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

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Re: GMAT CLUB OLYMPICS: If k 0 and (k - 1/k) [#permalink]
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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.
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Re: GMAT CLUB OLYMPICS: If k 0 and (k - 1/k) [#permalink]
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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
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Re: GMAT CLUB OLYMPICS: If k 0 and (k - 1/k) [#permalink]
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