Question : If the perimeter of circle A is equal to the perimeter of semicircle B, what is the ratio of their areas?
Option 1: $\left (x+ 2\right)^{2} : 2\pi ^{2}$
Option 2: $2\pi ^{2} : \left ( x+2 \right)^{2}$
Option 3: $\left (\pi +2 \right )^{2} : 4\pi ^{2}$
Option 4: $4\pi ^{2}:\left ( \pi +2 \right)^{2}$
Correct Answer: $\left (\pi +2 \right )^{2} : 4\pi ^{2}$
Solution : Let the radius of circle A be r1 units and the radius of semicircle B is r2 units. According to the question, 2$\pi$r1 = $\pi$r2 + 2r2 ⇒ 2$\pi$r1 = r2($\pi$+2) ⇒ $\frac{r_{1}}{r_{2}}=\frac{\pi+2}{2\pi}$ $\therefore$ Ratio of their areas = $\frac{\pi(r_{1})^{2}}{\pi(r_{2})^{2}} = \frac{(\pi+2)^{2}}{(2\pi)^{2}}=(\pi+2)^{2} : 4\pi^{2}$ Hence, the correct answer is $(\pi+2)^{2} : 4\pi^{2}$.
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Question : If the perimeter of the circle is 13.2 cm, then the radius of the circle is___________. $\left(\right.$ Take $\left.\pi=\frac{22}{7}\right)$
Option 1: 6.6 cm
Option 2: 4.2 cm
Option 3: 2.1 cm
Option 4: 3.3 cm
Question : If $x^2-3 x+1=0$, then the value of $\left(x^4+\frac{1}{x^2}\right) \div\left(x^2+1\right)$ is:
Option 1: 5
Option 2: 6
Option 3: 9
Option 4: 7
Question : If $\frac{1}{x^{2}}+x^{2}$ represents the radius of circle $P$ and $\frac{1}{x}+x=17$, which of the following best approximates the circumference of circle $P$?
Option 1: $287\pi$
Option 2: $547\pi$
Option 3: $574\pi$
Option 4: $278\pi$
Question : If $x^2-5 x+1=0$, then the value of $\left(x^4+\frac{1}{x^2}\right) \div\left(x^2+1\right)$ is:
Option 1: 21
Option 2: 22
Option 3: 25
Option 4: 24
Question : What is $\frac{\left (x^{2}-y^{2} \right)^{3}+\left (y^{2}-z^{2} \right )^{3}+\left (z^{2}-x^{2} \right )^{3}}{\left (x-y \right)^{3}+\left (y-z \right )^{3}+\left (z-x \right)^{3}}?$
Option 1: $\frac{(x+y)(y+z)}{(x+z)}$
Option 2: $(x+y)^3(y+z)^3(z+x)^3$
Option 3: $(x+y)(y+z)(z+x)$
Option 4: $(x+y)(y+z)$
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