Question

Consider the following three statements.
Statement-1: Volume of a cylinder is equal to the $3$ times of the volume of a cone.

Statement-2: Total surface area of a cone $=$ curved surface area of the cone $+$ area of its base.

Statement-3 : Total surface area of a hemisphere of radius $r=2\pi r^2$.

Which of the following is correct?

• A. Statements 1 and 3 are true and statement 2 is false.
• B. Statements 2 and 3 are true and statement 1 is false.
• C. Statements 1 and 2 are true and statement 3 is false.
• D. Statements 2 and 3 are false and statement 1 is true.

Answer 1

The correct option is C

Statements 1 and 2 are true and statement 3 is false.

Explanation for correct option

C:

Statement 1:

Step 1: Given data

We have to prove that,

Volume of a cylinder is equal to the $3$ times of the volume of a cone

Step 2: Using the formula

Volume of cylinder$={\mathrm{\pi r}}^{2}h$

Volume of cone$=\frac{1}{3}{\mathrm{\pi r}}^{2}h$

Step 3: Evaluating the given expression

Dividing volume of cylinder by volume of cone,

$$\begin{gathered} \Rightarrow \frac{Volumeofcylinder}{Volumeofcone}=\frac{{\mathrm{\pi r}}^2\mathrm{h}}{{\displaystyle \frac{1}{3}}{\mathrm{\pi r}}^2\mathrm{h}}=3 \\ \Rightarrow Volumeofcylinder=3\times Volumeofcone \end{gathered}$$

So, volume of cylinder is equal to $3$ times the volume of cone

Thus, LHS$=$RHS

Statement 1 is correct.

Statement 2:

Step 1: Given data

We have to prove that,

Total surface area of a cone $=$ curved surface area of the cone $+$ area of its base.

Step 2: Using the formula

Total surface area of cone$={\mathrm{\pi r}}^2+\mathrm{\pi rl}$

Curved surface area of cone$=\mathrm{\pi rl}$

Area of circle$={\mathrm{\pi r}}^2$

Step 3: Evaluating the given expression

LHS$=$Total surface area of cone$={\mathrm{\pi r}}^2+\mathrm{\pi rl}$

We know that, base of a cone is a circle.

Area of base of cone$={\mathrm{\pi r}}^2$

Curved surface area of cone$=\mathrm{\pi rl}$

RHS$=\mathrm{\pi rl}+{\mathrm{\pi r}}^2=$LHS

Thus, LHS$=$RHS

Statement 2 is correct.

Explanation for incorrect options

A,B,D:

Statement 3:

Step 1: Given data

We have to prove that,

Total surface area of a hemisphere of radius $r=2\pi r^2$

Step 2: Using the formula

Curved surface area of hemisphere$=2{\mathrm{\pi r}}^2$

Area of circle$={\mathrm{\pi r}}^2$

Step 3: Evaluating the given expression

LHS$=$Total surface area of hemisphere$=$Curved surface area of hemisphere$+$Area of base of hemisphere

We know that, base of a hemisphere is a circle.

Area of base of hemisphere$={\mathrm{\pi r}}^2$

Curved surface area of hemisphere$=2{\mathrm{\pi r}}^2$

So, Total surface area of hemisphere$=2{\mathrm{\pi r}}^2+{\mathrm{\pi r}}^2=3{\mathrm{\pi r}}^2$

RHS$=2{\mathrm{\pi r}}^2$

Thus, LHS$\ne$RHS

Statement 3 is incorrect.

Statements 1 and 2 are true and statement 3 is false.

Hence, option C is correct.