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26 Jun 2021, 07:24
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D
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
55%
(hard)
Question Stats:
51% (01:55) correct
49%
(02:14)
wrong
based on 67
sessions
History
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Time
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Not Attempted Yet
If \(k > 0\), what is the value of \( k\)?
(1) \(k^2 − \frac{k\sqrt{2}}{4} − \frac{1}{4} = 0\)
(2) \(k^2 − k\sqrt{2} + \frac{1}{2} = 0\)
(1) \(k^2 − \frac{k\sqrt{2}}{4} − \frac{1}{4} = 0\)
(2) \(k^2 − k\sqrt{2} + \frac{1}{2} = 0\)
27 Jun 2021, 00:32
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sjuniv32
(1) \(k^2 − \frac{k\sqrt{2}}{4} − \frac{1}{4} = 0………(k)^2-\frac{k}{2\sqrt{2}}-\frac{1}{4}=0\)
\(k^2-\frac{k}{\sqrt{2}}+\frac{1}{2\sqrt{2}}-\frac{1}{2\sqrt{2}}*\frac{1}{\sqrt{2}}=0…………(k-\frac{1}{\sqrt{2}})(\frac{1}{2\sqrt{2}}=0\)
So k can be \(\frac{1}{\sqrt{2}}\) or \(\frac{-1}{2\sqrt{2}}\)
But k>0, so k = \(\frac{1}{\sqrt{2}}\)
Sufficient
If you know Descartes’ rule of sign although not tested, the answer is immediate.
Descartes’ rule of sign says that change of signs in the polynomial f(x) when terms in it are placed in order of degree of x will give the number of positive roots.
Here change in signs are + to - and then - to - : +\(k^2 − \frac{k\sqrt{2}}{4} − \frac{1}{4} = 0\) Thus one positive root.
Sufficient as we can find out the value.
(2) \(k^2 − k\sqrt{2} + \frac{1}{2} = 0…………(k)^2-2*k*\frac{1}{\sqrt{2}}+(\frac{1}{\sqrt{2}})^2=0……(k-\frac{1}{\sqrt{2}})^2=0\)
Sufficient
D
27 Jun 2021, 06:35
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sjuniv32
When we factor the quadratic in Statement 1, our factorization will look like:
(k + some number)(k - some number) = 0
because we want our two numbers to give us a negative product of -1/4. If that's what our factorization looks like, one solution will be positive, the other negative, and since we know k > 0, Statement 1 will give us just one solution for k. So it's sufficient.
When we factor the quadratic in Statement 2, our factorization will look like:
(k - some number)(k - some number) = 0
because we have a positive product (+1/2) and a negative sum (-√2). Most of the time, you'll get two different positive solutions to this equation, because most of the time the two factors will be different, and each will give a different solution for k. You'll only get exactly one solution for k if both factors are identical. So that's the only way a Statement like this could be sufficient if we want to pin down an exact value of k, and that's all we need to check. If the two numbers are positive and identical, and multiply to 1/2, they must each be √(1/2) = √2/2. If the factorization was indeed:
(k - √2/2)(k - √2/2)
then multiplying out, we get a sum of -√2/2 - √2/2 = -2√2/2 = √2 in front of the 'k' term, so this does turn out to be the correct factorization, and again we get only one solution so the answer is D.
