Last visit was: 24 Jul 2024, 05:19 It is currently 24 Jul 2024, 05:19
Close
GMAT Club Daily Prep
Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track
Your Progress

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History
Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.
Close
Request Expert Reply
Confirm Cancel
SORT BY:
Date
Tags:
Show Tags
Hide Tags
Math Expert
Joined: 02 Sep 2009
Posts: 94605
Own Kudos [?]: 643497 [13]
Given Kudos: 86734
Send PM
Most Helpful Reply
Intern
Intern
Joined: 02 Feb 2019
Posts: 39
Own Kudos [?]: 84 [7]
Given Kudos: 14
Location: Uzbekistan
GMAT 1: 640 Q50 V25
GMAT 2: 670 Q51 V28
GPA: 3.4
Send PM
Math Expert
Joined: 02 Sep 2009
Posts: 94605
Own Kudos [?]: 643497 [0]
Given Kudos: 86734
Send PM
General Discussion
GMAT Club Legend
GMAT Club Legend
Joined: 03 Jun 2019
Posts: 5312
Own Kudos [?]: 4248 [3]
Given Kudos: 161
Location: India
GMAT 1: 690 Q50 V34
WE:Engineering (Transportation)
Send PM
Re: GMAT CLUB OLYMPICS: If k 0 and (k - 1/k) [#permalink]
2
Kudos
1
Bookmarks
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

Originally posted by Kinshook on 20 Aug 2021, 09:37.
Last edited by Kinshook on 21 Aug 2021, 09:18, edited 1 time in total.
Intern
Intern
Joined: 08 Aug 2021
Posts: 31
Own Kudos [?]: 119 [2]
Given Kudos: 3
Location: India
Send PM
Re: GMAT CLUB OLYMPICS: If k 0 and (k - 1/k) [#permalink]
1
Kudos
1
Bookmarks
Bunuel wrote:
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. \(\sqrt{5}+1\)


 


This question was provided by GMAT Club
for the GMAT Club Olympics Competition

Win over $40,000 in prizes such as Courses, Tests, Private Tutoring, and more

 




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
Senior Manager
Senior Manager
Joined: 23 Oct 2015
Posts: 297
Own Kudos [?]: 271 [1]
Given Kudos: 33
Location: United States (NH)
Concentration: Leadership, Technology
Schools: Wharton '25
WE:Information Technology (Non-Profit and Government)
Send PM
Re: GMAT CLUB OLYMPICS: If k 0 and (k - 1/k) [#permalink]
1
Kudos


Attachment:
SquareRootProblem.png
SquareRootProblem.png [ 125.04 KiB | Viewed 2335 times ]
User avatar
Non-Human User
Joined: 09 Sep 2013
Posts: 34061
Own Kudos [?]: 853 [0]
Given Kudos: 0
Send PM
Re: GMAT CLUB OLYMPICS: If k 0 and (k - 1/k) [#permalink]
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
GMAT Club Bot
Re: GMAT CLUB OLYMPICS: If k 0 and (k - 1/k) [#permalink]
Moderator:
Math Expert
94605 posts