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Mathematics

Assertion (A) : The mean proportion of 3−2 and 3+2\sqrt{3} - \sqrt{2} \text{ and } \sqrt{3} + \sqrt{2} is 1.

Reason (R) : Mean proportion of x = (a - b) and y = (a + b) is 12(x+y)\dfrac{1}{2}(x + y).

  1. A is true, R is false.

  2. A is false, R is true.

  3. Both A and R are true.

  4. Both A and R are false.

Ratio Proportion

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Answer

Let mean proportion of 3−2 and 3+2\sqrt{3} - \sqrt{2} \text{ and } \sqrt{3} + \sqrt{2} be x.

∴3−2x=x3+2⇒x2=(3−2)(3+2)⇒x2=3+6−6−2⇒x2=1⇒x=1.\therefore \dfrac{\sqrt{3} - \sqrt{2}}{x} = \dfrac{x}{\sqrt{3} + \sqrt{2}} \\[1em] \Rightarrow x^2 = (\sqrt{3} - \sqrt{2})(\sqrt{3} + \sqrt{2}) \\[1em] \Rightarrow x^2 = 3 + \sqrt{6} - \sqrt{6} - 2 \\[1em] \Rightarrow x^2 = 1 \\[1em] \Rightarrow x = 1.

∴ Assertion (A) is true.

Let mean proportion between x and y be z.

∴xz=zy⇒z2=xy⇒z=xy⇒z=(a−b)(a+b)⇒z=a2−b2.\therefore \dfrac{x}{z} = \dfrac{z}{y} \\[1em] \Rightarrow z^2 = xy \\[1em] \Rightarrow z = \sqrt{xy} \\[1em] \Rightarrow z = \sqrt{(a - b)(a + b)} \\[1em] \Rightarrow z = \sqrt{a^2 - b^2}.

According to reason (R) :

Mean proportion between x and y is

12(x+y)=12[(a−b)+(a+b)]=12×2a\dfrac{1}{2}(x + y) = \dfrac{1}{2}[(a - b) + (a + b)] = \dfrac{1}{2} \times 2a = a.

∴ Reason (R) is false.

Hence, Option 1 is the correct option.

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