Question

Choose the correct alternative answer for each of the following questions.

(1) The ratio of circumference and area of a circle is 2:7. Find its circumference.
(A) $14\mathrm{\pi }$ (B) $\frac{7}{\mathrm{\pi }}$ (C) 7$\mathrm{\pi }$ (D) $\frac{14}{\mathrm{\pi }}$

(2) If measure of an arc of a circle is 160^° and its length is 44 cm, find the circumference of the circle.

(A) 66 cm (B) 44 cm (C) 160 cm (D) 99 cm

(3) Find the perimeter of a sector of a circle if its measure is 90^° and radius is 7 cm .

(A) 44 cm (B) 25 cm (C) 36 cm (D) 56 cm

(4) Find the curved surface area of a cone of radius 7 cm and height 24 cm.

(A) 440 cm² (B) 550 cm² (C) 330 cm² (D) 110 cm²

(5) The curved surface area of a cylinder is 440 cm 2 and its radius is 5 cm. Find its height.
(A) $\frac{44}{\mathrm{\pi }}$cm (B) 22$\mathrm{\pi }$cm (C) 44$\mathrm{\pi }$cm (D) $\frac{22}{\mathrm{\pi }}$ cm

(6) A cone was melted and cast into a cylinder of the same radius as that of the base of the cone. If the height of the cylinder is 5 cm, find the height of the cone.

(A) 15 cm (B) 10 cm (C) 18 cm (D) 5 cm

(7) Find the volume of a cube of side 0.01 cm.

(A) 1 cm³ (B) 0.001 cm³ (C) 0.0001 cm³ (D) 0.000001 cm³

(8) Find the side of a cube of volume 1 m³ .

(A) 1 cm (B) 10 cm (C) 100 cm (D)1000 cm

Answer 1

(1)
Let r be the radius of the circle.

Circumference of the circle : Area of the circle = 2 : 7

$$\begin{gathered} \Rightarrow \frac{2\mathrm{\pi }r}{\mathrm{\pi }{r}^2}=\frac{2}{7} \\ \Rightarrow \frac{1}{r}=\frac{1}{7} \\ \Rightarrow r=7 \end{gathered}$$
∴ Circumference of the circle = $2\mathrm{\pi }r=2\mathrm{\pi }\times 7=14\mathrm{\pi }$

Hence, the correct answer is option (A).

(2)
Measure of arc of circle, θ = 160º

Let r be the radius of the circle.

Length of the arc = 44 cm

$$\begin{gathered} \therefore \frac{\theta }{360°}\times 2\mathrm{\pi }r=44 \\ \Rightarrow \frac{160°}{360°}\times 2\mathrm{\pi }r=44 \\ \Rightarrow 2\mathrm{\pi }r=\frac{44\times 360°}{160°}=99\mathrm{cm} \end{gathered}$$
Thus, the circumference of the circle is 99 cm.

Hence, the correct answer is option (D).

(3)
Here, r = 7 cm and θ = 90º.

∴ Perimeter of the sector = $\frac{\theta }{360°}\times 2\mathrm{\pi }r+2r=\frac{{\displaystyle 90°}}{{\displaystyle 360°}}\times 2\times \frac{22}{7}\times 7+2\times 7$ = 11 + 14 = 25 cm

Hence, the correct answer is option (B).

(4)
Radius of the cone, r = 7 cm

Height of the cone, h = 24 cm

Let l be the slant height of the cone.

$l=\sqrt{r^2+h^2}=\sqrt{7^2+{24}^2}=\sqrt{49+576}=\sqrt{625}$ = 25 cm

∴ Curved surface area of the cone = $\mathrm{\pi }rl=\frac{22}{7}\times 7\times 25$ = 550 cm²

Hence, the correct answer is option (B).

(5)
Radius of the cylinder, r = 5 cm

Let the height of the cylinder be h cm.

Curved surface area of the cylinder = 440 cm²

$$\begin{gathered} \therefore 2\mathrm{\pi }rh=440 \\ \Rightarrow h=\frac{440}{2\mathrm{\pi }r} \\ \Rightarrow h=\frac{440}{2\mathrm{\pi }\times 5}=\frac{44}{\mathrm{\pi }}\mathrm{cm} \end{gathered}$$
Thus, the height of the cylinder is $\frac{44}{\mathrm{\pi }}$ cm.

Hence, the correct answer is option (A).

(6)
Radius of the cone = Radius of the cylinder = r cm (Say)

Height of the cylinder, h = 5 cm

Let the height of the cone be H cm.

It is given that the cone melted and recasted into a cylinder.

∴ Volume of the cone = Volume of the cylinder

$$\begin{gathered} \Rightarrow \frac{1}{3}\mathrm{\pi }{r}^{2}H=\mathrm{\pi }{r}^{2}h \\ \Rightarrow H=3h \\ \Rightarrow H=3\times 5=15\mathrm{cm} \end{gathered}$$
Thus, the height of the cone is 15 cm.

Hence, the correct answer is option (A).

(7)
Side of the cube = 0.01 cm

∴ Volume of the cube = (Side)³ = (0.01 cm)³ = 0.000001 cm³

Hence, the correct answer is option (D).

(8)
Volume of the cube = 1 m³

∴ (Side)³ = 1 m³ = (1 m)³

⇒ Side = 1 m = 100 cm

Thus, the side of the cube is 100 cm.

Hence, the correct answer is option (C).