Mensuration (मोजमाप) MCQ Test — Class 8 Maharashtra SSC

Solution:

The area of a trapezium is given by A = 1/2 × h × (sum of parallel sides). — A = 1/2 × h × (a + b)

The term (a+b)/2 represents the average length of the parallel sides. If we imagine stretching the shorter parallel side to match the longer, or vice versa, the 'effective' base length becomes the average. Multiplying this average base by the height 'h' indeed gives the area, just like a rectangle with that average base and height 'h'.

Thus, Ravi's explanation correctly captures the essence of how the formula works conceptually.

Solution:

A regular hexagon can be divided into simpler shapes. If divided into two identical trapeziums and one rectangle, you'd need the dimensions for each of these shapes.

The side length of the hexagon would give you the parallel sides of the trapeziums and one side of the rectangle. To find the area of the trapeziums, you would also need their height (the perpendicular distance between their parallel sides).

This height is typically derived from the hexagon's geometry, often related to its apothem or the overall width. Therefore, the side length and the height of the trapeziums are essential.

Solution:

Surface area is the total area of all the faces (surfaces) of a three-dimensional object. For a cube, it has 6 identical square faces.

Statement A is false: When a cube is cut into two cuboids, new surfaces are exposed (the cut faces), increasing the total surface area.

Statement B is true: By definition, the total surface area of a cube is the sum of the areas of its six square faces. If 'a' is the side length, TSA = 6a².

Statement C is false: Surface area is measured in square units (e.g., cm²), while volume is in cubic units (e.g., cm³); they cannot be equal.

Statement D is false: If the side length 'a' is doubled to '2a', the new surface area becomes 6(2a)² = 6(4a²) = 24a², which is four times the original surface area (6a²), not double.

Solution:

The quadrilateral ABCD can be divided into two triangles by the diagonal AC: triangle ABC and triangle ADC.

Area of triangle ABC = 1/2 × base × height = 1/2 × AC × (perpendicular from B to AC) = 1/2 × 24 m × 8 m = 96 m².

Area of triangle ADC = 1/2 × base × height = 1/2 × AC × (perpendicular from D to AC) = 1/2 × 24 m × 13 m = 156 m².

Total area of the field = Area of triangle ABC + Area of triangle ADC = 96 m² + 156 m² = 252 m².

Solution:

Original volume of the cuboid (V₁) = length × breadth × height = 10 cm × 5 cm × 4 cm = 200 cm³.

New length = 2 × original length = 2 × 10 cm = 20 cm.

New height = original height / 2 = 4 cm / 2 = 2 cm.

New breadth remains the same = 5 cm.

New volume of the cuboid (V₂) = new length × new breadth × new height = 20 cm × 5 cm × 2 cm = 200 cm³.

Wait, let me re-calculate: New length = 20 cm, New breadth = 5 cm, New height = 2 cm. V₂ = 20 × 5 × 2 = 200 cm³.

My calculation for V₂ was 200 cm³. Original V₁ = 200 cm³. This means the volume remains the same.

Let me re-check my calculations and the question. Original L=10, B=5, H=4. V1 = 10*5*4 = 200. New L=2*10=20. New H=4/2=2. New B=5. V2 = 20*5*2 = 200. So the volume remains the same. The correct answer should be D. My previous correct answer was wrong.

Let's re-evaluate. Original V = l × b × h. New V' = (2l) × b × (h/2) = 2 × l × b × h / 2 = l × b × h. So, the new volume is equal to the original volume.

Original volume (V_original) = 10 cm × 5 cm × 4 cm = 200 cm³.

New length (L') = 2 × 10 cm = 20 cm.

New height (H') = 4 cm / 2 = 2 cm.

New breadth (B') = 5 cm (unchanged).

New volume (V_new) = L' × B' × H' = 20 cm × 5 cm × 2 cm = 200 cm³.

Comparing V_new and V_original, we find that V_new = V_original. Therefore, the new volume remains the same as the original volume.

Solution:

A closed cylinder has three distinct surfaces: two flat circular bases (top and bottom) and one curved lateral surface.

The area of each circular base is given by πr², where 'r' is the radius.

The area of the curved surface is given by 2πrh, where 'r' is the radius and 'h' is the height.

Therefore, the Total Surface Area (TSA) of a closed cylinder is the sum of the areas of the two circular bases and the curved surface area: TSA = 2(πr²) + 2πrh = 2πr(r + h).

Solution:

Length is a one-dimensional measurement, typically expressed in meters (m).

Area is a two-dimensional measurement (length × width), which means its unit will be the unit of length squared.

Volume is a three-dimensional measurement (length × width × height), and its unit will be the unit of length cubed.

Since the field is 2-dimensional (length and width), its area should be expressed in square meters (m²). Hectares are also units of area, but m² is the direct standard unit derived from meters.

Solution:

Volume of the large rectangular box (V_large) = length × width × height = 60 cm × 40 cm × 30 cm = 72,000 cm³.

Volume of one small cubic box (V_small) = side × side × side = 10 cm × 10 cm × 10 cm = 1,000 cm³.

Number of small boxes that can be packed = V_large / V_small = 72,000 cm³ / 1,000 cm³ = 72.

Solution:

The area of the rectangle is correctly calculated as length × width = 10 cm × 4 cm = 40 cm².

The formula for the area of a trapezium is correctly used: 1/2 × height × (sum of parallel sides).

Deepak's calculation for the trapezium is 1/2 × 4 × (4 + 6). Here, he used '4' as the height of the trapezium. However, the problem states 'trapezium attached to its side (parallel sides 4 cm and 6 cm, height 3 cm)'. Deepak incorrectly substituted 4 (likely the width of the rectangle) for the actual height of the trapezium, which is given as 3 cm.

The correct calculation for the trapezium's area should be 1/2 × 3 × (4 + 6) = 1/2 × 3 × 10 = 15 cm².

Therefore, Deepak made a mistake in identifying the correct height for the trapezium.

Solution:

The formula for the volume of a cube is V = a³, where 'a' is the length of one side.

We are given the volume V = 512 cm³.

So, a³ = 512 cm³.

To find 'a', we need to calculate the cube root of 512. We can test the options or recall common cube roots: 6³ = 216, 7³ = 343, 8³ = 512, 9³ = 729.

Therefore, a = 8 cm.