Dubey

Algebrathon The Last Word on Algebra By: Santosh Dubey

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 ,[object Object],[object Object],[object Object],[object Object]

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 ,[object Object],[object Object],[object Object],[object Object]

If x satisfies I x 2 – 3x + 2 I + I x-1 I = x – 3 then ,[object Object],[object Object],[object Object],[object Object]

Let f(x) = α x / (x +1) , x ≠ - 1, then for what values of α is f [ { f(x)} ] = x ,[object Object],[object Object],[object Object],[object Object]

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 ,[object Object],[object Object],[object Object],[object Object]

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)] ,[object Object],[object Object],[object Object],[object Object]

If f(x) = ( a – x n ) 1/n then find f [f(x)] ,[object Object],[object Object],[object Object],[object Object]

If g(x) = 1 + √x and f[g(x)] = 3 + 2 √x + x then f(x) = ,[object Object],[object Object],[object Object],[object Object]

If f(x) = (2 x + 2 -x )/2 then f(x+y).f(x-y) = ,[object Object],[object Object],[object Object],[object Object]

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

Which of the following is an even function ,[object Object],[object Object],[object Object],[object Object]

Let f(x) = x 2 + 1/x 2 and x ≠ 0 then f(x) is equal to ,[object Object],[object Object],[object Object],[object Object]

If 2f(x) – 3f(1/x) = x 2 and x ≠ 0 then f(2) is equal to ,[object Object],[object Object],[object Object],[object Object]

Range of (x 2 + x +2) / (x 2 +x +1) is ,[object Object],[object Object],[object Object],[object Object]

If s is the set of all real x such that (2x - 1) / (2x 3 +3x 2 + x) is positive then s contains ,[object Object],[object Object],[object Object],[object Object]

If f(x) = log[(1+x)/(1+x)], and g(x) = [ (3x + x 3 ) / (1+3x 2 ) ] then f [ g(x) ] = ,[object Object],[object Object],[object Object],[object Object]

If x ε R and P = x 2 / (x 4 - 2x 2 + 4) then P lies in the interval ,[object Object],[object Object],[object Object],[object Object]

The minimum value of (x 2 – 3) 3 +27 2 ,[object Object],[object Object],[object Object],[object Object]

If f(x) = 4 x / (4 x +2) then f(1/1997) + f(2/1997) + f(3/1997) + f(4/1997) ………f(1996/1997 ) ,[object Object],[object Object],[object Object],[object Object]

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 ,[object Object],[object Object],[object Object],[object Object]

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 ,[object Object],[object Object],[object Object],[object Object]

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 ,[object Object],[object Object],[object Object],[object Object]

Let f(x) = 64x 3 + 1/x 3 and a and b are the roots of equation (4x) + (1/x) = 3 then ,[object Object],[object Object],[object Object],[object Object]

If f( x+y , x-y) = xy ,then the arithmetic mean of f(x ,y) and f(y , x) is ,[object Object],[object Object],[object Object],[object Object]

If f(x) = (1- x)/(1+ x) x ≠ 0 then f[f(x)] + f[f(1/x)] ,[object Object],[object Object],[object Object],[object Object]

If f(n+1) = [2f(n) + 1]/2 , n = 1,2,3…… and f(1) = 2 then f(101) equals ,[object Object],[object Object],[object Object],[object Object]

a,b,c are three distinct real numbers in G.P.If a+b+c = xb, then ,[object Object],[object Object],[object Object],[object Object]

The sum of all numbers of the form n 3 which lie between 100 and 10,000 ,[object Object],[object Object],[object Object],[object Object]

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 ,[object Object],[object Object],[object Object],[object Object]

Sum of n terms of 1.2.3 + 2.3.4 + 3.4.5……… will be ,[object Object],[object Object],[object Object],[object Object]

In a G.P (m +n) th term is 9 and (m-n) th term is 4 then m th term ,[object Object],[object Object],[object Object],[object Object]

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 ,[object Object],[object Object],[object Object],[object Object]

The product of (32)(32) 1/6 (32) 1/36 … ∞ ,[object Object],[object Object],[object Object],[object Object]

Sum of all terms of a G.P is five times the sum of the odd terms.The common ratio is ,[object Object],[object Object],[object Object],[object Object]

The sum of the series 1 + 2.2 + 3.2 2 + 4.2 3 + 5.2 4 +….+100.2 99 ,[object Object],[object Object],[object Object],[object Object]

The 20 th term of the series 2x4 + 4x6 + 6x8……………………. ,[object Object],[object Object],[object Object],[object Object]

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 ,[object Object],[object Object],[object Object],[object Object]

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 ,[object Object],[object Object],[object Object],[object Object]

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 = ,[object Object],[object Object],[object Object],[object Object]

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

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 ,[object Object],[object Object],[object Object],[object Object]

Find the sum of the product of every pair of first n natural number

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 )]…………….

Three successive terms of a G.P will form a triangle if the common ratio lies between

Two consecutive numbers from 1,2,3 ………..n are removed, arithmetic mean of the remaining numbers is 105/4. Find the removed numbers

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

100 th term of he series 1 + 3 + 7 + 15……. ,[object Object],[object Object],[object Object],[object Object]

Sum to 20 terms of series 1.3 2 +2.5 2 + 3.7 2 …………….. ,[object Object],[object Object],[object Object],[object Object]

Sum to 16 terms of (1 3 /1) + (1 3 + 2 3 )/3 + (1 3 + 2 3 + 3 3 )/1 + 3 + 5 ……………… ,[object Object],[object Object],[object Object],[object Object]

The largest value of positive integer k for which n k + 1 divides 1 + n + n 2 + ………..+ n 127 is divisible by ,[object Object],[object Object],[object Object],[object Object]

If x + y + z = 1 and x,y,z are positive numbers such that (1 – x) (1 – y) (1 – z) ≥ kxyz then k = ,[object Object],[object Object],[object Object],[object Object]

Number of terms in the sequence 1,3,6,10,15……………5050 ,[object Object],[object Object],[object Object],[object Object]

If a + b + c = 3 a > 0 , b >0 , c > 0 then the greatest value of a 2 b 3 c 2 ,[object Object],[object Object],[object Object],[object Object]

The largest term in the sequence (1/503) , (4/524) , (9/581), (16/692) ………….is ,[object Object],[object Object],[object Object],[object Object]

If T n = n x n! then Σ T n (n = 1 to20) is equal to ,[object Object],[object Object],[object Object],[object Object]

[object Object],[object Object],[object Object]

[object Object],[object Object]

[object Object],[object Object],[object Object],[object Object],[object Object]

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

[object Object],[object Object],[object Object],[object Object]

How many ordered pair of Positive integers satisfy the condition p+q+12 =pq

[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

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

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

NUMBER SYSTEMS PROBLEM SOLVING SESSION - 1

.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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