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Mathematics

Assertion (A): The ratio between radius and area of a circle is 7 : 44. If the radius of this circle is doubled, the ratio between the new radius and area of the resulting circle will be 7 : 176.

Reason (R): Given : rπr2=744\dfrac{r}{πr^2} = \dfrac{7}{44}

∴ 2rr(2r)2=r2πr2=12×744=7:88\dfrac{2r}{r(2r)^2} = \dfrac{r}{2πr^2} = \dfrac{1}{2} \times \dfrac{7}{44} = 7 : 88

  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.

Mensuration

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Answer

A is true, R is false.

Explanation

Let r be the radius of the circle.

rπr2=744\dfrac{r}{πr^2} = \dfrac{7}{44}

⇒1πr=744⇒1227×r=744⇒r=44×722×7⇒r=2⇒ \dfrac{1}{πr} = \dfrac{7}{44}\\[1em] ⇒ \dfrac{1}{\dfrac{22}{7} \times r} = \dfrac{7}{44}\\[1em] ⇒ r = \dfrac{44 \times 7}{22 \times 7}\\[1em] ⇒ r = 2

If the radius of this circle is doubled,

r' = 2r = 2 x 2 = 4

=new radiusnew area=r′πr′2=4227×42=4×722×16=7176= \dfrac{\text{new radius}}{\text{new area}}\\[1em] = \dfrac{r'}{πr'^2}\\[1em] = \dfrac{4}{\dfrac{22}{7} \times 4^2}\\[1em] = \dfrac{4 \times 7}{22 \times 16}\\[1em] = \dfrac{7}{176}\\[1em]

∴ Assertion (A) is true.

Given : rπr2=744\dfrac{r}{πr^2} = \dfrac{7}{44}

2rr(2r)2=2r2πr2=12×744=788\dfrac{2r}{r(2r)^2} = \dfrac{2r}{2πr^2}\\[1em] = \dfrac{1}{2} \times \dfrac{7}{44}\\[1em] = \dfrac{7}{88}\\[1em]

∴ Reason (R) is false.

Hence, Assertion (A) is true, Reason (R) is false.

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