GMAT CLUB OLYMPICS: If k 0 and (k - 1/k)

Question

If k ≠ 0 and  \sqrt{k - \frac{1}{k}} + \sqrt{1 - \frac{1}{k}} = k , then what is the value of k ?

A.  \frac{1-\sqrt{5}}{2}

B.  \frac{\sqrt{5}-1}{2}

C.  \sqrt{5}-1

D.  \frac{1+\sqrt{5}}{2}

E.  \sqrt{5}+1

M37-33

Bunuel · Accepted Answer

Official Solution:

If  k ≠ 0  and  \sqrt{k - \frac{1}{k}} + \sqrt{1 - \frac{1}{k}} = k , then what is the value of  k  ?

A.  \frac{1-\sqrt{5}}{2} 
B.  \frac{\sqrt{5}-1}{2} 
C.  \sqrt{5}-1 
D.  \frac{1+\sqrt{5}}{2} 
E.  \sqrt{5}+1

Step 1: Define variables 
 
Let  \sqrt{k - \frac{1}{k}}=x  and  \sqrt{1 - \frac{1}{k}}=y . We will use these variables to simplify the given equation.
 
 Step 2: Write two equations 
 
From the given equation, we have:
 
 x+y=k ;
 
Multiply both sides by  (x-y)  to obtain  (x-y)(x+y)=k(x-y) . 
 
 Step 3: Simplify the equation 
 
 x^2-y^2=k(x-y) ; 
 
 (k - \frac{1}{k})-(1 - \frac{1}{k})=k(x-y) ; 
 
 k-1=k(x-y) ;
 
 x-y=\frac{k-1}{k} . 
 
 Step 4: Solve the system of equations 
 
We now have two equations:  x+y=k  and  x-y=\frac{k-1}{k} .
 
Add these two equations:  2x=k+\frac{k-1}{k}=k+1-\frac{1}{k} . 
 
 Step 5: Substitute and solve for  x  
 
Substitute  \sqrt{k - \frac{1}{k}}=x  into  2x=(k-\frac{1}{k})+1 :
 
 2x=x^2+1 ;
 
 (x-1)^2=0 ;
 
 x=1 .
 
 Step 6: Solve for  k  
 
Now we have  \sqrt{k - \frac{1}{k}}=x=1 ;
 
Squaring both sides:  k - \frac{1}{k}=1 ;
 
 k^2 - k-1=0 ;
 
This gives us two possible values for  k :  k = \frac{1-\sqrt{5}}{2}  or  k = \frac{1+\sqrt{5}}{2} .
 
 Step 7: Determine the valid value of  k  
 
The first root is negative, which is not valid because the sum of two non-negative values ( \sqrt{k - \frac{1}{k}}  and  \sqrt{1 - \frac{1}{k}} ) cannot equal a negative number ( k ). Therefore, the valid value for  k  is  k = \frac{1+\sqrt{5}}{2} .

Answer: D

Berdiyor · Answer

Hello everyone

Here is my solution 
The answer is D

Kinshook · Answer

Asked: If k ≠ 0 and \sqrt{k - \frac{1}{k}} + \sqrt{1 - \frac{1}{k}} = k , then what is the value of k ? \sqrt{k - \frac{1}{k}} + \sqrt{1 - \frac{1}{k}} = k \sqrt{k - \frac{1}{k}} = k - \sqrt{1 - \frac{1}{k}} Squaring both sides k - 1/k = k^2 + (1-1/k) - 2k\sqrt{1 - \frac{1}{k}} k^2 - k + 1 = 2k\sqrt{1 - \frac{1}{k}} (k^2 - k + 1)^2 = 4k^2 (1-1/k) = 4k^2 - 4k = 4k(k-1) (k(k-1) + 1)^2 = 4k(k-1) Let k(k-1) = t (t+1)^2 = 4t t^2 + 2t + 1 = 4t t^2 - 2t + 1 = 0 (t - 1)^2 = 0 t = 1 k(k-1) = 1 k^2 - k - 1 = 0 k = \frac{(1 +- \sqrt{1 + 4})}{2} k = \frac{(1 + - \sqrt{5})}{2} Since (k - 1/k) >=0 & (1- 1/k) >=0 (1- 1/k) >=0 (k - 1)/k >=0 k >=1 or k<0 (k - 1/k) >=0 (k^2 - 1)/k >=0 (k-1)(k+1)/k >=0 k>=1 or -1<=k<0 Combining k>=1 or -1<=k<0 k = \frac{1-\sqrt{5}}{2} = -.61 k = \frac{1+\sqrt{5}}{2} = 1.61>1 Since summation of 2 square roots is a positive value, k = \frac{1+\sqrt{5}}{2} = 1.61>1 IMO D

cs15811581 · Answer

The best way is to substitute the values.

A.  \frac{1-\sqrt{5}}{2}  will be a negative value. Discard

B.  \frac{\sqrt{5}-1}{2}=1.2/2=0.6 
 \sqrt{0.6 - \frac{1}{0.6}} + \sqrt{1 - \frac{1}{0.6}} = 0.6 
Discard as the second root comes out to be negative.

C.  \sqrt{5}-1=1.2 
 \sqrt{1.2 - \frac{1}{1.2}} + \sqrt{1 - \frac{1}{1.2}} = 1.2 
 \sqrt{1.2 - 0.83 }+ \sqrt{1 -0.83} = k 
 \sqrt{0.36 }+ \sqrt{0.16} = 1.2 
 0.6+0.4=1.2 ….Discard

D.  \frac{1+\sqrt{5}}{2} 
 \sqrt{1.6 - \frac{1}{1.6}} + \sqrt{1 - \frac{1}{1.6}} = 1.6 
 \sqrt{1.6 - 0.625}+ \sqrt{1 -0.625} = 1.6 
 \sqrt{1}+ \sqrt{0.4} = 1.6 
 1+0.62=1.6 ….
Almost the same
Correct

E.  \sqrt{5}+1

D

sivatx2 · Answer

https://gmatclub.com/forum/download/file.php?mode=view&id=63218 SquareRootProblem.png

bumpbot · Answer

Automated notice from GMAT Club BumpBot:

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

This post was generated automatically.

GMAT CLUB OLYMPICS: If k 0 and (k - 1/k)

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

GMAT CLUB OLYMPICS: If k 0 and (k - 1/k)

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards