Question 1

If a river $2.5 	ext{ m}$ deep and $45 	ext{ m}$ wide is flowing at the rate of $3.6 	ext{ km}$ per hour, then the amount of water that runs into the sea per minute is

Accepted Answer

Concept & Formula: The volume of water flowing through a cross-section in a given time equals the area of the cross-section multiplied by the distance the water travels in that time (speed). Volume per minute = Width × Depth × Speed per minute Step-by-Step Solution: 1. Convert the speed to meters per minute: Speed = 3.6 km/hr = 3600 m / 60 min = 60 m/min 2. Calculate the cross-sectional area of the river: Area = 2.5 m (depth) × 45 m (width) = 112.5 m^2 3. Calculate the volume per minute: Volume = Area × Speed per minute Volume = 112.5 m^2 × 60 m/min = 6750 m^3 Exam Strategy & Shortcut: Recognize that 3.6 km/hr = 1 m/s. Since we need the volume per minute, the speed is simply 1 m/s × 60 s = 60 m/min. Multiply 45 × 2.5 × 60 \implies 45 × 150 = 6750. Common Pitfall: Failing to convert the time and distance units uniformly before multiplying (e.g., mixing km/hr with meters). Final Answer: Therefore, the correct answer is 6750 cu. m.

Question 2

The sum of the length, breadth and depth of a cuboid is $19 	ext{ cm}$ and its diagonal is $5\sqrt{5} 	ext{ cm}$. It surface area is

Accepted Answer

Concept & Formula: This problem relies on the algebraic identity connecting the sum of three terms, the sum of their squares, and their product sums. (l+b+h)^2 = l^2 + b^2 + h^2 + 2(lb + bh + hl) Where 2(lb + bh + hl) is the Total Surface Area, and \sqrt{l^2 + b^2 + h^2} is the diagonal. Step-by-Step Solution: 1. Identify the given values: Sum of dimensions: l + b + h = 19 cm Diagonal: \sqrt{l^2 + b^2 + h^2} = 5\sqrt{5} cm 2. Square the diagonal to find the sum of squares: l^2 + b^2 + h^2 = (5\sqrt{5})^2 = 25 × 5 = 125 3. Use the algebraic identity: (l+b+h)^2 = l^2 + b^2 + h^2 + Total Surface Area (19)^2 = 125 + Total Surface Area 361 = 125 + Total Surface Area 4. Solve for the Total Surface Area: Total Surface Area = 361 - 125 = 236 cm^2 Exam Strategy & Shortcut: Memorize the relationship: Surface Area = (Sum of sides)^2 - (Diagonal)^2. Plug in the numbers instantly: 19^2 - (5\sqrt{5})^2 = 361 - 125 = 236. Common Pitfall: Forgetting the square root in the diagonal formula and equating l^2+b^2+h^2 directly to 5\sqrt{5}, or forgetting to square the 5 when expanding (5\sqrt{5})^2. Final Answer: Therefore, the correct answer is 236 cm^2.

Question 3

The sum of perimeters of the six faces of a cuboid is $72 	ext{ cm}$ and the total surface area of the cuboid is $16 	ext{ cm}^2$. Find the longest possible length that can be kept inside the cuboid

Accepted Answer

Concept & Formula: The longest possible length inside a cuboid is its spatial diagonal. We use the same algebraic identity as the previous problem, relating the sum of dimensions to the diagonal and surface area. Diagonal = \sqrt{l^2 + b^2 + h^2} (l+b+h)^2 = l^2 + b^2 + h^2 + Surface Area Step-by-Step Solution: 1. Analyze the sum of perimeters: A cuboid has 6 faces: 2 of (l × b), 2 of (b × h), and 2 of (h × l). Perimeters are 2(l+b), 2(b+h), and 2(h+l). Sum of all 6 perimeters = 2[2(l+b) + 2(b+h) + 2(h+l)] = 4l + 4b + 4b + 4h + 4h + 4l = 8(l+b+h). 8(l+b+h) = 72 \implies (l+b+h) = 9 cm. 2. Use the surface area given: 2(lb + bh + hl) = 16 cm^2. 3. Apply the algebraic identity to find the sum of squares: (l+b+h)^2 = (l^2 + b^2 + h^2) + 2(lb + bh + hl) 9^2 = (l^2 + b^2 + h^2) + 16 81 = (l^2 + b^2 + h^2) + 16 \implies (l^2 + b^2 + h^2) = 65 4. Calculate the longest length (diagonal): Diagonal = \sqrt{65} \approx 8.062 cm. The closest given option is 8.05 cm. Exam Strategy & Shortcut: Recognize that "sum of perimeters of 6 faces" always equals 8 × (sum of length, breadth, and height). Thus, Sum of sides = (72/8) = 9. Then, Diagonal = \sqrt{9^2 - 16} = \sqrt{65} \approx 8.06. Common Pitfall: Incorrectly mapping the sum of the perimeters. Some students assume the sum of the 12 edges is 4(l+b+h) and equate that to 72, which leads to a completely wrong dimension sum. Final Answer: Therefore, the correct answer is 8.05 cm.

Question 4

Length of a rectangular solid is increased by $10\%$ and breadth is decreased by $10\%$. Then the volume of the solid

Accepted Answer

Concept & Formula: When two multiplying variables are changed by +x% and -x%, the net effect on their product is always a decrease of (x^2/100)%. Since height is unmentioned, it is assumed constant, meaning the volume scales directly with the base area. Net Change = ( (x^2/100) ) % Decrease Step-by-Step Solution: 1. Set up the initial variables: Let the original dimensions be L, B, and H. Original Volume V_1 = L × B × H. 2. Determine the new dimensions: New Length L_2 = L + 0.10L = 1.1L New Breadth B_2 = B - 0.10B = 0.9B New Height H_2 = H (remains unchanged) 3. Calculate the new volume: V_2 = (1.1L) × (0.9B) × (H) = 0.99 LBH 4. Determine the percentage change: Change = 0.99 V_1 - V_1 = -0.01 V_1 A negative 0.01 multiplier means a 1% decrease. Exam Strategy & Shortcut: Use the successive percentage formula: A + B + (AB/100). Here, A = +10 and B = -10. Net effect = 10 - 10 + (10 × -10/100) = 0 - (100/100) = -1%. The negative sign indicates a decrease. Common Pitfall: Assuming that an equal percentage increase and decrease cancel each other out, leading to the incorrect "remains unchanged" option. Final Answer: Therefore, the correct answer is decreases by 1%.

Question 5

An open box is made by cutting the congruent squares from the corners of a rectangular sheet of cardboard of dimensions 20 cm $	imes$ 15 cm. If the side of each square is 2 cm, the total outer surface area of the box is

Accepted Answer

Concept & Surface Area of an Open Box: When an open box is formed by cutting corners from a sheet and folding up the flaps, the outer surface area of the resulting box is exactly equal to the area of the original sheet minus the area of the cut-out pieces. Formula: Surface Area = Area of Sheet - Area of Cut Squares Step-by-Step Solution: 1. Given: Dimensions of the rectangular cardboard sheet = 20 cm × 15 cm. Side of each cut square = 2 cm. 2. Calculate Original Area: The area of the entire rectangular sheet is 20 × 15 = 300 cm^2. 3. Calculate Cut Area: 4 squares are cut from the corners. The area of one square is 2 × 2 = 4 cm^2. The area of all 4 squares is 4 × 4 = 16 cm^2. 4. Calculate Final Surface Area: The remaining area forms the box, which is 300 - 16 = 284 cm^2. Exam Strategy & Shortcut: Instead of calculating the dimensions of the folded box (16 × 11 × 2) and applying the surface area formula lb + 2(bh + hl), simply subtract the area of the removed squares from the total sheet area. It is conceptually identical and much faster. Common Pitfall: Calculating the inner volume or needlessly calculating the final 3D box dimensions, which takes more time and introduces calculation errors. Final Answer: Therefore, the correct answer is 284 cm^2.

Question 6

A covered wooden box has the inner measures as 115 cm, 75 cm and 35 cm and the thickness of wood is 2.5 cm. Find the volume of the wood.

Accepted Answer

Concept & Volume of Material in a Cuboid: The volume of the material (wood) used to make a hollow object is the difference between the external volume and the internal volume. Volume of Material = External Volume - Internal Volume Step-by-Step Solution: 1. Given: Inner dimensions: l = 115 cm, b = 75 cm, h = 35 cm. Thickness t = 2.5 cm. It is a "covered" (closed) box. 2. Calculate Outer Dimensions: Since the box is closed, add 2t = 5 cm to all inner dimensions. Outer length L = 115 + 5 = 120 cm. Outer breadth B = 75 + 5 = 80 cm. Outer height H = 35 + 5 = 40 cm. 3. Calculate Volumes: External Volume = 120 × 80 × 40 = 384000 cu.cm. Internal Volume = 115 × 75 × 35 = 301875 cu.cm. 4. Calculate Volume of Wood: 384000 - 301875 = 82125 cu.cm Exam Strategy & Shortcut: To quickly compute 115 × 75 × 35, break it down or use digit sum/last digit logic if options are suitably spaced. Here, ending digits are 5, so exact calculation is safest. 115 × 75 = 8625, then 8625 × 35 = 301875. Common Pitfall: Mistaking "inner measures" for outer measures and subtracting the thickness instead of adding it to find the missing dimensions. Final Answer: Therefore, the correct answer is 82125 cu.cm.

Question 7

An open box is made of wood 3 cm thick. Its external dimensions are 1.46 m, 1.16 m and 8.3 dm. The cost of painting the inner surface of the box at 50 paise per 100 sq. cm is

Accepted Answer

Concept & Inner Surface Area of an Open Box: To find the inner surface area of an open box, you must calculate the area of the four inner walls plus the inner base. Formula: Inner Surface Area = (l × b) + 2(bh + hl) Step-by-Step Solution: 1. Given: External dimensions: L = 1.46 m = 146 cm, B = 1.16 m = 116 cm, H = 8.3 dm = 83 cm. Thickness t = 3 cm. 2. Calculate Internal Dimensions: Since it is an open box: l = 146 - 2(3) = 140 cm. b = 116 - 2(3) = 110 cm. h = 83 - 3 = 80 cm (no top thickness). 3. Calculate Inner Surface Area: Area of base = 140 × 110 = 15400 cm^2. Area of walls = 2(110 × 80 + 80 × 140) = 2(8800 + 11200) = 2(20000) = 40000 cm^2. Total Inner Surface Area = 15400 + 40000 = 55400 cm^2. 4. Calculate Cost: Rate is 50 paise (0.5 Rupees) per 100 cm^2. Total Cost = (55400/100) × 0.5 = 554 × 0.5 = 277 Rupees. Exam Strategy & Shortcut: Ensure all units are converted to centimeters first to match the given painting rate of 100 sq. cm. Be careful to convert dm (decimeters) to cm correctly (1 dm = 10 cm). Common Pitfall: Failing to unify the units before calculating the dimensions, particularly misinterpreting 8.3 dm. Final Answer: Therefore, the correct answer is ₹ 277.

Question 8

A cistern of capacity 8000 litres measures externally 3.3 m by 2.6 m by 1.1 m and its walls are 5 cm thick. The thickness of the bottom is

Accepted Answer

Concept & Internal Capacity of a Cistern: The internal capacity (volume) is found using the internal dimensions. 1 litre equals 1000 cm^3. Capacity = l × b × h Step-by-Step Solution: 1. Given: External dimensions: L = 3.3 m = 330 cm, B = 2.6 m = 260 cm, H = 1.1 m = 110 cm. Wall thickness t = 5 cm. Capacity = 8000 litres. 2. Convert Capacity to cm^3: 8000 L = 8000 × 1000 = 8000000 cm^3 3. Calculate Internal Length and Breadth: l = 330 - 2(5) = 320 cm. b = 260 - 2(5) = 250 cm. 4. Calculate Internal Height (h): l × b × h = 8000000 320 × 250 × h = 8000000 80000h = 8000000 \implies h = 100 cm 5. Determine Bottom Thickness: A cistern is open at the top. The difference between external and internal height represents the thickness of the bottom. Bottom thickness = H - h = 110 - 100 = 10 cm. 6. Convert Units: 10 cm equals 1 decimeter (dm). Exam Strategy & Shortcut: Recognize that a cistern is typically open at the top, which means only the bottom contributes to the height difference. Common Pitfall: Assuming the cistern is a closed tank and setting the bottom thickness to half the difference of the heights. Final Answer: Therefore, the correct answer is 1 dm.

Question 9

The cost of the paint is ₹ 36.50 per kg. If 1 kg of paint covers 16 square feet, how much will it cost to paint outside of a cube having 8 feet each side?

Accepted Answer

Concept & Logic: To find the cost of painting the outside of a cube, we first need to determine the total surface area to be painted. Then, find out how many kilograms of paint are required by dividing the area by the coverage per kg. Finally, multiply the required kg by the cost per kg. Surface Area = 6a^2 Total Cost = ((Surface Area/Coverage per kg)) × Cost per kg Step-by-Step Solution: 1. Calculate Surface Area: Side a = 8 feet. Surface Area = 6 × 8^2 = 6 × 64 = 384 square feet. 2. Calculate Paint Required: 1 kg covers 16 sq. feet. Paint required = (384/16) = 24 kg. 3. Calculate Total Cost: Cost per kg = 36.50. Total Cost = 24 × 36.50 Total Cost = 24 × (36 + 0.50) = 864 + 12 = 876. Exam Strategy & Shortcut: Break down the multiplication for faster mental math: 24 × 36.5 = 24 × (73/2) = 12 × 73. 12 × 70 = 840 and 12 × 3 = 36. 840 + 36 = 876. Common Pitfall: Calculating the volume (8^3 = 512) instead of the surface area (6 × 8^2 = 384). Painting always relates to surface area, not volume. Final Answer: Therefore, the correct answer is ₹ 876.

Question 10

The cost of painting the whole surface area of a cube at the rate of 13 paise per sq. cm is ₹ 343.98. Then the volume of the cube is :

Accepted Answer

Concept & Strategy: The total cost of painting is the product of the total surface area and the rate per square unit. By dividing the total cost by the rate, we can find the surface area. Then, deduce the side length and compute the volume. Total Surface Area = (Total Cost/Rate) Surface Area = 6a^2 Volume = a^3 Step-by-Step Solution: 1. Convert Units: Total Cost = ₹ 343.98 = 34398 paise. Rate = 13 paise per sq. cm. 2. Find Surface Area: Surface Area = (34398/13) Surface Area = 2646 cm^2. 3. Find Side Length (a): 6a^2 = 2646 a^2 = (2646/6) = 441 a = \sqrt{441} = 21 cm. 4. Calculate Volume: Volume = 21^3 Volume = 21 × 21 × 21 = 9261 cm^3. Exam Strategy & Shortcut: Once you determine that a = 21, you need to find 21^3. The unit digit of 21^3 is 1 × 1 × 1 = 1. Looking at the options, only 9261 ends in 1. You can confidently select this without calculating 21 × 21 × 21. Common Pitfall: Failing to align units. The rate is given in paise, but the total cost is in rupees. If you don't convert ₹ 343.98 to 34398 paise (or 13 paise to ₹ 0.13), your surface area calculation will be off by a factor of 100. Final Answer: Therefore, the correct answer is 9261 cm^3.

Question 11

If the areas of the three adjacent faces of a cuboidal box are 120 cm$^2$, 72 cm$^2$ and 60 cm$^2$ respectively, then find the volume of the box.

Accepted Answer

Concept & Volume from Adjacent Faces: As derived from the geometric properties of a cuboid, if x, y, and z are the areas of three adjacent faces, the volume V is the square root of their product. V = \sqrt{x × y × z} Step-by-Step Solution: 1. Given: x = 120, y = 72, z = 60. 2. Apply Formula: V = \sqrt{120 × 72 × 60} 3. Simplify the Calculation: Instead of multiplying everything out, factorize to find perfect squares. 120 × 60 = 7200 V = \sqrt{7200 × 72} V = \sqrt{100 × 72 × 72} V = 10 × 72 = 720 cm^3 Exam Strategy & Shortcut: Never multiply large numbers into a massive single integer when taking a square root. Group them efficiently. Spotting 120 × 60 = 72 × 100 immediately pairs with the other 72 to make extraction trivial. Common Pitfall: Getting lost in multiplying 120 × 72 × 60 = 518400 and then struggling to calculate the square root of a large number without a calculator. Final Answer: Therefore, the correct answer is 720 cm^3.

Question 12

If the areas of three adjacent faces of a rectangular block are in the ratio of 2 : 3 : 4 and its volume is 9000 cu. cm; then the length of the shortest side is

Accepted Answer

Concept & Finding Dimensions from Face Areas: Using the standard identity V = \sqrt{xyz} where x, y, z are the areas of adjacent faces, we can find the actual areas given their ratio. Once the face areas are known, individual sides can be found since l = V / (bh), b = V / (lh), and h = V / (lb). Step-by-Step Solution: 1. Given: Ratio of areas x : y : z = 2 : 3 : 4. Volume V = 9000. Let the areas be 2k, 3k, 4k. 2. Find the Constant k: V = \sqrt{x × y × z} 9000 = \sqrt{2k × 3k × 4k} 9000 = \sqrt{24k^3} Square both sides: 81000000 = 24k^3 k^3 = (81000000/24) = 3375000 k = \sqrt[3]{3375 × 1000} = 15 × 10 = 150 3. Calculate Face Areas: The areas are x = 300, y = 450, z = 600. 4. Find the Shortest Side: The side lengths are (V/Area). The shortest side corresponds to the largest face area. Shortest side = (V/Largest Area) = (9000/600) = 15 cm Exam Strategy & Shortcut: To find the shortest side directly, divide the volume by the largest face area. You don't need to calculate all three sides. Common Pitfall: Calculating k but forgetting to apply it back to find the largest area, or mistakenly dividing the volume by the smallest area to find the shortest side (inverse relationship). Final Answer: Therefore, the correct answer is 15 cm.

Question 13

The dimensions of a certain machine are 48" $	imes$ 30" $	imes$ 52". If the size of the machine is increased proportionately until the sum of its dimensions equals 156", what will be the increase in the shortest side?

Accepted Answer

Concept & Proportional Scaling: When an object is increased proportionately, all of its dimensions are multiplied by the same scale factor. The sum of the dimensions will also scale by this exact same factor. Step-by-Step Solution: 1. Given: Original dimensions are 48", 30", 52". New sum of dimensions = 156". 2. Calculate Original Sum: Sum = 48 + 30 + 52 = 130" 3. Find the Scale Factor: Scale Factor = \frac{New Sum}{Original Sum} = (156/130) = (6/5) = 1.2 4. Identify the Shortest Side: The shortest original side is 30". 5. Calculate New Shortest Side: New side = 30 × (6/5) = 36" 6. Calculate the Increase: Increase = 36 - 30 = 6" Exam Strategy & Shortcut: Instead of finding the new dimension and subtracting, calculate the proportional increase directly: The scale factor is 1.2, meaning a 20% increase. 20% of 30 is simply 6. Common Pitfall: Trying to find the volume and scaling that, forgetting that linear dimensions scale directly with their sum, while volume scales cubically. Final Answer: Therefore, the correct answer is 6".

Question 14

If a metal slab of size 1 m $	imes$ 20 cm $	imes$ 1 cm is melted to another slab of 1 mm thickness and 1 m width, then the length of the new slab thus formed will be

Accepted Answer

Concept & Conservation of Volume: When a solid is melted and recast into a new shape, the total volume of material remains constant. V_{old} = V_{new} Step-by-Step Solution: 1. Given: Old slab dimensions: L_1 = 1 m, B_1 = 20 cm, H_1 = 1 cm. New slab dimensions: H_2 = 1 mm, B_2 = 1 m. 2. Unify Units: Convert everything to centimeters (cm) to avoid decimal errors. L_1 = 1 m = 100 cm B_1 = 20 cm H_1 = 1 cm H_2 = 1 mm = 0.1 cm B_2 = 1 m = 100 cm 3. Calculate Original Volume: V_1 = 100 × 20 × 1 = 2000 cm^3 4. Set Up Equation for New Slab: V_2 = L_2 × B_2 × H_2 2000 = L_2 × 100 × 0.1 2000 = 10 × L_2 L_2 = (2000/10) = 200 cm Exam Strategy & Shortcut: Observe the dimensions: L_1 and B_2 are both 1m, so they cancel each other out in the volume equation. You just need to solve 20 cm × 1 cm = L_2 × 0.1 cm, making L_2 = 20 / 0.1 = 200. Common Pitfall: Failing to convert millimeters (mm) to centimeters (cm) correctly. Mixing m, cm, and mm in a single equation guarantees an incorrect order of magnitude. Final Answer: Therefore, the correct answer is 200 cm.

Question 15

The length of an edge of a hollow cube open at one face is $\sqrt{3}$ metres. What is the length of the largest pole that it can accommodate?

Accepted Answer

Concept & Formula: The length of the largest pole that can be accommodated in a cube (whether hollow, open at one face, or closed) is equal to its main body diagonal. The opening on one face does not change the internal maximum distance from one corner to the opposite corner. Diagonal = a\sqrt{3} Where a is the length of the edge. Step-by-Step Solution: 1. Given: The edge length of the cube, a = \sqrt{3} metres. 2. Calculation: Substitute the value of a into the body diagonal formula. 3. Diagonal = (\sqrt{3}) × \sqrt{3} 4. Diagonal = 3 metres. Exam Strategy & Shortcut: Don't let the phrase "hollow cube open at one face" trick you into calculating something different like a face diagonal or subtracting anything. The longest internal distance in any rectangular prism or cube is always the main diagonal. Simply multiply the side by \sqrt{3}. Common Pitfall: A common mistake is thinking the "open face" means you can only place a pole along the open surface or confusing the body diagonal with the face diagonal (a\sqrt{2}). Final Answer: Therefore, the correct answer is 3 m.

Question 16

If the total length of diagonals of a cube is 12 cm, then what is the total length of the edges of the cube?

Accepted Answer

Concept & Logic: A cube has 4 main body diagonals that pass through its center, connecting opposite vertices. The "total length of diagonals" refers to the sum of these 4 lengths. A cube also has 12 equal edges. Total\ Diagonals = 4 × (a\sqrt{3}) Total\ Edges = 12 × a Step-by-Step Solution: 1. Find the length of one diagonal: Total length of 4 diagonals = 12 cm. Length of one diagonal = (12/4) = 3 cm. 2. Find the edge length (a): a\sqrt{3} = 3 a = (3/\sqrt{3)} Rationalizing the denominator: a = \frac{3\sqrt{3}}{3} = \sqrt{3} cm. 3. Calculate total length of edges: Total edges = 12 × a Total edges = 12 × \sqrt{3} = 12\sqrt{3} cm. Exam Strategy & Shortcut: Notice the relationship algebraically: Total diagonals = 4a\sqrt{3}. We want 12a. To get 12a from 4a\sqrt{3}, multiply by \sqrt{3}: (4a\sqrt{3}) × \sqrt{3} = 4a × 3 = 12a. Since the total diagonals sum is 12, just multiply it by \sqrt{3} \rightarrow 12\sqrt{3}. Common Pitfall: Assuming "diagonals" refers to the face diagonals (there are 12 of them, each a\sqrt{2}) instead of the 4 main body diagonals. The phrasing "diagonals of a cube" conventionally implies the body diagonals. Final Answer: Therefore, the correct answer is 12\sqrt{3} cm.

Question 17

From a cube of side 8m, a square hole of 3m side is hollowed from end to end. What is the volume of the remaining solid?

Accepted Answer

Concept & Strategy: When a hole is drilled through a solid, the volume of the remaining solid is the original volume minus the volume of the material removed. Since the hole is square and goes "from end to end," the removed material forms a cuboid. Remaining Volume = Volume_{cube} - Volume_{hole} Step-by-Step Solution: 1. Calculate the Original Volume: Side of the cube, a = 8 m. Volume_{cube} = 8^3 = 8 × 8 × 8 = 512 m^3. 2. Calculate the Volume of the Hole: The hole has a square cross-section of 3 m × 3 m. Because it goes from end to end of the 8m cube, its length (height) is 8 m. Volume_{hole} = L × B × H = 3 × 3 × 8 = 72 m^3. 3. Calculate Remaining Volume: Remaining Volume = 512 - 72 = 440 m^3. Exam Strategy & Shortcut: Visualize the removed piece as a prism with a base area of 3^2=9. The volume of any uniform prism is Base Area × Length. 9 × 8 = 72. Subtracting 72 from 8^3 (512) takes only seconds. Common Pitfall: Assuming the hole is a smaller cube (i.e., subtracting 3^3 = 27 instead of 3 × 3 × 8 = 72). The phrase "from end to end" indicates the hole spans the entire 8m depth of the original cube. Final Answer: Therefore, the correct answer is 440 m^3.

Question 18

The volume of a cube is numerically equal to the sum of its edges. What is its total surface area in square units?

Accepted Answer

Concept & Formula: A cube consists of 12 equal edges. By setting the formula for volume equal to the formula for the sum of all edges, we can solve for the edge length (or its square), which directly leads to the surface area. Volume = a^3 Sum\ of\ edges = 12a Total\ Surface\ Area = 6a^2 Step-by-Step Solution: 1. Set up the given relation: Volume = Sum\ of\ edges a^3 = 12a 2. Solve for a^2: Since a physical edge a must be greater than 0, divide both sides by a. a^2 = 12 (Note: We don't need to find a = \sqrt{12} because the surface area formula only requires a^2.) 3. Calculate Total Surface Area: Surface Area = 6a^2 Substitute a^2 = 12 directly: Surface Area = 6 × 12 = 72 square units. Exam Strategy & Shortcut: Recognize that you do not always need to isolate the variable completely. Finding a^2 = 12 is enough since the target expression (6a^2) uses a^2. Avoiding the square root saves time and reduces calculation errors. Common Pitfall: Mistakenly thinking the "sum of its edges" refers to just the perimeter of one face (4a) rather than the total framework of the cube (12 edges \rightarrow 12a). Setting a^3 = 4a leads to an incorrect surface area of 24. Final Answer: Therefore, the correct answer is 72.

Question 19

Except for one face of a given cube, identical cubes are glued through their faces to all the other faces of the given cube. If each side of the given cube measures 3 cm, then what is the total surface area of the solid body thus formed?

Accepted Answer

Concept & Surface Area Logic: When identical cubes are glued to the faces of a central cube, the total surface area changes. Each glued face covers up surface area on both the central cube and the attached cube. Total Exposed Faces = Exposed Faces on Central Cube + Exposed Faces on Attached Cubes Step-by-Step Solution: • Given: A central cube has identical cubes attached to 5 of its 6 faces (since one face is excepted). • Total cubes = 1 central + 5 attached = 6 cubes. • Let's count the exposed faces on the central cube: Since 5 faces are glued, only 1 face remains exposed. • Let's count the exposed faces on the attached cubes: Each of the 5 attached cubes has 1 face glued to the central cube, leaving 5 faces exposed per attached cube. • Total exposed faces = 1 + (5 × 5) = 1 + 25 = 26 faces. • The side length of each cube is 3 cm. Area of one face = 3 × 3 = 9 cm^2. • Total surface area = 26 × 9 = 234 cm^2. Exam Strategy & Shortcut: Instead of calculating the total initial area and subtracting hidden faces, directly count the exposed faces of the final 3D figure. 1 central face + 5 × 5 attached faces = 26 faces. Multiply by face area (9) to get 234 instantly. Common Pitfall: A common mistake is assuming all 6 faces of the central cube have attached cubes, or miscalculating the covered faces by subtracting incorrectly from the aggregate area. Final Answer: Therefore, the correct answer is 234 cm².

Question 20

A cube of length 1 cm is taken out from a cube of length 8 cm. What is the weight of the remaining portion?

Accepted Answer

Concept & Volume Proportionality: For uniform solid objects, weight is directly proportional to volume. The fraction of the remaining weight will be equal to the fraction of the remaining volume. Volume of a Cube = side^3 Step-by-Step Solution: • Find the volume of the original cube: Side = 8 cm. Volume = 8^3 = 512 cm^3. • Find the volume of the removed cube: Side = 1 cm. Volume = 1^3 = 1 cm^3. • Calculate the remaining volume: 512 - 1 = 511 cm^3. • Determine the weight ratio: Since weight correlates with volume, the remaining weight is (511/512) of the original weight. Exam Strategy & Shortcut: Just look at the cubes of the dimensions: 1^3 vs 8^3. The volumes are 1 and 512. The remainder is 512 - 1 = 511. The ratio is automatically (511/512). Common Pitfall: Mistaking the relationship between linear dimensions and volume/weight. Some might just subtract the lengths (8 - 1 = 7) and incorrectly choose (7/8). Final Answer: Therefore, the correct answer is (511/512) of the weight of the original cube.

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