Question 1

A sphere of maximum volume is cut out from a solid hemisphere of radius $r$. The ratio of the volume of the hemisphere to that of the cut out sphere is :

Accepted Answer

Concept & Inscribed Sphere in Hemisphere: To fit a sphere of maximum volume inside a hemisphere of radius r, the sphere must sit perfectly within the height and width constraints. The maximum possible diameter of this sphere equals the radius of the hemisphere r. Therefore, the radius of the new sphere is r/2. Volume of hemisphere: V = (2/3)\pi r^3 Volume of sphere: V = (4/3)\pi R^3 Step-by-Step Solution: 1. The original solid is a hemisphere with radius r. Its volume is V_{hemi} = (2/3)\pi r^3. 2. The largest sphere that can be carved out will have its diameter equal to the hemisphere's height (which is r). So, the radius of this inscribed sphere is R = (r/2). 3. Calculate the volume of this new sphere: V_{sphere} = (4/3)\pi ((r/2))^3 = (4/3)\pi ((r^3/8)) = (1/6)\pi r^3. 4. Find the required ratio of the hemisphere's volume to the cut-out sphere's volume: \frac{V_{hemi}}{V_{sphere}} = (\frac{2/3)\pi r^3}{(1/6)\pi r^3}. 5. Cancel out \pi r^3 and simplify the fraction: (2/3/1/6) = (2/3) × (6/1) = (12/3) = 4. 6. The ratio is 4 : 1. Exam Strategy & Shortcut: Visualize the dimensions. A sphere inscribed in a hemisphere has half the radius. Because volume scales with the cube of the radius, halving the radius means the sphere's volume is (1/8) of a *full* sphere of radius r. Since a hemisphere is half a full sphere, the ratio is (1/2/1/8) = (1/2) × 8 = 4. Common Pitfall: Students often guess that the inscribed sphere has a radius of r divided by \sqrt{2} or make an error assuming it shares the same base area. Recognizing that the height r strictly limits the diameter is key. Final Answer: Therefore, the correct answer is 4 : 1.

Question 2

What is the volume in cubic cm of a pyramid whose area of the base is 25 sq cm and height 9 cm?

Accepted Answer

Concept & Volume of a Pyramid: The volume of any pyramid is one-third of the product of its base area and its perpendicular height, regardless of the shape of the base (square, triangular, etc.). Volume Formula: V = (1/3) × Base Area × Height Step-by-Step Solution: 1. Extract the given values from the problem: Base Area (A) = 25 sq cm Height (h) = 9 cm 2. Apply the volume formula for a pyramid: V = (1/3) × A × h. 3. Substitute the given numbers into the formula: V = (1/3) × 25 × 9. 4. Simplify the expression by multiplying (1/3) with 9 first: V = 25 × 3. 5. Calculate the final result: V = 75 cubic cm. Exam Strategy & Shortcut: You don't need to know what kind of polygon forms the base; the formula V = (Ah/3) applies universally. Always look to divide the height by 3 first to make mental math trivial (9/3 = 3, 25 × 3 = 75). Common Pitfall: A common mistake is treating the pyramid like a prism and forgetting to divide by 3 (which would incorrectly yield 225). Final Answer: Therefore, the correct answer is 75.

Question 3

A right pyramid is on a regular hexagonal base. Each side of the base is $10	ext{ m}$ and the height is $60	ext{ m}$. The volume of the pyramid is

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Concept & Formula: The area of a regular hexagon can be found by treating it as six equilateral triangles. Base Area = 6 × \frac{\sqrt{3}}{4} × a^2 Volume = (1/3) × Base Area × h Step-by-Step Solution: 1. Given: - Side of hexagonal base, a = 10 m - Height, h = 60 m 2. Calculate Base Area: - Base Area = 6 × \frac{\sqrt{3}}{4} × (10)^2 - Base Area = 6 × \frac{\sqrt{3}}{4} × 100 = 6 × 25\sqrt{3} = 150\sqrt{3} m^2 3. Calculate Volume: - Volume = (1/3) × 150\sqrt{3} × 60 - Volume = 50\sqrt{3} × 60 = 3000\sqrt{3} 4. Convert to Decimal: - Since \sqrt{3} \approx 1.732 - Volume = 3000 × 1.732 = 5196 m^3 Exam Strategy & Shortcut: Instead of multiplying large numbers, group them logically: (1/3) × 60 = 20. Then 20 × 150\sqrt{3} = 3000\sqrt{3}. You can mentally shift the decimal place of 1.732 three places to the right to get 1732, and then multiply by 3 to get 5196. Common Pitfall: A common mistake is calculating the perimeter instead of the area of the hexagon, or forgetting the (1/3) in the pyramid volume formula. Final Answer: Therefore, the correct answer is 5196 m^3.

Question 4

A right pyramid has an equilateral triangular base of side $4	ext{ units}$. If the number of square units of its whole surface area is three times the number of cubic units of its volume, find its height.

Accepted Answer

Concept & Formula: The problem asks us to equate the numeric value of the total surface area to three times the volume. Total Surface Area = Base Area + Lateral Area Volume = (1/3) × Base Area × Height *(Note: There is a known typographical error in standard textbook printings of this question where none of the options perfectly match the mathematical derivation of 8 units. We illustrate the structural logic below.)* Step-by-Step Solution: 1. Base Parameters: - Side a = 4 units. - Base Area (B) = \frac{\sqrt{3}}{4} × 4^2 = 4\sqrt{3}. - Inradius r (distance from center to side) = (a/2\sqrt{3)} = (4/2\sqrt{3)} = (2/\sqrt{3)}. 2. Volume (V): - V = (1/3) × B × h = (1/3) × 4\sqrt{3} × h = \frac{4\sqrt{3}}{3}h. 3. Surface Area (S): - Slant height l = \sqrt{h^2 + r^2} = \sqrt{h^2 + (4/3)}. - Perimeter P = 3 × 4 = 12. - Lateral Area (L) = (1/2) × P × l = 6\sqrt{h^2 + (4/3)}. - S = 4\sqrt{3} + 6\sqrt{h^2 + (4/3)}. 4. Equating S = 3V: - 4\sqrt{3} + 6\sqrt{h^2 + (4/3)} = 3 × (\frac{4\sqrt{3}}{3}h) - 4\sqrt{3} + 6\sqrt{h^2 + (4/3)} = 4\sqrt{3}h - Dividing by 2\sqrt{3}: 2 + 3\sqrt{(3h^2+4/9)} = 2h \implies \sqrt{3h^2+4} = 2h - 2 - Squaring both sides: 3h^2 + 4 = 4h^2 - 8h + 4 - h^2 - 8h = 0 \implies h = 8. - Mathematically, the height is 8 units. In many competitive exams, typographical errors occur in the options. Exam Strategy & Shortcut: When setting up algebraic equations equating area to volume, isolate the radical before squaring to prevent messy polynomial expansions. If your calculated answer (8) isn't in the options, re-read carefully to ensure no misinterpretations (like confusing total vs lateral area). Common Pitfall: Forgetting the base area when calculating the "whole" surface area. If the problem meant "lateral surface area is three times the volume", 6\sqrt{h^2 + 4/3} = 4\sqrt{3}h \implies h=2, which is also not an option. Final Answer: Therefore, mathematically the height is 8 units (none of the options accurately reflect this standard textbook misprint).

Question 5

The radius of a hemispherical bowls is $6	ext{cm}$. The capacity of the bowl is $\left(	ext{Take } \pi=\frac{22}{7}ight)$

Accepted Answer

Concept & Formula: The capacity of a bowl is synonymous with its internal volume. For a hemisphere, the volume is exactly half of a full sphere. Volume of Hemisphere = (2/3)\pi r^3 Step-by-Step Solution: 1. Given: - Radius r = 6cm - \pi = (22/7) 2. Apply the Formula: - Volume = (2/3) × (22/7) × (6)^3 - Volume = (2/3) × (22/7) × 216 3. Simplify and Calculate: - Divide 216 by 3 to get 72. - Volume = 2 × (22/7) × 72 - Volume = (44 × 72/7) = (3168/7) 4. Convert to Decimal: - 3168 \div 7 = 452.5714... - Rounding to two decimal places, we get 452.57 cm^3. Exam Strategy & Shortcut: When multiplying 44 × 72, you can split it as 44 × (70 + 2) = 3080 + 88 = 3168. When dividing by 7, notice that 7 × 400 = 2800 and 7 × 50 = 350, so the result must be slightly above 450. Option (b) is the only reasonable match without fully completing the long division. Common Pitfall: Using the formula for the volume of a full sphere ((4/3)\pi r^3) instead of a hemisphere, which would yield exactly double the correct answer. Final Answer: Therefore, the correct answer is 452.57cm^3.

Question 6

A hemisphere and a cone have equal bases. If their heights are also equal, then the ratio of their curved surfaces will be

Accepted Answer

Concept & Geometry Integration: To find the ratio of their curved surfaces, we first need to express the dimensions of both shapes in terms of a single variable. For a hemisphere, its height is inherently equal to its base radius. Hemisphere CSA = 2\pi r^2 Cone CSA = \pi rl (where l = \sqrt{r^2 + h^2}) Step-by-Step Solution: 1. Establish Dimensions: - Equal bases mean they share the same radius, r. - The height of a hemisphere is equal to its radius (r). - Since their heights are equal, the height of the cone (h) is also r. (h = r) 2. Calculate Slant Height of Cone (l): - l = \sqrt{r^2 + h^2} - Substitute h = r: l = \sqrt{r^2 + r^2} = \sqrt{2r^2} = r\sqrt{2} 3. Calculate Both Curved Surface Areas: - Hemisphere CSA = 2\pi r^2 - Cone CSA = \pi r (r\sqrt{2}) = \sqrt{2}\pi r^2 4. Find the Ratio: - \frac{Hemisphere CSA}{Cone CSA} = (2\pi r^2/\sqrt{2)\pi r^2} - Ratio = (2/\sqrt{2)} = \frac{\sqrt{2} × \sqrt{2}}{\sqrt{2}} = \sqrt{2} - Ratio = \sqrt{2} : 1 Exam Strategy & Shortcut: Recognize that a cone with equal height and radius forms an isosceles right triangle in cross-section. Its slant edge ratio is \sqrt{2}. The hemisphere's CSA multiplier is 2, and the cone's is \sqrt{2}. The ratio (2/\sqrt{2)} simplifies to \sqrt{2}:1 directly. Common Pitfall: A common misstep is failing to realize that the height of a hemisphere is strictly equal to its radius. Without this deduction, the problem appears unsolvable due to missing information. Final Answer: Therefore, the correct answer is \sqrt{2} : 1.

Question 7

If the radius of a sphere is increased by $10\%$, then the volume will be increased by

Accepted Answer

Concept & Successive Percentage Change: The volume of a sphere is given by: V = (4/3) \pi r^3 Since V \propto r^3, a percentage increase in the radius acts as three successive percentage increases on the volume. Step-by-Step Solution: • Given: • Increase in radius = 10% • Calculation (Method 1: Formula): 1. Let the initial radius be r=10 units. Initial Volume \propto 10^3 = 1000. 2. New radius r' = 10 + 10% of 10 = 11 units. 3. New Volume \propto 11^3 = 1331. 4. Increase in volume = 1331 - 1000 = 331. 5. Percentage increase = ((331/1000)) × 100 = 33.1%. Exam Strategy & Shortcut: Use the successive percentage change formula a + b + (ab/100) for the three factors of 10%: First two dimensions: 10 + 10 + (10 × 10/100) = 21%. Combine this result with the third dimension: 21 + 10 + (21 × 10/100) = 31 + 2.1 = 33.1%. Alternatively, memorize that 1.1^3 = 1.331, directly indicating a 33.1% increase. Common Pitfall: Adding the percentages directly (10% × 3 = 30%) is the most common mistake. This ignores the compounding effect of the volume expanding in three dimensions simultaneously. Final Answer: Therefore, the correct answer is 33.1%.

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