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05 Dec 2018, 09:22
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E
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Difficulty:
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Which of the following are pairs of numbers that are reciprocals of each other?
(I) \(\frac{-5}{12}\) and \(\frac{12}{5}\)
(II) \(\frac{\sqrt{7}}{2}\) and \(\frac{2\sqrt{7}}{7}\)
(III) \(\sqrt{17}\) - 4 and \(\sqrt{17}\) + 4
(A) II only
(B) III only
(C) I and II
(D) I and III
(E) II and III
(I) \(\frac{-5}{12}\) and \(\frac{12}{5}\)
(II) \(\frac{\sqrt{7}}{2}\) and \(\frac{2\sqrt{7}}{7}\)
(III) \(\sqrt{17}\) - 4 and \(\sqrt{17}\) + 4
(A) II only
(B) III only
(C) I and II
(D) I and III
(E) II and III
Chethan92
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05 Dec 2018, 09:34
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A can be eliminated as one is positive and the other is negative.
Now \(\sqrt{7}/2\) = reciprocal of \(2\sqrt{7}/7\) = 1/\(2\sqrt{7}/7\) = \(7/2\sqrt{7}\) = \(\sqrt{7}\sqrt{7}/2\sqrt{7}\) = \(\sqrt{7}/2\)
Hence 2 is true.
Now rationalizing 3,
\(1/\sqrt{17}+4\) = \(\sqrt{17}-4/17-16\) = \(\sqrt{17}-4\)
Hence 3 is also true.
E is the answer.
Now \(\sqrt{7}/2\) = reciprocal of \(2\sqrt{7}/7\) = 1/\(2\sqrt{7}/7\) = \(7/2\sqrt{7}\) = \(\sqrt{7}\sqrt{7}/2\sqrt{7}\) = \(\sqrt{7}/2\)
Hence 2 is true.
Now rationalizing 3,
\(1/\sqrt{17}+4\) = \(\sqrt{17}-4/17-16\) = \(\sqrt{17}-4\)
Hence 3 is also true.
E is the answer.
05 Dec 2018, 19:13
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CAMANISHPARMAR
One number is the reciprocal of another if their product equals 1
(I) \(\frac{-5}{12}\) and \(\frac{12}{5}\)
\((\frac{-5}{12}*\frac{12}{5})=\frac{-60}{60}=-1\)
NO. The product is \(-1\)
(II) \(\frac{\sqrt{7}}{2}\) and \(\frac{2\sqrt{7}}{7}\)
\((\frac{\sqrt{7}}{2}* \frac{2\sqrt{7}}{7})=\frac{2*\sqrt{7}*\sqrt{7}}{2*7}=\frac{2*7}{2*7}=1\)
That works.
(III) \(\sqrt{17}-4\) and \(\sqrt{17}+4\)
• Product = 1?
Difference of squares:\((a+b)(a-b)=a^2-b^2\)
\((\sqrt{17}-4)(\sqrt{17}+4)=(17-16)=1\)
• OR: \((n * \frac{1}{n})=1\)?
\((\sqrt{17}+4)*\frac{1}{(\sqrt{17}+4)}=\)
Rationalize the denominator of the second term
\((\sqrt{17}+4)*(\frac{1}{(\sqrt{17}+4)}*\frac{\sqrt{17}-4}{\sqrt{17}-4})=\)
\((\sqrt{17}+4)*(\frac{\sqrt{17}-4}{17-16})=\)
\(\frac{(\sqrt{17}-4)(\sqrt{17}+4)}{1}=(\sqrt{17}-4)(\sqrt{17}+4)=(17-16)=1\)
That works.
Roman numerals II and III are reciprocal pairs.
Answer E
