Dubey

1. Algebrathon The Last Word on Algebra By: Santosh Dubey

2. The function f(x) = log( x - 1) log( x - 2), andg(x) = log(x-1/x-2) are identical when x lies in the interval

(a) [ 1 , 2 ]

(b) [ 2 , ]

(c) ( 2 , )

(d) (- ,)

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

None of these

4. If x satisfiesI x 2 3x + 2 I+Ix-1 I= x 3 then

X[1,2]

X[3 , )

X(- , )

5. Let f(x) = x / (x +1) , x - 1, then for what values of is f [ { f(x)} ] = x

6. Let g(x) = 1 + x [x] and f(x)=-1 ,x < 0 0 ,x = 01, x > 0then for all x f [g(x)] is

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)]

8. If f(x) = ( a x n ) 1/nthen find f [f(x)]

9. If g(x) = 1 + x and f[g(x)] = 3 + 2 x + x then f(x) =

1+ 2x 2

2 + x 2

10. If f(x) = (2 x+ 2 -x )/2 then f(x+y).f(x-y) =

[f(2x)+ f(2y)]

1/4 [f(2x)+f(2y)]

[f(2x) - f(2y)]

1/4 [f(2x)- f(2y)]

11. If f(x) = log[(1-x)/(1+x)], then f [ 2x/(1+x 2 ) ] =

12. Which of the following is an even function

F(x) = (x)[(a x -1)/(a x +1)]

f(x) = [(a x +a -x )/(a x a -x )]

f(x) = [(a x+ 1)/(a x -1)]

f(x) = log2(x + (x 2+ 1)

13. Let f(x) = x 2+ 1/x 2and x 0 then f(x) is equal to

X 2- 1

X 2- 2

None of these

14. If 2f(x) 3f(1/x) = x 2and x 0 then f(2) is equal to

None of these

15. Range of (x 2+ x +2)/(x 2+x +1) is

(a) ( 1 , )

(b) ( 1 , 7/3)

(c) ( 1 , 7/5)

(d) ( 1 ,11/7)

16. Ifsis the set of all real x such that(2x - 1)/(2x 3+3x 2+ x)is positive thenscontains

(a) ( -1/4 , 1/2 )

(b) ( 1/2 , 3)

(c) ( -3/2 , 1/4)

(d) ( -3/2 ,1/2)

17. If f(x) = log[(1+x)/(1+x)], andg(x) = [ (3x + x 3 ) / (1+3x 2 ) ] then f [ g(x) ] =

[f(x)] 3

18. If x R and P =x 2 / (x 4- 2x 2+ 4)then P lies in the interval

(a) ( 0 , 1/2 )

(b) ( 3/4 , 4/5)

(c) ( 0 , 1/3)

(d) ( 0 ,1/4)

19. The minimum value of(x 2 3) 3+272

None of these

20. If f(x) = 4 x/ (4 x+2) thenf(1/1997) + f(2/1997) + f(3/1997) + f(4/1997) f(1996/1997 )

None of these

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

X 2- 1

-X 2 +1

1- X 2

X 2 +1

22. Let f(x) = 1 / (1- x) , g(x) = f { f(x)} andh(x) = f [ f { f(x)}] then f(x)g(x)h(x) is

None of these

23. If f(x + y) = f(x) + f(y) xy 1for all x,y R andf(1) = 1then the number of solutions to f(n) = n

No solution

None of these

24. Let f(x) = 64x 3+ 1/x 3and a and b are the roots of equation (4x) + (1/x) = 3then

f(a) = 12

f(b) = 11

f(a) = f(b)

None of these

25. If f( x+y , x-y) = xy ,then the arithmetic mean of f(x ,y) and f(y , x) is

None of these

26. If f(x) = (1- x)/(1+ x) x 0 thenf[f(x)] + f[f(1/x)]

x< -3 or x>2

x< -4 or x>3

None of these

29. The sum of all numbers of the form n 3which lie between 100 and 10,000

None of these

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

31. Sum of n terms of1.2.3 + 2.3.4 + 3.4.5 will be

n(n+1)(n + 2)(n + 3)/4

n(n+1)(n + 2)(n + 3)/3

(n+1)(n + 2)(n + 3)/4

n(n - 1)(n - 1)(n - 3)/4

32. In a G.P (m +n) thterm is 9 and (m-n) thterm is 4 then m thterm

None of these

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

-( 5 + 1)/2

(1 - 5)/2

( 5 - 1)/2

( 5 + 1)/2

34. The product of (32)(32) 1/6(32) 1/36

35. Sum of all terms of a G.P is five times the sum of the odd terms.The common ratio is

36. The sum of the series1 + 2.2 + 3.2 2+ 4.2 3+ 5.2 4 +.+100.2 99

99.2 100+ 1

100.2 100

99.2 100

99.2 100- 1

37. The 20 thterm of the series2x4 + 4x6 + 6x8.

38. The sum of the first n terms of1 + 2.2 2+ 3 2+ 2.4 2+ 5 2+ 5.6 2 ..is n(n+1) 2when n is even what is the sum when n is odd

n 2 (n+1)/2

n(n+1) 2 /2

[n(n+1)] 2 /2

n(n+1)/2

39. In a G.P the (p + q) thterm is m and (p - q) thterm is n, then what is the p thterm

(m + n)/2

40. Let Tr be the r thterm 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 =

1/m + 1/n

41. 1.f (x) =x 42. 2.f (x) =| x | 43. 3.f (x) =x 2 44. 4.f (x) =1/ x 45. 5.f (x) = x 3 46. 6.f (x) = x + | x | 47. 7.f (x) = x - | x | 48. 8.f (x) = -1, x, 1 49. 10.f (x) = x+1, x, x-1 50. 11.f (x) = sinx 51. 12.f (x) = cosx 52. 13.f (x) = tanx 53. 14.f (x) = cotx 54. 15.f (x) = secx 55. 15.f (x) = cosecx 56. If a 1 ,a 2 ,a 3are 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 1is satisfied 57. If x = 1+ a + a 2+ a 3+ a 4..andy = 1+ b + b 2+ b 3+ b 4..thenthe value of 1 + ab + a 2 b 2+ a 3 b 3+ a 4 b 4 .. if 0 < a < 1 and 0 < b 0 , b >0 , c > 0 then the greatest value of a 2 b 3 c 2

(3 10 .2 4 )/7 7

(3 9 .2 4 )/7 7

(3 8 .2 4 )/7 7

None of these

70. The largest term in the sequence (1/503) , (4/524) , (9/581), (16/692) .is

16/692

49/1529

None of these

71. If T n= n x n! thenT n (n = 1 to20)is equal to

21! -1

20! - 1

21! +1

None of these

If a is positive and the roots of the quadratic equation

(a 2 +b 2 )x 2+2(ab +bc)x +(b 2 + c 2 )=0 are real,

find the minimum possible value of quadratic expression ax 2 +bx+c

Solution set for

|x-1| + |x-2| + |x-3| > 6

a= 1/3

Then find a

How many positive integral solutions are there for (x,y,z) given that

X + (-1) zy= z and x,y,z5

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.

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

a) x=3y/2

b) y=3z/2

c) x=9z/4

d) y=9x/4

Integers a,b,c and d are such that

-5 a 4, -2b 6, -3 c 8, -4 d 6.

If E= ab + bc+ cd + da

then find the least value of E.

79. How many ordered pair ofPositive integers satisfy the condition p+q+12 =pq 80.

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

x 2+ cx + d = 0,

which of the following relationship holds true

b 2 bd = ad 2 cd

b 2 bd = cd 2 ad 2

b 2 b 2 d = c 2 d a 2 d

b 2 b 2 d = a 2 d c 2 d

IF |2x 5| 9 and |4y 7| 21,

what is the max value of |x| - |y| ?

82. Ifp + q + r = 7, p 2+ q 2+ r2= 35, and p 3+ q3+ r 3= 151, then Find the value of pq + qr + rpFind the value of pqr Find the value of p 4+ q 4+ r 4 83.

Ifx,y, z are integers andx + y + z = 3,

then find the minimum value of

1/x + 1/y +1/z

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.

If p 3 +q 3 +r 3= 48,and pqr = 16, then

find the value of(p 2+q 2+r 2 ) /(pq +qr +rp)

The equation

x 4 px 3+ qx + rx + 1 = 0 has four integral roots.

Which of the following could be the value of expression pq+ qr +rp?

(a) -32 (b) 16 (c) 32 (d) 0

A= min [24 -8x-x 2 , x 2+ 9x 6]. What is the max value of A

If x is real the max value of

(x + 2)/(2x 2+ 4x + 8)

If x,y,z are distinct positive integers, how many ordered triplets (x,y,z) satisfy the condition

z+xy+xyz = 59 (1+z)(x+y)?

Solution set of inequality

|x 3- 6x 2+ 12x 6| (x- 2) 3

The solutions of the system of the equation |x| - x + |y| + y = 12 and

|x| - x + |y| - y = 6 are ordered pairs of the form (xi, yi).

Findx i+y i

92. NUMBER SYSTEMS PROBLEM SOLVING SESSION - 1 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?????

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