Please solve this Q from CL - FLT

[sntrao]

(1)

x^a + x^b = 1 --- eq 1

x > 0. hence taking log on both sides of eq 1 we get

a.log(x) + b.log(x) = log(1)
=> (a+b).log(x) = 0
=> either log(x) = 0 or (a+b) = 0

take log(x) = 0. this gives x = 1 which when substituted in eq 1 gives 1+1 = 1 which is absurd. hence (a+b) = 0. or, a = - b

therefore, a.b cannot be > 0. statement 1 is ruled out; which goes to say that options a, b, c cannot be true. hence my answer would be option 'd'.

(2)

anupam, it is given that |x|, |y| < 1. hence, using a = |x| and b = |y| and solving as u explained we get

sum of series = (1+a)/(1-ab)

But, a = |x| and b = |y|. choice a is true only when both x and y are > 0. so, i wud choose 'd'.

Regds
TK

[anupam will return]

Originally Posted by sntrao

(2)

anupam, it is given that |x|, |y| < 1. hence, using a = |x| and b = |y| and solving as u explained we get

sum of series = (1+a)/(1-ab)

But, a = |x| and b = |y|. choice a is true only when both x and y are > 0. so, i wud choose 'd'.

Regds
TK

Actually if you read the question a bit carefully you'll see the given data as:
[x],[y] < 1 and not |x|,|y| <1.
Here [x] stands for the greatest integer function ( defined as the greatest integer greater than or equal to x )
My answer remains 1

cheers,

[sntrao]

Originally Posted by anupam will return

Actually if you read the question a bit carefully you'll see the given data as:
[x],[y] < 1 and not |x|,|y| <1.
Here [x] stands for the greatest integer function ( defined as the greatest integer greater than or equal to x )
My answer remains 1

cheers,

yes, u r rite. i thot 'catquery' did not find the mod symbol in his keyboard and hence used [] to represent ||.

Regds
TK

[harikishore]

i think i solved the problem.

answer is : (4/10)*(3/9)*(2/9) = 1/30

[mohitkumar17]

Hi All,

Sorry for the delay in posting questions. This question had appeared in CL FLT -10. I am not able to understand the solution. Could anybody explain me the solution?

The lines y=3x and x+y=40 and the x-axis bound the triangular area. Find the total number of points on or inside the triangle with integral number of co-ordinates.
a. 600
b. 641.
c. 655
d. 662

Thanks,
Mohit

[Apprentice]

Originally Posted by harikishore

i think i solved the problem.

answer is : (4/10)*(3/9)*(2/9) = 1/30

Brilllliantttt...If u can come up with solutions without ne help frm the question...

[anupam will return]

Originally Posted by Apprentice

Brilllliantttt...If u can come up with solutions without ne help frm the question...

[Mohit102]

Originally Posted by catquery

1. let x^a + x^b = 1 , x > 0 Consider three statements :

1 ab > 0
2 a(x-1) > 0
3 a(x-1) < 0

Which of the following is necessarily true ?

a. 1 only b. 1 and 2 c. 1 and 3 d. None of these

2. t_1, t_2, t_3........... are infinite terms of a series defined as follows :

t_1 = 1
t_n = X * t_n-1 for all n>= 2 and n being even
t_n = Y * t_n-1 for all n>= 2 and n being odd
where x and y are constants such that [x] < 1 and [y] < 1
What is the sum to infinite terms ?

a. 1+x/ 1-xy b. 1+xy/ 1-xy+x+y c. 1-xy/ 1+xy-x-y d. None of these

For first problem,

x^a + x^b = 1 , x > 0

If either a or b is 0 that would mean 1 + x ^ b or x ^ a + 1 which is not possible...
Now again for x^a + x^b = 1 both terms x^a and x^b should be fraction. that would mean a * b > 0

Second statement states that a > 0 Not possible (Refer to above)

Third statement states that a < 0 which is true and x < 1 which should be an issue here...
Hence answer is C.

[Mohit102]

Originally Posted by mohitkumar17

Hi All,

Sorry for the delay in posting questions. This question had appeared in CL FLT -10. I am not able to understand the solution. Could anybody explain me the solution?

The lines y=3x and x+y=40 and the x-axis bound the triangular area. Find the total number of points on or inside the triangle with integral number of co-ordinates.
a. 600
b. 641.
c. 655
d. 662

Thanks,
Mohit

Is it 600... with zeros included 641.

[thakarsagar]

Originally Posted by Mohit102

Is it 600... with zeros included 641.

Soln - 641

Proof:


Using the line y=3x we have the following solutions...
0-> 0
1-> 0,1,2,3
2-> 0,1,2,3,4,5,6
.
.
.
10-> 0 ,1 ... , 30

Hence, 1+4+7+....31 = 176 -----------------------------(1)

Using the line x+y = 40, we have the following solutions.
40-> 1
39-> 0,1
38-> 0,1,2
.
.
.
11->0,1,2,...29
Hence, it's 1+2 +3 +...+ 29 = 465 ------------------------(2)
Adding (1) & (2) we get 641.