Question 1

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 2

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.

Question 3

How many cubes of 10 cm edge can be put in a cubical box of 1 m edge?

Accepted Answer

Concept & Volume Division: To find how many smaller 3D objects fit into a larger one perfectly, divide the volume of the larger object by the volume of the smaller object, provided their dimensions align evenly. Number of Objects = \frac{Total Volume}{Volume of One Object} Step-by-Step Solution: • Convert dimensions to the same unit: The larger cubical box has an edge of 1 m. Convert this to centimeters: 1 m = 100 cm. • Calculate the volume of the larger box: 100 cm × 100 cm × 100 cm = 1,000,000 cm^3. • Calculate the volume of one smaller cube: 10 cm × 10 cm × 10 cm = 1,000 cm^3. • Divide the volumes: (1,000,000/1,000) = 1000 cubes. • Alternatively, divide the edges: \frac{100 cm}{10 cm} = 10 cubes per side. Total cubes = 10 × 10 × 10 = 1000. Exam Strategy & Shortcut: Find how many small cubes fit along one edge: \frac{1 m}{10 cm} = (100/10) = 10. Since it's a cube, cube this number: 10^3 = 1000. Common Pitfall: Failing to convert meters to centimeters correctly before cubing, or squaring instead of cubing the linear ratio. Final Answer: Therefore, the correct answer is 1000.

Question 4

A 4 cm cube is cut into 1 cm cubes. The total surface area of all the small cubes is

Accepted Answer

Concept & Multiplied Surface Area: When a solid is cut into smaller pieces, its total volume remains constant, but the total surface area increases because new faces are exposed. Total Surface Area = Number of Cubes × Surface Area of One Cube Step-by-Step Solution: • Determine the total number of small cubes: Volume of large cube = 4^3 = 64 cm^3. Volume of one small cube = 1^3 = 1 cm^3. Total cubes = (64/1) = 64. • Calculate the surface area of one small cube: A cube has 6 faces. Surface area = 6 × (side)^2 = 6 × 1^2 = 6 cm^2. • Calculate the total surface area of all small cubes combined: 64 × 6 cm^2 = 384 cm^2. Exam Strategy & Shortcut: If a cube of side n × a is cut into cubes of side a, the number of cubes is n^3. The surface area of each is proportional to (1/n^2). The total surface area increases by a factor of n. Here, n = 4. Original surface area = 6 × 4^2 = 96. Total new surface area = 96 × 4 = 384. Common Pitfall: Calculating the surface area of the original 4 cm cube (96 cm^2) and selecting that as the answer, forgetting that cutting the cube exposes new surfaces. Final Answer: Therefore, the correct answer is 384 cm².

Question 5

A rectangular block with a volume of 250 cm³ was sliced into two cubes of equal volume. How much greater (in sq. cm) is the combined surface area of the two cubes then the original surface area of the rectangular block?

Accepted Answer

Concept & Slicing Solids: When a solid is sliced into two pieces, the total volume remains identical, but the combined surface area increases because new faces are created at the slice. Specifically, slicing a block into two exposes two new faces. Increase in Area = 2 × Area of the Cross-Section Step-by-Step Solution: • The total volume of the rectangular block is 250 cm^3. • It is sliced into two identical cubes. Therefore, the volume of one cube = (250/2) = 125 cm^3. • Find the side length of the cube: a^3 = 125 \Rightarrow a = 5 cm. • The slicing creates two new faces, one on each of the newly formed cubes. • These new faces are squares with side length a = 5 cm. • Area of one new face = 5 × 5 = 25 cm^2. • Since the slice separates the block into two, it exposes two such faces. • Increase in total surface area = 2 × 25 cm^2 = 50 cm^2. Exam Strategy & Shortcut: Recognize that slicing any shape into two adds exactly two cross-sectional faces to the total surface area. Since the halves are 125 cm^3 cubes, their faces are 5 × 5 = 25. Two new faces mean 2 × 25 = 50 extra surface area. No need to calculate total areas! Common Pitfall: Wasting time calculating the total surface area of the original block and the total surface area of the two cubes, then subtracting them. While this works, it is prone to arithmetic errors and takes much longer. Final Answer: Therefore, the correct answer is 50.

Question 6

A rectangular box measures internally 1.6 m long, 1 m broad and 60 cm deep. The number of cubical blocks each of edge 20 cm that can be packed inside the box is

Accepted Answer

Concept & 3D Packing: To find out how many blocks can fit inside a box, ensure the dimensions of the blocks divide evenly into the dimensions of the box. If they do, simply divide the total volume by the volume of one block. Number of Blocks = \frac{Internal Volume of Box}{Volume of One Block} Step-by-Step Solution: • Convert all dimensions of the rectangular box to the same unit (centimeters): • Length = 1.6 m = 160 cm • Breadth = 1 m = 100 cm • Depth = 60 cm • Check divisibility to ensure perfect packing: • Length: (160/20) = 8 blocks • Breadth: (100/20) = 5 blocks • Depth: (60/20) = 3 blocks • Since all dimensions perfectly accommodate the 20 cm edges, no space is wasted. • Multiply the number of blocks along each dimension to find the total: 8 × 5 × 3 = 120 blocks. Exam Strategy & Shortcut: Avoid calculating massive volumes. Instead, divide each dimension of the large box by the side length of the cube: (160/20) × (100/20) × (60/20) = 8 × 5 × 3 = 120. This keeps the numbers small and manageable. Common Pitfall: Calculating the volumes first (160 × 100 × 60 = 960,000 and 20^3 = 8,000) creates large numbers which increase the likelihood of a division error (e.g., mistaking the number of zeros). Final Answer: Therefore, the correct answer is 120.

Question 7

How many cubes of 3 cm edge can be cut out of a cube of 18 cm edge?

Accepted Answer

Concept & Volume Ratio: When smaller objects are cut from a larger solid, the number of objects is found by dividing the total volume by the volume of one smaller object. Number of Cubes = \frac{Volume of Large Cube}{Volume of Small Cube} Step-by-Step Solution: • Volume of the larger cube with 18 cm edge = 18^3 cm^3. • Volume of the smaller cube with 3 cm edge = 3^3 cm^3. • Divide the volumes: (18^3/3^3) = ((18/3))^3. • Simplify the ratio: 6^3 = 216. Exam Strategy & Shortcut: Instead of calculating 18^3, simply divide the linear edges first: (18/3) = 6. Then cube this linear ratio: 6^3 = 216. This avoids large numbers entirely. Common Pitfall: Squaring the ratio instead of cubing it (e.g., doing 6^2 = 36) because of confusion with surface area formulas. Final Answer: Therefore, the correct answer is 216.

Question 8

Shobhraj takes a cube of 1 m edge-length and meticulously cuts smaller cubes, each of edge-length 1 mm from the parent cube. He joins these small cubes end-to-end. Thus, the total length of this 'cube-robe' will be

Accepted Answer

Concept & Conservation of Volume: The total volume of the cut smaller cubes is exactly equal to the volume of the original large cube. The total length of the joined cubes depends on how many small cubes are formed. Total Length = Number of Cubes × Edge Length of Small Cube Step-by-Step Solution: • Convert the parent cube's edge to millimeters: 1 m = 1000 mm. • Calculate the total volume in mm^3: Volume = (1000 mm)^3 = 1,000,000,000 mm^3 (or 10^9 mm^3). • Since each small cube has a volume of 1 mm^3, the number of small cubes is 10^9. • Joining them end-to-end means multiplying the number of cubes by the edge length of one cube (1 mm): Total length = 10^9 × 1 mm = 10^9 mm. • Convert millimeters back to meters: 10^9 mm = 10^6 meters. • Convert meters to kilometers: 10^6 m = 1000 km. Exam Strategy & Shortcut: Number of cubes = (1000)^3 = 10^9. Length in mm = 10^9 mm. To convert mm to km directly, shift the decimal 6 places (3 for meters, 3 for km). 10^9 / 10^6 = 10^3 = 1000 km. Common Pitfall: Mistaking the conversion factors, such as assuming 1 m = 100 mm instead of 1000 mm, leading to an answer of 1 km or 10 km. Final Answer: Therefore, the correct answer is 1000 km.

Question 9

How many small cubes, each of 96 cm surface area, can be formed from the material obtained by melting a larger cube of 384 cm surface area?

Accepted Answer

Concept & Surface Area to Volume Relation: To find the number of smaller cubes formed by melting a larger one, we must compare their volumes. We first need to derive the side lengths (and subsequently volumes) from the given surface areas. Surface Area of Cube = 6a^2 Step-by-Step Solution: • Find the side length of the larger cube: 6A^2 = 384 \Rightarrow A^2 = 64 \Rightarrow A = 8 cm. • Calculate the volume of the larger cube: Volume = 8^3 = 512 cm^3. • Find the side length of the small cube: 6a^2 = 96 \Rightarrow a^2 = 16 \Rightarrow a = 4 cm. • Calculate the volume of one small cube: Volume = 4^3 = 64 cm^3. • Divide the total volume by the volume of one small cube: (512/64) = 8 cubes. Exam Strategy & Shortcut: Surface areas are in a ratio of 384:96 = 4:1. Since surface area is proportional to the square of the side length (a^2), the ratio of the side lengths is \sqrt{4}:\sqrt{1} = 2:1. Volume is proportional to the cube of the side length, so the ratio of volumes is 2^3:1^3 = 8:1. Therefore, 8 small cubes can be formed. Common Pitfall: Directly dividing the surface areas (384 / 96 = 4) and incorrectly assuming 4 cubes can be formed. Surface area does not scale linearly with volume! Final Answer: Therefore, the correct answer is 8.

Question 10

A cuboidal block of 6 cm x 9 cm x 12 cm is cut up into an exact number of equal cubes. The least possible number of cubes will be

Accepted Answer

Concept & HCF for Minimum Divisions: To cut a block into the least possible number of identical cubes, the cubes must be as large as possible. The maximum edge length of such a cube is the Highest Common Factor (HCF) of the block's dimensions. Maximum Edge Length = HCF(L, B, H) Step-by-Step Solution: • Dimensions of the cuboidal block: Length = 12 cm, Breadth = 9 cm, Height = 6 cm. • Find the HCF of 12, 9, and 6: The largest number that divides all three evenly is 3. So, the maximum edge of the cube is 3 cm. • Find the volume of one such cube: 3 × 3 × 3 = 27 cm^3. • Find the volume of the block: 6 × 9 × 12 = 648 cm^3. • Divide the block's volume by the cube's volume: (648/27) = 24 cubes. Exam Strategy & Shortcut: Instead of multiplying volumes, divide each dimension by the HCF (3) and multiply the results: ((6/3)) × ((9/3)) × ((12/3)) = 2 × 3 × 4 = 24. Common Pitfall: Taking the LCM instead of the HCF, which is the strategy used for building a large solid from smaller blocks (like the next question type), rather than cutting a solid down. Final Answer: Therefore, the correct answer is 24.

Question 11

The size of a wooden block is 5 x 10 x 20 cm. How many such blocks will be required to construct a solid wooden cube of minimum size?

Accepted Answer

Concept & LCM for Constructing Solids: To construct the smallest possible solid cube from identical rectangular blocks, the side length of the cube must be the Least Common Multiple (LCM) of the block's dimensions. Minimum Edge Length = LCM(L, B, H) Step-by-Step Solution: • Dimensions of the given wooden block: 5 cm, 10 cm, and 20 cm. • Find the LCM of 5, 10, and 20: The smallest number divisible by all three is 20. So, the edge of the minimum solid cube is 20 cm. • Calculate the volume of this constructed cube: 20 × 20 × 20 = 8000 cm^3. • Calculate the volume of one wooden block: 5 × 10 × 20 = 1000 cm^3. • Divide the total volume by the block volume to find the count: (8000/1000) = 8 blocks. Exam Strategy & Shortcut: Find how many blocks fit along each axis of the constructed 20 cm cube: (20/5) = 4 blocks, (20/10) = 2 blocks, (20/20) = 1 block. Total blocks = 4 × 2 × 1 = 8. Common Pitfall: Finding the HCF instead of LCM, which would mistakenly apply the logic of cutting a block rather than building up to a cube. Final Answer: Therefore, the correct answer is 8.

Question 12

An iron cube of side 10 cm is hammered into a rectangular sheet of thickness 0.5 cm. If the sides of the sheet are in the ratio 1 : 5, the sides are

Accepted Answer

Concept & Conservation of Volume: When a solid object is reshaped (hammered, melted, recast), its total volume remains constant regardless of the new shape. Volume of Cube = Volume of Rectangular Sheet Step-by-Step Solution: • Calculate the volume of the iron cube: 10^3 = 1000 cm^3. • Let the dimensions of the rectangular sheet be x and 5x based on the given 1:5 ratio. The thickness (height) is 0.5 cm. • Formulate the volume equation for the sheet: Length × Breadth × Thickness = 5x × x × 0.5 = 2.5x^2. • Equate the volumes: 2.5x^2 = 1000. • Solve for x^2: x^2 = (1000/2.5) = 400. • Solve for x: x = \sqrt{400} = 20 cm. • Calculate the other side: 5x = 5(20) = 100 cm. Exam Strategy & Shortcut: Test the options directly into the volume formula. For option (a), 10 × 50 × 0.5 = 250 (too small). For option (b), 20 × 100 × 0.5 = 1000. This perfectly matches the cube's volume of 1000, making (b) the immediate correct choice. Common Pitfall: Misplacing the decimal when dividing by 0.5 or 2.5, leading to an incorrect root like \sqrt{40} instead of \sqrt{400}. Final Answer: Therefore, the correct answer is 20 cm, 100 cm.

Question 13

There is a cube of volume 216 cm³. It is to be moulded into a cuboid having one edge equal to 6 cm. The number of ways that it can be done so that the edges have different integral values is

Accepted Answer

Concept & Volume Dimensions: The volume of a cuboid is the product of its three dimensions (length, breadth, and height). When a shape is remoulded, its volume remains constant. V = l × b × h Step-by-Step Solution: 1. Given: Volume of the original cube = 216 cm^3. This means the new cuboid also has a volume of 216 cm^3. 2. One edge of the cuboid is given as 6 cm. Let the other two edges be l and b. 3. Thus, l × b × 6 = 216. 4. Dividing by 6, we get l × b = 36. 5. We need to find pairs of integers (l, b) whose product is 36. The possible factor pairs of 36 are: - (1, 36) - (2, 18) - (3, 12) - (4, 9) - (6, 6) 6. The question specifies that the edges must have different integral values. The three edges are l, b, and 6. 7. This means neither l nor b can be 6, which rules out the (6, 6) pair (where edges would be 6, 6, 6). 8. Checking the remaining pairs against the third edge (6): - 1, 36, 6 (all different) - Valid - 2, 18, 6 (all different) - Valid - 3, 12, 6 (all different) - Valid - 4, 9, 6 (all different) - Valid 9. There are 4 valid ways to form the cuboid. Exam Strategy & Shortcut: To quickly solve constraint-based factor problems, instantly list out all factor pairs of the remaining product area (36). Then apply constraints: eliminate pairs with duplicates (like 6 × 6) or pairs matching the constant given edge. Count what remains. Common Pitfall: A common mistake is failing to check the generated pairs against the fixed edge of 6 cm or forgetting that (6,6) creates a shape with identical edges, violating the "different integral values" condition. Final Answer: Therefore, the correct answer is 4.

Question 14

If three cubes of copper, each with an edge of 6 cm 8 cm and 10 cm respectively are melted to form a single cube, then the diagonal of the new cube will be

Accepted Answer

Concept & Cube Transformation: When multiple objects are melted to form a new object, the total volume is conserved. The volume of a cube with edge a is a^3, and its space diagonal is given by: Diagonal = a\sqrt{3} Step-by-Step Solution: 1. Calculate the volume of the original three cubes: - Volume of first cube = 6^3 = 216 cm^3 - Volume of second cube = 8^3 = 512 cm^3 - Volume of third cube = 10^3 = 1000 cm^3 2. Find the total volume: - Total Volume = 216 + 512 + 1000 = 1728 cm^3 3. Determine the edge of the new cube: - Let the edge of the new cube be A. - A^3 = 1728 - A = \sqrt[3]{1728} = 12 cm 4. Calculate the diagonal of the new cube: - Diagonal = A\sqrt{3} = 12\sqrt{3} - Knowing \sqrt{3} \approx 1.732, we get 12 × 1.732 = 20.784 cm. 5. Rounding to one decimal place gives 20.8 cm. Exam Strategy & Shortcut: Memorize the cubes of numbers from 1 to 15. The sum of 6^3 + 8^3 + 10^3 equals 12^3. This is a well-known Pythagorean-like relationship for cubes: 6, 8, 10 \rightarrow 12. Recognizing this instantly gives you the new side 12. Then calculate 12 × 1.732. Common Pitfall: A frequent error is assuming the diagonal of the new cube is simply the sum of the diagonals of the three smaller cubes, ignoring the volume conservation principle. Final Answer: Therefore, the correct answer is 20.8 cm.

Question 15

Five equal cubes, each of side 5 cm, are placed adjacent to each other. The volume of the new solid formed will be

Accepted Answer

Concept & Additive Volume: When identical standard shapes are stacked or placed adjacently, their combined volume is simply the sum of their individual volumes. Forming a new cuboid by placing cubes adjacent to each other increases length while keeping breadth and height constant. V = Number of cubes × Volume of one cube Step-by-Step Solution: 1. Method 1: Direct Volume Addition - Volume of one cube = side^3 = 5^3 = 125 cm^3. - Since there are 5 identical cubes, the total volume = 5 × 125 = 625 cm^3. 2. Method 2: Cuboid Dimensions - Placing 5 cubes of side 5 cm in a row forms a cuboid. - Length (L) = 5 × 5 = 25 cm. - Breadth (B) = 5 cm. - Height (H) = 5 cm. - Volume of cuboid = L × B × H = 25 × 5 × 5 = 625 cm^3. Exam Strategy & Shortcut: Do not recalculate physical dimensions unless asked for surface area. For total volume of identical adjacent objects, always use direct multiplication: N × a^3. It is much faster and less error-prone. Common Pitfall: Calculating surface area instead of volume, or mistakenly cubing the combined length (25^3) instead of properly multiplying length, breadth, and height. Final Answer: Therefore, the correct answer is 625 cm³.

Question 16

If three equal cubes are placed adjacently in a row, then the ratio of the total surface area of the new cuboid to the sum of the surface areas of the three cubes will be

Accepted Answer

Concept & Obscured Surface Area: When solid shapes are joined, the faces that touch each other become "hidden" and are no longer part of the total exterior surface area. For n cubes placed in a row, n-1 pairs of faces are hidden. TSA_{cuboid} = 2(lb + bh + hl) Step-by-Step Solution: 1. Let the side of each cube be a. - Surface area of one cube = 6a^2. - Sum of surface areas of three distinct cubes = 3 × 6a^2 = 18a^2. 2. Determine the dimensions of the new cuboid: - Three cubes placed in a row make a cuboid of length l = 3a. - Breadth b = a. - Height h = a. 3. Calculate the Total Surface Area (TSA) of the new cuboid: - TSA = 2(lb + bh + hl) - TSA = 2((3a)(a) + (a)(a) + (a)(3a)) - TSA = 2(3a^2 + a^2 + 3a^2) = 2(7a^2) = 14a^2. 4. Determine the ratio: - Ratio = TSA_{cuboid} : Sum of TSA_{cubes} - Ratio = 14a^2 : 18a^2 = 14 : 18. - Simplifying by dividing by 2 yields 7 : 9. Exam Strategy & Shortcut: Instead of formulas, simply count exposed square faces. - 3 independent cubes have 3 × 6 = 18 faces. - When 3 cubes are joined in a line, 2 internal joints hide 2 × 2 = 4 faces. - Exposed faces = 18 - 4 = 14. - The ratio is instantly 14 : 18 = 7 : 9. Common Pitfall: Students often assume the surface area remains exactly the same when objects are joined, forgetting to subtract the internal, hidden faces where the cubes touch. Final Answer: Therefore, the correct answer is 7 : 9.

Question 17

The volumes of two cubes are in the ratio 8 : 27. The ratio of their surface areas is

Accepted Answer

Concept & Multidimensional Scaling: To find the ratio of surface areas from volumes, we first step down to 1D lengths (edges) by taking the cube root of the volume ratio, and then step up to 2D areas by squaring the edge ratio. (A_1/A_2) = ( \sqrt[3]{(V_1/V_2)} )^2 Step-by-Step Solution: 1. Find the ratio of their edges: - Let the edges of the two cubes be a_1 and a_2. - Ratio of volumes = (a_1^3/a_2^3) = (8/27). - Taking the cube root, the ratio of edges = (a_1/a_2) = \sqrt[3]{(8/27)} = (2/3). 2. Find the ratio of their surface areas: - Surface area of a cube = 6a^2. - Ratio of surface areas = (6a_1^2/6a_2^2) = (a_1^2/a_2^2) = ((a_1/a_2))^2. - Substituting the edge ratio: ((2/3))^2 = (4/9). 3. Determine the final ratio: - The ratio of their surface areas is 4 : 9. Exam Strategy & Shortcut: Memorize the scaling rule for similar solids: - Edge ratio = a : b - Area ratio = a^2 : b^2 - Volume ratio = a^3 : b^3 Given V \rightarrow 8:27, take cube roots \rightarrow a:b = 2:3, square them for area \rightarrow 4:9. Common Pitfall: A frequent error is taking the ratio (8/27) and dividing it by integers or arbitrarily squaring/cubing without extracting the base edge ratio first. Final Answer: Therefore, the correct answer is 4 : 9.

Question 18

If each edge of a cube is doubled, then its volume:

Accepted Answer

Concept & Volumetric Scaling: The volume of a cube depends on the cube of its edge length. When the 1D dimension is scaled by a factor k, the 3D volume scales by k^3. V = a^3 Step-by-Step Solution: 1. Initial Volume: - Let the original edge of the cube be a. - The original volume is V_1 = a^3. 2. New Volume: - The edge is doubled, so the new edge is 2a. - The new volume is V_2 = (2a)^3. 3. Compare Volumes: - V_2 = 8 × a^3. - Since a^3 = V_1, we have V_2 = 8 × V_1. 4. Conclusion: - The volume becomes 8 times the original volume. Exam Strategy & Shortcut: Always remember the dimensional rule: scaling a length by N scales the area by N^2 and the volume by N^3. If the edge is doubled (N=2), the volume increases by 2^3 = 8 times. Common Pitfall: Students often confuse volume scaling with length or area scaling and mistakenly answer "is doubled" or "becomes 4 times". Final Answer: Therefore, the correct answer is becomes 8 times.

Question 19

A cube of edge 20 cm is completely immersed in a rectangular vessel containing water. If the dimensions of the base of the vessel are 20 cm by 40 cm, the rise in water level will be

Accepted Answer

Concept & Displacement of Fluid: Archimedes' principle of volume displacement states that when a solid object is completely immersed in a liquid, it displaces a volume of liquid equal to its own volume. The displaced liquid raises the water level forming a "water cuboid". V_{immersed object} = Base Area of Vessel × Rise in Height Step-by-Step Solution: 1. Calculate the volume of the immersed cube: - Edge of the cube, a = 20 cm. - Volume of the cube = a^3 = 20 × 20 × 20 = 8000 cm^3. 2. Set up the displacement equation: - Let the rise in the water level be h cm. - The volume of the displaced water forms a cuboid with the vessel's base. - Volume of displaced water = Length × Breadth × h = 20 × 40 × h = 800h cm^3. 3. Equate the volumes: - 800h = 8000 4. Solve for h: - h = (8000/800) = 10 cm. Exam Strategy & Shortcut: Skip computing full volumes if factors cancel out. Write the equation directly: 20 × 20 × 20 = 20 × 40 × h Cancel the common 20: 400 = 40 × h \implies h = 10. Common Pitfall: A common mistake is adding the initial volume of water in the vessel to the calculation, which is unnecessary since we only care about the *change* (rise) in the water volume. Final Answer: Therefore, the correct answer is 10 cm.

Question 20

If the radius and height of a right circular cylinder are 21 cm and 35 cm respectively, then the total surface area of the cylinder is

Accepted Answer

Concept & Total Surface Area of Cylinder: The total surface area (TSA) of a solid right circular cylinder encompasses the curved surface area plus the area of the two circular bases (top and bottom). TSA = 2\pi rh + 2\pi r^2 = 2\pi r(r + h) Step-by-Step Solution: 1. Identify given dimensions: - Radius, r = 21 cm. - Height, h = 35 cm. 2. Apply the TSA formula: - TSA = 2 × (22/7) × 21 × (21 + 35) 3. Simplify the expression: - Calculate the sum in the brackets: (21 + 35) = 56. - Cancel the 7 with 21: (21/7) = 3. - TSA = 2 × 22 × 3 × 56 4. Calculate final value: - TSA = 132 × 56 - 132 × 50 = 6600 - 132 × 6 = 792 - 6600 + 792 = 7392 sq cm. Exam Strategy & Shortcut: Use unit digit estimation if options allow it. The expression is 2 × 22 × 3 × 56 \rightarrow focus on unit digits: 2 × 2 × 3 × 6 = 4 × 18 = 72. The unit digit must be 2. Unfortunately, all options end in 2, so full multiplication or factoring is required. Breaking 132 × 56 into (132 × 100)/2 + 132 × 6 is a reliable fast method. Common Pitfall: A frequent mistake is calculating only the Curved Surface Area (2\pi rh) instead of the Total Surface Area, or incorrectly adding the radius and height. Final Answer: Therefore, the correct answer is 7392 sq cm.

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