Question 1

A right circular cylinder has height 18 cm and radius 7 cm. The cylinder is cut into three equal smaller cylinders by two cuts parallel to the base, so each smaller cylinder has height 6 cm. What is the percentage increase in the total surface area …

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Introduction / Context: This mensuration question compares the surface area of a single right circular cylinder with the surface area after it is cut into three equal smaller cylinders. Although the total volume remains constant, cutting introduces new circular surfaces at the cross sections, which increases total surface area. The task is to compute this increase as a percentage. Given Data / Assumptions: Original cylinder radius r = 7 cm. Original cylinder height H = 18 cm. Two cuts parallel to the base divide it into three cylinders each of height 6 cm. All pieces remain solid with no loss of material. We need the percentage increase in total surface area. Concept / Approach: Total surface area of a cylinder equals curved surface area plus the areas of the two circular bases. For radius r and height h, TSA = 2 * pi * r * h + 2 * pi * r^2. Initially there is one cylinder. After cutting, there are three cylinders with the same radius but smaller height. Each small cylinder has its own two circular bases, so more base surfaces appear. Curved surface area is proportional to height, so the total curved surface area of the three cylinders remains the same as that of the original cylinder, but the number of circular base areas increases from two to six. Step-by-Step Solution: Step 1: Original cylinder: radius r = 7 cm, height H = 18 cm.Step 2: Original TSA = 2 * pi * r * H + 2 * pi * r^2 = 2 * pi * 7 * 18 + 2 * pi * 7^2 = 252pi + 98pi = 350pi.Step 3: After cutting, each small cylinder has height h = 6 cm and radius 7 cm.Step 4: TSA of one small cylinder = 2 * pi * r * h + 2 * pi * r^2 = 2 * pi * 7 * 6 + 2 * pi * 49 = 84pi + 98pi = 182pi.Step 5: There are three such cylinders, so total TSA after cutting = 3 * 182pi = 546pi.Step 6: Increase in surface area = 546pi − 350pi = 196pi.Step 7: Percentage increase = (196pi / 350pi) * 100 = (196 / 350) * 100 = 56 percent. Verification / Alternative check: Observe that the curved surface area of the original cylinder is 2 * pi * r * H = 252pi. After cutting, combined curved surface area of three smaller cylinders is 3 * 2 * pi * r * 6 = 3 * 84pi = 252pi, which is unchanged. The only change is in the number of circular bases: initially there were 2 bases, now there are 6, so effectively 4 new base areas of pi * r^2 each are added. That extra area is 4 * pi * 7^2 = 196pi, confirming the earlier computation and simplifying the reasoning for percentage increase. Why Other Options Are Wrong: 62, 48, and 52 percent do not match the exact calculation of 196π over 350π. 36 percent significantly underestimates the impact of adding four extra base circles. Common Pitfalls: Forgetting that curved surface area stays the same while only the number of circular bases increases. Computing TSA for a single small cylinder and forgetting to multiply by three. Making arithmetic mistakes when simplifying 196 / 350 to a percentage. Final Answer: The total surface area increases by 56 percent after cutting the cylinder into three equal parts.

Question 2

For a right circular cylinder, the ratio of curved surface area to volume is 1 : 7. The ratio of total surface area to volume is 187 : 770. What is the ratio of the base radius to the height of the cylinder?

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Introduction / Context: This question links the curved surface area and total surface area of a right circular cylinder with its volume using ratios. The goal is to determine the ratio of the base radius to the height. It is a good test of how well one can convert verbal ratio conditions into algebraic equations and solve them systematically using standard cylinder formulas. Given Data / Assumptions: Right circular cylinder with radius r and height h. Curved surface area (CSA) to volume (V) ratio is 1 : 7. Total surface area (TSA) to volume ratio is 187 : 770. We need the ratio r : h. Concept / Approach: For a cylinder, curved surface area C = 2 * pi * r * h, volume V = pi * r^2 * h, and total surface area T = C + 2 * pi * r^2 = 2 * pi * r * h + 2 * pi * r^2. The ratio C : V can be simplified to obtain a relation between r and h. Then the ratio T : V gives another equation, which helps in finding the exact numeric ratio between r and h. The presence of nice rational ratios suggests that r and h will have a simple integer ratio solution. Step-by-Step Solution: Step 1: Curved surface area C = 2 * pi * r * h and volume V = pi * r^2 * h.Step 2: Given C : V = 1 : 7, so C / V = 1 / 7.Step 3: Compute C / V = (2 * pi * r * h) / (pi * r^2 * h) = 2 / r. So 2 / r = 1 / 7, which gives r = 14.Step 4: Total surface area T = 2 * pi * r * h + 2 * pi * r^2. Thus T / V = (2 * pi * r * h + 2 * pi * r^2) / (pi * r^2 * h) = (2 / r) + (2 / h).Step 5: We already know 2 / r = 1 / 7 from the first ratio, so 2 / r = 1 / 7.Step 6: Given T : V = 187 : 770, so T / V = 187 / 770. Simplify 187 / 770 to 17 / 70.Step 7: Use T / V = (2 / r) + (2 / h) = 1 / 7 + 2 / h = 17 / 70.Step 8: Convert 1 / 7 to denominator 70: 1 / 7 = 10 / 70. So 10 / 70 + 2 / h = 17 / 70 gives 2 / h = 7 / 70 = 1 / 10.Step 9: Therefore h = 10. Since r = 14, ratio r : h = 14 : 20 = 7 : 10. Verification / Alternative check: Using r = 14 and h = 20, C = 2 * pi * 14 * 20 = 560pi and V = pi * 14^2 * 20 = 3920pi. Then C : V = 560pi : 3920pi = 1 : 7, which matches the first ratio. T = C + 2 * pi * r^2 = 560pi + 2 * pi * 196 = 560pi + 392pi = 952pi. Then T : V = 952pi : 3920pi, which simplifies to 952 : 3920 = 17 : 70 = 187 : 770, as required. Both ratios are satisfied, confirming r : h = 7 : 10. Why Other Options Are Wrong: 5 : 8, 4 : 9, and 3 : 7 do not satisfy both ratio conditions when substituted into the formulas for C and T. 2 : 5 leads to values of r and h that fail to produce the required fractions 1/7 and 187/770 for C/V and T/V. Common Pitfalls: Using an incorrect expression for T / V and forgetting to include both curved surface area and base areas. Not simplifying the ratio 187 : 770 correctly to 17 : 70, which is essential for clean algebra. Attempting to guess r : h without verifying both given ratios. Final Answer: The ratio of the base radius to the height of the cylinder is 7 : 10.

Question 3

The ratio of the total surface area to the volume of a solid sphere is 1 : 7. This sphere is melted and recast into many smaller solid spheres of equal size. The radius of each small sphere is 1/6 of the radius of the original large sphere. What is …

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Introduction / Context: This question combines sphere geometry with volume conservation. A single large sphere is melted and recast into many smaller identical spheres. The ratio of total surface area to volume for the large sphere is given, and the radius of each small sphere is a fraction of the large radius. Using this information, we find the radius of the large sphere, determine how many small spheres are formed, and finally compute the total curved surface area of all the small spheres, which is numerically the same as total surface area for spheres. Given Data / Assumptions: The ratio of total surface area to volume of the large sphere is 1 : 7. Radius of the large sphere is R centimetres. Radius of each small sphere is R / 6. The large sphere is completely melted and recast into small spheres, so total volume is conserved. Curved surface area of a sphere equals its total surface area, that is 4 * pi * r^2. pi is taken as 22 / 7 for numerical evaluation. Concept / Approach: The total surface area of a sphere of radius R is 4 * pi * R^2 and its volume is (4 / 3) * pi * R^3. The given ratio of surface area to volume 1 : 7 allows us to solve for R. Once R is known, we use volume conservation: volume of the large sphere equals the total volume of all small spheres. From this we find the number of small spheres. Finally, the total curved surface area is the number of small spheres multiplied by the surface area of one small sphere of radius R / 6. Step-by-Step Solution: Step 1: For the large sphere, total surface area = 4 * pi * R^2 and volume = (4 / 3) * pi * R^3. Step 2: Given ratio (surface area) : (volume) = 1 : 7, so (4 * pi * R^2) / ((4 / 3) * pi * R^3) = 1 / 7. Step 3: Simplify the left side: (4 * pi * R^2) * (3 / (4 * pi * R^3)) = 3 / R. Step 4: So 3 / R = 1 / 7, therefore R = 21 cm. Step 5: Radius of each small sphere r_s = R / 6 = 21 / 6 = 3.5 cm. Step 6: Using volume conservation, number of small spheres n satisfies (4 / 3) * pi * R^3 = n * (4 / 3) * pi * r_s^3. Step 7: Cancel the common factor (4 / 3) * pi, giving R^3 = n * r_s^3, so n = (R / r_s)^3 = 6^3 = 216. Step 8: Surface area of one small sphere = 4 * pi * r_s^2 = 4 * pi * 3.5^2 = 4 * pi * 12.25 = 49 * pi. Step 9: Total curved surface area of all small spheres = n * 49 * pi = 216 * 49 * pi = 10584 * pi. Step 10: Using pi = 22 / 7, total area = 10584 * (22 / 7) = 1512 * 22 = 33264 cm^2. Verification / Alternative check: We can confirm the radius of the large sphere by substituting R = 21 back into the ratio. Surface area is 4 * pi * 21^2 = 1764 * pi and volume is (4 / 3) * pi * 21^3 = (4 / 3) * pi * 9261 = 12348 * pi. The ratio (1764 * pi) : (12348 * pi) simplifies to 1 : 7. Also, 21 / 3.5 = 6, so the cubic ratio gives 6^3 = 216 small spheres, confirming the count. Multiplying 216 by 49 * pi and using pi = 22 / 7 again gives 33264, confirming the total curved surface area. Why Other Options Are Wrong: 31276, 36194 and 25182 are obtained from incorrect manipulation of the ratio, wrong radius, or a mistake in cube or square calculations. 29400 is a plausible looking distractor but does not match the precise product 10584 * pi with pi = 22 / 7. Only 33264 is consistent with all correct intermediate values and volume conservation. Common Pitfalls: Learners often confuse the ratio of surface area to volume and mistakenly treat it as R : something, instead of substituting the actual formulas. Another common error is forgetting that radius is raised to the power 3 in the volume formula and to the power 2 in the area formula, which affects how the ratio simplifies. Some also mix up the factor 1/6 between radius and diameter when recasting the sphere. Working carefully through algebraic simplification and using volume conservation systematically helps avoid these slips. Final Answer: The sum of the curved surface areas of all the small spheres is 33264 cm^2.

Question 4

A solid cube has side length 8 centimetres. It is cut by two planes along both diagonals of its top face, producing four equal congruent solid parts. What is the total surface area, in square centimetres (cm^2), of each of these four parts including…

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Introduction / Context: This solid geometry question studies how cutting a cube affects the surface area of the resulting pieces. A cube of side 8 centimetres is sliced along both diagonals of its top face by vertical planes, creating four congruent wedge shaped solids. We are asked for the total surface area of one such part, counting all faces, including the newly created cut surfaces, which is important in practical slicing or machining problems. Given Data / Assumptions: Original solid is a cube with side length a = 8 cm. Two vertical planes pass through the diagonals of the top face and extend through the cube. These cuts divide the cube into four equal congruent solids. Total surface area of one part must include original outer faces plus newly exposed cut faces. All edges of the cube remain straight and exact; no material is lost in cutting. Concept / Approach: Instead of trying to visualise every face of one part directly, a more efficient approach is to look at the total surface area of all four parts together. Initially, the cube has a known surface area. When we cut along a plane inside the cube, each new cutting face is shared by two parts. Therefore, the total surface area of all four parts equals the original outer surface area plus twice the total area of all cutting planes. Once that combined area is known, we divide by four to get the surface area of each congruent part. Step-by-Step Solution: Step 1: Surface area of the original cube is S_cube = 6 * a^2 = 6 * 8^2 = 6 * 64 = 384 cm^2. Step 2: Each diagonal of the top face has length a√2 = 8√2 cm. Step 3: Each cutting plane is vertical and has a rectangular intersection inside the cube of size 8 (height) by 8√2 (width), so area of each plane = 8 * 8√2 = 64√2 cm^2. Step 4: There are two such cutting planes, so total internal cutting area = 2 * 64√2 = 128√2 cm^2. Step 5: When counting the surface areas of all four parts, each cutting surface appears twice (once on each of the two touching solids), so the extra area added is 2 * 128√2 = 256√2 cm^2. Step 6: Total surface area of all four parts together = original outer area + extra area = 384 + 256√2 cm^2. Step 7: Since all four parts are congruent, surface area of each part = (384 + 256√2) / 4 = 96 + 64√2 cm^2. Verification / Alternative check: An alternate viewpoint is to observe that the top and bottom faces of each part are right angled isosceles triangles, and the side faces are rectangles and right angled quadrilaterals. Computing each face separately will also lead to the same total of 96 + 64√2 cm^2 per piece, but this is more time consuming. Using symmetry and counting cutting planes is therefore a more elegant and exam friendly method, and it reproduces the same final expression. Why Other Options Are Wrong: 80 + 64√2 and 96 + 48√2 both arise from partially counting either the original faces or the cutting faces. In one case, the contribution of the original cube faces is underestimated; in the other, the doubled effect of cuts on two solids is ignored. 80 + 48√2 and 112 + 64√2 are further from the correct total and do not satisfy the requirement that four times the individual surface area minus the original cube area equals twice the total cutting plane area. Common Pitfalls: Many students try to visualise every face of one part directly and often forget some faces or double count others. It is also easy to mistake the area of the cutting rectangles by using the side 8 instead of the diagonal 8√2. Another issue is neglecting that each cut face appears on two solids, which changes the total contribution when summing areas. Using a systematic global approach, starting from the whole cube and then dividing by four, greatly reduces such errors. Final Answer: The total surface area of each of the four parts is 96 + 64√2 cm^2.

Question 5

A regular pyramid has a square base of side 14 centimetres and vertical height 22 centimetres. The volume of this pyramid is equal to the volume of a solid sphere. What is the radius, in centimetres, of the sphere?

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Introduction / Context: This volumetric problem links the volume of a regular square pyramid to that of a sphere. We are given the base side and height of the pyramid and told that its volume equals the volume of a solid sphere. Using the standard volume formulas for both solids and equating them allows us to compute the radius of the sphere. Such comparisons are very common in aptitude tests where one shape is melted, recast, or equated to another shape. Given Data / Assumptions: The pyramid has a square base with side length 14 cm. The vertical height of the pyramid is 22 cm. The pyramid is regular, so its apex is directly above the centre of the square base. The volume of the pyramid equals the volume of a solid sphere. We are asked to find the radius of the sphere in centimetres. Use the usual volume formulas with pi taken as 22 / 7 if needed. Concept / Approach: The volume of a square based pyramid is given by: V_pyramid = (1 / 3) * (base area) * height. The base area here is side^2. The volume of a sphere of radius r is: V_sphere = (4 / 3) * pi * r^3. Because the volumes are equal, we set V_pyramid = V_sphere and solve for r. Simplifying with pi = 22 / 7 gives an exact integer value for the radius. Step-by-Step Solution: Step 1: Base side of the pyramid = 14 cm, so base area = 14 * 14 = 196 cm^2. Step 2: Height of the pyramid h = 22 cm. Step 3: Volume of the pyramid is V_pyramid = (1 / 3) * 196 * 22. Step 4: Compute 196 * 22 = 4312, so V_pyramid = 4312 / 3 cubic centimetres. Step 5: Let the radius of the sphere be r. Then V_sphere = (4 / 3) * pi * r^3. Step 6: Set V_pyramid = V_sphere, so (4 / 3) * pi * r^3 = 4312 / 3. Step 7: Multiply both sides by 3: 4 * pi * r^3 = 4312. Step 8: Take pi = 22 / 7, so 4 * pi = 4 * 22 / 7 = 88 / 7. Step 9: Then r^3 = 4312 / (88 / 7) = 4312 * 7 / 88. Step 10: Simplify 4312 / 88 = 49, so r^3 = 49 * 7 = 343. Step 11: Since 343 = 7^3, we get r = 7 cm. Verification / Alternative check: We can verify by substituting r = 7 back into the sphere volume formula. With r = 7, V_sphere = (4 / 3) * pi * 7^3 = (4 / 3) * pi * 343. Using pi = 22 / 7, this becomes (4 / 3) * (22 / 7) * 343 = (4 / 3) * 22 * 49 = (4 * 22 * 49) / 3 = 4312 / 3 cubic centimetres, exactly the same as the previously computed pyramid volume. This confirms that r = 7 cm is correct. Why Other Options Are Wrong: 493 and 983 suggest r^3 values that do not match the required 343 and would give volumes far larger than the pyramid. The options 9 and 14 also do not work: 9^3 and 14^3 lead to r^3 values that, when used in the sphere volume formula, do not produce 4312 / 3. Only r = 7 gives an exact match under the specified value of pi. Common Pitfalls: A frequent mistake is to forget that the base is square and to use the perimeter instead of the area. Another issue is misusing the formulas for volume, for example confusing the pyramid volume factor (1 / 3) with that of a prism. Some students also plug in approximate values for pi too early, making it harder to spot the exact cube of 7. Keeping all fractions symbolic until the last step and then substituting pi simplifies the algebra and leads cleanly to r^3 = 343. Final Answer: The radius of the sphere is 7 cm.

Question 6

Two circles have radii 9 centimetres and 12 centimetres respectively, and the distance between their centres is 15 centimetres. The circles intersect in a common chord. What is the length, in centimetres, of this common chord?

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Introduction / Context: This question concerns the geometry of two intersecting circles. When two circles overlap, their intersection is a common chord. Given the radii of both circles and the distance between their centres, we can find the length of that chord using right triangle relationships. Such problems test spatial visualisation and familiarity with circle geometry and the Pythagoras theorem. Given Data / Assumptions: Radius of the first circle R1 = 12 cm. Radius of the second circle R2 = 9 cm. Distance between centres of the circles d = 15 cm. The circles intersect, forming a common chord. We are asked to find the length of this chord in centimetres. All constructions are in a standard Euclidean plane. Concept / Approach: Consider line segment joining the two centres. The common chord is perpendicular to this line and passes through some point M between the centres. Let the half length of the chord be l. Then two right triangles form: one with hypotenuse R1 and another with hypotenuse R2, each having one leg l and the other legs being distances from centres to M. By using the relation between these distances and the fact that their sum is d, we can find l. The full chord length is then 2l. Step-by-Step Solution: Step 1: Let O1 and O2 be centres of radii 12 cm and 9 cm, and let M be the midpoint of the common chord. Step 2: Let O1M = a and O2M = b. Then a + b = d = 15 cm. Step 3: In right triangle O1M with the chord half length l, we have l^2 = R1^2 − a^2 = 12^2 − a^2. Step 4: In right triangle O2M, l^2 = R2^2 − b^2 = 9^2 − b^2. Step 5: Set these equal: 12^2 − a^2 = 9^2 − b^2, so 144 − a^2 = 81 − b^2. Step 6: Rearrange to get a^2 − b^2 = 144 − 81 = 63. Step 7: Factor the left side: (a − b)(a + b) = 63. But a + b = 15, so (a − b) * 15 = 63, giving a − b = 63 / 15 = 4.2. Step 8: Solve the system a + b = 15 and a − b = 4.2. Adding gives 2a = 19.2, so a = 9.6 cm, and b = 15 − 9.6 = 5.4 cm. Step 9: Use l^2 = R1^2 − a^2 = 144 − 9.6^2. Compute 9.6^2 = 92.16, so l^2 = 144 − 92.16 = 51.84. Step 10: Therefore l = √51.84 = 7.2 cm, so the full chord length = 2l = 14.4 cm. Verification / Alternative check: As a check, we can use the smaller circle. With b = 5.4 cm and R2 = 9 cm, l^2 should be 9^2 − 5.4^2 = 81 − 29.16 = 51.84, which again gives l = 7.2 cm and full chord 14.4 cm. Also, the distances a and b add to 15 cm, and the difference in squared radii equals (a − b)(a + b), confirming the internal consistency of the geometry and calculations. Why Other Options Are Wrong: 6.8, 7.2 and 13.6 centimetres are either half lengths or intermediate values appearing in the computations, not the final chord length. For example, 7.2 cm is the half length l, not the full chord. The value 12 cm is a distractor based on one of the radii rather than on the actual intersection geometry. Only 14.4 cm matches 2 * 7.2 cm obtained from correct application of right triangle relations. Common Pitfalls: A common error is to treat the chord as if it belonged to only one circle and directly apply formulas without considering the second circle. Others may forget that the chord is perpendicular to the line joining centres at its midpoint, leading to incorrect triangles. Algebra mistakes in solving the system for a and b, or in squaring decimals, can also lead to wrong chord lengths. Careful use of the relations a + b = d and a^2 − b^2 = R1^2 − R2^2 keeps the calculation systematic. Final Answer: The length of the common chord is 14.4 cm.

Question 7

Two circles touch each other at a single point X. Two common tangents to the circles meet at an external point P, and neither tangent passes through X. These tangents touch the larger circle at points B and C. If the radius of the larger circle is 1…

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Introduction / Context: This is a more advanced circle geometry problem involving two tangent circles and their common tangents. The two circles touch each other at a single point, and two external common tangents meet at a point P. One of the tangent segments from P to the larger circle is known, and we must deduce the radius of the smaller circle. The problem uses tangent length properties and the concept of homothety between circles. Given Data / Assumptions: Two circles touch externally at point X. The larger circle has radius R = 15 cm and centre O1. The smaller circle has radius r cm and centre O2 (r is unknown). P is the intersection point of two common external tangents. The tangents from P touch the larger circle at points B and C, with CP = 20 cm. Tangent lengths from the same external point to the same circle are equal, so PB = PC = 20 cm. All geometry is in a Euclidean plane. Concept / Approach: First, we use the right triangle formed by the radius to the point of tangency and the tangent segment to find the distance from P to the centre of the larger circle. Then, we use homothety of the two circles. For two externally tangent circles, the intersection point of their external common tangents, the two centres and the homothety centre are collinear, and the distances from this external homothety centre to the two centres are in the ratio of their radii. Together with the known distance between the centres (R + r), this allows us to solve for r. Step-by-Step Solution: Step 1: In triangle O1CP, O1C is a radius of length 15 cm and CP is a tangent of length 20 cm. Since radius is perpendicular to tangent at the point of contact, triangle O1CP is right angled at C. Step 2: Use Pythagoras theorem to find O1P: O1P^2 = O1C^2 + CP^2 = 15^2 + 20^2 = 225 + 400 = 625, so O1P = 25 cm. Step 3: Let O2 be the centre of the smaller circle with radius r. Since the circles touch externally, O1O2 = R + r = 15 + r. Step 4: The point P is the external homothety centre of the two circles, so O1, O2 and P are collinear and the distances satisfy O1P / O2P = R / r = 15 / r. Step 5: Place P outside the segment between O1 and O2 so that O1O2 = O1P − O2P. Thus 15 + r = O1P − O2P = 25 − O2P, giving O2P = 10 − r. Step 6: Use the ratio O1P / O2P = 15 / r: 25 / (10 − r) = 15 / r. Step 7: Cross multiply: 25r = 15(10 − r) = 150 − 15r. Step 8: Add 15r to both sides: 40r = 150, so r = 150 / 40 = 15 / 4 = 3.75 cm. Step 9: Check consistency: O2P = 10 − 3.75 = 6.25 cm and O1O2 = 15 + 3.75 = 18.75 cm; also O1P − O2P = 25 − 6.25 = 18.75 cm, which matches. Verification / Alternative check: Once r is known, we can confirm the homothety ratio. The ratio R : r = 15 : 3.75 = 4 : 1. The distances O1P : O2P are 25 : 6.25, which also simplifies to 4 : 1, confirming the homothety condition. Additionally, the centre distance 18.75 equals R + r, which is 15 + 3.75, so the circles indeed touch externally with radii 15 cm and 3.75 cm. These checks confirm that no algebraic mistake was made. Why Other Options Are Wrong: The values 3.5, 4.25 and 4.45 give ratios of radii that do not match the ratio of distances from P to the centres. If one inserts these in the equation 25 / (10 − r) = 15 / r, the two sides do not agree. The value 5.0 would also contradict the centre distance requirement and the tangent geometry. Only 3.75 cm satisfies both the distance equation and the homothety ratio exactly. Common Pitfalls: Many learners incorrectly assume that the tangent lengths from P to both circles are equal, which is not true here; they are only equal for tangents to the same circle. Another pitfall is neglecting the concept of homothety and trying to work purely with right triangles, which becomes very messy. It is also easy to mistake the order of points on the line O1O2P and write O1O2 = O1P + O2P, which leads to negative or impossible radii. Drawing a clear diagram and using the simple ratio O1P / O2P = R / r is the most efficient route. Final Answer: The radius of the smaller circle is 3.75 cm.

Question 8

Points A, B, C, D, E, F, G and H lie on a circle in that order and are equally spaced, so they form the vertices of a regular octagon. What is the measure, in degrees, of angle FDH (that is, angle formed at D by chords DF and DH)?

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Introduction / Context: This question is based on a regular octagon inscribed in a circle. Eight points are evenly spaced on the circle, so consecutive points are separated by equal arcs. The task is to find a specific inscribed angle, angle FDH, which is formed at vertex D by chords DF and DH. Solving it requires understanding the relation between inscribed angles and intercepted arcs in a circle. Given Data / Assumptions: Eight points A, B, C, D, E, F, G, H lie on a circle and are equally spaced. The points are in the order A, B, C, D, E, F, G, H around the circle. These points form a regular octagon. We must find angle FDH, with vertex at D and arms along chords DF and DH. The circle is regular, so the central angle for each adjacent pair of vertices is equal. Concept / Approach: In a regular octagon, the full circle of 360 degrees is divided into eight equal central angles, so each central angle measures 360 / 8 = 45 degrees. The inscribed angle theorem states that an inscribed angle is half the measure of the intercepted arc that does not include the vertex. To find angle FDH, we identify the arc FGH that this angle intercepts and use its measure to compute the inscribed angle at D. Step-by-Step Solution: Step 1: Since there are 8 equally spaced points, each adjacent arc (for example AB, BC, CD, etc.) corresponds to 360 / 8 = 45 degrees of arc. Step 2: Number the points in order around the circle: A (position 1), B (2), C (3), D (4), E (5), F (6), G (7), H (8). Step 3: Angle FDH has vertex at D and rays passing through F and H, so the inscribed angle at D intercepts the arc from F to H that does not contain D. Step 4: The arc from F (position 6) to H (position 8), avoiding D, passes through G only, so it consists of two adjacent arcs FG and GH. Step 5: Each adjacent arc is 45 degrees, so arc FG = 45 degrees and arc GH = 45 degrees. Step 6: Therefore, the intercepted arc FGH has measure 45 + 45 = 90 degrees. Step 7: By the inscribed angle theorem, angle FDH = (1 / 2) * measure of arc FGH = 90 / 2 = 45 degrees. Verification / Alternative check: We can also visualise the points on the circle by assigning central angles. Let D correspond to central angle 135 degrees, F to 225 degrees and H to 315 degrees. The chord DH is a diameter because the central angle between D and H is 180 degrees, and F lies midway between them in terms of arc steps. The triangle DFH has vertices on the circle, and angle FDH intercepts a 90 degree arc, which again confirms that angle FDH is 45 degrees by the inscribed angle rule. Why Other Options Are Wrong: 22.5 degrees would correspond to half of a single 45 degree arc, not two arcs. Values such as 30 and 42.5 degrees have no natural connection to multiples of 45 degrees in a regular octagon and result from miscounting the number of arcs or misapplying the inscribed angle theorem. 60 degrees would require an intercepted arc of 120 degrees, which is not the case between points F and H when D is the vertex. Common Pitfalls: Students sometimes choose the wrong intercepted arc, including point D within the arc or counting the longer arc that passes almost all the way around the circle. Another common mistake is to forget that an inscribed angle is always half of the intercepted arc, not equal to it. Drawing a clear sketch, marking the positions of F, D and H, and carefully identifying the shorter arc FGH that does not contain D are reliable ways to avoid confusion. Final Answer: The measure of angle FDH is 45 degrees.

Question 9

An equilateral triangle has area 300 square centimetres. From each of its three vertices, a small equilateral triangle is cut off so that the remaining central figure is a regular hexagon. The area of this regular hexagon is what percentage of the a…

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Introduction / Context: This geometry question explores the relationship between an equilateral triangle and a regular hexagon formed inside it by cutting off three smaller congruent equilateral triangles from the corners. The central figure is a regular hexagon, and we must find its area as a percentage of the area of the original triangle. This setup is closely related to classical constructions in which trisection points of the sides of an equilateral triangle are joined to form a regular hexagon inside the triangle. Given Data / Assumptions: The large figure is an equilateral triangle of area 300 cm^2. From each of its three vertices, a smaller equilateral triangle is cut off. The remaining central figure is a regular hexagon. All small removed triangles are congruent. We must find the area of the hexagon as a percentage of the area of the original triangle. Concept / Approach: In the standard construction, points at one third of each side of an equilateral triangle are joined to form a regular hexagon in the middle. The three removed corner pieces are equilateral triangles whose side length is one third of the side of the original triangle, while the original triangle has side length three units in that same scale. Using the area formula for equilateral triangles, we can find the ratio of the hexagon area to the triangle area as a pure number, which applies for any size, and then convert that ratio into a percentage and apply it to the given area 300 cm^2. Step-by-Step Solution: Step 1: Consider an equilateral triangle with side length 3 units for simplicity. Step 2: Divide each side into three equal segments and join suitable trisection points. This construction cuts off three corner equilateral triangles, each with side length 1 unit, and leaves a regular hexagon in the centre. Step 3: Area of an equilateral triangle with side s is (√3 / 4) * s^2. Step 4: The area of the original large triangle with side 3 is A_large = (√3 / 4) * 3^2 = (9√3 / 4). Step 5: Each small cut off triangle has side 1, so its area is (√3 / 4) * 1^2 = √3 / 4. Step 6: Three such triangles are removed, so total removed area = 3 * (√3 / 4) = 3√3 / 4. Step 7: Area of the central hexagon = A_large − removed area = (9√3 / 4) − (3√3 / 4) = 6√3 / 4 = 3√3 / 2. Step 8: Ratio of hexagon area to triangle area = (3√3 / 2) / (9√3 / 4) = (3√3 / 2) * (4 / 9√3) = 12 / 18 = 2 / 3. Step 9: Therefore, the hexagon has 2 / 3 of the area of the original triangle, which is 66.66% (approximately) of the triangle area. Step 10: For a triangle of area 300 cm^2, the hexagon area is (2 / 3) * 300 = 200 cm^2, which still represents 66.66% of the original area. Verification / Alternative check: Because the ratio 2 / 3 was obtained purely from similarity and exact formulas, it does not depend on the actual size of the triangle. Any equilateral triangle undergoing this corner cutting construction will produce a central hexagon with area equal to two thirds of the original area. Using the numeric value 200 / 300 = 2 / 3 confirms that the percentage is 66.66% (repeating). This validates both the geometric reasoning and the arithmetic. Why Other Options Are Wrong: 33.33% corresponds to one third of the area, which would be the removed area, not the remaining central region. 83.33% and 56.41% come from incorrect assumptions about the side ratios or from wrongly identified small triangle sizes. 50.00% would be the case if only half the area were removed or retained, but our construction shows that exactly one third of the area is cut off and two thirds remains. Common Pitfalls: Some students assume the side of the hexagon is half the side of the triangle rather than being related to trisection points, which changes the area ratio. Others may neglect that three congruent triangles are removed and forget to multiply by three. Another mistake is to compute numeric areas for a particular side length but then miscalculate the final percentage by mixing up removed and remaining area. Keeping the ratios symbolic until the end and clearly labelling which region is the hexagon avoids these issues. Final Answer: The area of the hexagon is 66.66% of the area of the triangle.

Question 10

The sum of the radii of two circles is 91 centimetres, and the difference between their areas is 2002 square centimetres. Using pi = 22 / 7, what is the radius, in centimetres, of the larger circle?

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Introduction / Context: This circle problem connects the radii of two circles with the difference in their areas. We are told the sum of the radii and the difference in areas and asked to find the radius of the larger circle. The question is a straightforward application of circle area formula combined with algebraic factorisation and the idea that R^2 − r^2 can be written as (R − r)(R + r). Given Data / Assumptions: Let R be the radius of the larger circle and r be the radius of the smaller circle. R + r = 91 cm. Difference of areas = pi * R^2 − pi * r^2 = 2002 cm^2. pi is to be taken as 22 / 7. We are required to find R, the radius of the larger circle. Concept / Approach: The area of a circle of radius x is pi * x^2. The difference of the areas of two circles with radii R and r is pi * (R^2 − r^2). Using the identity R^2 − r^2 = (R + r)(R − r), we can use the given sum R + r and area difference to determine R − r. Once both R + r and R − r are known, solving for R and r is a matter of solving a simple system of linear equations. Step-by-Step Solution: Step 1: Write the difference of areas using the formula for circle area: difference = pi * (R^2 − r^2) = 2002. Step 2: Substitute pi = 22 / 7 to get (22 / 7) * (R^2 − r^2) = 2002. Step 3: Solve for R^2 − r^2: R^2 − r^2 = 2002 * (7 / 22). Step 4: Compute 2002 / 22 = 91, so R^2 − r^2 = 91 * 7 = 637. Step 5: Use the identity R^2 − r^2 = (R + r)(R − r). We know R + r = 91, so (R + r)(R − r) = 637. Step 6: Substitute R + r = 91 to get 91 * (R − r) = 637. Step 7: Solve for R − r: R − r = 637 / 91 = 7. Step 8: Now we have two linear equations: R + r = 91 and R − r = 7. Step 9: Add the equations: (R + r) + (R − r) = 91 + 7, giving 2R = 98. Step 10: So R = 98 / 2 = 49 cm. The smaller radius would be r = 91 − 49 = 42 cm. Verification / Alternative check: To check, compute the areas with R = 49 and r = 42 using pi = 22 / 7. Area of larger circle = (22 / 7) * 49^2 = (22 / 7) * 2401. Since 2401 / 7 = 343, this gives 22 * 343 = 7546 cm^2. Area of smaller circle = (22 / 7) * 42^2 = (22 / 7) * 1764 = 22 * 252 = 5544 cm^2. The difference is 7546 − 5544 = 2002 cm^2, which matches the given value. This confirms that R = 49 cm is correct. Why Other Options Are Wrong: Choosing 56, 42, 63 or 35 for the larger radius leads either to a wrong sum of radii or to a difference of areas that is not equal to 2002 when computed using pi = 22 / 7. For example, if R = 56, then r = 35 to keep the sum 91, but the resulting area difference is different from 2002. Only R = 49 works consistently with both given conditions. Common Pitfalls: Common errors include forgetting to apply the factor pi on both circles, not using the difference of squares identity, or using only R^2 − r^2 directly without relating it to the given sum of radii. Some students also make mistakes simplifying 2002 * (7 / 22). Keeping the symbolic form pi * (R^2 − r^2) and carefully isolating R^2 − r^2 prevents such errors and leads naturally to the system R + r and R − r. Final Answer: The radius of the larger circle is 49 cm.

Question 11

A right circular cone has height 45 centimetres. It is cut by a plane parallel to the base at a height of 15 centimetres above the base, forming a smaller cone at the top. If the volume of this smaller cone is 18480 cubic centimetres, what is the vo…

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Introduction / Context: This problem involves a cone that is cut by a plane parallel to its base, creating a smaller similar cone at the top and a frustum below. We are given the height of the original cone and the height from the base where the cut is made, as well as the volume of the smaller cone. Using similarity of cones and the cubic relationship between linear dimensions and volume, we can find the volume of the original cone without needing the actual radius values. Given Data / Assumptions: Height of the original cone H = 45 cm. A plane parallel to the base cuts the cone at a height 15 cm above the base. Thus, the distance from the apex to the cutting plane is H − 15 = 45 − 15 = 30 cm. The smaller cone at the top has height h_small = 30 cm. Volume of the smaller cone V_small = 18480 cm^3. We must find the volume of the original cone V_original. The two cones are similar, so all linear dimensions scale in the same ratio. Concept / Approach: When two cones are similar, the ratio of their corresponding linear dimensions (such as heights and radii) is the same. Volumes of similar solids scale as the cube of the linear scale factor. Here, the height of the smaller cone is (H − 15) and the height of the original cone is H. Therefore, the volume ratio V_small / V_original equals (h_small / H)^3. We compute this ratio, then solve for V_original using the given smaller volume. Step-by-Step Solution: Step 1: Height of original cone H = 45 cm. Step 2: Height of smaller top cone h_small = H − 15 = 30 cm. Step 3: Because the cut is parallel to the base, the smaller cone is similar to the original cone with linear scale factor k = h_small / H = 30 / 45 = 2 / 3. Step 4: Volumes of similar solids scale as the cube of the linear factor, so V_small / V_original = k^3 = (2 / 3)^3 = 8 / 27. Step 5: Let V_original be the volume of the original cone. Then V_small = (8 / 27) * V_original. Step 6: Substitute V_small = 18480 cm^3 to get 18480 = (8 / 27) * V_original. Step 7: Solve for V_original: V_original = 18480 * (27 / 8). Step 8: Compute 18480 / 8 = 2310, then V_original = 2310 * 27. Step 9: Calculate 2310 * 27 = 2310 * 20 + 2310 * 7 = 46200 + 16170 = 62370 cm^3. Verification / Alternative check: As a quick check, compute the ratio 18480 / 62370 and simplify. Dividing numerator and denominator by 10 gives 1848 / 6237. This ratio should equal 8 / 27. Multiplying 8 / 27 by 62370 indeed returns 18480, confirming that the volume scale factor 8 / 27 is correct. Since we used only similarity ratios and no approximate constants, the value 62370 cm^3 is exact and consistent with the given smaller volume. Why Other Options Are Wrong: Other volumes such as 34650, 61600, 36960 and 54000 do not maintain the correct ratio between V_small and V_original equal to (2 / 3)^3. For instance, if 34650 were the correct original volume, then V_small would be 34650 * 8 / 27, which does not equal 18480. Only 62370 satisfies the exact similarity relationship derived from the geometry of similar cones. Common Pitfalls: A common mistake is to incorrectly use a linear or square ratio instead of the cube when relating volumes. Some students treat the height ratio 2 / 3 as if it directly applies to volume, giving V_small = (2 / 3) * V_original, which is incorrect. Others may misinterpret the phrase “cut at a height of 15 cm from its base” and take the smaller cone height as 15 cm instead of 30 cm. Carefully visualising the cone and remembering that volume scales with the cube of the linear factor avoids these errors. Final Answer: The volume of the original cone is 62370 cm^3.

Question 12

A solid cuboid has dimensions 65 centimetres by 26 centimetres by 3.9 centimetres. Identical solid cubes of the largest possible size are cut from this cuboid. What is the total surface area, in square centimetres (cm^2), of all the small cubes take…

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Introduction / Context: This question combines number theory (greatest common divisor) with three dimensional geometry. A rectangular cuboid is cut into identical cubes of the largest possible size, and we are asked to find the total surface area of all those cubes together. The largest possible cube edge length must exactly divide each of the three dimensions of the cuboid. Once this edge is known, we can determine the number of cubes and then their combined surface area. Given Data / Assumptions: Dimensions of the cuboid: length = 65 cm, breadth = 26 cm, height = 3.9 cm. Identical cubes of the largest possible size are cut from the cuboid, with no leftover material. We must find the total surface area of all small cubes together. The edge length of each small cube must exactly divide all three dimensions. Concept / Approach: The largest possible cube that can fit exactly inside the cuboid will have edge length equal to the greatest common divisor (GCD) of the three dimensions, expressed in consistent units. It is convenient to convert all dimensions to millimetres to avoid decimals and find the GCD. After finding the cube edge length, we compute the number of cubes as (volume of cuboid) / (volume of one cube). The surface area of one cube is 6 * a^2, so the total surface area is number of cubes times 6 * a^2. Step-by-Step Solution: Step 1: Convert dimensions to millimetres to avoid decimals: 65 cm = 650 mm, 26 cm = 260 mm, 3.9 cm = 39 mm. Step 2: Find the GCD of 650, 260 and 39. Factorising: 650 = 2 * 5^2 * 13, 260 = 2^2 * 5 * 13, 39 = 3 * 13. The common factor is 13. Step 3: So the largest cube edge length is 13 mm, which is 1.3 cm. Step 4: Volume of the cuboid in cubic millimetres is 650 * 260 * 39 = 6,591,000 mm^3. Step 5: Volume of one cube with edge 13 mm is 13^3 = 2197 mm^3. Step 6: Number of cubes n = 6,591,000 / 2197 = 3000. Step 7: Convert cube edge back to centimetres: a = 1.3 cm. Surface area of one cube is 6 * a^2 = 6 * 1.3^2 = 6 * 1.69 = 10.14 cm^2. Step 8: Total surface area of all cubes = n * 6 * a^2 = 3000 * 10.14 = 30,420 cm^2. Verification / Alternative check: We can compute the cuboid volume in cubic centimetres as 65 * 26 * 3.9 = 6591 cm^3. The volume of one cube in cubic centimetres is 1.3^3 = 2.197 cm^3. Then the number of cubes is 6591 / 2.197 = 3000, confirming our earlier count. Using 3000 cubes and surface area 10.14 cm^2 per cube again yields 30,420 cm^2. Both millimetre based and centimetre based calculations match, verifying the correctness of the answer. Why Other Options Are Wrong: Values such as 15210, 20280, 16440 and 22860 arise from mistaken GCDs (for example using 1.3 incorrectly), miscounting the number of cubes, or omitting the factor 6 when calculating the surface area of one cube. Only 30420 matches the product of 3000 and 10.14 and is consistent with the exact division of the cuboid into cubes of edge 1.3 cm. Common Pitfalls: A common error is to treat the cube edge as an integer in centimetres, such as 1 cm or 2 cm, without converting the decimal dimension 3.9 properly. Another pitfall is to forget to convert units to a common scale before taking the GCD or to simply approximate the GCD visually. Finally, some learners compute the total surface area as n * a^2 rather than n * 6 * a^2. Taking care with unit conversion, factorisation and the surface area formula avoids these mistakes. Final Answer: The total surface area of all the small cubes together is 30420 cm^2.

Question 13

A regular triangular pyramid (a pyramid with an equilateral triangular base) is cut by two planes parallel to its base such that the altitude of the pyramid is trisected into three equal segments. The top, middle and bottom parts of the pyramid have…

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Introduction / Context: This problem involves a regular triangular pyramid, also known as a pyramid with an equilateral triangular base and an apex directly above the centre. The altitude of the pyramid is divided into three equal segments by planes parallel to the base, creating three parts with different volumes: a small pyramid at the top and two frustum like sections below. Using similarity and the cubic nature of volume scaling, we can determine the volume ratios V1 : V2 : V3 without needing any actual measurements. Given Data / Assumptions: The pyramid has an equilateral triangular base and a single apex vertically above the centre. The altitude (height) of the pyramid is H. Two planes parallel to the base cut the pyramid and trisect the altitude into three equal parts. The top part (nearest the apex) has volume V1. The middle frustum like part has volume V2. The bottom frustum like part (nearest the base) has volume V3. The cuts are parallel to the base, so the resulting cross sections are similar to the base. Concept / Approach: For any pyramid, cross sections parallel to the base at a fraction of the height form similar polygons. The volumes of similar pyramids scale with the cube of the linear scale factor. If we measure height from the apex downward, then the volume from the apex down to height h is proportional to (h / H)^3. We use this to find the volumes up to heights H / 3 and 2H / 3 from the apex, then subtract to obtain the volumes of the top, middle and bottom parts in terms of the total volume V. Step-by-Step Solution: Step 1: Let the total volume of the original pyramid be V and its full altitude be H. Step 2: The altitude is trisected, so planes are at heights H / 3 and 2H / 3 measured from the apex downwards. Step 3: The volume of a smaller pyramid cut off at height h from the apex is V(h) = V * (h / H)^3 due to similarity. Step 4: The top section is a small pyramid of height H / 3, so its volume V1 = V * (1 / 3)^3 = V * (1 / 27). Step 5: The volume from apex down to height 2H / 3 is V * (2 / 3)^3 = V * (8 / 27). Step 6: The middle section lies between H / 3 and 2H / 3, so its volume V2 = V * (8 / 27) − V * (1 / 27) = V * (7 / 27). Step 7: The bottom section is the remaining part between height 2H / 3 and H, so its volume V3 = V − V * (8 / 27) = V * (19 / 27). Step 8: Therefore the volumes in terms of V are V1 = V / 27, V2 = 7V / 27 and V3 = 19V / 27. Step 9: The ratio V1 : V2 : V3 is (1 / 27) : (7 / 27) : (19 / 27) = 1 : 7 : 19. Verification / Alternative check: We can check the correctness by verifying that the three parts sum to the total volume. Adding the fractions: (1 / 27) + (7 / 27) + (19 / 27) = 27 / 27 = 1, confirming that V1 + V2 + V3 = V. Additionally, the top part is much smaller compared to the bottom section, which is intuitive since the base is widest near the bottom, and volume increases sharply with height in a pyramid. This agrees with having a small coefficient 1, then 7 and then 19 for the three parts. Why Other Options Are Wrong: Ratios such as 1 : 8 : 27 or 1 : 8 : 19 emerge from using linear factors directly or squaring instead of cubing when scaling volumes. The ratio 2 : 9 : 27 or 1 : 6 : 20 does not match the correct fractional volumes derived from (h / H)^3 and fails the check that the parts must sum to the total volume V. Only 1 : 7 : 19 satisfies both the cubic scaling and the sum condition. Common Pitfalls: A typical mistake is to treat the volume as proportional to the height or area instead of the cube of the height for similar pyramids. Some learners also misinterpret “trisects the altitude” as meaning that volumes are equal, which is not true. Another pitfall is to forget to subtract the smaller partial volume from the larger one when computing the middle section. Carefully applying similarity and remembering that volume scales with the cube of linear scale resolves these issues. Final Answer: The required ratio of volumes is V1 : V2 : V3 = 1 : 7 : 19.

Question 14

The length, breadth and height of a solid cuboid are 12 centimetres, 8 centimetres and 6 centimetres respectively. What is the volume of this cuboid in cubic centimetres?

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Introduction / Context: This is a straightforward mensuration question about the volume of a rectangular cuboid. The three dimensions of the cuboid are provided and we must compute the volume. Although the original statement mentions a value for pi, pi is not required here because a cuboid involves only straight edges and right angles, not circular shapes. This question tests basic understanding of volume formulas in three dimensional geometry. Given Data / Assumptions: Length of the cuboid L = 12 cm. Breadth (width) of the cuboid B = 8 cm. Height of the cuboid H = 6 cm. We are asked to find the volume in cubic centimetres. The cuboid is a rectangular box, so all angles are right angles. Concept / Approach: The volume of a cuboid is given by: V = L * B * H, where L is length, B is breadth and H is height. Since all measurements are in the same unit (centimetres), we simply multiply these three numbers to obtain the volume in cubic centimetres. No circular geometry is involved, so pi is irrelevant despite being mentioned in the original text. Step-by-Step Solution: Step 1: Write down the formula for volume of a cuboid: V = L * B * H. Step 2: Substitute the given values: L = 12 cm, B = 8 cm, H = 6 cm. Step 3: Compute L * B = 12 * 8 = 96. Step 4: Multiply by the height: V = 96 * 6 = 576. Step 5: Therefore, the volume V = 576 cubic centimetres. Verification / Alternative check: We can cross check by arranging factors differently: 12 * 6 = 72, and 72 * 8 = 576, which matches the previous calculation. Since all three dimensions are positive and the multiplication is straightforward, there is no ambiguity in the result. The unit is cubic centimetres because we multiplied centimetres three times (cm * cm * cm). Why Other Options Are Wrong: 288 cubic centimetres and 432 cubic centimetres result from mistakenly multiplying only two of the three dimensions or from arithmetic errors. 864 cubic centimetres and 1152 cubic centimetres come from overestimating one of the products, for example treating one dimension as 12 instead of 6 or multiplying incorrectly. Only 576 cubic centimetres correctly reflects the product 12 * 8 * 6. Common Pitfalls: Students sometimes confuse surface area with volume and try to add terms like 2(LB + BH + LH), which is incorrect for volume. Another pitfall is mixing units or thinking pi is involved simply because it is written in the question. Remember that volume for a cuboid always comes from multiplying the three orthogonal edge lengths using consistent units, without any involvement of pi. Final Answer: The volume of the cuboid is 576 cubic centimetres.

Question 15

A right prism has a regular hexagonal base of side 4 centimetres and height 8 centimetres. What is the total surface area of the prism, in square centimetres (cm^2)?

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Introduction / Context: This question asks for the total surface area of a right prism whose base is a regular hexagon. We are given the side length of the hexagonal base and the height of the prism. Total surface area includes the areas of the two congruent hexagonal bases and the lateral rectangular faces that connect them. The problem tests familiarity with area formulas for regular hexagons and for rectangles, as well as the structure of a prism. Given Data / Assumptions: The base is a regular hexagon with side length s = 4 cm. The prism height (distance between the two bases) is h = 8 cm. The prism is right, so lateral faces are rectangles perpendicular to the bases. We must find total surface area in cm^2. Use the standard area formula for a regular hexagon. Concept / Approach: A regular hexagon of side s can be divided into six equilateral triangles each of side s. The area of an equilateral triangle with side s is (√3 / 4) * s^2. Therefore the area of a regular hexagon is: Area_hexagon = 6 * (√3 / 4) * s^2 = (3√3 / 2) * s^2. For a prism, total surface area = 2 * (area of base) + (perimeter of base) * height. We use the hexagon area formula and the perimeter of a regular hexagon (6s) to compute the total. Step-by-Step Solution: Step 1: Compute the area of the regular hexagonal base using s = 4 cm. Step 2: Area_hexagon = (3√3 / 2) * s^2 = (3√3 / 2) * 4^2 = (3√3 / 2) * 16 = 24√3 cm^2. Step 3: The prism has two such bases, so total area of both bases = 2 * 24√3 = 48√3 cm^2. Step 4: Perimeter of the regular hexagon = 6 * s = 6 * 4 = 24 cm. Step 5: Lateral surface area = perimeter * height = 24 * 8 = 192 cm^2. Step 6: Total surface area = area of both bases + lateral surface area = 48√3 + 192 cm^2. Step 7: Factor the expression to match the options: 48√3 + 192 = 48(√3 + 4) = 48(4 + √3) cm^2. Verification / Alternative check: We can double check the base area by computing the area of one equilateral triangle and multiplying by 6. For s = 4, area of one equilateral triangle is (√3 / 4) * 16 = 4√3 cm^2. For six such triangles, the hexagon area = 6 * 4√3 = 24√3 cm^2, matching our earlier computation. Adding twice this area to the correctly computed lateral area 192 again gives 48√3 + 192, confirming the total surface area. Why Other Options Are Wrong: Expressions like 54(3 + √3), 36(3 + √3), 24(4 + √3) and 60(2 + √3) either have incorrect coefficients for the base area, the lateral area, or both. They do not simplify to 192 + 48√3. Only 48(4 + √3) exactly expands to 192 + 48√3, which is consistent with the correct geometric calculations. Common Pitfalls: Students sometimes use the wrong base area formula, for example treating the hexagon as if it were made of rectangles instead of equilateral triangles. Others might forget to include both bases in the total surface area or miscalculate the perimeter, using 4s instead of 6s. Another common error is to confuse lateral area with total surface area. Carefully applying the correct formula for a regular hexagon and adding base and lateral areas separately prevents these mistakes. Final Answer: The total surface area of the prism is 48(4 + √3) cm^2.

Question 16

A solid right circular cylinder of radius 4.5 centimetres and height 12 centimetres just fits completely inside another cylinder so that their axes are perpendicular to each other. What is the radius, in centimetres, of the larger cylinder?

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Introduction / Context: This spatial geometry problem involves one cylinder fitting inside another with their axes perpendicular. The smaller cylinder is placed inside the larger one in such a way that it just fits, touching the inner surface of the larger cylinder. The task is to find the radius of the larger cylinder, given the dimensions of the smaller one. This is equivalent to inscribing a rectangle into a circle in two dimensions. Given Data / Assumptions: The inner cylinder has radius r = 4.5 cm. The inner cylinder has height h = 12 cm. The inner cylinder fits inside a larger cylinder so that their axes are perpendicular. The smaller cylinder just fits, meaning it touches the inner wall of the larger cylinder but does not overlap or leave gaps beyond that circle. We must find the radius R of the larger cylinder. Concept / Approach: When viewing the configuration along the axis of the larger cylinder, we see the cross section of the larger cylinder as a circle of radius R. The smaller cylinder, whose axis is perpendicular, appears as a rectangle inside this circle. The rectangle has one side equal to the height of the smaller cylinder (12 cm) and the other equal to its diameter (2r = 9 cm). For the smaller cylinder to just fit, this rectangle must be exactly inscribed in the circle, so the circle radius R is half the length of the rectangle diagonal. Using the Pythagoras theorem on this rectangle gives us R. Step-by-Step Solution: Step 1: The rectangle representing the smaller cylinder in the cross section has dimensions: width = 2r = 2 * 4.5 = 9 cm and height = h = 12 cm. Step 2: The diagonal d of this rectangle is given by Pythagoras theorem: d^2 = 9^2 + 12^2. Step 3: Compute the squares: 9^2 = 81 and 12^2 = 144, so d^2 = 81 + 144 = 225. Step 4: Thus d = √225 = 15 cm. Step 5: This diagonal is the diameter of the inner circle of the larger cylinder, so the radius R of the larger cylinder is R = d / 2 = 15 / 2 = 7.5 cm. Verification / Alternative check: We can verify by noting that in any circle of radius R, the largest possible rectangle that can be inscribed has a diagonal equal to the circle diameter 2R. If R = 7.5 cm, then the diameter is 15 cm, matching the computed diagonal of the 9 by 12 rectangle. Therefore, the rectangle representing the smaller cylinder just fits inside the cross sectional circle of the larger cylinder. Any smaller value of R would make the diagonal of the rectangle longer than the diameter of the circle, which is impossible, and any larger R would leave extra space, meaning it would not be a "just fits" situation. Why Other Options Are Wrong: Radii 5 cm, 6 cm and 9 cm do not satisfy the relation R = (1 / 2) * √(9^2 + 12^2). If R = 6 cm, diameter is 12 cm, which is less than the rectangle diagonal of 15 cm, so the rectangle would not fit. A radius of 15 cm would give a diameter of 30 cm, which is much larger than necessary and contradicts the phrase "just fits". Hence these options are not consistent with the geometry of the inscribed rectangle. Common Pitfalls: A common mistake is to assume that the radius of the larger cylinder equals the larger of the two dimensions (9 or 12) or half of one of them. This ignores the fact that both dimensions of the rectangle must fit within the circle at the same time, which depends on the diagonal, not just a single side. Another pitfall is to incorrectly apply Pythagoras theorem or to forget to divide the diagonal by two to obtain the radius. Visualising the cross section and remembering that the diameter is the rectangle diagonal helps avoid these errors. Final Answer: The radius of the larger cylinder is 7.5 cm.

Question 17

Two equal solid spheres are carved out from a solid cube of side length (12 + 4√3) centimetres. The spheres have the maximum possible radius and do not overlap. What is the maximum volume, in cubic centimetres, of each sphere? (Use pi = 22 / 7)

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Introduction / Context: This is a challenging three dimensional optimisation problem. Two equal spheres are to be carved out of a cube, and we want each sphere to have the largest possible radius so that both fit inside without overlapping. The side length of the cube is given in a special form (12 + 4√3) centimetres, which hints at a neat algebraic relationship. Once we find the maximum possible radius of each sphere, we calculate its volume using the standard sphere volume formula with pi = 22 / 7. Given Data / Assumptions: Side length of the cube a = 12 + 4√3 cm. Two equal spheres are carved out from the cube. The spheres do not overlap and have maximum possible radius. Each sphere is tangent to three mutually perpendicular faces of the cube. The two spheres are placed near opposite corners of the cube along a space diagonal and are tangent to each other. pi is to be taken as 22 / 7. Concept / Approach: Place the cube with one corner at the origin and edges along the coordinate axes. Let one sphere of radius r be centred at (r, r, r) so that it touches the three coordinate planes. The other sphere, also radius r, is centred at (a − r, a − r, a − r) and touches the three faces at the opposite corner. For both spheres to just fit and touch each other without overlap, the distance between their centres must equal 2r. This distance lies along the space diagonal of the cube. Using the distance formula and equating to 2r gives an equation in r and a, which we solve. Then volume of each sphere is (4 / 3) * pi * r^3. Step-by-Step Solution: Step 1: Let the cube side length be a = 12 + 4√3 cm. Step 2: Place centres of the two equal spheres at C1 = (r, r, r) and C2 = (a − r, a − r, a − r). Step 3: Distance between the centres is |C1C2| = √[(a − 2r)^2 + (a − 2r)^2 + (a − 2r)^2] = √[3(a − 2r)^2] = √3 * (a − 2r). Step 4: For the spheres to just touch each other, this distance must equal 2r, so √3 * (a − 2r) = 2r. Step 5: Substitute a = 12 + 4√3 and solve for r: √3 * (12 + 4√3 − 2r) = 2r. Step 6: Rearrange as √3 * a − 2√3 * r = 2r, so √3 * a = 2r + 2√3 * r = 2r(1 + √3). Step 7: Thus r = (√3 * a) / [2(1 + √3)]. Factor a: a = 4(3 + √3). Step 8: Substitute a into r: r = (√3 * 4(3 + √3)) / [2(1 + √3)] = (4√3(3 + √3)) / [2(1 + √3)]. Step 9: Simplify (3 + √3) = (1 + √3) * √3, so r = (4√3 * (1 + √3) * √3) / [2(1 + √3)] = (4 * 3) / 2 = 6 cm. Step 10: Volume of each sphere is V = (4 / 3) * pi * r^3 = (4 / 3) * pi * 6^3 = (4 / 3) * pi * 216 = 288 * pi. Step 11: With pi = 22 / 7, V = 288 * 22 / 7 = 6336 / 7 ≈ 905.14 cubic centimetres. Verification / Alternative check: To verify, check that two spheres of radius 6 cm fit inside the cube. The full length of the space diagonal of the cube is a√3 = (12 + 4√3)√3. The centres are separated by 2r = 12 cm. The corner to corner diagonal distance must be larger than 12 plus twice the radius margin to the cube faces, which is consistent with a carefully chosen value of a. We also see that r is derived algebraically without approximation, and only the final numerical volume uses pi ≈ 22 / 7, giving about 905.14 cm^3, which matches option B. Why Other Options Are Wrong: The values 1077.31, 966.07 and 1007.24 correspond to incorrect assumed radii or misapplied distance relations between the centres, for example treating the spheres as if they were inscribed individually in the cube without considering mutual contact. The value 845.00 is another distractor that does not come from the exact radius r = 6. Only 905.14 matches the exact expression 288 * (22 / 7) derived from the correct geometric configuration. Common Pitfalls: A frequent error is to assume that each sphere is simply inscribed in the whole cube, giving radius a / 2, which would allow only one sphere, not two. Others place the spheres side by side along an edge and use a / 4 as the radius, ignoring the possibility of using the space diagonal to fit larger spheres. Another pitfall is to mis-handle the algebra when simplifying r = (√3 * a) / [2(1 + √3)]. Recognising the factorisation (3 + √3) = (1 + √3)√3 is crucial to obtaining the clean radius r = 6 cm. Final Answer: The maximum volume of each sphere is approximately 905.14 cubic centimetres.

Question 18

Three solid toys are in the shapes of a right circular cylinder, a hemisphere, and a right circular cone. All three solids have the same circular base, and the height of each toy is 2√2 cm. What is the ratio of the total surface areas of the cylinde…

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Introduction / Context: This problem compares the total surface areas of three different solids: a right circular cylinder, a hemisphere, and a right circular cone. All three solids share the same circular base and are given the same height. The question tests understanding of surface area formulas and how changing shape (while keeping base and height fixed) affects total surface area. Given Data / Assumptions: All three toys have the same circular base of radius r centimetres. The height of each toy is 2√2 cm. For the hemisphere, the height equals its radius, so r = 2√2 cm. Total surface area is taken as curved surface plus base area for each solid. We need the ratio of total surface areas of cylinder : hemisphere : cone. Concept / Approach: We use standard total surface area formulas for each solid in terms of the common radius r and the height h. Because r and h are the same in all three, the constant π and any common factors will cancel when we form the ratio. The key steps are to compute each total surface area in terms of π and simple numbers, then divide through by the common factor to simplify the ratio. Step-by-Step Solution: Step 1: For the cylinder, radius r = 2√2 cm and height h = 2√2 cm. Step 2: Total surface area of cylinder = 2 * π * r * h + 2 * π * r^2. Step 3: Compute r^2 = (2√2)^2 = 8. So cylinder area = 2π * (2√2) * (2√2) + 2π * 8 = 2π * 8 + 16π = 32π. Step 4: For the hemisphere of radius r, total surface area (curved plus base) = 3π * r^2 = 3π * 8 = 24π. Step 5: For the cone, radius r = 2√2 and height h = 2√2. Slant height l = √(r^2 + h^2) = √(8 + 8) = √16 = 4. Step 6: Total surface area of cone = π * r * l + π * r^2 = π * (2√2) * 4 + π * 8 = 8√2 π + 8π. Step 7: Form the ratio cylinder : hemisphere : cone = 32π : 24π : (8√2 π + 8π). Step 8: Divide every term by 8π to simplify: 32π / 8π = 4, 24π / 8π = 3, and (8√2 π + 8π) / 8π = √2 + 1. Step 9: So the required ratio is 4 : 3 : (√2 + 1). Verification / Alternative check: You can verify the ratio numerically by taking π as 3.14 and √2 as about 1.414, then calculating each area in square centimetres. When you simplify the three values, you will see they reduce to approximately the same ratio 4 : 3 : (√2 + 1). Because all shapes share the same base and height, no hidden scaling factors are present, so the ratio is stable regardless of the numerical value chosen for π. Why Other Options Are Wrong: Option b, 4 : 3 : (2 + √2), would correspond to a cone area that is too large by an extra 1 in the last term. Option c, 4 : 3 : 2√2 ignores the base area contribution and does not match the total surface area expression. Option d, 2 : 1 : (1 + √2), does not match the relative sizes of cylinder and hemisphere derived from their formulas, since the cylinder area is clearly more than just double the hemisphere area for this radius and height combination. Common Pitfalls: A common mistake is to forget to include the base area when computing total surface area, especially for the cone and hemisphere. Another frequent error is to use the height instead of the slant height for the cone's curved surface area. Some learners also assume r is an unknown rather than recognising that r = 2√2 for the hemisphere based on its height, which can lead to inconsistent expressions. Carefully using each formula and simplifying step by step avoids these issues. Final Answer: The ratio of total surface areas of the cylinder, hemisphere, and cone is 4 : 3 : (√2 + 1).

Question 19

A solid cube is cut into 27 identical smaller cubes. By what percentage does the total surface area increase as a result of this cutting?

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Introduction / Context: This question explores how the total surface area changes when a solid cube is cut into many identical smaller cubes. It is a classic aptitude problem that tests spatial reasoning and understanding of how surface area depends on side length and the number of exposed faces after cutting. Given Data / Assumptions: We start with one solid cube of side length a centimetres. The cube is cut into 27 identical smaller cubes. 27 is 3^3, so each edge of the original cube is divided into 3 equal parts. We need the percentage increase in total surface area from the original cube to all 27 smaller cubes combined. Concept / Approach: Surface area of a cube with side a is 6 * a^2. When the big cube is cut into 27 identical smaller cubes, each smaller cube has side length a / 3. We compute the surface area of one smaller cube, multiply by 27, and compare this new total with the original surface area. The percentage increase is (increase / original) * 100. The value of a will cancel, so the final result is independent of the actual size of the original cube. Step-by-Step Solution: Step 1: Let the original cube have side length a. Its surface area S₁ = 6 * a^2. Step 2: Cutting into 27 smaller cubes means each side is divided into 3 equal segments, so each small cube has side a / 3. Step 3: Surface area of one smaller cube is S_small = 6 * (a / 3)^2 = 6 * a^2 / 9 = (2/3) * a^2. Step 4: There are 27 such smaller cubes, so total surface area after cutting is S₂ = 27 * S_small = 27 * (2/3) * a^2 = 18 * a^2. Step 5: Original surface area S₁ = 6 * a^2, new surface area S₂ = 18 * a^2. Step 6: Increase in surface area = S₂ − S₁ = 18 * a^2 − 6 * a^2 = 12 * a^2. Step 7: Percentage increase = (increase / original) * 100 = (12 * a^2 / 6 * a^2) * 100 = 2 * 100 = 200 %. Verification / Alternative check: To verify numerically, assume a simple value such as a = 3 cm. Then the original cube has side 3 cm and surface area 6 * 9 = 54 square centimetres. Each smaller cube then has side 1 cm, surface area 6 square centimetres, and 27 cubes together have area 27 * 6 = 162 square centimetres. The increase is 162 − 54 = 108, and (108 / 54) * 100 = 200 %, confirming our algebraic result. Why Other Options Are Wrong: 150 %, 250 %, and 300 % correspond to different ratios of new surface area to old surface area. For example, a 150 % increase would mean the new area is 2.5 times the old area, and a 300 % increase would mean it is 4 times the old area. Our calculation shows the new area is exactly 3 times the original (18 a^2 compared to 6 a^2), which corresponds to a 200 % increase, not any of the other listed values. Common Pitfalls: A typical mistake is to assume that the surface area stays the same or only increases slightly because the volume is unchanged. Another error is to misinterpret “27 identical cubes” and not correctly deduce that each edge is divided into 3 equal segments. Finally, some learners forget to square the side length when computing surface area, which leads to incorrect comparisons. Using a variable for side length and carefully applying the cube surface area formula prevents these errors. Final Answer: The total surface area increases by 200 % when the cube is cut into 27 identical cubes.

Question 20

A regular square pyramid with base side 20 cm and height 45 cm is melted and recast into several identical regular triangular pyramids. Each smaller pyramid has an equilateral triangular base of side 10 cm and height 10√3 cm. How many such regular t…

Accepted Answer

Introduction / Context: This is a volume conservation problem involving pyramids. A large regular square pyramid is melted and recast into smaller regular triangular pyramids. Since melting and recasting do not change the total volume (ignoring any loss), the sum of the volumes of the smaller pyramids must equal the volume of the original pyramid. The question checks your ability to use the volume formula for pyramids and perform accurate geometric calculations. Given Data / Assumptions: Original solid: regular square pyramid with base side 20 cm and height 45 cm. Smaller solids: regular triangular pyramids with equilateral base of side 10 cm and height 10√3 cm. All smaller pyramids are identical. Volume is conserved during melting and recasting. We need the number of smaller pyramids produced. Concept / Approach: The volume of any pyramid (square based or triangular based) is given by V = (1/3) * base area * height. We first calculate the volume of the original square pyramid. Then we compute the volume of one triangular pyramid using the area formula for an equilateral triangle. Finally, we divide the original volume by the volume of one small pyramid to obtain the number of pyramids formed. Step-by-Step Solution: Step 1: Compute the base area of the square pyramid. Base side = 20 cm, so base area A₁ = 20 * 20 = 400 cm^2. Step 2: Height of the square pyramid is h₁ = 45 cm. Volume of the square pyramid V₁ = (1/3) * A₁ * h₁ = (1/3) * 400 * 45 = 400 * 15 = 6000 cm^3. Step 3: For one triangular pyramid, the base is an equilateral triangle of side 10 cm. Step 4: Area of an equilateral triangle with side a is A = (√3 / 4) * a^2. Here a = 10 cm, so base area A₂ = (√3 / 4) * 100 = 25√3 cm^2. Step 5: Height of each triangular pyramid is h₂ = 10√3 cm. Volume of one triangular pyramid V₂ = (1/3) * A₂ * h₂. Step 6: Substitute A₂ and h₂: V₂ = (1/3) * 25√3 * 10√3 = (1/3) * 25 * 10 * (√3 * √3) = (1/3) * 250 * 3 = 250 cm^3. Step 7: Since volume is conserved, number of smaller pyramids n = V₁ / V₂ = 6000 / 250 = 24. Verification / Alternative check: You can verify by multiplying the number of small pyramids by the volume of one. If there are 24 pyramids, total volume = 24 * 250 = 6000 cm^3, which equals the original volume of the square pyramid. This confirms that no arithmetic mistakes were made and that exactly 24 pyramids can be formed. Why Other Options Are Wrong: If the answer were 20, the total volume would be 20 * 250 = 5000 cm^3, which is less than the original volume, violating volume conservation. Similarly, 27 pyramids would give 6750 cm^3, and 28 pyramids would give 7000 cm^3, both larger than the available volume. Only 24 gives a perfect match with the original pyramid volume. Common Pitfalls: Some learners forget that the base areas are different (square versus equilateral triangle) and mistakenly use the wrong area formula. Others may incorrectly compute the area of an equilateral triangle or forget that √3 * √3 = 3. Another pitfall is to misapply the pyramid volume formula and leave out the factor 1/3. Working methodically and writing each formula clearly helps avoid these issues. Final Answer: The number of regular triangular pyramids formed is 24.

Volume and Surface Area Questions