Question 1

A solid right circular cylinder of radius 7 cm and height 21 cm is melted and recast into small identical bullets. Each bullet consists of a right circular cylinder surmounted by a hemisphere on one base. The total height of each bullet is 3.5 cm an…

Accepted Answer

Introduction / Context: This question combines volume of a cylinder and hemisphere and the idea of melting and recasting a solid. The original solid is a large cylinder, and it is converted into smaller composite solids shaped like bullets. Each bullet consists of a cylindrical part plus a hemispherical cap. To find how many bullets can be formed, we compare the total volume of the original cylinder with the volume of one bullet and then take the integer number of complete bullets. Given Data / Assumptions: Original solid is a right circular cylinder with radius R = 7 cm and height H = 21 cm. Each bullet has radius r = 2.1 cm at the base. Each bullet has total height 3.5 cm and consists of a cylinder plus a hemisphere. The hemispherical part has radius 2.1 cm. Height of cylindrical part of each bullet = 3.5 cm minus 2.1 cm = 1.4 cm. Take pi = 22 / 7 for calculations. Concept / Approach: First, we find the volume of the large cylinder using V = pi * R^2 * H. Next, we compute the volume of one bullet as the sum of the volume of its cylindrical part and the volume of its hemispherical part. That means V_bullet = pi * r^2 * h_cyl + (2 / 3) * pi * r^3. Dividing the total original volume by the volume of one bullet gives the number of bullets. Because only complete bullets are counted, we take the integer part of this ratio. Step-by-Step Solution: Volume of original cylinder: V_big = pi * R^2 * H = (22 / 7) * 7^2 * 21. Compute V_big: (22 / 7) * 49 * 21 = (22 / 7) * 1029 = 22 * 147 = 3234 cubic centimetres. For each bullet, height of cylindrical part h_cyl = 3.5 cm - 2.1 cm = 1.4 cm. Volume of cylindrical part: V_cyl = pi * r^2 * h_cyl = (22 / 7) * 2.1^2 * 1.4. Volume of hemispherical part: V_hemi = (2 / 3) * pi * r^3 = (2 / 3) * (22 / 7) * 2.1^3. Performing the calculation gives V_bullet = V_cyl + V_hemi = approximately 38.808 cubic centimetres. Number of bullets = V_big / V_bullet = 3234 / 38.808 which is about 83.33. Therefore, the number of complete bullets that can be formed is 83. Verification / Alternative check: We can quickly check by multiplying 83 by the bullet volume: 83 * 38.808 is slightly less than 3234, so the material is sufficient for 83 complete bullets. If we tried 84 bullets, the required volume would exceed the original 3234 cubic centimetres, so 84 is not possible. This confirms that 83 is the maximum number of complete bullets that can be formed. Why Other Options Are Wrong: 89 and 92 would require more volume than available in the original cylinder and so are impossible. 74 and 79 are smaller than the correct integer count and would leave some usable volume unused, which contradicts the idea of forming as many complete bullets as possible. Common Pitfalls: A common mistake is to ignore the hemispherical part and treat the bullet as a simple cylinder, which underestimates its volume and leads to a larger incorrect count. Another error is to take the total height of the bullet as the height of the cylindrical part and forget to subtract the radius reserved for the hemisphere. Finally, rounding pi or intermediate values too early can slightly distort the final answer, so it is better to keep enough precision until the end and then choose the closest valid integer. Final Answer: Approximately 83 complete bullets can be made from the original cylinder.

Question 2

A solid cuboid of dimensions 50 cm × 40 cm × 30 cm is cut into 8 identical smaller cuboids by making 3 cuts, each cut parallel to one of its faces. What is the total surface area, in square centimetres, of all the 8 smaller cuboids together?

Accepted Answer

Introduction / Context: This problem tests understanding of how cutting a solid affects total surface area. A large cuboid is divided into smaller identical cuboids by planes parallel to its faces. Although the total volume remains the same, new internal faces become external surfaces of the smaller pieces, which changes the sum of their surface areas. Such questions are common in aptitude exams because they involve both geometry and spatial reasoning. Given Data / Assumptions: Original cuboid dimensions: 50 cm, 40 cm and 30 cm. The cuboid is cut into 8 identical smaller cuboids. Three cuts are made, each parallel to a face, so that the three dimensions are each divided by 2. No material is lost in cutting. We must find the total surface area of all 8 smaller cuboids combined. Concept / Approach: To obtain 8 equal parts from a cuboid, one cut is made along each dimension, halving length, breadth and height. This yields smaller cuboids with dimensions equal to half of each original dimension. The surface area of a cuboid with dimensions l, b and h is given by 2 * (l * b + b * h + h * l). First, determine the dimensions of each smaller cuboid and compute its surface area. Then multiply by 8 for the total surface area of all parts. Step-by-Step Solution: Original dimensions: length L = 50 cm, breadth B = 40 cm, height H = 30 cm. To get 8 parts, each dimension is halved: L is cut into two segments of 25 cm, B into two segments of 20 cm, and H into two segments of 15 cm. So each small cuboid has dimensions: l = 25 cm, b = 20 cm, h = 15 cm. Surface area of one small cuboid: S_one = 2 * (l * b + b * h + h * l). Compute products: l * b = 25 * 20 = 500; b * h = 20 * 15 = 300; h * l = 15 * 25 = 375. Sum inside brackets: 500 + 300 + 375 = 1175. S_one = 2 * 1175 = 2350 square centimetres. There are 8 identical cuboids, so total surface area S_total = 8 * 2350 = 18800 square centimetres. Verification / Alternative check: We can compare with the surface area of the original cuboid. Original surface area S_original = 2 * (50 * 40 + 40 * 30 + 30 * 50) = 2 * (2000 + 1200 + 1500) = 2 * 4700 = 9400 square centimetres. After cutting, the total surface area 18800 is exactly double. This makes sense because new faces appear inside the original cuboid that become external surfaces of the smaller pieces, increasing the overall surface area. Why Other Options Are Wrong: 11750 and 14100 correspond to miscalculations where one or two face products are omitted or not doubled correctly. 23500 would arise if you mistakenly took the surface area of one small cuboid as the total or miscounted the number of parts. 16400 is another incorrect intermediate value that does not match any consistent application of the surface area formula and the number of pieces. Common Pitfalls: Learners sometimes assume the total surface area remains unchanged after cutting, which is true for volume but not for surface area. Another mistake is to divide the original surface area by 8 instead of recalculating for each smaller cuboid. Misapplying the formula for surface area and confusing l, b and h values also leads to errors. It is important to recompute based on the new dimensions and then multiply by the correct number of identical parts. Final Answer: The total surface area of all the 8 smaller cuboids is 18800 square centimetres.

Question 3

Ratio of the height of the cone and cylinder is $3 : 2$. Ratio of the radius of the cone and cylinder is $3 : 4$. Ratio of the volume of the cone and cylinder is $m : n$. The length of the cuboid is $(m + 11)$ and the breadth of the cuboid is $(n - …

Accepted Answer

Concept & Volume Proportions: The key is establishing the relationship between the volumes of a cone and a cylinder using their given dimension ratios. The volume of a cone is V_{cone} = (1/3)\pi r^2 h and the volume of a cylinder is V_{cylinder} = \pi r^2 h. The ratio of their volumes is: \frac{V_{cone}}{V_{cylinder}} = (\frac{1/3)\pi r_1^2 h_1}{\pi r_2^2 h_2} Step-by-Step Solution: • Establish Ratios: Let the radii of the cone and cylinder be 3x and 4x respectively. Let the heights of the cone and cylinder be 3y and 2y respectively. • Calculate Volume Ratio (m : n): (m/n) = (\frac{1/3) \pi (3x)^2 (3y)}{\pi (4x)^2 (2y)} (m/n) = (1/3) × (9x^2/16x^2) × (3y/2y) (m/n) = (1/3) × (9/16) × (3/2) = (9/32) Assuming the simplest integer values for the ratio, m = 9 and n = 32. • Calculate Cuboid Dimensions: Length of cuboid L = m + 11 = 9 + 11 = 20. Breadth of cuboid B = n - 11 = 32 - 11 = 21. • Find Height of Cuboid (H): Total volume of cuboid = L × B × H = 11340. 20 × 21 × H = 11340 420 × H = 11340 \implies H = 27 • Evaluate Statements: I. 3m = 3(9) = 27 (Matches) II. (n/4) = (32/4) = 8 (Does not match) III. n - m - 15 = 32 - 9 - 15 = 8 (Does not match) Only statement I yields the correct height. Exam Strategy & Shortcut: When variables expressed directly from ratios (like m and n forming m+11) are used as absolute lengths, it strongly implies that the base simplified ratio values (9 and 32) are the actual intended absolute values. Plug k=1 in immediately to save time, and quickly verify if the resulting numbers divide cleanly (like 11340 \div 420 = 27). Common Pitfall: Forgetting the (1/3) factor in the cone's volume formula, which would drastically alter the m:n ratio and lead to incorrect cuboid dimensions. Final Answer: Therefore, the correct answer is Only I.

Question 4

Directions: Read the information carefully and answer the following questions.  A cube of side $C$ cm is placed inside the sphere of radius $R$ cm in such a way that sphere touches all the vertex of the cube. A cone of radius $\sqrt{3}R$ cm and heig…

Accepted Answer

Concept & Mensuration of 3D Figures: When a cube is inscribed inside a sphere (touching all vertices), the body diagonal of the cube is exactly equal to the diameter of the sphere. Body Diagonal = \sqrt{3} × C Diameter = 2 × R Step-by-Step Solution: 1. Establish the relation between C and R: - Since the sphere touches all vertices of the cube, the longest distance inside the cube (body diagonal) spans across the sphere's center. - \sqrt{3}C = 2R - Squaring both sides: 3C^2 = 4R^2 \Rightarrow C^2 = (4/3)R^2 2. Calculate Total Surface Area (TSA) of the sphere: - TSA_{sphere} = 4\pi R^2 = 4 × (22/7) × R^2 = (88/7)R^2 3. Calculate Lateral Surface Area (LSA) of the cube: - The lateral surface area comprises 4 vertical faces. - LSA_{cube} = 4C^2 - Substituting C^2 = (4/3)R^2: LSA_{cube} = 4 × ((4/3)R^2) = (16/3)R^2 4. Find the Ratio: - Ratio = \frac{TSA_{sphere}}{LSA_{cube}} = (\frac{88/7)R^2}{(16/3)R^2} - Ratio = (88/7) × (3/16) = (11/7) × (3/2) = (33/14) Exam Strategy & Shortcut: Always leave \pi as 22/7 when ratios are involved because standard options will often perfectly cancel out common factors like 11 or 4. Also, write out formulas symbolically before plugging any values in to avoid redundant or nested calculations. Common Pitfall: A very common mistake is confusing Total Surface Area (6 faces) with Lateral Surface Area (4 faces) of the cube. The question explicitly asks for the lateral surface area (4C^2). Final Answer: Therefore, the correct answer is 33 : 14.

Question 5

Directions: Read the information carefully and answer the following questions.  A cube of side $C$ cm is placed inside the sphere of radius $R$ cm in such a way that sphere touches all the vertex of the cube. A cone of radius $\sqrt{3}R$ cm and heig…

Accepted Answer

Concept & Algebraic Substitution in Geometry: By finding a common variable (in this case, the sphere's radius R) between the cube and the cone, we can link the given volume of the cone directly to the dimensional side of the cube. Volume of Cone = (1/3)\pi r^2 h Step-by-Step Solution: 1. Analyze the Cone's Volume: - Radius of cone = \sqrt{3}R - Height of cone = H - Volume = 3168 cm^3 - (1/3) × \pi × (\sqrt{3}R)^2 × H = 3168 2. Simplify the Cone Equation: - (1/3) × (22/7) × 3R^2 × H = 3168 - The 3s cancel out: (22/7) × R^2 × H = 3168 - R^2H = 3168 × (7/22) - 3168 / 22 = 144. So, R^2H = 144 × 7 = 1008. 3. Relate R back to C: - From the initial conditions (cube inscribed in sphere), the body diagonal equals the diameter: \sqrt{3}C = 2R. - Squaring both sides: 3C^2 = 4R^2 \Rightarrow R^2 = (3C^2/4). 4. Substitute and Solve for Relation: - Substitute R^2 into the simplified volume equation: ((3C^2/4)) × H = 1008 - 3C^2H = 1008 × 4 - 3C^2H = 4032 - C^2H = (4032/3) = 1344 - C^2 = (1344/H) Exam Strategy & Shortcut: Rather than calculating 144 × 7 immediately, keep the terms in factored form as 144 × 7 × (4/3) when bridging the two formulas. 144 / 3 = 48, and 48 × 28 = 1344. Delayed multiplication reduces arithmetic errors and speeds up the simplification process. Common Pitfall: Forgetting to properly square the \sqrt{3} coefficient when substituting the radius of the cone into the volume formula, which fundamentally alters the entire factor chain. Final Answer: Therefore, the correct answer is C^2 = (1344/H).

Question 6

The area of the four walls of a room is $120 	ext{ m}^2$ and the length is twice the breadth. If the height of the room is $4	ext{ m}$, then the area of the floor is

Accepted Answer

Concept & Surface Area of Walls: The area of four walls of a room is given by the lateral surface area formula. Area of 4 walls = 2(l + b)h Step-by-Step Solution: • Given height h = 4 m and the area of the four walls = 120 m^2. • Let the breadth of the room be b. According to the problem, the length is l = 2b. • Substitute the known values into the area formula: 2(2b + b) × 4 = 120. • Simplify the equation: 2(3b) × 4 = 120 \Rightarrow 24b = 120. • Solving for breadth gives b = 5 m. • Since length is twice the breadth, l = 2 × 5 = 10 m. • The area of the floor is given by l × b. Thus, Floor Area = 10 × 5 = 50 m^2. Exam Strategy & Shortcut: Notice that the floor area is l × b = (2b) × b = 2b^2. By simplifying the wall area formula to 24b = 120, we immediately find b=5. We just need 2(5^2) = 50, avoiding the need to write down every individual dimension. Common Pitfall: A frequent error is confusing the total surface area of a cuboid with the area of its four walls. Remember that the four walls formula explicitly excludes the floor and ceiling. Final Answer: Therefore, the correct answer is 50 m^2.

Question 7

The length of a hall is 20 metres and the width is 16 metres. The sum of the areas of the floor and roof is equal to the sum of the areas of the four walls. Find the volume of the hall.

Accepted Answer

Concept & Volume of a Cuboid: To find the volume of the hall, we first need to determine its height. The problem states a clear relationship between the areas of the top/bottom and the lateral surfaces (walls). Volume = Length × Breadth × Height Area\_of\_Floor\_and\_Roof = 2 × (Length × Breadth) Area\_of\_Four\_Walls = 2 × Height × (Length + Breadth) Step-by-Step Solution: 1. Given Data: - Length (L) = 20 m - Width / Breadth (B) = 16 m 2. Calculate Area of Floor and Roof: - Area = 2 × (20 × 16) = 2 × 320 = 640 m² 3. Set Up the Equation: - The area of the four walls is 2H(L + B) = 2H(20 + 16) = 2H(36) = 72H - According to the condition: 72H = 640 4. Find the Height (H): - H = 640 / 72 = 80 / 9 m 5. Calculate Volume: - Volume = L × B × H = 20 × 16 × (80 / 9) - Volume = 320 × (80 / 9) = 25600 / 9 - Volume ≈ 2844.44 m³ Exam Strategy & Shortcut: When you reach the fraction 25600 / 9, realize that 25 + 6 = 11, which isn't divisible by 9, meaning it will be a recurring decimal. 25600 / 9 = 2844.44... Knowing the fraction breakdown instantly guides you to the correct choice without exhaustive division. Common Pitfall: A frequent error is forgetting to multiply the floor area by 2 when equating it to the walls. "Floor and roof" implies two identical rectangular areas. Final Answer: Therefore, the correct answer is 2844.4 m³.

Question 8

It is required to construct a big rectangular hall to accommodate 500 persons, allowing 22.5 m³ space per person. The height of the hall is to be kept at 7.5 m, while the total inner surface area of the walls must be 1200 sq. m. Then the length and …

Accepted Answer

Concept & System of Equations in Geometry: This problem creates a system of two equations based on volume capacity and lateral surface area. Total\_Volume = Number\_of\_Persons × Volume\_per\_Person Area\_of\_Walls = 2H(L + B) Step-by-Step Solution: 1. Find Total Required Volume: - Volume = 500 persons × 22.5 m³/person = 11250 m³ 2. Form Equation 1 (Volume): - Volume = L × B × H - 11250 = L × B × 7.5 - L × B = 11250 / 7.5 = 1500 3. Form Equation 2 (Area of Walls): - Area = 2 × 7.5 × (L + B) = 15(L + B) - We are given Area = 1200 sq. m. - 15(L + B) = 1200 - L + B = 80 4. Solve for L and B: - We need two numbers that add up to 80 and multiply to 1500. - The numbers are 50 and 30. Exam Strategy & Shortcut: Once you deduce that L + B = 80 and L × B = 1500, immediately scan the options. (a) 40+30=70 (Reject) (b) 45+35=80, but 45×35 = 1575 (Reject) (c) 50+30=80, and 50×30 = 1500 (Accept) (d) 60+20=80, but 60×20 = 1200 (Reject) Common Pitfall: Using the total surface area formula 2(LB + BH + HL) instead of the area of the four walls 2H(L + B). "Total inner surface area of the walls" strictly excludes the floor and ceiling. Final Answer: Therefore, the correct answer is 50 m and 30 m.

Question 9

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 10

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 11

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 12

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 13

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 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

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 16

Three rectangles $A_1$, $A_2$ and $A_3$ have the same area. Their lengths $a_1$, $a_2$ and $a_3$ respectively are such that $a_1 < a_2 < a_3$. Cylinders $C_1$, $C_2$ and $C_3$ are formed from $A_1$, $A_2$ and $A_3$ respectively by joining the parall…

Accepted Answer

Concept & Logical Deduction: When a rectangle is formed into a cylinder by joining the sides along its breadth, the length (a) forms the circumference of the cylinder's base, and the breadth (b) becomes the height (h). Volume of cylinder: V = \frac{Circumference^2 × Height}{4\pi} Step-by-Step Solution: • Let the constant area of each rectangle be K. So, a_i × b_i = K. • Circumference C_i = a_i, Height h_i = b_i. • Volume V_i = (a_i^2 b_i/4\pi). • Since a_i b_i = K, we can rewrite the volume as: V_i = (a_i (a_i b_i)/4\pi) = (a_i K/4\pi) • This shows that for a constant area K, the volume is directly proportional to the length a_i (V_i \propto a_i). • Given a_1 < a_2 < a_3, it directly follows that V_1 < V_2 < V_3. Exam Strategy & Shortcut: Recognize that volume depends on the square of the radius (derived from length) but only linearly on height (breadth). Squaring a larger quantity produces a much larger result. Hence, rolling along the longer side always produces a cylinder with a larger volume. Since a_3 is the longest, C_3 has the maximum volume. Common Pitfall: Misinterpreting "joining the parallel sides along the breadth" as using the breadth for the circumference. Even so, confusing whether maximizing radius or height impacts volume more is a common trap. Final Answer: Therefore, the correct answer is C_3 will enclosed maximum volume.

Question 17

The ratio of total surface area to lateral surface area of a cylinder whose radius is 20 cm and height 60 cm, is

Accepted Answer

Concept & Ratio Simplification: Instead of calculating the full numerical areas, simplify the ratio of their formulas first. Total Surface Area (TSA): 2\pi r(h + r) Lateral (Curved) Surface Area (LSA): 2\pi rh Step-by-Step Solution: • Given: Radius (r) = 20 cm, Height (h) = 60 cm. • Write the ratio expression: (TSA/LSA) = (2\pi r(h + r)/2\pi rh) • Cancel the common terms (2\pi r): (TSA/LSA) = (h + r/h) • Substitute the given values for h and r: (TSA/LSA) = (60 + 20/60) (TSA/LSA) = (80/60) • Simplify the fraction: (80/60) = (4/3) • The ratio is 4 : 3. Exam Strategy & Shortcut: Always reduce formulas algebraically before plugging in numbers. Recognizing that (TSA/LSA) = 1 + (r/h) makes it an instant mental calculation: 1 + (20/60) = 1 + (1/3) = (4/3). Common Pitfall: Wasting time computing 2 × \pi × 20 × 80 and 2 × \pi × 20 × 60, which opens up opportunities for arithmetic mistakes. Final Answer: Therefore, the correct answer is 4 : 3.

Question 18

$1496$ cm³ of metal is used to cast a pipe of length $28$ cm. If the internal radius of the pipe is $8$ cm, the outer radius of the pipe is

Accepted Answer

Concept & Finding Unknown Radius: When the volume, length, and one radius of a hollow cylinder are known, we can isolate the unknown radius using the volume formula. V = \pi × h × (R^2 - r^2) Step-by-Step Solution: - Given: - Volume V = 1496 cm^3. - Length h = 28 cm. - Inner radius r = 8 cm. - Let the outer radius be R. - Calculation: - Substitute the known values into the formula: - 1496 = (22/7) × 28 × (R^2 - 8^2) - Simplify the constants: - 1496 = 22 × 4 × (R^2 - 64) - 1496 = 88 × (R^2 - 64) - Isolate the radius term: - R^2 - 64 = (1496/88) - R^2 - 64 = 17 - R^2 = 17 + 64 = 81 - R = \sqrt{81} = 9 cm. Exam Strategy & Shortcut: Notice that 1496 \div 88 must end in 7 because 8 × 7 = 56. This makes division faster: 88 × 10 = 880, and the rest follows naturally to yield 17. Common Pitfall: Forgetting to take the square root at the very end after isolating R^2. Final Answer: Therefore, the correct answer is 9 cm.

Question 19

The curved surface of a right circular cone of height $84$ cm and base diameter $70$ cm is

Accepted Answer

Concept & Curved Surface Area of a Cone: The curved surface area (CSA) of a cone requires the slant height (l), which is found using the height (h) and radius (r). l = \sqrt{r^2 + h^2} CSA = \pi × r × l Step-by-Step Solution: - Given: - Height h = 84 cm. - Base diameter = 70 cm \implies Radius r = 35 cm. - Calculation: - First, find the slant height l: - Both 35 and 84 are multiples of 7. (35 = 7 × 5, 84 = 7 × 12). - This is a multiple of the common 5, 12, 13 right triangle. - Therefore, l = 7 × 13 = 91 cm. - Now, find the CSA: - CSA = (22/7) × 35 × 91 - Cancel 7 with 35: CSA = 22 × 5 × 91 - CSA = 110 × 91 = 10010 cm^2. Exam Strategy & Shortcut: Always look for Pythagorean triplets! Recognizing that (35, 84) scales down to (5, 12) saves you from calculating \sqrt{35^2 + 84^2} = \sqrt{1225 + 7056} = \sqrt{8281}, which is time-consuming. Common Pitfall: Using the height (h = 84) instead of the slant height (l = 91) in the curved surface area formula. Final Answer: Therefore, the correct answer is 10010 cm².

Question 20

A hollow spherical metallic ball has an external diameter $6$ cm and is $\frac{1}{2}$ cm thick. The volume of metal used in the ball is

Accepted Answer

Concept & Volume of a Hollow Sphere: The volume of material in a hollow sphere is the difference between the external volume and the internal volume. V = (4/3)\pi (R^3 - r^3) Where R is the external radius and r is the internal radius. Internal radius r = R - thickness. Step-by-Step Solution: • Given: External diameter = 6 cm \Rightarrow External radius R = 3 cm. • Thickness = 0.5 cm. • Internal radius r = 3 - 0.5 = 2.5 cm = (5/2) cm. • Volume of metal = (4/3)\pi (R^3 - r^3) • = (4/3) × (22/7) × (3^3 - ((5/2))^3) • = (4/3) × (22/7) × (27 - (125/8)) • = (4/3) × (22/7) × ((216 - 125/8)) • = (4/3) × (22/7) × (91/8) • Simplify the expression: 91 \div 7 = 13, and 4 \div 8 = (1/2). • Volume = (1/3) × 22 × 13 × (1/2) = (11 × 13/3) = (143/3) • Converting to a mixed fraction: 143 \div 3 = 47 with a remainder of 2. • Volume = 47 (2/3) cm^3. Exam Strategy & Shortcut: Keep terms in fraction format (like 5/2) rather than decimals (like 2.5) when substituting into volume equations. Cubing and subtracting fractions is structurally easier to simplify with factors like 22/7. Common Pitfall: Using the diameter (6 cm) directly in the formula instead of the radius (3 cm), or subtracting the thickness from the diameter instead of the radius to find the inner dimension. Final Answer: Therefore, the correct answer is 47 (2/3) cm^3.

Volume and Surface Area Questions

Questions

Practice smarter