Practice the questions given in the worksheet on mean proportional
1. Find the mean proportional of the following sets of positive numbers:
(i) x\(^{3}\)y, xy\(^{3}\)
(ii) (x  y)\(^{2}\), (x + y)\(^{3}\)
2. Find the mean proportional of the following:
(i) 9, 16
(ii) 4\(\frac{4}{7}\), 3\(\frac{1}{2}\)
(iii) (a + b)(a  b)\(^{3}\), (a + b)\(^{3}\)(a  b)
(iv) \(\frac{x^{2}}{4ab}\), \(\frac{a}{by^{2}}\)
3. Find the mean proportional between
(i) 5 and 45
(ii) 0.04 and 0.0036
(iii) 0.25 and 6.25
4. Find the third proportional of the following:
(i) 0.5, 0.25
(ii) a\(^{2}\)b, ab\(^{2}\)
(iii) \(\frac{x}{y}\) + \(\frac{y}{x}\), \(\frac{x}{y}\)
5. (i) If the mean proportional of a and c is b then prove that a, c, a\(^{2}\) + b\(^{2}\) and b\(^{2}\) + c\(^{2}\) are proportional.
(ii) If b is the mean proportional of a and c, prove that the mean proportional of a\(^{2}\) + b\(^{2}\) and b\(^{2}\) + c\(^{2}\) is ab + bc.
(iii) If b is the mean proportional of a and c, prove that
\[\left ( \frac{ab + bc + ca}{a + b + c} \right )^{3} = abc\]
Answers for the worksheet on mean proportional are given below.
Answers
1. (i) x\(^{2}\)y\(^{2}\)
(ii) x\(^{2}\)  y\(^{2}\)
2. (i) 12
(ii) 4
(iii) \((a^{2}  b^{2})^{2}\)
(iv) \(\frac{x}{2by}\)
3. (i) 15
(ii) 0.012
(iii) 1.25
4. (i) 0.125
(ii) b\(^{3}\)
(iii) \(\frac{x^{3}}{y(x^{2} + y^{2})}\)
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