)(n-2)/8 b) n(n-1)(n-2)(n -3)/6 c) n(n-1)(n-2)(n -3)/8 d) none
99) The straight lines I₁, I₂, I₃ are parallel and lie in the same plane. A total number of m points are taken on I₁; n Points on I₂; k points on I₃. The maximum number of triangles formed with vertices at these points is
a) ᵐ⁺ⁿ⁺ᵏC₃ b) ᵐ⁺ⁿ⁺ᵏC₃ - ᵐC₃ - ⁿ⁺C₃ - ᵏC₃ c) ᵐC₃ + ⁿC₃ + ᵏC₃ d) none
100) A is set containing n elements. A subset P of A is chosen. The set A is reconstructed by replacing the elements of P. A subset Q of A is again chosen. The number of ways of choosing P and Q so that P ∩Q = φ is
a) 2²ⁿ - ²ⁿCₙ b) 2ⁿ c) 2ⁿ -1 d) 3ⁿ.
101) the number of ways of selecting 10 balls out of an unlimited number of white, red, blue and green balls is:
a) 270 b) 84 c) 286 d) 86
102) Seven different lectures are to deliver lectures in seven periods of a class on a particular day. X, Y and Z are three of the lectures. The number of ways in which a routine for the day can be made such that X delivers his lecture before Y and Y before Z, is
a) 420 b) 120 c) 210 d) none
103) In the next World Cup there will be 12 teams, divided equally in two groups. Teams of each group will play a match against each other. From each group, 3 top teams will qualify for the next round. In this round, each team will play against others once. Four top teams of this round will qualify for the semi-final round, where each team will play against the others once. Two top teams of this round will go to the final round, where they will play the best of three matches. The minimum number of matches in the next World Cup will be:
a) 54 b) 53 c) 38 d) none
104) A shop keeper sells three verities of perfumes and he has a large number of bottles of the same size of each variety in his stock. There are 5 places in a row in his showcase. The number of different ways of displaying the three varieties of perfumes in the showcase is:
a) 6 b) 50 c) 150 d) none
105) How many different nine-digit numbers can be formed from the number 223355888 by rearranging its digits so that the odd digits occupy even positions?
a) 16 b) 36 c) 60 d) 180
106) Two straight lines intersect at a point O. Points P₁, P₂, .......Pₙ are taken on one line and points Q₁, Q₂,.....Qₙ on the other. If the point O is not to be used, the number of triangles that can be drawn using these points as vertices is:
a) n(n -1) b) n(n -1)² c) n²(n -1) d) n²(n -1)².
107) The number of positive integers satisfying the inequality ⁿ⁺¹Cₙ₋₂ - ⁿ⁺¹Cₙ₋₁ ≤ 100 is:
a) five b) six c) eight d) nine
108) On a railway there are 20 stations. The number of different tickets required in order that it may be possible to travel from every station to every station is:
a) 195 b) 105 c) 210 d) 225
10ô) If the (n +1) numbers A, B, C, D,..... will be all different and each of them is a prime numbers, then the number of different factors (other than 1) of Aᵐ.B. C. D.....is:
a) m - 2ⁿ b) (m +1)2ⁿ c) (m +1)2ⁿ -1 d) none
110) The sum of all 4 digit numbers that can be formed by using the digits 2,4,6,8 (repetition of digits not allowed) is:
a) 133320 b) 533280 c) 53328 d) none
111) Passengers are used to travel by a double decker bus which can accommodate 13 in upper deck and 7 in the lower deck. The number of ways that they can be distributed if 5 refuse to sit in the upper deck and 8 refuse to sit in the lower deck is:
a) 25 b) 21 c) 18 d) 15
112) There are p, q, r points on three parallel lines L₁, L₂, and L₃ respectively, all of which lie in one plane. The number of triangles which can be formed with vertices at these points is:
a) ᵖ⁺ᑫ⁺ʳC₃ b) ᵖ⁺ᑫ⁺ʳC₃ - ᵖC₃ - ᑫC₃ - ʳC₃ c) ᵖC₃ + ᑫC₃ + ʳC₃ d) none
113) Let p be a prime number such that p≥ 111. Let n= p! + 1. The number of points in the list:
n+1, n+2, n+3,......n+ p -1 is
a) p -1 b) 2 c) 1 d) none
114) The greatest common divisors of: ³¹C₁₆, ³¹C₁₇, ......³¹C₃₀ is:
a) ³¹C₁₆ b) 31 c) 465 d) none
115) Mr. P has m children from his first wife and Mrs. Q has m+1 children by her first husband. They marry and have children of their own. The whole family has 10 children. Assuming that two children of the same parents do not fight, the maximum number of fight ps that can take place among children is:
a) 33 b) 35 c) 38 d) none
116) For x∈ R, let [x] denote the largest integer less than or equal to x, then the value of
E= [1/3] + [1/3 + 1/100] + [1/3 + 2/100] + .....+ [1/3 + 99/100] is
a) 30 b) 31 c) 32 d) 33
117) Five distinct letter ps are to be transmitted through a communication channel. A total number of 15 blanks is to be inserted between the two letters with atleast three between every two. The number of ways in which this can be done is
a) 1200 b) 1800 c) 2400 d) 3000
118) All the letters of the word EAMCET are arranged in all possible ways. The number of such arrangements in which two vowels are adjacent to each other is:
a) 360 b) 144 c) 72 d) 54
119) The different letters of an alphabet are given. Words with five letters are formed from these given letters. The number of words which have atleast one letter repeated is:
a) 69760 b) 30240 c) 99748 d) 43929
120) The number of ways in which four particular persons A, B, C and D and six more persons can stand in a queue so that A always stands before B, B before C and C before D, is
a) 7!4! b) 10! - 7! 4! c) 10!/4! d) none
121) The total number of arrangements of the letters in the expansion a³b²c⁴ when written at full length is
a) 1260 b) 2520 c) 610 d) none
122) The number of distinct rational numbers n such that 0< n < 1 and x= p/q, where p, q belongs to {1,2,3,4,5,6} is:
a) 15 b) 13 c) 12 d) 11
123) The number of integers that can be formed with 10 points as vertices n of them being collinear, is 110. Then n is:
a) 3 b) 4 c) 5 d) 6
124) The number of proper divisors of 1800. Which are also divisible by 10, is
a) 18 b) 34 c) 27 d) none
125) In a test there were n questions. In test 2ⁿ⁻ᶦ students gave wrong answers to i questions, where i= 1,2,3,.....n. if the total number of wrong answers given us 2047, then n is
a) 12 b) 11 c) 10 d) none
126) If a,b,c are three rational numbers in AP and a+ npb + c = 21, then the possible number of values of the ordered triplet (a,b,c) is
a) 15 b) 14 c) 13 d) none
127) Let aₙ = 10ⁿ/n! for n≥ 1. Then aₙ takes the greatest value when
a) n= 9 b) n= 8 c) n= 11 d) n= 12
128) If n< p < 2n and p is prime arN= ²ⁿCₙ, the pn:
a) p | N b) p is not N c) p² | N d) none
129) Let Tₙ denotes the number of triangles which can be formed using the vertices of a regular polygon of n sides. If Tₙ₊₁ - Tₙ = 21, then n equals
a) 5 b) 7 c) 6 d) 4
130) The number of arrangements of the letters of the word BANANA in which the two N's do not appear adjascently is
a) 40 b) 60 c) 80 d) 100
131) The number of ways in which 6 men and 5 women can dine at a round table if no two women are to sit together is given by:
a) 30 b) 5! x 4! c) 7! x 5! d) 6! x 5!
132) A student is to answer 10 out of 13 questions in an examination such that he must choose atleast 4 from the first five questions. The number of choices available to him is:
a) 196 b) 280 c) 346 d) 140
133) How many ways are there to arrange the letters in the word GARDEN with the vowels in alphabetical order?
a) 120 b) 240 c) 360 d) 480
134) If ⁿ⁻¹Cᵣ = (r¹ -3)ⁿCᵣ₊₁, then k belongs
a) (-∞,-2) b) (2,∞) c) (-√3,√3) d) (√3,2)
1a 2c 3b 4c 5d 6c 7c 8d 9b 10c 11d 12b 13b 14c 15b 16d 17b 18c 19a 20d 21d 22c 23a 24b 25a 26c 27d 28c 29b 30c 31b 32d 33a 34c 35a 36a 37b 38a 39c 40b 41c 42d 43a 44c 45d 46c 47c 48b 49d 50a 51d 52b 53c 54b 55b 56a 57a 58c 59d 60d 61b 62b, 62c 63c 64b 65c 66a 67b 68b 69b 70c 71a 72b 73c 74c 75a 76d 77c 78d 79b 80d 81c 82c 83b 84b 85a 86b 86d 88a 89c 90d 91a 92c 93b 94a 95b 96b 97c 97c 99b 100d 101c 102d 103b 104c 105c 106c 107c 108c 109c 110a 111b 112b 113d 114b 115a 116d 117c 118c 119a 120c 121a 122d 123c 124a 125b 126c 127a 128a 129b 130a 131d 132a 133a 134d
PROGRESSIONS
1) If for three real numbers a, b, c the number (b + c)/a, (c + a)/b, (a+ b)/c are in AP and a+ b + c ≠ 0, then a, b, c are in
a) AP b) HP c) GP d) none
2) If b(≠0) is the AM between a and c and ab + cd + ad = 3bc, then, between b and d, the number c is
a) the AM b) a GM c) the HM d) none
3) (aⁿ⁺¹ + bⁿ⁺¹)/(aⁿ + bⁿ) is the AM between two positive unequal real numbers a and b if n=
a) 1 b) 0 c) 2 d) none of these
4) If a₁ = 2 and aₙ - aₙ₋₁ = 2n (n≥2), then aₙ=
a) n(n+1)/2 b) 2(n +1) c) n(n +1) d) none of these
5) For any positive integer n, (n +1)²+ (n +2)²+ ....+ (2n)²=
a) (2n(2n +1)+4n +1))/6 b) {2n(2n +1)/2}² c) (n/6) (2n+1)(7n +1) d) none
6) The ratio of the sums of p AM's and q AM's between two numbers is
a) q: p b) (p+2): (q+2) c) (p+1): (q+ 1) d) p:q
7) If for some nonzero integer r, 7r is the GM between+3r +1) and (19r +8), then r=
a) 1 b) -1 c) 2 d) none
8) If 1/(a+ b), 1/2b, 1/+b + c) are in AP, then a, b, c are in
a) GP b) AP c) HP d) none
9) The ratio of HM to AM of two positive numbers is 24: 25. Then the ratio of the numbers is
a) 4:1 b) 3:4 c) 4:3 d) 2:3
10) In a GP of alternatively positive and negative terms, each term is the AM of the next two terms. Then the common ratio of the GP is
a) 2 b) -2 c) 3 d) -3
11) In a GP of alternatively positive and negative terms, 9 times each term is the HM of the next two terms. Then the common ratio of the GP is
a) 2/3 b) -3 c) (9+3√17)/4 d) none
12) (aⁿ⁻² + bⁿ⁻²)/(aⁿ⁻³ + bⁿ⁻³) is the HM between two unequal real numbers a and b if n=
a) 0 b) 3 c) 2 d) none
13) The AM, GM, HM between two unequal positive real numbers are in
a) GP b) AP c) HP d) none
14) Let p be the nth of (2n -1) AM's and q be the nth of (2n -1) GM 's between two unequal positive real numbers a and b. If r be the HM between a and b, then the roots of px²+ qx + r = 0 are
a) real and equal b) real and unequal c) imaginary d) none
15) If a₁, a₂, a₃,....a₂ₙ are 2n AM's, b₁, b₂, ....b₂ₙ are 2n GM 's and c be the HM between two unequal positive real numbers, then ⁿᵢ₌₁∑ (aᵢ + a₂ₙ₋ᵢ₊₁)/(bᵢ b₂ₙ₋ᵢ₊₁)=
a) 2nc b) 2n/c c) c/n d) nc
16) The AM of two unequal positive whole numbers is 2. If the greater number is increased by 1, their positive GM becomes the AM of the given numbers. Then the HM of these two numbers is
a) 2/3 b) 3/2 c) 1/2 d) none
17) For two unequal real numbers a and b of the same sign, a, p, b are in AP, a, q, b are in GP and a, r, b are in HP. If r= 25p, then
a) 2q= p+ r b) q= 13p c) r= 5q d) |q|= 5|p|
18) If a,b,c,d be four unequal positive real numbers such that the first three are in HP and the last three are in AP, then
a) the GM of first and fourth= the GM of second and third
b) the GM of first and third = the GM of second and fourth
c) the GM of first and second = the GM of third and fourth d) none
19) If a, b, c are in GP (a> 1, b >1, c> 1), then for any real number x (with x> 0, x ≠1), theyligₐx, logᵥx, log꜀x are in
a) GP b) AP c) HP d) none
20) Three unequal positive real numbers a, b, c are such that a, b, c are in GP while log(5c/2a), log(7b/5c), log(2a/7b) are in AP. Then a,b,c are the lengths of the sides of
a) an isosceles triangle b) an equilateral triangle c) a scalene triangle d) none
21) {tₖ} is a sequence of positive integers such that t₁ = 1 and tₖ (k > 1)= 3ⁿ⁺¹ whenever, for some non-negative integyn, 1+2+2² + 2³ + ....+ 2ⁿ< k ≤ 1+ 2² + 2³ +...+ 2ⁿ⁺¹. Then, t₁₁₅₅ =
a) 3¹¹ b) 3¹⁰ c) 3⁹ d) none
22) If three distinct numbers x, y, z are in GP and are such that 4x, 12y, 11z are in AP, then the common ratio of the GP is
a) 3 b) 1/11 c) -2 d) 2 or 2/11
23) If the first term and the third term of an AP are 7/3 and 11/5, then the largest value of r for which tᵣ > 0 is
a) 35 b) 18 c) 36 d) 34
24) If x be the GM of two unequal positive real numbers a and b, then a+ x, 2x, b+ x are in
a) AP b) HP c) GP d) none
25) Given an AP, if S₅ = 25r (for some positive integer r) and if S'₇ = 97r where S'ₙ is the sum of the first n terms of an AP each of whose term is twice the corresponding term of the given AP, then Sᵣ =
a) 35r b) 12r c) r³ d) 12r²
26) The first term and the common ratio of a GP are respectively 1 and 2. Then Sₙ, S₂ₙ - Sₙ, S₃ₙ - S₂ₙ are in
a) GP b) HP c) AP d) none
27) If b is the AM between a and c, d is the HM between c and e and e is the GM between b and d, then, between a and e, the number c is the
a) AM b) GM c) HM d) none
28) The number first term of an increasing AP is zerr. Then, among the two GM's between the fifth and the tenth of the AP.
a) none is a term of the AP
b) one is the second term of the AP
c) one is a term between the fifth and the tenth
d) one is the third term of the AP
29) The number of terms which are equal in the two progression 1,3,9,...to 100th term and 1,3,5,7,9,....to 100th terms is
a) 3 b) 5 c) 2 d) 4
30) If xᵃ = yᵇ = vᶜ , where a, b, c are in GP, then logᵥy =
a) logₓy b) logᵧx c) b/c d) c/a
31) If the HM between the numbers 1/3¹⁺²ˣ and 3²ˣ⁻¹ is 1/5, then the set of all possible values of x is
a) {x ∈ R : 4x² -1=0} b) {3,1/3} c) {9, 1/9} d) {-1/2}
32) The lengths of three edges of a rectangular parallelopiped are in GP. The volume and the total surface area of the parallelopiped are respectively 64 cubic meter and 112 square metre. The length of the smallest edge is
a) 4m b) 8m c) 3m d) 2m
33) If x be the AM and y, z be two GM's between two unequal positive real numbers, then the HM between these numbers is
a) 4x/yz b) x/yz c) yz/x d) y/zx
34) if x and y are two AM's and z and t are two GM's between two unequal positive real numbers, then 3t/(x + y) is
a) the HM between the given numbers.
b) the GM between the given numbers.
c) the AM between the given numbers
d) thrice the HM between the given numbers.
35) If a, b, c are in HP, then a/bc, 1/c, 1/b are in
a) AP b) HP c) GP d) none
36) Suppose tᵣ denotes the rth term of an AP. Let three natural numbers m,n, p be such that n- m = p - n. Then tₙ is
a) the HM between tₘ and tₚ
b) the GM between tₘ and tₚ
c) the AM between tₘ and tₚ d) none
37) If in an AP t₂₉ = log a, t₃₈ = log b and t₄₇ = log c, then a, b, c are in
a) GP b) HP c) AP d) none
38) If n!, 5(n!), (n+1)! are in GP, then 3(n!), 14(n!), (n +1)! are in
a) GP b) AP c) HP d) none
39) If a, b, c, d are four distinct positive real numbers such that
(a² + b²+ c²)x² - 2(ab+ bc+ cd)x + ( b² + c² + d²)≤ 0 for some real values of x, then
a) AM of a, d = AM of b, c
b) AM of a, d = GM of b, c
c) GM of a, d = AM of b, c
d) GM of a, d = GM of b, c
40) On R, Let f(x)= 2x +3 be a function, such that f(x), f(3x), f(2x) are distinct and are in GP. Then, x =
a) 0 b) 1 c) 2 d) none
41) For three distinct prime numbers a, b, c, the numbers √a, √b, √c are
a) in an AP
b) not in AP but three non-consecutive terms of an AP
c) not at all any three terms of an AP
d) are in GP
42) The minimum number of terms of 2,5,8,11,.... that add up to a number exceeding 1550 is
a) 32 b) 15 c) 30 d) 34
43) The sum of all divisors (> 1) of 59049 is
a) 88572 b) 88573 c) 29524 d) 29523
44) The sum of all odd factors (> 1) of 590490 is
a) 88572 x 6 b) 88573 x 6 - 1 c) 88572 x 5 d) 88573 x 5 - 1
45) In a GP of alternatively positive and negative terms, |t₇| : |t₅|= 4:1 and t₂ + t₅ = 252. Then the first term is
a) 14 b) -18 c) 12 d) none of these
46) The only common term of the two AP's 5,16,27,.... and 69,72,75,...is
a) the 8th term b) the 7th term c) the 9th term d) none
47) The nth term of the progression 1,4,12,32,80,,....is
a) n.2ⁿ b) n.2ⁿ⁻¹ c) (n +1)2ⁿ d) (n +1)2ⁿ⁻¹
48) Three consecutive terms of a progression are 15,12,10. The next term of the progression is
a) 60/7 b) 7 c) 7/60 d) none
49) If there positive real numbers a,b, c are in AP, then (2b - a)/ab, 1/b, (2b - c)/bc are in
a) HP b) GP c) AP d) none
50) Let the three unequal positive numbers a, b,c in HP. Then the numbers a- b/2, b/2, c - b/2 are in
a) GP b) HP c) AP d) none
51) The sum of the 16th and 17th terms of the sequence
1, 5/x, 1, 9/x², 2/x, 13/x³, 3/x², 17/x⁴, 4/x³,.....is
a) (33+8x)/x⁹ b) (37 +9x)/x⁸ c) (33+8x)/x⁸ d) (37+9x)/x⁹
52) The sum of the 18th and 19th terms of the sequence
1, 3/2, 1, 5/4, 1, 7/8, 3/4, 9/16, 1/12,.....is
a) 41/2⁹ b) 37/2¹⁰ c) 41/2¹⁰ d) 37/2⁹
53) The sum of the first 12 terms of a GP is 10 times the sum of the odd terms before the 12th term. Then the common ratio of the GP is
a) 10 b) 9 c) 11 d) none
54) The coefficient of x¹³ in the product
(1- 3x)+1- 3²x)(1- ₃2²x)(1- ₃2³x)....(1- ₃2¹²x) is
a) - ₃2¹³ -1 b) ₃2¹³ -1 c) -₃2¹² -1 d) none
55) The coefficient of x¹⁰ in the product
(1- 2²x)+1+ 2⁵x)(1- 2⁸x)+1+ 2¹¹x)....(1- 2²⁶x)+1+ 2²⁹x) is
a) 2¹⁵⁵ b) -2¹⁵⁵ c) -2³¹⁰ d) none
56) The coefficient of x¹⁶ in the product (3x -2)+3x -5)(3x -8)...(3x - 50) is
a) 17 x 26x 3¹⁶ b) -17x 26x 3¹⁷ c) -17 x 26x 3¹⁶ d) 16 x 26 x 3¹⁶
57) Let the first terms and the twenty first terms of an AP a GP and an HP be y. Also their eleventh terms are x, y and z respectively. Then
a) zx - y² = 0 b) x + z = 3y c) xy+ yz - 2zx = 0 d) x = y = z
58) The 7th term of the serr0.3+0.006+0.00009+ 0.0000012... is
a) 21/10⁷ b) 24/10¹² c) 24/10⁷ d) 21/10¹³
59) ᵐₚ₌₁∑ p² - ᵐₙ₌₁∑ ⁿₚ₌₁∑p=
a) ᵐₚ₌₁∑ p² + ᵐₚ₌₁∑ p
b) (1/2) +ᵐₚ₌₁∑ p² + ᵐₚ₌₁∑ p)
c) ᵐₚ₌₁∑ p² - ᵐₚ₌₁∑ p d) 0
60) 1/log₃9 + 1/log₉9 + 1/log₂₇ô + ....+ 1/log₃ⁿ 9=
a) n(n+1)(2n +1)/12 b) n(n +1)/2 c) n(n+1)/4 d) n²(n+1)²/16
1b 2a 3b 4c 5c 6d 7c 8a 9d 10b 11d 12c 13a 14c 15b 16b 17d 18a 19c 20c 21b 22d 23a 24b 25c 26a 27b 28c 29b 30b 31a 32d 33c 34d 35b 36c 37a 38b 39d 40c 41c 42a 43a 44b 45d 46c 47b 48a 49b 50a 51c 52d 53b 54a 55b 56c 57a 58d 59b 60c
Series
1) If log₃2, log₃(2ˣ -5) and lig₃(2ˣ - 7/2) are in AP, then x is equal to
a) 2 b) 3 c) 4 d) 2,3
2) If the numbers a, b, c, d, e form an AP, then the value of A- 4b + 6c - 4d + e is:
a) 1 b) 2 c) 0 d) none
3) 110 logs are stached such that 15 are in the bottom row, 14 in the next and 13 in the third row from the bottom; so on. The number of rows is:
a) 13 b) 12 c) 11 d) 18
4) The 10th common term between the series 3+7+11+ and 1+ 6+11+...is
a) 191 b) 193 c) 211 d) none
5) The number of terms common to two AP's 3,7,11,....407 and 2,9,16,.....,709 is
a) 14 b) 21 c) 28 d) none
6) Let Sₙ denotes the sum of first n terms of an AP. If S₂ₙ= 3Sₙ, then the ratio S₃ₙ/Sₙ is equal to
a) 4 b) 6 c) 8 d) 10
7) Consider an AP with first term a and common difference d. Let Sₖ denote the sum of first k terms. If Sₖₓ/Sₓ is independent of x, then
a) a= 2d b) a= d c) 2a = d d) none
8) If the pth term of an AP is q and qth term is p, then the rth term is
a) q- p+ r b) p- q+ r c) p+ q+ r d) p- q- r
9) The sum of integers from 1 to 200, which are divisible by 2 or 5, is
a) 3000 b) 3010 c) 3150 d) 3050
10) If a₁, a₂, a₃,....aₙ are in AP, aᵢ> 0 for all i, then the value of
1/(√a₁ + √a₂) + 1/(√a₂ + √a₃) + .....+ 1/(aₙ₋₁ + √aₙ) is
a) 1/(√a₁ + √aₙ)
b) 1/(√a₁ - √aₙ)
c) n/(√a₁ - √aₙ)
d) (n -1)/(√a₁ + √aₙ)
11) If aᵣ> 0, r ∈ N and a₁, a₂, a₃,....a₂ₙ are in AP., then
(a₁+ a₂ₙ)/(√a₁ + √a₂) + (a₂ + a₂ₙ₋₁)/(√a₂ + √a₃) + ....+ (aₙ + aₙ₊₁)/(√aₙ + √aₙ₊₁) is equal to
a) n-1 b) n(a₁ + a₂ₙ)/(√a₁ + √aₙ₊₁)
c) (n -1)/(√a₁ + √aₙ₊₁) d) none
12) If the roots of the equation x³ - 12x² + 39x - 28= 0 are in AP then their common difference is
a) ±1 b) ±2 c) ±3 d) ±4
13) If S₁, S₂ and S₃ denote the sum of first n₁, n₂ and n₃ terms respectively of an AP., then
(S₁/n₁) (n₂ - n₃)+ (S₂/n₂) (n₃ - n₁)+ (S₃/n₃) (n₁ - n₂) is equal to
a) 0 b) 1 c) S₁S₂S₃ d) n₁n₂n₃
14) The largest common to the sequences 1,11,21,31,.....to 100 terms and 31,36,41,46,....to 100 terms is
a) 381 b) 471 c) 281 d) none
15) The interior angles of a convex polygon are in AP, the common difference being 5°. If the smallest angle is 2π/3, then the number of sides is
a) 9 b) 16 c) 7 d) none
16) In the sequence 1,2,2,3,3,3,4,4,4,4,......, where n consecutive terms have the value n, the 150th term is
a) 17 b) 16 c) 18 d) none
17) Given two numbers a and b. Let A denote the single AM between them and S denote the sum of n AM's between them. Then S/A dependens upon.
a) n,a,b b) n,a c) n,b d) n
18) cosx= b. For what b do the roots of the equation form an AP?
a) -1 b) 1/2 c) √3/2 d) none
19) The fourth, seventh and tenth terms of a GP are p,q,r respectively, then
a) p² = q² + r² b) q² = pr c) p² = qr d) pqr+ pq+1=0
20) The third term of a GP is 4. The product of first five terms is:
a) 4³ b) 4⁵ c) 4⁴ d) none
21) Fifth term of a GP is 2, then the product of its 9 terms is:
a) 256 b) 512 c) 1024 d) none
22) If the sum of the roots of the equation ax² + bx + c=0 is equal to the sum of the squares of their reciprocals, then a/c, b/a and c/b are in
a) GP b) HP c) arithmetic - geometric progression d) AP
23) If f(x) is a polynomial function of second degree. If f(1)= f(-1) and a,b,c are in AP, then f'(a), f'(b), f'(c) are in
a) GP b) HP c) arithmetic - geometric progression d) AP
24) The sequence aₙ= sin(π/4 + nπ) for n= 1,2,3,....is:
a) 5 b) 6 c) 7 d) 8
25) if the sum of the first n positive integers is 1/5 times the sum of their squares, then n equals
a) 5 b) 6 c) 7 d) 8
26) The product (32)(32)¹⁾⁶(32)¹⁾³⁶.....∞ is
a) 16 b) 64 c) 32 d) 0
27) Sum of n terms of the series 1/2+3/4+7/8+15/16+....is equal to:
a) 2ⁿ - n -1 b) 1- 2⁻ⁿ c) n+ 2⁻ⁿ -1 d) 2ⁿ +1
28) If a,b,c are in AP and a², b², c² are in HP, then
a) a= b = c b) 2b= 3a+ c d) b² √(ac/8) d) none
29) Two AM's A₁ and A₂, two GM's G₁ and G₂ and two HM's H₁ and H₂ are inserted between any two numbers, then H₁⁻¹ + H₂⁻¹ equals
a) A₁⁻¹ + A₂⁻² b) G₁⁻¹ + G₂⁻² c) (G₁G₂)/(A₁ + A₂) d) (A₁+ A₂)/G₁G₂)
30) Let Tᵣ be the rth term of an AP for r= 1,2,3,..... If for some positive integers m,n have Tₘ= 1/n and Tₙ= 1/m, then Tₘₙ equals
a) 1/mn b) 1/m + 1/n c) 1 d) 0
31) The value of the sum ¹³ₙ₌₁∑ (iⁿ + iⁿ⁺¹), where i=√-1, equals
a) I b) i -1 c) - I d) 0
32) The harmonic mean of the roots of the equation:
(5+√2)x² - (4+√5)x + 8+ 2√5=0 is
a) 2 b) 4 c) 6 d) 8
33) Consider an infinite geometric series with first term a and common ratio r. If its sum is 4 and the second term is 3/4, then
a) a=7/4, r=3/7 b) a=2 r=3/8 c) a=3/2 ,r= 1/2 d) a=3, r=1/4
34) An AP consist of n(odd terms) and its middle term is m. Then the sum of the AP is
a) 2mn b) mn/2 c) mn d) mn²
35) If x> 1, then (1/x)ᵃ, (1/x)ᵇ, (1/x)ᶜ are in GP, then a,b,c are in
a) AP b) GP c) HP d) none
36) If a,b,c are the sides of a ∆ ABC, which are in AP then cot(C/2) equals
a) 3 tan(A/2) b) 3 tan(B/2) c) 3 cot(A/2) d) 3 cot(B/2)
37) if the sum of n positive integer is 1/5 times the sum of their squares , then n equals
a) 5 b) 6 c) 7 d) 8
38) If S denotes the tsum to infinity and Sₙ, the sum of n terms of the series 1+ 1/2+ 1/4+ 1/8+.... such that S - Sₙ< 1/1000, then the least value of n is
a) 8 b) 9 c) 10 d) 11
39) Sum to infinity of the series 1+ 4/5+7/5²+ 10/5³+....is
a) 16/35 b) 11/8 c) 35/16 d) 8/11
40) The sum of the series 1+ 2x + 3x²+ 4x³+....to ∞ when x lies between 0 and 1 i.e., 0< x < 1 is
a) 1/(1+ x) b) 1/(1- x) c) 1/(1- 2x) d) 1/(1- x)²
41) 1²+ +1+ 2²+ 2+ 3²+ 3+ ....n²+ n is equal to:
a) n(n+1)/2 b) (n(n+1)/2)² c) n(n+1)(n+2)/3 d)n(n+1)(n+2)(n+3)/4
42) The product of n positive integer is 1, then their sum is a positive integer, that is:
a) equals to 1 b) equals to n+ n² c) divisible by n d) never less than n
43) If Sₙ= nP+ n(n+1)Q/1, where Sₙ denotes the sum of the first n terms of an AP, the common difference is:
a) P+ Q b) 2P+ 3Q c) 2Q d) Q
44) If the roots of the equation:
(b- c)x² + (c- a)x + (A- b)=0 are equal, then a,b,c are in
a) AP b) GP c) HP d) none
45) If a> 0, b> 0, c> 0 are in GP, then logₐx, logᵥx, log꜀x are in
a) AP b) GP c) HP d) none
46) (a+ bx)/(a- bx) =(b+ cx)/(b - x) =(c+ dx)/(c- dx), (x≠ 0), then a, b, c, d are in
a) AP b) GP c) HP d) none
47) aˣ = bʸ = cᶻ = dᵗ and a, b, c, d are in GP then x, y, z, t are in
a) AP b) GP c) HP d) none
48) if the arithmetic and geometric means of two distinct, positive numbers are A and G respectively, then their harmonic mean is
a) A/G b) G/A c) G²/A d) A²/G
49) If H is the harmonic mean between P and Q, then H/P + H/Q is
a) 2 b) (P+ Q)/PQ c) PQ/(P+ Q) d) none
50) The AM, GM, HM of two numbers are x, y and z respectively, then which of the following is true ?
a) z< y< x b) y< x< z c) x< y< z d) z< x< y
51) If (m+1)th, (n+1)th, (r+1)th terms of an AP are in GP and m,n, r in HP, then ratio of the first term of the AP to its common difference in terms ofn is
a) n/2 b) -n/2 c) n/3 d) - n/3
52) If log(a+ c), log(c - a), log(a - 2b + c) are in AP then
a) a,b,c are in AP
b) a²,b²,c² are in AP
c) a,b,c are in GP
d) a,b,c are in HP
53) If a, b, c, d are in HP then
a) a+b> c+ d b) a+c > b+ d c) a+ d> c+ b d) none
54) If a, b, c are in AP, p, q, r are in HP and ap, bq, cr are in GP then p/r+ r/p equal
a) a/c - c/a b) a/c + c/a c) b/q + q/b d) b/q - q/b
55) If log2, log(2ˣ -1) and log(2ˣ+3) are in AP then 2, 1ˣ -1, 2ˣ+3 are in
a) AP b) HP c) GP d) none
56) Log₃2, log₆2, log₁₂2 are in
a) AP b) GP c) HP d) none
57) The consecutive numbers 1/(1+√n), 1/(1- n), 1/(1-√n) of a series are in
a) HP b) GP c) AP d) AP, GP
58) The HM of two numbers is 4 and AM. A and GM. G satisfy the relation 2A+ G²= 27, The numbers are
a) 6,3 b) 5,4 c) 5, -2.5 d) -3,11
59) If a, b, c are in HP then correct statement is:
a) a²+ c²> b² b) a²+ c²> 2b² c) a²+ c²< 2b² d) a²+ c²= 2 b²
60) If a₂a₃/a₁a₄ = (a₂+ a₃)/(a₁+ a₄)= 3{(a₂- a₃)/(a₁ - a₄)}, then a₁, a₂, a₃, a₄ are in
a) AP b) GP c) HP d) none
61) Let x, y, z be three positive prime numbers. The progression in which √x, √y, √z can be three terms (not necessary consecutive) is:
a) AP b) GP c) HP d) none
62) The sum of the products of the 10 numbers ±1, ±2, ±3, ±4, ±5, taking two at a time, is
a) 165 b) -55 c) 55 d) none
63) If n is an odd integer greater than or equal to 1, then the value of:
n³ - (n -1)³+ (n -2)³- ....+ (-1)ⁿ⁻¹.1³ is
a) (n+1)²(2n -1)/4
b) (n -1)²(2n -1)/4
c) (n+1)²(2n+1)/4 d) none
64) If A, G and H are respectively the AM, the GM and the HM between two positive numbers a and b then
a) A= G²H b) G²= AH c) A²= GH d) A= GH²
65) Let Aₙ be the nth term of a GP of positive integers. Let ¹⁰⁰ₙ₌₁∑a₂ₙ = α and ¹⁰⁰ₙ₌₁∑a= ₂ₙ₊₁= β such that α≠ β. Then the common ratio is
a) α/β b) β/α c) (α/β)¹⁾² d) (β/α)¹⁾²
66) If x= ∞ₙ₌₀∑ aⁿ, y= ∞ₙ₌₀∑ bⁿ, z=∞ₙ₌₀∑ (ab)ⁿ, where a, b < 1, then
a) XYZ= x+ y+ z b) xz+ yz= xy+ z c) xy+ yz= xz+ y d) xy+ xz= yz + x
67) If x¹⁸ = y²¹ = z²⁸, the pn 3, 3logᵧx, 3 logᵥy, 7 logₓv are in
a) AP b) GP c) HP d) none
68) The sum of the square of three distinct real numbers which are in GP is S². If their sum is kS, then:
a) 1/3≤ k²≤ 3 b) 1/3 <k²<3 c) 1 <k <3 d) 1/3 <k <1
69) The sum of all two digit odd numbers is:
a) 2475 b) 2530 c) 4905 d) 5049
70) If x= 1+ a+ a² + .....to ∞
and y= 1+ b+ b² + .....to ∞, where a and b are proper fractions, then 1+ ab+ a²b²+.....∞ equals
a) xy/(y+ x -1) b) (x+ y)/(x- y) c) (x²+ y²)/(x+ y) d) none
71) The co-efficient of x⁹⁹ in the expansion of:
(x-1)(x- 2)....(x -100) is
a) 5050 b) 5000 c) -5050 d) -5000
72) The sum of first 10 terms of the series:
(x + 1/x)²+ (x² + 1/x²)² + (x³ + 1/x³)² +.....is
a) {(x²⁰ -1)/(x²- 1)}{(x²²+1)/(x²⁰)} + 20
b) {(x¹⁸-1)/(x²-1)}{(x¹¹+1)/x⁹}+ 20
c) {(x¹⁸ -1)/(x² -1)}{(x¹¹-1)/x⁹}+ 20 d) none
73) if the sides of a right angle triangle form an AP, then the sines of the acute angles are:
a) 3/5,4/5 b) √3, 1/√3
c) [√{(√5-1)/2}, √{(√5+1)/2}]
d) [√{(√3-1)/2}, √{(√3+1)/2}]
74) Let α, β be the roots of x²- x + p= 0 and γ, δ be the roots of x² - 4x + q=0. If α, β, γ, δ are in GP, then the integral values of p and q respectively, are
a) -2,-32 b) -2,3 c) -6,3 d) -6,-32
75) If x≠ 1, y≠ 1, x≠ y, then the sum to n terms of the series:
(x+ y)+ (x²+ xy+ y²)+ (x³ + x²y + xy²+ y³)+ ...is
a) (xⁿ - yⁿ)/(x - y)
b) (xⁿ - yⁿ + x - y)/(x- y)xy
c) (xⁿ⁺¹ - yⁿ⁺¹)+x + y)/(x- y)x²y² d) none
76) The minimum sum of the series
20+ 58/3+ 56/3+.....is
a) 310 b) 300 c) 320 d) none
77) If a, b, c, d and p are distinct nonzero real numbers such that:
(a²+ b²+ c²)p² - 2(ab + bc + cd)p + (b² + c² + d²)≤ 0, then a,b,c,d:
a) are in AP b) are in GP c) are in HP d) satisfy ad= bc
78) If a, b, c are in GP, then the equation ax² + 2bx + c=0 and dx²+!²ex + f=0 have a common root if d/a, e/b, f/c are in
a) AP b) GP c) HP d) none
79) If x> 1, y > 1, z> 1 are in GP, then
1/(1+ logx), 1/(1+ logy), 1/(1+ logz) are in
a) AP b) HP c) GP d) none
80) Let a₁, a₂, ....,a₁₀ be in AP and h₁, h₂, ....,h₁₀ be in HP
If a₁= h₁= 2 and a₁₀= h₁₀= 3, then a₄h₇ is
a) 2 b) 3 c) 5 d) 6
81) The sum of
(x+2)ⁿ⁻¹+ (x+2)ⁿ⁻² (x+1)+ (x+2)ⁿ⁻³(x+1)²+......(x+1)ⁿ⁻¹ equals:
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
82) If (a,b), (c,d), (e,f) are the vertices of a triangle such that a, c, e are in GP with common ratio r and b, d, f are in GP with common ratio s, then the area of the triangle is
a) (ab/2) (r+1)(s+2)(s+ r)
b) (ab/2) (r-1)(s-1)(s- r)
c) (ab/2) (r-1)(s-1)(s+r)
d) (ab/2) (r+1)(s+2)(s- r)
83) The value of ⁿᵣ₌₁∑ lof(aʳ/bʳ⁻¹) is
a) (n/2) log(aⁿ/bⁿ)
b) (n/2) log(aⁿ⁺¹/bⁿ)
c) (n/2) log(aⁿ⁺¹/bⁿ)
d) (n/2) log(aⁿ⁺¹/bⁿ⁻¹)
84) (666....6)²+ (888...8) is equal to:
ⁿ-ᵈᶦᵍᶦᵗˢ ⁿ-ᵈᶦᵍᶦᵗˢ
a) (4/9) (10ⁿ -1)
b) (4/9) (10²ⁿ -1)
c) (4/9) (10ⁿ -1)² d) none
85) The sum of first n terms of the series
1²+ 2.2² + 3² + 2.4² + 5² + 2.6²+.... is n(n +1)²/2 when n is even. When, n is odd, the sum is
a( n(n+1)/2 b) n²(n +1)/2 c) n(n+1)²/4 d) n(n +1)/2)²
86) If a, b, c are three unequal numbers such that a,b,c are in AP and b- a, c- b, a are in GP, then a:b: c is
a) 1:2:3 b) 1:3:5 c) 2:3:4 d) 1:2:4
87) if x₁, x₂, .......xₙ are n nonzero real numbers such that (x₁² + x₂² + .....+ xₙ₋₁²)(x₂² + x₃² +....+xₙ²)≤ (x₁x₂+ x₂x₃ + .....+xₙ₋₁xₙ)², then x₁, x₂,.....xₙ are in
a) AP b) GP c) HP d) none
88) If (1.05)⁵⁰= 11.658, then ⁴⁹ₙ₌₁∑(1.05)ⁿ equals:
a) 208.34 b) 212.12 c) 212.16 d) 213.16
89) If a₁, a₂, a₃,.....are in GP, then the value of determinant
∆= logaₙ log aₙ₊₁ log aₙ₊₂
log aₙ₊₃ log aₙ₊₄ log aₙ₊₅
log aₙ₊₆ log aₙ₊₇ log aₙ₊₈
is
a) 0 b) log(aₙaₙ₊₈) - 2 logaₙ c) loga₁a₃ - 2 log a₂ d) -1
90) For 0< θ<π/2 , if x= ∞ₙ₌₀∑cos²ⁿθ, y= ∞ₙ₌₀∑sin²ⁿθ, z= ∞ₙ₌₀∑cos²ⁿθ sin²ⁿθ, then
a) xyz= xz+ y b) xyz= xy + z c) xyz= x + z+ y d) xyz= yz+ x
91) If a₁, a₂, a₃,.....is an AP such that:
a₁ + a₅ + a₁₀ + a₁₅+ a₂₀ + a₂₄ = 225 then
a₁+ a₂+ a₃+....+a₂₃+ a₂₄ is
a) 909 b) 75 c) 750 d) 900
92) Let Sₙ = 1/1³ + (1+2)/(1³+2³) + .....+ (1+2+...+n)/(1³+ 2³+ .....+n³); n= 1,2,3,.... Then Sₙ is not greater than
a) 1/2 b) 1 c) 2 d) 4
93) 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 ⁿₓ₌₁∑ d(x)= 120. Then the value of n is
a) 4 b) 5 c) 6 d) none
94) If a₁, a₂, a₃,....aₙ are in AP with common difference d, then the sum of the series:
sin d [cosec a₁ cosec a₂ + cosec a₂ cosec a₃ + ....+ cosec aₙ₋₁ cosec aₙ] is
a) sec a₁ - sec aₙ
b) cosec a₁ - cosec aₙ
c) cot a₁ - cot aₙ
d) tan a₁ - tan aₙ
95) If x₁, x₂, x₃, as well as y₁, y₂, y₃ are in GP with the same common ratio, then the points (x₁, y₁),(x₂, y₂) and (x₃, y₃):
a) lie on straight line
b) lie on an ellipse
c) lie on circle
d) are vertices of the triangle
96) If sin²α , 1, sin⁴α ,sin⁵α, (-π α<π) are in AP then α lies in the interval:
a) (-π/2,π/2) b) (-π/³,π/3) c) (-π/6,π/6) d) none
97) The value of (0.2)^ log√₅ +1/4+ 1/8+ 1/16+....) is
a) 1 b) 2 c) 1/2 d) 5
98) If A, G and H are respectively A.M, G.M, H.M of three positive numbers a, b and c, then the equation whose roots are a, b, c is given by:
a) x³ - 3Ax² + 3G³x - G³ = 0
b) x³ - 3Ax² + 3(G³/H)x - G³ = 0
c) x³ + 3Ax² + 3(G³/H)x - G³ = 0
d) x³ - 3Ax² - 3(G²/H)x - G³ = 0
99) If the 4th, 7th, and 10th term of a GP be a, b, c respectively, then the relation between a, b, c is
a) b= (a+ c)/2 b) a²= bc c) b²= ac d) c²= ab
100) If a₁, a₂, a₃,....aₙ are in HP, then a₁a₂ + a₂a₃ +....aₙ₋₁aₙ will be equal to
a) a₁aₙ b) na₁aₙ c) (n -1)a₁aₙ d) none
101) x+ y+ z= 15 if 9, x, y, z, a are is AP, while 1/x+ 1/y + 1/z = 5/3 if 9, x, y, z,a is HP, then the value of a will be
a) 1 b) 2 c) 3 d) 9
102) If ⁿᵣ₌₁∑tᵣ = ⁿₖ₌₁∑ ᵏⱼ₌₁∑ ʲᵢ₌₁∑2, then ⁿᵣ₌₁∑ 1/tᵣ equals:
a) (n+1)/n b) n/(n+1) c) (n-1)/n d) n/(n-1)
103) If the ratio of H. M and G. M between two numbers a and b is 4:5, then the ratio of the two numbers will be
a) 1:2 b) 2:1 c) 4:1 d) none
104) Let n(>1) be a positive integer, the largest integer m such that (nᵐ +1) divides (1+ n + n²+....+n¹²⁷) is
a) 32 b) 63 c) 64 d) 127
105) If p, q, r are in AP and are positive, the roots of the equation ox²+ qx + r=0 are all real for:
a) |r/p -7| ≥ 4√3 b) |p/r -7| < 4√3 c) all p and r d) no p and r
106) The odd numbers are divided as follows:
1 3
5 7 9 11
13 15 17 19 21 23
_____________________________
_____________________________
then the sum of nth row is
a) 2ⁿ⁻¹ [2ⁿ + 2ⁿ⁻¹ -1]
b) (1/2) (2n+1) c) 2n d) 4n³
107) Let aⁿ be the nth term of the GP of positive numbers. Let ¹⁰⁰ₙ₌₁∑a₂ₙ = α and ¹⁰⁰ₙ₌₁∑a₂ₙ₋₁ = β, such that α≠β, then the common ratio is
a) α/β b) β/α c) √(α/β) d) √(β/α)
108) Let f(x) be a function satisfying f(x + y) = f(x) f(y) for all x, y N such that f(1)= 3 and ⁿₓ₌₁∑f(x) = 120. Then the value of n is
a) 4 b) 5 c) 6 d) none
109) For any odd integer n≥ 1,
n³ - (n -1)³+ ....+(-1)ⁿ⁻¹ 1³ equal to
a) (1/2) (n-1)²(2n -1)
b) (1/4) (n-1)²(2n -1)
c) (1/2) (n+1)²(2n -1)
d) (1/4) (n+1)²(2n -1)
110) If x₁, x₂, x₃ as well as y₁, y₂, y₃ are in GP with the same common ratio, then the points (x₁, y₁),(x₂, y₂) and (x₃, y₃):
a) lie on straight line
b) lie on an ellipse
c) lie on circle
d) are vertices of a triangle
111) For a real number x, [x] denotes the integral part of x, the value of:
a) [1/2]+ [1/2+1/100]+ [1/2+ 2/100] +....+ [1/2+ 99/100] is
a) 49 b) 50 c) 48 d) 51
112)
Let Tᵣ be the rth term of an AP for r= 1,2,3,.... If for some positive integers m,n we have
Tₘ= 1/n and Tₙ = 1/m, then Tₘₙ equal to
a) 1/mn b) 1/m + 1/n c) 1 d) 0
113) If lₙ(a+ c), lₙ(c - a), lₙ(A- 2b + c) are in AP then
a) a,b,c are in AP
b) a², b², c² are in AP
c) a,b,c are in GP
d) a,b,c are in HP
114) The sum of the series:
1+ 1/5+ (1.3)/+5.10)+ (1.3.5)/(5,10.15)+ .....is equal to
a) 1/√5 b) 1/√2 c) √3 d) √5
115) if a, b, c are digits, then the rational number represented by 0.c ab ab ab .....is
cab/990 b) (990+ ab)/990 c) (99c + 10a + b)/99 d) (99c + 10a + b)/990
116) If Iₙ= ⁿ₀∫ (1- sin2nx)/(1- cos2n) dx, then I₁, I₂, I₃, .....are in:
a) AP b) GP c) HP d) none
117) If p, q, r three positive real numbers are in AP, then the roots of px² + qx + r=0 are all real for:
a) |(r/p) -7|≥4√3
b) |(p/r) -7| < 4√3
c) all p and r d) no p and r
118) If a₁ ,a₂, a₃,....aₙ are in HP. Then:
a₁/(a₂+ a₃+ ....aₙ) , a₂/(a₁+ a₃+....+aₙ),....., aₙ/(a₁+ a₂+....aₙ₋₁) are in
a) AP b) GP c) HP d) AGP
119) If <aₙ> is an AP and a₁+ a₄+ a₇ +....+a₁₆= 147, then a₁+ a₆ + a₁₁+ a₁₆ equals to
a) 96 b) 97 c) 100 d) none
120) In the sequence 1,2,2,3,3,3,4,4,4,4,...., where n consecutive terms have the value n, then 150th term is
a) 17 b) 16 c) 18 d) none
121) The value of ⁿᵣ₌₁∑ log{aʳ/bʳ⁻¹} is
a) (n/2) log(aⁿ/bⁿ) b) (n/2) log(ⁿ⁺¹/bⁿ)
c) (n/2) log(aⁿ⁺¹/bⁿ⁻¹) d) (n/2) log(aⁿ⁺¹/bⁿ⁺¹)
122) If the roots of the equation
(b - c)x² + (c - a)x+ (a - b)= 0 are equal, then a, b, c are in
a) AP b) GP c) HP d) none
123) If a₁, a₂,....aₙ are nonzero real numbers such that
(a₁²+ a₂² +......aₙ₋₁²)(a₂² + a₃² +....aₙ²)≤ (a₁a₂ + a₂ + a₃+.....+aₙ₋₁aₙ)², a₁, a₂, aₙ are in
a) AP b) GP c) HP d) none
124) The value of the sum ¹³ₙ₌₁∑(iⁿ + iⁿ⁺¹), where i=√(-1), equal to
a) I b) i -1 c) - I d) 0
125) If 5¹⁺ˣ + 5¹⁻ˣ , a/2, 25ˣ+ 25⁻ˣ are in AP then the set of values of a for such an AP to exist is:
a) (6, ∞) b) [12, ∞) c) (18,∞) d) (-12,∞)
126) For a positive integer n, let
aₙ= 1+ 1/2+ 1/3+ 1/4+ ....+ 1/(2ⁿ-1). Then:
a) a(100)≤ 100 b) a(200)≤ 100 c) a(200)> 100 d) none
127) Let S₁, S₂,....be square such that for each n> 1, the length of a side of Sₙ equals the length of a diagonal or Sₙ₊₁. If the length of a side of S₁ is 10cm, then for which of the following values of n is the area of Sₙ less than 1 sq cm ?
a) 6 b) 8 c) 5 d) 4
128) The sum of n terms of the series
1/(1.2.3.4) + 1/(2.3.4.5)+ 1/(3.4.5.6) + ....is
a) (n³+6)/{18(n+1)+n+2)(n+3)}
b)(1/18) - 1/{3(n+1)+n+2)(n+3)}
c) 1/66 - 1/{2(n+1)+n+2)(n+3)} d) none
129) If ∑n, (√10/3) ∑n², ∑n³ are in GP, then the value of n is
a) 3 b) 4 c) 2 d) does not exist
130) If ∞ᵣ₌₁∑ tᵣ= n(n+1)(n+2)(n+3)/8, where tᵣ denotes the rth term of a series, then
lim ₙ→∞ ∞ᵣ₌₁∑ 1/rʳ is
a) 1/8 b) 1/4 c) 1/2 d) 1
131) ⁿᵢ₌₁∑ ⁱⱼ₌₁∑ ʲₖ₌₁∑1 equal to
a) n(n+1)(n+2)/3 b) ∑n² c) ⁿC₃ x) ⁿ⁺²C₃
132) If logₓa, aˣ⁾² and logᵥx are in GP, then x equal to
a) logₐ(logᵥa) b) - logₐ(logₐv)
c) logₐ(logₑb)- logₐ(log꜀a) d) none
133) If the ratio of AM between two positive real numbers a and b to their H. M is m: n; then a: b is equal to
a) {√(m+ n) + √n}/{√(m- n) - √n}
b) {√(m+ n) + √n}/{√(n) - √(m-n}
c) {(√m + √(m+ n)}/{√(m) - √(m- n}
d) {(√m - √(m- n)}/{√(m) + √(m- n}
134) The number 1,4,16 can be three terms (not necessarily consecutive) of:
a) no AP b) only one GP
c) infinite number of AP's d) none
135) If a, b, c, d, e, f are in AP, then e- c is equal to:
a) 2(c - a) b) 2(f - d) c) 2(d - c) d) d- e
136) A student read Common difference of an AP as -2 instead of 2 and got the sum of first five terms as -5, the actual sum of first five terms is
a) 25 b) -25 c) -35 d) 35
137) The value of n for which
(aⁿ⁺¹ + bⁿ⁺¹)/(aⁿ + bⁿ) is the A. M between a and b is
a) 0 b) -1/2 c) 1 d) -1
138) If a, b, c are in AP then 3ᵃ, 3ᵇ, 3ᶜ are in:
a) AP b) GP c) HP d) none
139) If a, b, c in AP then
10ᵃˣ⁺¹⁰, 10ᵇˣ⁺¹⁰, 10ᶜˣ⁺¹⁰, x≠ 10 are in
a) AP b) GP only when x> 0 c) GP for all x d) GP only when x< 0
140) If the pth, qth, rth terms of a GP are l, m, n respectively, then lᑫ⁻ʳ mʳ⁻ᵖ nᵖ⁻ᑫ is
a) 0 b) 1 c) pqr d) lmn
141) If aˣ= bʸ = cᶻ and a, b, c in GP then x, y, z are in
a) AP b) GP c) HP d) none
142) The first term of a geometrical progression is 32 and the fifth is 162. Its sixth term is:
a) 243 b) 324 c) 256 d) 262
143) If the sum of n natural numbers is one sixth of their squares, then n is
a) 6 b) 7 c) 8 d) none
144) The sum of areas of squares having sides 3,4,....10 is
a) 360 b) 380 c) 340 d) 350
145) If a,b, c, d are in HP then
a) a+ b > c+ d
b) a+ c> b+ d
c) a+ d> b+ c
d) none
146) If a, b, c are in HP then correct statement is
a) a²+ c²> b²
b) a²+ c²> 2b²
c) a²+ c²<2 b²
d) a²+ c²= 2b²
147) If P,Q,R be the AM, GM, HM respectively between any two rational numbers a and b, then P- Q is
a) (a- b)/a b) (a+ b)/c c) 2ab/(a+ b) d) {(√a - √b)/√2}²
148) Let two numbers have arithmetic mean 9 and geometric mean 4. Then these numbers are the roots of the equation:
a) x²+ 18x + 16=0
b) x²- 18x + 16=0
c) x²+ 18x - 16=0
d) x²- 18x - 16=0
149) Let Tᵣ be the rth term of an AP whose first term is a and common ris d. If for some positive integers, m, n, m≠ n, Tₘ= 1/n and Tₙ= 1/m, then A- d equal to
a) 0 b) 1 c) 1/mn d) 1/m + 1/n
150) The sides a, b, c of ∆ ABC are in GP, where log a - log 2b, log 2b - log 3c, log 3c - log a, are in AP then ∆ ABC is:
a) acute angled
b) obtuse angled
c) right angled d) none
151) If sinα, 1, sin⁴α, sin⁵α (-π<α<π) are in AP then α lies in the interval:
a) (-π/2,π/2) b) (-π/3,π/3) c) (-π/6,π/6) d) none
152) Let the positive numbers a, b, c, d be in AP. Then abc, abd, acd, bcd are:
a) not in AP/GP/HP b) in AP c) in GP d) in HP
153) If the sum of the first 2n terms of the AP 2,5,8,....is equal to the sum of the first n terms of the AP 57,59,61,...., then n equals
a) 10 b) 12 c) 11 d) 13
154) Suppose a, b, c are in AP and a², b², c² are in GP if a< b < c and a+ b + c= 3/2, then the value of a is:
a) 1/2√2 b) 1/2√3 c) 1/2-1/√3 d) 1/2-1/√2
155) The value of 2¹⁾⁴. 4¹⁾⁸. 8¹⁾¹⁶......∞ is
a) 1 b) 2 c) 3 d) 4
156) If 1, (1/2) log₃(3¹⁻ˣ+2), log₃(4.3ˣ-1) are in AP then x equals
a) log₃4 b) 1- log₃4 c) 1- log₄3 d) log₄3
157) An infinite GP has first term x and sum 5 then x belongs to
a) x<-10 b) -10< x < 0 c) 0< x < 10 d) x > 10
1d 2c 3c 4a 5a 6b 7c 8d 9d 10d 11b 12c 13a 14d 15a 16a 17d 18a 19b 20b 21b 22b 23d 24b 25c 26b 27c 28a 29d 30c 31b 32b 33d 34c 35a 36a 37c 38d 39c 40d 41c 42d 43d 44a 45c 46b 47c 48c 49a 50a 51b 52d 53c 54b 55c 56c 57c 58a 59b 60c 61d 62b 63b 64b 65b 66b 67a 68a 69a 70a 71c 72a 73a 74a 75d 76a 77b 78a 79b 80d 81c 82b 83d 84b 85b 86a 87b 88c 88a 99b 91d 92c 93a 94c 95a 96d 96d 97b 99c 100c 101a 102b 103c 104c 105a 106d 106a 108a 109d 110a 111b 112c 113d 114c 115d 116a 117a 118c 119b 120a 121c 122a 123b 124b 125b 126a 127b 128b 129a 130c 131d 132a 133c 134c 135c 136d 137c 138b 139c 140b 141c 142a 143d 144b 145c 146b 147d 148b 149a 150b 151d 152d 153c 154d 155b 156b 157c
