Mensuration — Statement I-II Type Questions

Statement I: An isosceles trapezium has parallel sides of 8 cm and 10 cm. If the distance between them is 6 cm, then the area of the trapezium is 54 cm²

Statement II: Area of a trapezium = $\dfrac{1}{2}$ × sum of parallel sides × distance between parallel sides.

1. Statement I is true but Statement II is false.

2. Statement I is false but Statement II is true.

3. Both Statement I and Statement II are true.

4. Both Statement I and Statement II are false.

Answer

Statement I

Parallel sides of the isosceles trapezium = 8 cm and 10 cm

Distance between the parallel sides = 6 cm

By formula,

Area of a trapezium = $\dfrac{1}{2}$ × sum of parallel sides × distance between the parallel sides

= $\dfrac{1}{2}$ × (8 + 10) × 6 cm²

= $\dfrac{1}{2}$ × 18 × 6 cm²

= 9 × 6 cm²

= 54 cm²

So, the area of the trapezium is 54 cm² and Statement I is true.

Statement II

Area of a trapezium = $\dfrac{1}{2}$ × sum of parallel sides × distance between parallel sides

This is the standard formula for the area of a trapezium, so Statement II is true.

Hence, option 3 is the correct option.

Statement I: The total surface area of a cylinder is always greater than its curved surface area.

Statement II: Total surface area of a cylinder = 2πrh

1. Statement I is true but Statement II is false.

2. Statement I is false but Statement II is true.

3. Both Statement I and Statement II are true.

4. Both Statement I and Statement II are false.

Answer

Statement I

By formula,

Curved surface area of a cylinder = 2πrh

And, total surface area of a cylinder = 2πrh + 2πr²

Since the radius r of a cylinder is always positive, 2πr² is always positive.

So, the total surface area of a cylinder is always greater than its curved surface area and Statement I is true.

Statement II

The expression 2πrh gives the curved surface area of a cylinder and not its total surface area.

The total surface area of a cylinder = 2πr(h + r)

So, Statement II is false.

Hence, option 1 is the correct option.

Statement I: In a cylinder, if the circumference of the base is 100 cm and the height of the cylinder is 10 cm, then the volume of the cylinder is 1000 cm³

Statement II: Volume of a cylinder = πr²h

1. Statement I is true but Statement II is false.

2. Statement I is false but Statement II is true.

3. Both Statement I and Statement II are true.

4. Both Statement I and Statement II are false.

Answer

Statement I

Circumference of the base of the cylinder = 100 cm

Height of the cylinder = 10 cm

Let the radius of the base of the cylinder be r cm.

By formula,

Circumference of a circle = 2πr

⇒ 2πr = 100

⇒ r = $\dfrac{100}{2\pi}$

⇒ r = $\dfrac{50}{\pi}$

By formula,

Volume of a cylinder = πr²h

$= \pi \times \left(\dfrac{50}{\pi}\right)^2 \times 10\\[1em] = \pi \times \dfrac{2500}{\pi^2} \times 10\\[1em] = \dfrac{25000}{\pi} \text{ cm}^3$

Taking $\pi = \dfrac{22}{7}$, the volume of the cylinder is about 7954.5 cm³, which is not equal to 1000 cm³.

So, Statement I is false.

Statement II

Volume of a cylinder = πr²h

This is the standard formula for the volume of a cylinder, so Statement II is true.

Hence, option 2 is the correct option.