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# Dubey

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### Dubey

1. 1. Algebrathon The Last Word on Algebra By: Santosh Dubey
2. 2. The function f(x) = log( x - 1) – log( x - 2), and g(x) = log(x-1/x-2) are identical when x lies in the interval <ul><li>(a) [ 1 , 2 ] </li></ul><ul><li>(b) [ 2 , ∞] </li></ul><ul><li>(c) ( 2 , ∞) </li></ul><ul><li>(d) (-∞ ,∞) </li></ul>
3. 3. For n = 1,2,3…… the value of [(n +1)/2 ] + [ (n +2)/4 ] + [ (n +4)/8 ]+ [ (n +8)/16 ]+……… ∞ where [ . ] represents the greatest integer function <ul><li>n – 1 </li></ul><ul><li>n </li></ul><ul><li>n +2 </li></ul><ul><li>None of these </li></ul>
4. 4. If x satisfies I x 2 – 3x + 2 I + I x-1 I = x – 3 then <ul><li>X ε {} </li></ul><ul><li>X ε [1,2] </li></ul><ul><li>X ε [3 , ∞) </li></ul><ul><li>X ε (-∞ , ∞) </li></ul>
5. 5. Let f(x) = α x / (x +1) , x ≠ - 1, then for what values of α is f [ { f(x)} ] = x <ul><li>√ 2 </li></ul><ul><li>-√2 </li></ul><ul><li>1 </li></ul><ul><li>-1 </li></ul>
6. 6. Let g(x) = 1 + x – [x] and f(x)= -1 ,x < 0 0 ,x = 0 1, x > 0 then for all x f [g(x)] is <ul><li>X </li></ul><ul><li>1 </li></ul><ul><li>f(x) </li></ul><ul><li>g(x) </li></ul>
7. 7. For real number x , [x] denotes the integral part of x then the value of [1/2] + [ (1/2) + (1/100)] + [(1/2) + (2/100)]….…[(1/2) + (99/100)] <ul><li>49 </li></ul><ul><li>50 </li></ul><ul><li>48 </li></ul><ul><li>51 </li></ul>
8. 8. If f(x) = ( a – x n ) 1/n then find f [f(x)] <ul><li>X 1/n </li></ul><ul><li>x n </li></ul><ul><li>a - x </li></ul><ul><li>x </li></ul>
9. 9. If g(x) = 1 + √x and f[g(x)] = 3 + 2 √x + x then f(x) = <ul><li>1+ 2x 2 </li></ul><ul><li>2 + x 2 </li></ul><ul><li>1 + x </li></ul><ul><li>2 + x </li></ul>
10. 10. If f(x) = (2 x + 2 -x )/2 then f(x+y).f(x-y) = <ul><li>½[f(2x)+ f(2y)] </li></ul><ul><li>1/4 [f(2x)+f(2y)] </li></ul><ul><li>½[f(2x) - f(2y)] </li></ul><ul><li>1/4 [f(2x)- f(2y)] </li></ul>
11. 11. If f(x) = log[(1-x)/(1+x)], then f [ 2x/(1+x 2 ) ] = <ul><li>f(x) </li></ul><ul><li>2f(x) </li></ul><ul><li>3f(x) </li></ul><ul><li>4f(x) </li></ul>
12. 12. Which of the following is an even function <ul><li>F(x) = (x)[(a x -1)/(a x +1)] </li></ul><ul><li>f(x) = [(a x +a -x )/(a x – a -x )] </li></ul><ul><li>f(x) = [(a x + 1)/(a x -1)] </li></ul><ul><li>f(x) = log2(x + √(x 2 + 1) </li></ul>
13. 13. Let f(x) = x 2 + 1/x 2 and x ≠ 0 then f(x) is equal to <ul><li>X 2 - 1 </li></ul><ul><li>X 2 - 2 </li></ul><ul><li>X 2 </li></ul><ul><li>None of these </li></ul>
14. 14. If 2f(x) – 3f(1/x) = x 2 and x ≠ 0 then f(2) is equal to <ul><li>-7/4 </li></ul><ul><li>5/2 </li></ul><ul><li>-1 </li></ul><ul><li>None of these </li></ul>
15. 15. Range of (x 2 + x +2) / (x 2 +x +1) is <ul><li>(a) ( 1 , ∞ ) </li></ul><ul><li>(b) ( 1 , 7/3) </li></ul><ul><li>(c) ( 1 , 7/5) </li></ul><ul><li>(d) ( 1 ,11/7) </li></ul>
16. 16. If s is the set of all real x such that (2x - 1) / (2x 3 +3x 2 + x) is positive then s contains <ul><li>(a) ( -1/4 , 1/2 ) </li></ul><ul><li>(b) ( 1/2 , 3) </li></ul><ul><li>(c) ( -3/2 , 1/4) </li></ul><ul><li>(d) ( -3/2 ,1/2) </li></ul>
17. 17. If f(x) = log[(1+x)/(1+x)], and g(x) = [ (3x + x 3 ) / (1+3x 2 ) ] then f [ g(x) ] = <ul><li>f(3x) </li></ul><ul><li>[f(x)] 3 </li></ul><ul><li>3f(x) </li></ul><ul><li>-f(x) </li></ul>
18. 18. If x ε R and P = x 2 / (x 4 - 2x 2 + 4) then P lies in the interval <ul><li>(a) ( 0 , 1/2 ) </li></ul><ul><li>(b) ( 3/4 , 4/5) </li></ul><ul><li>(c) ( 0 , 1/3) </li></ul><ul><li>(d) ( 0 ,1/4) </li></ul>
19. 19. The minimum value of (x 2 – 3) 3 +27 2 <ul><li>1 </li></ul><ul><li>2 </li></ul><ul><li>2 27 </li></ul><ul><li>None of these </li></ul>
20. 20. If f(x) = 4 x / (4 x +2) then f(1/1997) + f(2/1997) + f(3/1997) + f(4/1997) ………f(1996/1997 ) <ul><li>1997 </li></ul><ul><li>998 </li></ul><ul><li>0 </li></ul><ul><li>None of these </li></ul>
21. 21. Let f(x) be a function defined on [-1 , 1].If the area of the equilateral triangle with two of its vertices at (0 , 0) and (x ,f(x)) is (√3)/4 then the function f(x) is <ul><li>√ X 2 - 1 </li></ul><ul><li>-√X 2 + 1 </li></ul><ul><li>±√1- X 2 </li></ul><ul><li>√ X 2 + 1 </li></ul>
22. 22. Let f(x) = 1 / (1- x) , g(x) = f { f(x)} and h(x) = f [ f { f(x)}] then f(x)g(x)h(x) is <ul><li>1 </li></ul><ul><li>-1 </li></ul><ul><li>0 </li></ul><ul><li>None of these </li></ul>
23. 23. If f(x + y) = f(x) + f(y) – xy – 1for all x,y ε R and f(1) = 1then the number of solutions to f(n) = n <ul><li>One </li></ul><ul><li>Two </li></ul><ul><li>No solution </li></ul><ul><li>None of these </li></ul>
24. 24. Let f(x) = 64x 3 + 1/x 3 and a and b are the roots of equation (4x) + (1/x) = 3 then <ul><li>f(a) = 12 </li></ul><ul><li>f(b) = 11 </li></ul><ul><li>f(a) = f(b) </li></ul><ul><li>None of these </li></ul>
25. 25. If f( x+y , x-y) = xy ,then the arithmetic mean of f(x ,y) and f(y , x) is <ul><li>x </li></ul><ul><li>y </li></ul><ul><li>0 </li></ul><ul><li>None of these </li></ul>
26. 26. If f(x) = (1- x)/(1+ x) x ≠ 0 then f[f(x)] + f[f(1/x)] <ul><li><2 </li></ul><ul><li>≥ 2 </li></ul><ul><li>= 2 </li></ul><ul><li>None of these </li></ul>
27. 27. If f(n+1) = [2f(n) + 1]/2 , n = 1,2,3…… and f(1) = 2 then f(101) equals <ul><li>52 </li></ul><ul><li>49 </li></ul><ul><li>48 </li></ul><ul><li>51 </li></ul>
28. 28. a,b,c are three distinct real numbers in G.P.If a+b+c = xb, then <ul><li>x< -1 or x>3 </li></ul><ul><li>x< -3 or x>2 </li></ul><ul><li>x< -4 or x>3 </li></ul><ul><li>None of these </li></ul>
29. 29. The sum of all numbers of the form n 3 which lie between 100 and 10,000 <ul><li>43261 </li></ul><ul><li>53261 </li></ul><ul><li>63261 </li></ul><ul><li>None of these </li></ul>
30. 30. If the first term of an infinite G.P series is 1 and its every term is the sum of next successive terms then its fourth term will be <ul><li>1/16 </li></ul><ul><li>1/8 </li></ul><ul><li>1/4 </li></ul><ul><li>1/2 </li></ul>
31. 31. Sum of n terms of 1.2.3 + 2.3.4 + 3.4.5……… will be <ul><li>n(n+1)(n + 2)(n + 3)/4 </li></ul><ul><li>n(n+1)(n + 2)(n + 3)/3 </li></ul><ul><li>(n+1)(n + 2)(n + 3)/4 </li></ul><ul><li>n(n - 1)(n - 1)(n - 3)/4 </li></ul>
32. 32. In a G.P (m +n) th term is 9 and (m-n) th term is 4 then m th term <ul><li>16 </li></ul><ul><li>1/6 </li></ul><ul><li>6 </li></ul><ul><li>None of these </li></ul>
33. 33. If each term of a G.P is positive and each term is sum of its two succeeding terms , then common ratio of the G.P <ul><li>-(√ 5 + 1)/2 </li></ul><ul><li>(1 - √ 5)/2 </li></ul><ul><li>(√ 5 - 1)/2 </li></ul><ul><li>(√ 5 + 1)/2 </li></ul>
34. 34. The product of (32)(32) 1/6 (32) 1/36 … ∞ <ul><li>64 </li></ul><ul><li>16 </li></ul><ul><li>32 </li></ul><ul><li>0 </li></ul>
35. 35. Sum of all terms of a G.P is five times the sum of the odd terms.The common ratio is <ul><li>5 </li></ul><ul><li>4 </li></ul><ul><li>3 </li></ul><ul><li>2 </li></ul>
36. 36. The sum of the series 1 + 2.2 + 3.2 2 + 4.2 3 + 5.2 4 +….+100.2 99 <ul><li>99.2 100 + 1 </li></ul><ul><li>100.2 100 </li></ul><ul><li>99.2 100 </li></ul><ul><li>99.2 100 - 1 </li></ul>
37. 37. The 20 th term of the series 2x4 + 4x6 + 6x8……………………. <ul><li>840 </li></ul><ul><li>420 </li></ul><ul><li>1680 </li></ul><ul><li>1600 </li></ul>
38. 38. The sum of the first n terms of 1 + 2.2 2 + 3 2 + 2.4 2 + 5 2 + 5.6 2 ………..is n(n+1) 2 when n is even what is the sum when n is odd <ul><li>n 2 (n+1)/2 </li></ul><ul><li>n(n+1) 2 /2 </li></ul><ul><li>[n(n+1)] 2 /2 </li></ul><ul><li>n(n+1)/2 </li></ul>
39. 39. In a G.P the (p + q) th term is m and (p - q) th term is n, then what is the p th term <ul><li>0 </li></ul><ul><li>mn </li></ul><ul><li>√ mn </li></ul><ul><li>(m + n)/2 </li></ul>
40. 40. Let Tr be the r th term of an A.P, for r = 1,2,3……..If for some positive integer m , n we have Tm = 1/m and Tn = 1/n then Tmn = <ul><li>1/mn </li></ul><ul><li>1/m + 1/n </li></ul><ul><li>1 </li></ul><ul><li>0 </li></ul>
41. 41. 1. f (x) = x
42. 42. 2. f (x) = | x |
43. 43. 3. f (x) = x 2
44. 44. 4. f (x) = 1/ x
45. 45. 5. f (x) = x 3
46. 46. 6. f (x) = x + | x |
47. 47. 7. f (x) = x - | x |
48. 48. 8. f (x) = -1, x, 1
49. 49. 10. f (x) = x+1, x, x-1
50. 50. 11. f (x) = sin x
51. 51. 12. f (x) = cos x
52. 52. 13. f (x) = tan x
53. 53. 14. f (x) = cot x
54. 54. 15. f (x) = sec x
55. 55. 15. f (x) = cosec x
56. 56. If a 1 ,a 2 ,a 3 are three consecutive terms of a G.P with common ratio k.find the values of k for which the inequality a 3 > 4a 2 – 3a 1 is satisfied
57. 57. If x = 1+ a + a 2 + a 3 + a 4 …..and y = 1+ b + b 2 + b 3 + b 4 …..then the value of 1 + ab + a 2 b 2 + a 3 b 3 + a 4 b 4 ….. if 0 < a < 1 and 0 < b <1 <ul><li>(x+ y)/x+y-1 </li></ul><ul><li>(x + y – 1)/xy </li></ul><ul><li>xy/(x +y-1) </li></ul><ul><li>None of these </li></ul>
58. 58. Find the sum of the product of every pair of first n natural number
59. 59. Sum to n terms of the series [1/(1 + 1 2 + 1 4 ) ] + [2/(1 + 2 2 + 2 4 )] + [3/(1 + 3 2 +3 4 )]…………….
60. 60. Three successive terms of a G.P will form a triangle if the common ratio lies between
61. 61. Two consecutive numbers from 1,2,3 ………..n are removed, arithmetic mean of the remaining numbers is 105/4. Find the removed numbers
62. 62. Let s k be the sum of the first k terms of an arithmetic progression. What must this progression be for s kx /s x to be independent of x
63. 63. 100 th term of he series 1 + 3 + 7 + 15……. <ul><li>2 50 - 1 </li></ul><ul><li>2 80 - 1 </li></ul><ul><li>2 100 - 1 </li></ul><ul><li>None of these </li></ul>
64. 64. Sum to 20 terms of series 1.3 2 +2.5 2 + 3.7 2 …………….. <ul><li>178090 </li></ul><ul><li>168090 </li></ul><ul><li>188090 </li></ul><ul><li>None of these </li></ul>
65. 65. Sum to 16 terms of (1 3 /1) + (1 3 + 2 3 )/3 + (1 3 + 2 3 + 3 3 )/1 + 3 + 5 ……………… <ul><li>450 </li></ul><ul><li>456 </li></ul><ul><li>446 </li></ul><ul><li>None of these </li></ul>
66. 66. The largest value of positive integer k for which n k + 1 divides 1 + n + n 2 + ………..+ n 127 is divisible by <ul><li>8 </li></ul><ul><li>16 </li></ul><ul><li>32 </li></ul><ul><li>64 </li></ul>
67. 67. If x + y + z = 1 and x,y,z are positive numbers such that (1 – x) (1 – y) (1 – z) ≥ kxyz then k = <ul><li>2 </li></ul><ul><li>4 </li></ul><ul><li>8 </li></ul><ul><li>16 </li></ul>
68. 68. Number of terms in the sequence 1,3,6,10,15……………5050 <ul><li>50 </li></ul><ul><li>75 </li></ul><ul><li>100 </li></ul><ul><li>125 </li></ul>
69. 69. If a + b + c = 3 a > 0 , b >0 , c > 0 then the greatest value of a 2 b 3 c 2 <ul><li>(3 10 .2 4 )/7 7 </li></ul><ul><li>(3 9 .2 4 )/7 7 </li></ul><ul><li>(3 8 .2 4 )/7 7 </li></ul><ul><li>None of these </li></ul>
70. 70. The largest term in the sequence (1/503) , (4/524) , (9/581), (16/692) ………….is <ul><li>16/692 </li></ul><ul><li>4/524 </li></ul><ul><li>49/1529 </li></ul><ul><li>None of these </li></ul>
71. 71. If T n = n x n! then Σ T n (n = 1 to20) is equal to <ul><li>21! -1 </li></ul><ul><li>20! - 1 </li></ul><ul><li>21! +1 </li></ul><ul><li>None of these </li></ul>
72. 72. <ul><li>If a is positive and the roots of the quadratic equation </li></ul><ul><li>(a 2 +b 2 )x 2 +2(ab +bc)x +(b 2 + c 2 )=0 are real, </li></ul><ul><li>find the minimum possible value of quadratic expression ax 2 +bx+c </li></ul>
73. 73. <ul><li>Solution set for </li></ul><ul><li>|x-1| + |x-2| + |x-3| > 6 </li></ul>
74. 74. <ul><li>√ a… </li></ul><ul><li>√ a </li></ul><ul><li>√ a </li></ul><ul><li>√ a = 1/3 </li></ul><ul><li>Then find a </li></ul>
75. 75. <ul><li>How many positive integral solutions are there for (x,y,z) given that </li></ul><ul><li>X + (-1) z y= z and x,y,z ≤ 5 </li></ul>
76. 76. If a,b,c & d are real numbers such that 2ac = b+d, then how many of the following statements are necessary true. x 2 + 2ax + b has real roots X 2 + 2cx + d has real roots
77. 77. <ul><li>If x,y,z are non zero real numbers and 4x 2 + 13 y 2 + 9 z 2 = 12y (x+z) which of the following is false </li></ul><ul><li>a) x=3y/2 </li></ul><ul><li>b) y=3z/2 </li></ul><ul><li>c) x=9z/4 </li></ul><ul><li>d) y=9x/4 </li></ul>
78. 78. <ul><li>Integers a,b,c and d are such that </li></ul><ul><li>-5 ≤ a≤ 4, -2≤b ≤6, -3 ≤c ≤8, -4 ≤d ≤6. </li></ul><ul><li>If E= ab + bc+ cd + da </li></ul><ul><li>then find the least value of E. </li></ul>
79. 79. How many ordered pair of Positive integers satisfy the condition p+q+12 =pq
80. 80. <ul><li>The equations x 2 + ax + b = 0 and x 2 + cx + d = 0 have a common root. The other root of x 2 +ax+b=0 is the square of the other root of </li></ul><ul><li>x 2 + cx + d = 0, </li></ul><ul><li>which of the following relationship holds true </li></ul><ul><li>b 2 – bd = ad 2 – cd </li></ul><ul><li>b 2 – bd = cd 2 – ad 2 </li></ul><ul><li>b 2 – b 2 d = c 2 d – a 2 d </li></ul><ul><li>b 2 – b 2 d = a 2 d – c 2 d </li></ul>
81. 81. <ul><li>IF |2x – 5| ≤ 9 and |4y – 7| ≤ 21, </li></ul><ul><li>what is the max value of |x| - |y| ? </li></ul>
82. 82. If p + q + r = 7, p 2 + q 2 + r 2 = 35, and p 3 + q 3 + r 3 = 151, then Find the value of pq + qr + rp Find the value of pqr Find the value of p 4 + q 4 + r 4
83. 83. <ul><li>If x,y, z are integers and x + y + z = 3, </li></ul><ul><li>then find the minimum value of </li></ul><ul><li>1/x + 1/y +1/z </li></ul><ul><li> </li></ul>
84. 84. If a, b, c are positive numbers satisfying a 2 + b 2 + c 2 = 12, then find the max/min value of a + b +c
85. 85. <ul><li>If p 3 +q 3 +r 3 = 48,and pqr = 16, then </li></ul><ul><li>find the value of (p 2 +q 2 +r 2 ) /(pq +qr +rp) </li></ul>
86. 86. <ul><li>The equation </li></ul><ul><li>x 4 – px 3 + qx + rx + 1 = 0 has four integral roots. </li></ul><ul><li>Which of the following could be the value of expression pq+ qr +rp? </li></ul><ul><li>(a) -32 (b) 16 (c) 32 (d) 0 </li></ul>
87. 87. <ul><li>A= min [24 -8x-x 2 , x 2 + 9x – 6]. What is the max value of A </li></ul>
88. 88. <ul><li>If x is real the max value of </li></ul><ul><li>(x + 2)/(2x 2 + 4x + 8) </li></ul>
89. 89. <ul><li>If x,y,z are distinct positive integers, how many ordered triplets (x,y,z) satisfy the condition </li></ul><ul><li>z+xy+xyz = 59 – (1+z)(x+y)? </li></ul>
90. 90. <ul><li>Solution set of inequality </li></ul><ul><li>|x 3 - 6x 2 + 12x – 6| ≥ (x- 2) 3 </li></ul>
91. 91. <ul><li>The solutions of the system of the equation |x| - x + |y| + y = 12 and </li></ul><ul><li>|x| - x + |y| - y = 6 are ordered pairs of the form (xi, yi). </li></ul><ul><li>Find Σ x i + Σ y i </li></ul>
92. 92. NUMBER SYSTEMS PROBLEM SOLVING SESSION - 1
93. 93. .abcabcabcabc…….=pq/rs.How many possible pairs of two digit numbers pq and rs are possible such that they are coprime to each other?????