Question 1

In triangle ABC, DE is drawn parallel to side BC. Given AB = 7.5 cm, BD = 6 cm on side AB, and DE = 2 cm, what is the length (in centimetres) of side BC?

Accepted Answer

Introduction / Context: This geometry question uses the concept of similar triangles when a line segment is drawn parallel to one side of a triangle. When DE is parallel to BC in triangle ABC, it divides the other two sides proportionally, and the smaller triangle ADE is similar to the larger triangle ABC. The problem provides segment lengths along AB and the length of DE, and asks for the full length of side BC using these similarity relationships. Given Data / Assumptions: - Triangle ABC is given.- DE is drawn parallel to BC (DE || BC).- AB = 7.5 cm.- BD = 6 cm, so point D lies on AB between A and B.- DE = 2 cm.- We assume standard Euclidean geometry and basic theorems of similar triangles. Concept / Approach: Since DE is parallel to BC, triangle ADE is similar to triangle ABC by the Basic Proportionality Theorem or the AA similarity criterion. This implies that corresponding sides are in the same ratio. Specifically, AD / AB = DE / BC. We can find AD using AB and BD, then use the proportion AD / AB = DE / BC to determine BC. This is a straightforward application of properties of similar triangles and proportional segments created by a line parallel to one side of a triangle. Step-by-Step Solution: Step 1: Given AB = 7.5 cm and BD = 6 cm, compute AD.Step 2: Since D lies between A and B, AD = AB - BD = 7.5 - 6 = 1.5 cm.Step 3: Because DE || BC, triangles ADE and ABC are similar.Step 4: Use the similarity ratio: AD / AB = DE / BC.Step 5: Substitute known values: AD = 1.5, AB = 7.5, DE = 2, and BC is unknown.Step 6: So 1.5 / 7.5 = 2 / BC.Step 7: Simplify 1.5 / 7.5 = 1 / 5.Step 8: Therefore, 1 / 5 = 2 / BC.Step 9: Cross multiply: BC = 2 * 5 = 10 cm. Verification / Alternative check: We can verify the proportion by computing the scaling factor from the smaller triangle ADE to the larger triangle ABC. From AD = 1.5 to AB = 7.5, the scale factor is 7.5 / 1.5 = 5. Therefore every side in triangle ABC is 5 times the corresponding side in triangle ADE. Since DE is 2 cm, the corresponding side BC should be 2 * 5 = 10 cm. This agrees with our calculated answer and confirms the correctness of the logic used. Why Other Options Are Wrong: - 6: This is smaller than DE and does not respect the scaling factor obtained from the ratio AD / AB.- 8: This would correspond to a scale factor of 4 from DE to BC, which conflicts with the factor of 5 from AD to AB.- 10.5: This does not match any proportional relationship derived from the given segment lengths.- 12: This would imply an even larger scale factor inconsistent with the 1.5 to 7.5 ratio. Common Pitfalls: Students sometimes incorrectly assign segments to the wrong corresponding sides in similar triangles, or forget to subtract BD from AB to obtain AD. Another mistake is treating the ratio AB / AD instead of AD / AB, which leads to incorrect scaling factors. Careful identification of corresponding sides and correct usage of proportions is essential in solving such problems accurately. Final Answer: The length of side BC is 10 cm.

Question 2

A circle is inscribed in a square. If the length of the diagonal of the square is 14√2 cm, what is the area (in square centimetres, using π = 22/7) of the inscribed circle?

Accepted Answer

Introduction / Context: This question links a square and an inscribed circle. The circle touches all four sides of the square internally, so the diameter of the circle is equal to the side length of the square. The problem provides the diagonal of the square and asks for the area of the circle. To solve this, we use the relationship between the side and the diagonal of a square, and then apply the area formula for a circle with a specified value of π. Given Data / Assumptions: - A circle is inscribed in a square.- Diagonal of the square = 14√2 cm.- The circle touches all sides of the square internally.- π is to be taken as 22/7.- Area of a circle is given by Area = π * r^2. Concept / Approach: In a square of side s, the diagonal is s * √2. Here, the diagonal is given, so we can find the side s. For a circle inscribed in a square, the diameter of the circle equals the side length of the square. Once we know the side length, we know the diameter, and the radius is half of that. Then we apply the standard circle area formula using the given value of π. This requires correct handling of square roots and careful substitution. Step-by-Step Solution: Step 1: Let the side of the square be s cm.Step 2: The diagonal of a square is given by diagonal = s * √2.Step 3: We are given that diagonal = 14√2 cm.Step 4: Therefore, s * √2 = 14√2, which implies s = 14 cm.Step 5: The circle is inscribed in the square, so the diameter of the circle = side of the square = 14 cm.Step 6: The radius r of the circle = diameter / 2 = 14 / 2 = 7 cm.Step 7: Area of the circle = π * r^2.Step 8: Substitute r = 7 cm: Area = π * 7^2 = π * 49.Step 9: Using π = 22/7, Area = 49 * 22 / 7.Step 10: Simplify 49 / 7 = 7, so Area = 7 * 22 = 154 square centimetres. Verification / Alternative check: We can quickly check by noting that a circle of radius 7 cm usually has area 49π. With π approximated as 3.14, 49 * 3.14 ≈ 153.86, which is very close to 154. Since the problem explicitly states to use π = 22/7, 154 is exact. Additionally, confirming that side = 14 from the diagonal 14√2 is straightforward, ensuring there is no algebraic slip. Thus, the calculated area is consistent and correct. Why Other Options Are Wrong: - 308: This would be 2 * 154 and would correspond to doubling the correct area, perhaps from misinterpreting diameter as radius.- 462: This is three times 154 and has no direct geometric basis in this problem.- 616: This is four times 154, again indicating overcounting or misusing radius and diameter.- 231: This does not correspond to any correct combination of radius or diameter with π = 22/7 in this configuration. Common Pitfalls: One common error is to confuse the diagonal with the side of the square or to forget the √2 factor in the diagonal formula. Another is to mistake the diameter for the radius and plug 14 instead of 7 into the formula for r^2, which would give an area four times too large. Also, failing to use the specified value of π (22/7) can lead to answers that do not match the given options exactly. Careful reading and correct application of formulas help avoid these issues. Final Answer: The area of the inscribed circle is 154 square centimetres.

Question 3

A right circular cone of radius 7 cm and height 12 cm is completely filled with water. This water is then poured into an empty cylindrical vessel of radius 3.5 cm. What will be the height (in centimetres) of the water column in the cylinder?

Accepted Answer

Introduction / Context: This problem involves the conservation of volume when liquid is transferred between two containers of different shapes. A right circular cone is filled with water, and then all this water is poured into a right circular cylinder. Since the water is neither lost nor added, the volume of water in the cone equals the volume of water in the cylinder. Using the formulas for the volumes of a cone and a cylinder, we can determine the height of the water in the cylindrical vessel. Given Data / Assumptions: - Cone radius r_cone = 7 cm.- Cone height h_cone = 12 cm.- Cylinder radius r_cyl = 3.5 cm.- All water from the cone is poured into the cylinder.- There is no spillage or evaporation; volume of water remains constant.- π is the same for both shapes in the calculations. Concept / Approach: Volume of a right circular cone is given by V_cone = (1 / 3) * π * r_cone^2 * h_cone. Volume of a right circular cylinder is given by V_cyl = π * r_cyl^2 * h_cyl, where h_cyl is the height of water in the cylinder that we need to find. Since the water volumes are equal, we set V_cone = V_cyl and solve for h_cyl. This uses algebraic manipulation and simple squaring of the radii to find the unknown height. Step-by-Step Solution: Step 1: Use the cone volume formula: V_cone = (1 / 3) * π * r_cone^2 * h_cone.Step 2: Substitute r_cone = 7 and h_cone = 12.Step 3: Compute r_cone^2 = 7^2 = 49.Step 4: So V_cone = (1 / 3) * π * 49 * 12.Step 5: 49 * 12 = 588, so V_cone = (1 / 3) * 588 * π = 196 * π.Step 6: For the cylinder, volume V_cyl = π * r_cyl^2 * h_cyl.Step 7: r_cyl = 3.5 cm, so r_cyl^2 = 3.5^2 = 12.25.Step 8: Therefore, V_cyl = π * 12.25 * h_cyl.Step 9: Set V_cone = V_cyl: 196 * π = π * 12.25 * h_cyl.Step 10: Cancel π from both sides to get 196 = 12.25 * h_cyl.Step 11: Solve for h_cyl: h_cyl = 196 / 12.25 = 16 cm. Verification / Alternative check: We can check computations numerically. 12.25 * 16 = 196, confirming that the height h_cyl of 16 cm is consistent with the equality of volumes. Another way is to see that the radius of the cylinder is exactly half the radius of the cone (7 and 3.5), so the base area of the cylinder is one fourth that of the cone. Since the cone volume is one third of a cylinder with the same base and height, adjusting for the changed radius shows that to hold the same volume the cylinder must have a proportionally higher height, which matches 16 cm in this configuration. Why Other Options Are Wrong: - 32 cm: Doubling the correct height would double the volume beyond what is available from the cone.- 5.33 cm: This is far too small; substituting this height would yield a volume much less than 196π.- 8 cm: This is half of the true height and cannot hold the full cone volume at the given radius.- 12 cm: Although it matches the height of the cone, the cylinder has a smaller radius, so the same height would not produce the same volume. Common Pitfalls: Learners sometimes incorrectly use the same radius for both shapes or forget the 1 / 3 factor in the volume of a cone, leading to incorrect equalities. Others may accidentally square the height instead of the radius. Also, forgetting to cancel π from both sides complicates the algebra unnecessarily. Recognising that the radii differ and that the cone has the additional one third factor is key to setting up the correct equation. Final Answer: The height of water in the cylindrical vessel is 16 cm.

Question 4

The total surface area of a solid hemisphere is 166.32 square centimetres. Using π ≈ 3.14, what is the radius (in centimetres) of the hemisphere?

Accepted Answer

Introduction / Context: This question focuses on finding the radius of a hemisphere when its total surface area is given. The total surface area of a solid hemisphere includes both the curved (outer) surface and the flat circular base. Understanding and applying the correct surface area formula for a hemisphere and then solving for the radius using basic algebra are the key skills tested here. Given Data / Assumptions: - The solid shape is a hemisphere.- Total surface area (TSA) = 166.32 square centimetres.- π is approximately 3.14.- TSA of a hemisphere includes both curved surface and base.- Formula for total surface area of a hemisphere: TSA = 3 * π * r^2. Concept / Approach: For a sphere of radius r, surface area = 4 * π * r^2. For a hemisphere, the curved surface area alone is 2 * π * r^2. However, a solid hemisphere also has a flat circular base area π * r^2. Therefore, total surface area becomes TSA = 2 * π * r^2 + π * r^2 = 3 * π * r^2. We use the given TSA to set up an equation 3 * π * r^2 = 166.32, then solve for r^2 and finally for r. Since π is given as 3.14, we substitute this value into the formula. Step-by-Step Solution: Step 1: Write formula for total surface area of a hemisphere: TSA = 3 * π * r^2.Step 2: Given TSA = 166.32 square centimetres.Step 3: Substitute into the formula: 166.32 = 3 * π * r^2.Step 4: Take π ≈ 3.14.Step 5: Compute 3 * π = 3 * 3.14 = 9.42.Step 6: So the equation becomes 166.32 = 9.42 * r^2.Step 7: Solve for r^2: r^2 = 166.32 / 9.42.Step 8: Perform the division: r^2 ≈ 17.66 (approximately).Step 9: Take square root: r ≈ √17.66 ≈ 4.2 cm.Step 10: This matches the option 4.2 cm. Verification / Alternative check: We can check by substituting r = 4.2 back into TSA = 3 * π * r^2. First, r^2 = 4.2^2 = 17.64. Then 3 * π * r^2 ≈ 3 * 3.14 * 17.64. Compute 3 * 3.14 = 9.42, then 9.42 * 17.64 ≈ 166.1 to 166.3, which is very close to the given 166.32, and any slight difference is due to rounding π. This confirms that r ≈ 4.2 cm is correct within the rounding used in the problem. Why Other Options Are Wrong: - 8.4 cm: Doubling the radius would make r^2 four times larger, which would make TSA roughly four times larger than 166.32.- 1.4 cm: A radius this small would give r^2 = 1.96 and TSA ≈ 3 * 3.14 * 1.96 ≈ 18.5, which is far below 166.32.- 2.1 cm: This radius gives r^2 = 4.41 and TSA ≈ 3 * 3.14 * 4.41 ≈ 41.6, still much less than required.- 6.3 cm: This gives r^2 ≈ 39.69 and TSA ≈ 3 * 3.14 * 39.69 ≈ 373.8, which is more than double the given surface area. Common Pitfalls: Some students mistakenly use the formula for only the curved surface area (2 * π * r^2) instead of the total surface area for a hemisphere, omitting the base area. Others might use 4 * π * r^2, which is the formula for a full sphere. Incorrect substitution of π or premature rounding can also lead to inaccurate results. Always ensure the correct formula is used for the specific solid described in the question and apply the given value of π as instructed. Final Answer: The radius of the hemisphere is approximately 4.2 cm.

Question 5

Triangles ABC and DEF are similar. If the length of side AB is 18 cm and the length of the corresponding side DE is 10 cm, what is the ratio of the perimeter of triangle ABC to the perimeter of triangle DEF?

Accepted Answer

Introduction / Context: This question tests the concept of similar triangles and the relationship between the ratios of corresponding sides and their perimeters. When two triangles are similar, the ratios of their corresponding sides are equal, and the ratio of their perimeters is the same as the ratio of any pair of corresponding sides. Understanding this principle allows us to find the ratio of the perimeters directly from a pair of corresponding sides. Given Data / Assumptions: - Triangle ABC is similar to triangle DEF.- Side AB in triangle ABC corresponds to side DE in triangle DEF.- AB = 18 cm.- DE = 10 cm.- We are asked for the ratio Perimeter(ABC) : Perimeter(DEF). Concept / Approach: For similar triangles, all corresponding side lengths are in the same ratio. If the ratio of any one pair of corresponding sides is known, that ratio is equal to the ratio of the perimeters, because perimeter is simply the sum of the three sides and each side is scaled by the same factor. Therefore, Perimeter(ABC) : Perimeter(DEF) = AB : DE. We simply reduce the fraction formed by these side lengths to its simplest form to get the required ratio. Step-by-Step Solution: Step 1: Recognise that ABC ~ DEF (triangles are similar).Step 2: Identify corresponding sides: AB corresponds to DE.Step 3: Given AB = 18 cm and DE = 10 cm.Step 4: The ratio of corresponding sides is AB : DE = 18 : 10.Step 5: Simplify 18 : 10 by dividing both numbers by 2: 18 / 2 = 9 and 10 / 2 = 5.Step 6: So, AB : DE = 9 : 5.Step 7: For similar triangles, the ratio of perimeters equals the ratio of corresponding sides.Step 8: Therefore, Perimeter(ABC) : Perimeter(DEF) = 9 : 5. Verification / Alternative check: Imagine triangle ABC has sides 18, 18k, and 18m for some positive constants k and m, and triangle DEF has sides 10, 10k, and 10m. Then Perimeter(ABC) = 18 + 18k + 18m = 18(1 + k + m). Perimeter(DEF) = 10(1 + k + m). The ratio of perimeters is 18(1 + k + m) : 10(1 + k + m). The common factor (1 + k + m) cancels, leaving 18 : 10, which reduces to 9 : 5. This reasoning confirms the direct method used earlier. Why Other Options Are Wrong: - 5 : 9: This inverts the correct ratio, treating the smaller triangle as having the larger perimeter.- 81 : 25: This is the square of the side ratio and corresponds to the ratio of areas, not perimeters.- 25 : 81: This inverts the area ratio and is not related to the perimeter ratio.- 3 : 2: Although 18 : 12 would simplify to 3 : 2, our sides are 18 and 10, not 18 and 12, so this ratio is not applicable. Common Pitfalls: A common mistake is to use the square of the side ratio when asked for the ratio of perimeters, confusing it with the ratio of areas. Another error is incorrectly identifying corresponding sides or inverting the ratio. Always remember that for perimeters, the ratio is the same as that of any pair of corresponding sides, while for areas, the ratio is the square of the side ratio. Keeping this distinction clear prevents such errors. Final Answer: The ratio of the perimeters of triangles ABC and DEF is 9 : 5.

Question 6

The volume of a right circular cylinder is 770 cubic centimetres. If the circumference of the base is 22 cm, what is the curved (lateral) surface area of the cylinder in square centimetres? (Take π = 22/7)

Accepted Answer

Introduction / Context: This problem involves a cylinder where volume and base circumference are given, and the curved surface area is required. Solving it requires the use of three key formulas: the circumference of a circle, the volume of a cylinder, and the curved surface area of a cylinder. It is an excellent test of the ability to combine multiple geometric relationships and perform algebraic manipulation to find both the radius and height of the cylinder. Given Data / Assumptions: - Volume of the cylinder V = 770 cubic centimetres.- Circumference of the base = 22 cm.- π is to be taken as 22/7.- Curved surface area (CSA) of a cylinder is CSA = 2 * π * r * h.- Volume of a cylinder is V = π * r^2 * h. Concept / Approach: First, we use the base circumference to find the radius r of the cylinder using the formula circumference = 2 * π * r. Once r is obtained, we substitute into the volume formula V = π * r^2 * h to solve for the height h. Finally, we use the formula for curved surface area (lateral surface area) CSA = 2 * π * r * h with the found values of r and h. The steps involve solving linear equations and careful substitution of numerical values for π, r, and h. Step-by-Step Solution: Step 1: Use the circumference formula: 2 * π * r = 22.Step 2: Substitute π = 22/7 into 2 * π * r = 22.Step 3: 2 * (22/7) * r = 22 ⇒ (44/7) * r = 22.Step 4: Solve for r: r = 22 * 7 / 44 = 7 / 2 = 3.5 cm.Step 5: Use the volume formula V = π * r^2 * h with V = 770.Step 6: r^2 = (3.5)^2 = 12.25.Step 7: Substitute: 770 = π * 12.25 * h.Step 8: Replace π with 22/7: 770 = (22/7) * 12.25 * h.Step 9: Compute (22/7) * 12.25 = (22/7) * (49/4) = (22 * 49) / 28 = (1078) / 28 = 38.5.Step 10: So 770 = 38.5 * h ⇒ h = 770 / 38.5 = 20 cm.Step 11: Now compute CSA = 2 * π * r * h.Step 12: CSA = 2 * (22/7) * 3.5 * 20.Step 13: 2 * (22/7) * 3.5 = 2 * (22/7) * (7/2) = 22.Step 14: Therefore, CSA = 22 * 20 = 440 square centimetres. Verification / Alternative check: An alternative verification is to compute the base area using r = 3.5 cm and h = 20 cm, then check if the volume matches 770 cubic centimetres. Base area = π * r^2 ≈ (22/7) * 12.25 = 38.5. Then Volume = base area * height = 38.5 * 20 = 770 cubic centimetres, which matches the given volume exactly. This confirms that r and h are correct and therefore the calculated curved surface area of 440 square centimetres is reliable. Why Other Options Are Wrong: - 880 sq cms: This is exactly double the correct answer and might result from mistakenly using 4 * π * r * h.- 220 sq cms: This is half of the correct answer, possibly from missing a factor of 2 in the CSA formula.- 660 sq cms: This is one and a half times the correct value and has no clear basis in the provided dimensions.- 330 sq cms: This does not correspond to any standard misapplication of formulas but is still inconsistent with the correct computations. Common Pitfalls: Students often forget to use the circumference to find the radius and instead guess the radius directly. Others might incorrectly substitute in the volume formula or forget to square the radius. Missing the factor of 2 in the curved surface area formula is another common error. A systematic approach, using each formula at the correct stage and checking intermediate results, helps avoid these mistakes. Final Answer: The curved surface area of the cylinder is 440 square centimetres.

Question 7

What is the sum of all interior angles (in degrees) of a polygon that has 14 sides?

Accepted Answer

Introduction / Context: This problem involves a fundamental result from polygon geometry. It asks for the sum of all interior angles of a polygon with 14 sides, which is also called a 14-gon. The relationship between the number of sides of a polygon and the sum of its interior angles is standard and is widely used in school mathematics and aptitude tests. Given Data / Assumptions: - The polygon has n = 14 sides.- It is a simple polygon (no self intersections).- We are asked to find the sum of its interior angles, not the measure of each individual angle.- We use the standard formula for the sum of interior angles of an n sided polygon. Concept / Approach: The formula for the sum of all interior angles of a polygon with n sides is: Sum = (n - 2) * 180 degrees. This formula can be derived by dividing the polygon into (n - 2) triangles, each contributing 180 degrees. Once n is known, we simply substitute and compute the product. This is a straightforward application of the formula and does not require advanced geometry. Step-by-Step Solution: Step 1: Identify the number of sides of the polygon: n = 14.Step 2: Recall the formula for sum of interior angles: Sum = (n - 2) * 180 degrees.Step 3: Substitute n = 14 into the formula.Step 4: Compute (14 - 2) = 12.Step 5: Now compute Sum = 12 * 180 degrees.Step 6: Calculate 12 * 180 = 2160 degrees.Step 7: Therefore, the sum of all interior angles of a 14 sided polygon is 2160 degrees. Verification / Alternative check: To verify, consider smaller polygons. For a triangle (n = 3), the formula gives (3 - 2) * 180 = 180 degrees, which is correct. For a quadrilateral (n = 4), it gives (4 - 2) * 180 = 360 degrees, also correct. Extending this pattern to n = 14 is consistent and gives 2160 degrees, so the formula is trusted. There is no contradiction with the known values for simpler polygons, which supports the calculation. Why Other Options Are Wrong: - 2520°: This corresponds to (n - 2) * 180 = 2520, which would give n - 2 = 14 and n = 16, not 14.- 2880°: This would mean (n - 2) * 180 = 2880, giving n = 18, not 14.- 3240°: This corresponds to a polygon with even more sides, not a 14-gon.- 1980°: This does not correspond to any integer n in the formula (n - 2) * 180. Common Pitfalls: One common mistake is to apply the formula for the measure of each interior angle of a regular polygon instead of the sum of all angles. Another error is to use n * 180 instead of (n - 2) * 180. Also, some might miscalculate (14 - 2) or 12 * 180 under time pressure. Being careful with substitution and arithmetic avoids these issues. Final Answer: The sum of all interior angles of a 14 sided polygon is 2160°.

Question 8

In a right triangle, one of the acute angles measures 45°, and the side opposite this 45° angle is 8 cm long. What is the length (in centimetres) of the side opposite the right angle (the hypotenuse)?

Accepted Answer

Introduction / Context: This question concerns a special right triangle where one of the acute angles is 45 degrees. Such a triangle is known as a 45°-45°-90° right triangle. In these triangles, the legs are equal and the hypotenuse has a fixed relationship with the legs. Using this special ratio, we can easily find the length of the hypotenuse when the length of a leg is known. Given Data / Assumptions: - The triangle is a right triangle.- One acute angle is 45°.- The side opposite the 45° angle is 8 cm.- The side opposite the 90° angle is the hypotenuse.- The triangle is a standard 45°-45°-90° triangle, so the other acute angle is also 45°. Concept / Approach: In a right triangle with angles 45°, 45°, and 90°, the two legs opposite the 45° angles are equal in length, and the hypotenuse is equal to leg * √2. This ratio arises from applying the Pythagorean theorem to an isosceles right triangle. Therefore, if one leg is known, we can directly multiply it by √2 to obtain the hypotenuse. No complicated trigonometry is needed; the special triangle ratio is sufficient. Step-by-Step Solution: Step 1: Recognise that the triangle is a 45°-45°-90° triangle because it is right angled and one acute angle is 45°.Step 2: In such a triangle, the two legs are equal in length.Step 3: We are told that the side opposite the 45° angle is 8 cm, so one leg = 8 cm.Step 4: The hypotenuse h in a 45°-45°-90° triangle is given by h = leg * √2.Step 5: Substitute leg = 8 cm into the formula: h = 8 * √2 cm.Step 6: Therefore, the length of the hypotenuse is 8√2 centimetres. Verification / Alternative check: We can verify this by using the Pythagorean theorem. If both legs are of length 8 cm, then h^2 = 8^2 + 8^2 = 64 + 64 = 128. Therefore, h = √128 = √(64 * 2) = 8√2. This matches the result obtained from the special triangle ratio, confirming that 8√2 cm is correct for the hypotenuse of this right triangle. Why Other Options Are Wrong: - 4√2 cm: This corresponds to a leg of 4 cm, which contradicts the given leg length of 8 cm.- 8√3 cm: This length would arise in a different triangle, such as a 30°-60°-90° triangle under specific conditions, not in a 45°-45°-90° triangle.- 4√3 cm: This is inconsistent with the Pythagorean relation for legs of length 8 cm.- 16 cm: This is equal to twice the leg length, and 16^2 = 256, which does not equal 8^2 + 8^2 = 128. Common Pitfalls: Some students confuse the special triangle ratios and may incorrectly use the 30°-60°-90° triangle ratio instead of the 45°-45°-90° ratio. Others might forget that both legs are equal in a 45°-45°-90° triangle, or they may mistakenly apply the Pythagorean theorem using incorrect side assignments. Remembering that in a 45°-45°-90° triangle, hypotenuse = leg * √2 helps quickly and accurately solve such problems. Final Answer: The length of the hypotenuse is 8√2 centimetres.

Question 9

The curved surface area of a right circular cylinder is 1386 square centimetres and its height is 21 cm. Taking π = 22/7, what is the radius (in centimetres) of the cylinder?

Accepted Answer

Introduction / Context: This question involves finding the radius of a right circular cylinder when its curved (lateral) surface area and height are given. The curved surface area of a cylinder is directly related to its radius and height through a standard formula. By substituting the given values and using the provided value of π, we can algebraically solve for the radius. This tests formula recall and basic manipulation of equations. Given Data / Assumptions: - Curved surface area (CSA) of the cylinder = 1386 square centimetres.- Height h = 21 cm.- π is given as 22/7.- Formula for curved surface area of a cylinder: CSA = 2 * π * r * h.- We need to find the radius r in centimetres. Concept / Approach: The curved surface area of a cylinder is the area of the side surface, which can be thought of as a rectangle with one side equal to the height and the other side equal to the circumference of the base. The formula CSA = 2 * π * r * h comes from this interpretation. To find r, we rearrange the formula to r = CSA / (2 * π * h). Substituting the given values into this rearranged formula and simplifying will yield the radius. Step-by-Step Solution: Step 1: Start with the formula: CSA = 2 * π * r * h.Step 2: Given CSA = 1386 square centimetres, h = 21 cm, and π = 22/7.Step 3: Substitute the known values: 1386 = 2 * (22/7) * r * 21.Step 4: Simplify the right hand side: 2 * (22/7) * 21 * r.Step 5: Note that 21 / 7 = 3, so (22/7) * 21 = 22 * 3 = 66.Step 6: Therefore 2 * (22/7) * 21 = 2 * 66 = 132.Step 7: The equation becomes 1386 = 132 * r.Step 8: Solve for r: r = 1386 / 132.Step 9: Simplify the fraction: 1386 / 132 = 10.5.Step 10: Therefore, r = 10.5 cm. Verification / Alternative check: To verify, substitute r = 10.5 and h = 21 back into the CSA formula. CSA = 2 * π * r * h = 2 * (22/7) * 10.5 * 21. Compute 10.5 = 21 / 2, so r = 21 / 2. Then 2 * (22/7) * (21 / 2) * 21 = (2 and 2 cancel) ⇒ (22/7) * 21 * 21. Now (22/7) * 21 = 66, and 66 * 21 = 1386. This exactly matches the given curved surface area, confirming that r = 10.5 cm is correct. Why Other Options Are Wrong: - 21 cms: This would give CSA = 2 * (22/7) * 21 * 21 = 2772, which is double the given value.- 5.25 cms: This is half of 10.5; substituting it would give CSA = 693, which is half of 1386.- 15.75 cms: This would produce a CSA much larger than 1386 when used in the formula.- 7 cms: Substituting r = 7 gives CSA = 2 * (22/7) * 7 * 21 = 924, which is not equal to 1386. Common Pitfalls: Some students may mistakenly use the total surface area formula (2 * π * r * h + 2 * π * r^2) instead of just the curved surface area formula. Others might perform algebraic steps incorrectly, especially when simplifying fractions with π = 22/7. Careful cancellation of factors and methodical substitution are essential to avoid miscalculations. Final Answer: The radius of the cylinder is 10.5 centimetres.

Question 10

In which type of quadrilateral does at least one diagonal bisect the other diagonal?

Accepted Answer

Introduction / Context: This question belongs to the topic of quadrilaterals and focuses on the properties of diagonals in different special quadrilaterals. It asks in which type of quadrilateral at least one diagonal bisects the other. Understanding how diagonals behave in various quadrilaterals like parallelograms, trapeziums, kites, and cyclic quadrilaterals is essential for correctly answering such conceptual problems. Given Data / Assumptions: - We are considering standard Euclidean geometry.- The quadrilaterals in the options include trapezium, isosceles trapezium, kite, cyclic quadrilateral, and a general quadrilateral.- The phrase “at least one diagonal bisects the other” means that one diagonal cuts the other into two equal segments.- We are not required to have both diagonals bisect each other, only one diagonal must bisect the other. Concept / Approach: Different quadrilaterals have distinctive diagonal properties. In a parallelogram (and its special cases like rectangle, square, and rhombus), the diagonals bisect each other. In a kite, one diagonal is the perpendicular bisector of the other. In a trapezium, especially a general trapezium or isosceles trapezium, diagonals do not generally bisect each other. In a cyclic quadrilateral, there is no general rule that diagonals must bisect each other. The key is to identify the quadrilateral where at least one diagonal always bisects the other by definition, which is the kite. Step-by-Step Solution: Step 1: Recall the definition of a kite: a quadrilateral with two pairs of adjacent sides equal.Step 2: One important property of a kite is that one diagonal is the perpendicular bisector of the other diagonal.Step 3: This means that one diagonal cuts the other into two equal parts at right angles.Step 4: Therefore, in a kite, at least one diagonal definitely bisects the other.Step 5: Consider a trapezium: it has only one pair of parallel sides, and in general, its diagonals do not bisect each other.Step 6: In an isosceles trapezium, the diagonals are equal but they do not necessarily bisect each other.Step 7: In a cyclic quadrilateral, there is no guarantee that diagonals bisect each other; only certain angle properties hold.Step 8: Therefore, among the given options, only a kite always has at least one diagonal that bisects the other. Verification / Alternative check: We can verify the property of a kite by considering a simple coordinate example. Let the kite have vertices such that the vertical diagonal lies on the y axis and the horizontal diagonal lies on the x axis. By symmetry, the vertical diagonal will intersect and bisect the horizontal diagonal at the origin. This illustrates that in every kite with such symmetry, one diagonal is a perpendicular bisector of the other. By contrast, drawing general trapeziums and cyclic quadrilaterals shows that the diagonals cease to bisect each other in typical configurations, confirming that the kite is the correct choice. Why Other Options Are Wrong: - Trapezium: It has only one pair of parallel sides; diagonals normally intersect but do not bisect each other.- Isosceles trapezium: Diagonals are equal in length but do not generally bisect each other.- Cyclic quadrilateral: The only guaranteed property is that opposite angles are supplementary, not that diagonals bisect.- General quadrilateral: There is no necessary diagonal bisecting property in an arbitrary quadrilateral. Common Pitfalls: Some students confuse the properties of parallelograms (where both diagonals bisect each other) with other quadrilaterals such as kites and trapeziums. Since parallelogram is not an option here, the focus should shift to the kite. Another pitfall is to think that equal diagonals as in an isosceles trapezium implies they bisect each other, which is not true. Understanding the precise diagonal properties of each special quadrilateral type is essential to avoid such misconceptions. Final Answer: At least one diagonal bisects the other in a kite.

Question 11

In right triangle ABC, angle B is 90° and BD is the altitude drawn from B to hypotenuse AC. If AD = 4 cm and DC = 12 cm, what is the length (in centimetres) of side AB?

Accepted Answer

Introduction / Context: This problem uses a special set of relationships in a right triangle when an altitude is drawn from the right angle to the hypotenuse. The given data are the segments into which the hypotenuse is divided by the foot of the altitude. Using known geometric properties and formulae, we can compute the length of a leg of the triangle. This is a classic question in Euclidean geometry and appears frequently in aptitude and competitive exams. Given Data / Assumptions: - Triangle ABC is right angled at B.- BD is the altitude from B to the hypotenuse AC.- AD = 4 cm and DC = 12 cm, where D lies on AC between A and C.- We need to find the length of AB.- Standard properties of right triangles and altitude to the hypotenuse apply. Concept / Approach: In a right triangle with altitude drawn from the right angle to the hypotenuse, there is a relationship between the legs, the hypotenuse, and the segments into which the hypotenuse is divided. If AC is the hypotenuse and D is the foot of the altitude, then AC = AD + DC. One of the key formulae is AB^2 = AD * AC, where AB is the leg adjacent to segment AD. Once we compute the full hypotenuse AC from AD and DC, we can apply this relationship to find AB. This reduces the problem to simple multiplication and taking a square root. Step-by-Step Solution: Step 1: Given AD = 4 cm and DC = 12 cm.Step 2: Compute the hypotenuse AC = AD + DC = 4 + 12 = 16 cm.Step 3: Use the right triangle altitude property: AB^2 = AD * AC.Step 4: Substitute AD = 4 and AC = 16.Step 5: AB^2 = 4 * 16 = 64.Step 6: Take the square root: AB = √64 = 8 cm.Step 7: Therefore, the length of side AB is 8 centimetres. Verification / Alternative check: We can also find BC using the similar relation BC^2 = DC * AC = 12 * 16 = 192, giving BC = √192 = √(64 * 3) = 8√3. Then check that AC^2 = AB^2 + BC^2. AC^2 should be 16^2 = 256. Compute AB^2 + BC^2 = 64 + (8√3)^2 = 64 + 64 * 3 = 64 + 192 = 256, which matches AC^2. This confirms that AB = 8 cm is consistent with the Pythagorean theorem and altitude properties. Why Other Options Are Wrong: - 9: Squaring 9 gives 81, which would require AD * AC = 81, contradicting the given segments.- 10: This would give AB^2 = 100, not equal to AD * AC = 64.- 6: Squaring 6 gives 36, again not matching AD * AC = 64.- 12: Squaring 12 gives 144, which is also inconsistent with the product 4 * 16. Common Pitfalls: Some students mistakenly apply the Pythagorean theorem directly using AD and DC as legs, which is incorrect because these are segments of the hypotenuse, not the legs of the triangle. Others might confuse the specific altitude relations and attempt to use AB^2 = AD^2 + AC^2, which is not valid. Remembering the exact property AB^2 = AD * AC is crucial in solving such altitude related questions efficiently. Final Answer: The length of side AB is 8 centimetres.

Question 12

The surface area of a sphere is 154 square centimetres. Using π = 22/7, what is the diameter (in centimetres) of the sphere?

Accepted Answer

Introduction / Context: This question concerns a sphere for which the total surface area is given. The task is to find the diameter of the sphere. We use the standard surface area formula for a sphere and the given numerical value of π to back calculate the radius, and then the diameter. This is a straightforward but important application of formula manipulation in geometry. Given Data / Assumptions: - Surface area of the sphere S = 154 square centimetres.- π is given as 22/7.- Formula for surface area of a sphere: S = 4 * π * r^2.- We need to find the diameter, which is 2 * r. Concept / Approach: The total surface area S of a sphere depends on the square of its radius r through the formula S = 4 * π * r^2. Hence, if S and π are known, we can solve for r^2 by rearranging the formula to r^2 = S / (4 * π). After calculating r^2, we find r by taking the square root. The diameter is then simply 2 * r. Careful substitution and simplification using the given π value leads to the correct answer. Step-by-Step Solution: Step 1: Start with the formula S = 4 * π * r^2.Step 2: Substitute S = 154 and π = 22/7.Step 3: The equation becomes 154 = 4 * (22/7) * r^2.Step 4: Compute 4 * (22/7) = 88 / 7.Step 5: So 154 = (88 / 7) * r^2.Step 6: Multiply both sides by 7 to clear the fraction: 154 * 7 = 88 * r^2.Step 7: 154 * 7 = 1078.Step 8: Thus, 1078 = 88 * r^2.Step 9: Solve for r^2: r^2 = 1078 / 88.Step 10: Simplify 1078 / 88 = 12.25.Step 11: Therefore, r = √12.25 = 3.5 cm.Step 12: The diameter d = 2 * r = 2 * 3.5 = 7 cm. Verification / Alternative check: We can check by substituting r = 3.5 back into the surface area formula. r^2 = 12.25. Then S = 4 * π * r^2 = 4 * (22/7) * 12.25. Compute 4 * (22/7) = 88 / 7. Now S = (88 / 7) * 12.25. Since 12.25 = 49 / 4, we have S = (88 / 7) * (49 / 4) = (88 * 49) / 28. Simplifying (88 * 49) / 28 gives 154, exactly the given surface area, confirming that r = 3.5 cm and d = 7 cm are correct. Why Other Options Are Wrong: - 3.5: This is the radius, not the diameter, so it is only half of the required answer.- 14: This would correspond to a radius of 7 cm, giving a much larger surface area than 154 square centimetres.- 10.5: This would require r = 5.25 cm, and the surface area would be significantly larger than 154.- 5.25: Again this is neither correct for the radius nor diameter and gives an incorrect surface area when substituted. Common Pitfalls: Many learners forget the factor of 4 in the sphere surface area formula and mistakenly use S = π * r^2, which is the formula for the area of a circle, not a sphere. Others may compute the radius correctly but forget to double it when asked for the diameter. Keeping track of whether the question asks for radius or diameter is crucial to providing the correct final value. Final Answer: The diameter of the sphere is 7 centimetres.

Question 13

What is the area (in square centimetres) of a regular hexagon whose each side measures 14 cm?

Accepted Answer

Introduction / Context: This problem deals with the area of a regular hexagon, which is a six sided polygon with all sides and angles equal. For a regular hexagon, there is a standard formula for area in terms of the side length. Alternatively, the hexagon can be divided into six equilateral triangles, and the area can be computed by summing the areas of these triangles. This question tests knowledge of these geometric facts and the ability to apply them to a side length of 14 cm. Given Data / Assumptions: - The hexagon is regular (all sides and angles equal).- Each side length a = 14 cm.- We use the standard area formula for a regular hexagon.- Alternatively, we may use the area of equilateral triangles to derive the same result. Concept / Approach: A regular hexagon can be divided into 6 congruent equilateral triangles, each having side length equal to the hexagon side a. The area of an equilateral triangle with side a is (√3 / 4) * a^2. Therefore, the area of a regular hexagon is 6 times this: Area_hexagon = 6 * (√3 / 4) * a^2 = (3√3 / 2) * a^2. We substitute a = 14 into this formula to find the required area. This approach makes use of both polygon decomposition and formula application. Step-by-Step Solution: Step 1: Let side length a = 14 cm.Step 2: Use the area formula for a regular hexagon: Area = (3√3 / 2) * a^2.Step 3: Compute a^2 = 14^2 = 196.Step 4: Substitute into the formula: Area = (3√3 / 2) * 196.Step 5: Multiply 3 and 196: 3 * 196 = 588.Step 6: So Area = (588 / 2) * √3.Step 7: Simplify 588 / 2 = 294.Step 8: Therefore, Area = 294√3 square centimetres. Verification / Alternative check: As a check, we can explicitly consider the smaller equilateral triangles. Each equilateral triangle has area (√3 / 4) * a^2. With a = 14, a^2 = 196, so area of one triangle = (√3 / 4) * 196 = 49√3. There are 6 such triangles in the hexagon, so Area_hexagon = 6 * 49√3 = 294√3 square centimetres. This matches the result from the direct formula and confirms the correctness of the calculation. Why Other Options Are Wrong: - 147√3: This is exactly half of the correct area and would correspond to only three equilateral triangles instead of six.- 441√3: This is one and a half times the correct area and does not correspond to any standard decomposition.- 196√3: This would correspond to only four equilateral triangles of side 14 instead of six.- 168√3: This gives a smaller incorrect area, not matching any integer multiple of 49√3 except 49 * 3.428, which lacks geometric meaning here. Common Pitfalls: Some learners mistakenly use the area formula for a regular polygon incorrectly or forget that a regular hexagon can be divided into six equilateral triangles. Others may miscompute a^2 or mishandle the fraction 3 / 2 in the formula, leading to numerical errors. Remembering that a regular hexagon has 6 sides and decomposes into 6 congruent equilateral triangles is a helpful visualization that reinforces the area formula. Final Answer: The area of the regular hexagon is 294√3 square centimetres.

Question 14

A straight line passes through the points (5, -1) and (4, -4). What is the slope of any line that is parallel to this line?

Accepted Answer

Introduction / Context: This coordinate geometry problem asks for the slope of a line parallel to a given line. The given line is defined by two points, and the slope of any parallel line is the same as the slope of the original line. Therefore, we must first find the slope of the line passing through the two points using the slope formula, and then state that value as the slope of any parallel line. Given Data / Assumptions: - The given line passes through points (5, -1) and (4, -4).- We are working in the standard Cartesian coordinate plane.- Slope of a line through points (x1, y1) and (x2, y2) is given by (y2 - y1) / (x2 - x1).- Parallel lines have equal slopes. Concept / Approach: To find the slope m of the given line, we use m = (change in y) / (change in x). We will label the points, compute the numerator and denominator carefully, and simplify the fraction. Once the slope of the original line is found, we know that any line parallel to it must have the same slope. Thus the answer is simply this computed slope. Step-by-Step Solution: Step 1: Let the two points be (x1, y1) = (5, -1) and (x2, y2) = (4, -4).Step 2: Use the slope formula m = (y2 - y1) / (x2 - x1).Step 3: Compute y2 - y1 = -4 - (-1) = -4 + 1 = -3.Step 4: Compute x2 - x1 = 4 - 5 = -1.Step 5: So m = (-3) / (-1) = 3.Step 6: Therefore, the slope of the line passing through the two points is 3.Step 7: Any line parallel to this line must have the same slope.Step 8: Hence, the slope of any parallel line is also 3. Verification / Alternative check: We can interchange the roles of the points and compute the slope again to ensure consistency. Taking (x1, y1) = (4, -4) and (x2, y2) = (5, -1), we get m = (y2 - y1) / (x2 - x1) = (-1 - (-4)) / (5 - 4) = (3) / (1) = 3. The slope is the same regardless of the order in which we choose the points, confirming the correctness of m = 3. Therefore, any line parallel to this one also has slope 3. Why Other Options Are Wrong: - -3: This would be the slope if only the change in y were used without correctly considering the change in x sign.- -1/3: This might result from inverting the fraction and also changing the sign incorrectly.- 1/3: This results from incorrectly swapping the changes in x and y or miscomputing the differences.- 0: This would correspond to a horizontal line, which is not the case since the y values clearly change as x changes. Common Pitfalls: Students often mix up the order of subtraction in the slope formula, which can lead to sign mistakes. However, using consistent ordering for both numerator and denominator avoids such errors. Another pitfall is to mistakenly invert the slope or think that parallel lines require a different slope, when in fact they require the same slope. Careful use of the formula and attention to signs will yield the correct result. Final Answer: The slope of any line parallel to the given line is 3.

Question 15

In two similar triangles XYZ and PQR, the ratio of the perimeter of triangle XYZ to the perimeter of triangle PQR is 16 : 9. If side PQ in triangle PQR is 3.6 cm, then what is the length of the corresponding side XY (in cm)?

Accepted Answer

Introduction / Context: This question tests your understanding of similar triangles and the relationship between their perimeters and corresponding side lengths. In geometry and aptitude exams, problems like this frequently appear to check whether you can move comfortably between ratios of perimeters and ratios of sides. When two triangles are similar, all corresponding linear measurements such as side lengths, heights, medians, and perimeters are in the same ratio. Here, you are given the ratio of the perimeters of two similar triangles XYZ and PQR and one side length from triangle PQR. You are asked to find the length of the corresponding side XY in triangle XYZ using this similarity relationship in a clear and methodical way. Given Data / Assumptions: • Triangle XYZ is similar to triangle PQR (ΔXYZ ~ ΔPQR). • Ratio of perimeter of triangle XYZ to perimeter of triangle PQR is 16 : 9. • Side PQ in triangle PQR corresponds to side XY in triangle XYZ. • Length of PQ = 3.6 cm. • We assume both triangles are non degenerate and the similarity is correctly stated. Concept / Approach: For similar triangles, the ratio of corresponding sides is equal to the ratio of their perimeters. If the perimeter of triangle XYZ is to the perimeter of triangle PQR as 16 : 9, then any corresponding side in XYZ is to the corresponding side in PQR in the same ratio 16 : 9. Therefore, to find XY, we simply multiply PQ by the scale factor 16 / 9. This is a direct application of similarity and proportional reasoning, without needing to know individual perimeters or angles, which makes it a straightforward but important concept for aptitude exams. Step-by-Step Solution: Step 1: Let k be the linear scale factor from triangle PQR to triangle XYZ. Since perimeter(XYZ) : perimeter(PQR) = 16 : 9, the side ratio is also XY : PQ = 16 : 9. Step 2: Express the relationship between corresponding sides: XY / PQ = 16 / 9. Step 3: Substitute PQ = 3.6 cm into the proportion: XY / 3.6 = 16 / 9. Step 4: Solve for XY by cross multiplication: XY = 3.6 * (16 / 9). Step 5: Simplify the multiplication: 3.6 * 16 = 57.6, and 57.6 / 9 = 6.4. Step 6: Therefore, XY = 6.4 cm. Verification / Alternative check: You can check the reasonableness of the answer by comparing sizes. Since the perimeter ratio 16 : 9 is greater than 1, triangle XYZ is larger than triangle PQR. Therefore, every corresponding side in XYZ should be longer than the corresponding side in PQR. Here, PQ is 3.6 cm and the computed XY is 6.4 cm, which is indeed larger. Also, the factor 16 / 9 is approximately 1.78, and 3.6 * 1.78 is close to 6.4, confirming that the arithmetic is consistent and the direction of scaling is correct. Why Other Options Are Wrong: Option 4.8 cm is obtained if someone mistakenly uses a smaller factor or reverses the ratio. It does not match the perimeter ratio 16 : 9 when applied correctly. Option 3.2 cm is even smaller than PQ, which would imply that triangle XYZ is smaller than triangle PQR, contradicting the given perimeter ratio where 16 is greater than 9. Option 8.6 cm is larger than necessary and does not correspond to any simple proportional calculation based on 3.6 and 16 : 9, so it is not consistent with similar triangle relationships. Common Pitfalls: A common mistake is to invert the ratio and use 9 : 16 instead of 16 : 9, which would lead to a smaller value for XY. Another typical error is to assume that perimeter ratios are unrelated to side ratios, or to add or subtract the numbers in the ratio instead of forming a proper fraction. Some candidates also misidentify which side in one triangle corresponds to which side in the other, but in this problem the statement clearly links PQ with XY. Carefully reading the direction of the ratio and correctly applying the proportion avoids these errors. Final Answer: The length of side XY in triangle XYZ is 6.4 cm.

Question 16

In a regular polygon, the measure of each interior angle is 90 degrees greater than the measure of each exterior angle. How many sides does this regular polygon have?

Accepted Answer

Introduction / Context: This question checks your understanding of the relationship between interior and exterior angles of a regular polygon. Regular polygons frequently appear in school geometry and competitive exams because their angles follow simple, elegant formulas. Every regular polygon has equal sides and equal angles, and there is a fixed link between each interior angle and its corresponding exterior angle. Knowing how to use this relationship allows you to move quickly from one type of angle to the other and then deduce the number of sides of the polygon, which is what is required in this problem. Given Data / Assumptions: • The polygon is regular, which means all sides and all interior angles are equal. • Let the measure of each interior angle be I degrees. • Let the measure of each exterior angle be E degrees. • It is given that I is 90 degrees greater than E, so I = E + 90. • For any polygon, each interior angle and its corresponding exterior angle form a linear pair, so I + E = 180 degrees. Concept / Approach: The key concepts are: (1) an interior angle and its adjacent exterior angle are supplementary and sum to 180 degrees, and (2) in a regular polygon, all exterior angles are equal and their sum is always 360 degrees. Once we find the measure of one exterior angle, we can use the formula number of sides n = 360 / E. The given condition that the interior angle is 90 degrees greater than the exterior angle provides a simple equation in terms of E, which we can solve to determine the angle values, and then the number of sides. Step-by-Step Solution: Step 1: Use the relationship between interior and exterior angles: I + E = 180. Step 2: Use the condition that I = E + 90. Step 3: Substitute I = E + 90 into I + E = 180 to get (E + 90) + E = 180. Step 4: Simplify the equation: 2E + 90 = 180, so 2E = 90 and E = 45 degrees. Step 5: The measure of each exterior angle of the regular polygon is therefore 45 degrees. Step 6: Use the formula for a regular polygon: number of sides n = 360 / E, so n = 360 / 45 = 8. Verification / Alternative check: We can verify the result by checking the interior angle. If E = 45 degrees, then I = 180 - 45 = 135 degrees from the linear pair relationship. The condition in the question says that the interior angle is 90 degrees more than the exterior angle. Indeed, 135 - 45 = 90, so the condition is satisfied. Also, the sum of all exterior angles is 360 degrees, and 8 * 45 = 360, which confirms that a regular polygon with 8 sides (an octagon) fits perfectly with the standard polygon angle properties. Why Other Options Are Wrong: If the polygon had 9 sides, the exterior angle would be 360 / 9 = 40 degrees, which would make the interior angle 140 degrees, and the difference would be 100 degrees, not 90. For 10 sides, the exterior angle would be 36 degrees and the interior angle 144 degrees, giving a difference of 108 degrees. For 12 sides, the exterior angle is 30 degrees and the interior angle 150 degrees, giving a difference of 120 degrees. None of these match the required 90 degree difference, so only 8 sides is correct. Common Pitfalls: Many learners forget that interior and exterior angles are supplementary in a simple convex polygon, and they may mistakenly try to use incorrect formulas. Others confuse the sum of interior angles with the sum of exterior angles, but remember that the sum of all exterior angles of any convex polygon is always 360 degrees. Another common error is to treat the 90 degree difference as I = 90 or E = 90 directly, which is not what the problem states. Carefully translating the statement into equations I = E + 90 and I + E = 180 avoids these mistakes. Final Answer: The regular polygon has 8 sides.

Question 17

Find the volume, in cubic centimetres, of a solid hemisphere whose diameter is 21 cm. Take π = 22/7 for the calculation.

Accepted Answer

Introduction / Context: This problem focuses on the volume of a hemisphere, which is a common three dimensional solid in mensuration and aptitude questions. A hemisphere is exactly half of a sphere, so its volume can be obtained by applying the standard sphere volume formula and then halving it. Competitive exams often test your ability to correctly identify which formula to use, substitute the given values accurately, and handle fractions and constants like π efficiently. Here you are given the diameter of the hemisphere and a specific approximate value for π, and you are asked to compute the volume in cubic centimetres. Given Data / Assumptions: • The solid is a hemisphere (half of a sphere). • Diameter of the hemisphere = 21 cm. • Therefore, radius r = 21 / 2 = 10.5 cm. • Use π = 22 / 7 as an approximation for calculations. • We need the volume expressed in cubic centimetres (cubic cm). Concept / Approach: The volume of a full sphere of radius r is given by V_sphere = (4 / 3) * π * r^3. A hemisphere is half of a sphere, so its volume is V_hemisphere = (1 / 2) * V_sphere = (2 / 3) * π * r^3. The key is to convert the diameter into the radius first, then substitute into the formula, and finally perform the arithmetic carefully. Using π = 22 / 7 helps produce a rational numerical result that matches the given options exactly. Step-by-Step Solution: Step 1: Compute the radius from the given diameter: r = 21 / 2 = 10.5 cm. Step 2: Recall the volume formula for a hemisphere: V = (2 / 3) * π * r^3. Step 3: Substitute π = 22 / 7 and r = 10.5 cm into the formula. Step 4: First compute r^3: 10.5 * 10.5 * 10.5 = 1157.625 cubic cm. Step 5: Now compute V = (2 / 3) * (22 / 7) * 1157.625. Step 6: Perform the multiplication and division stepwise; the simplified exact result is 2425.5 cubic cm. Verification / Alternative check: To verify, you can approximate using π ≈ 3.14 and see if the result is close to the chosen option. With π ≈ 3.14 and r ≈ 10.5, the value of (2 / 3) * π * r^3 will also be around 2400 to 2500 cubic cm, which confirms that 2425.5 cubic cm is reasonable. Additionally, volumes of spheres and hemispheres grow with the cube of the radius, so a radius slightly above 10 cm would indeed yield a volume in the low thousands of cubic centimetres, which supports the magnitude of the answer. Why Other Options Are Wrong: The option 2235.5 cubic cm is somewhat smaller and can come from an arithmetic slip, such as incorrectly computing r^3 or forgetting part of the fraction (2 / 3). The options 2040 and 1860 cubic cm are significantly lower and would result from either using π incorrectly (for example, approximating it as 3) or mistakenly using the formula for a cylinder or another shape. Only 2425.5 cubic cm matches the correct substitution into the hemisphere volume formula with π = 22 / 7 and radius 10.5 cm. Common Pitfalls: Common mistakes include using the diameter instead of the radius directly in the formula, forgetting that the volume is for a hemisphere and not a full sphere, or misapplying the formula for the volume of a cylinder or cone. Another source of error is rushing through the computation of r^3 and making multiplication errors. To avoid these pitfalls, always convert from diameter to radius first, double check that you are using V = (2 / 3) * π * r^3 for a hemisphere, and perform the arithmetic in clear steps, preferably with intermediate checks or estimation to ensure the result is in a sensible range. Final Answer: The volume of the hemisphere is 2425.5 cubic cm.

Question 18

In triangle ABC, O is the orthocentre. If the angle ∠BOC formed at the orthocentre by lines OB and OC is 110°, then what is the measure of angle ∠BAC (in degrees)?

Accepted Answer

Introduction / Context: This geometry question tests your knowledge of special angle relationships associated with the orthocentre of a triangle. The orthocentre is the point where the three altitudes of a triangle intersect. For an acute triangle, there is a well known relationship between the angle at the orthocentre formed by two vertices and the opposite vertex angle in the original triangle. Understanding this relationship allows you to convert between an angle at the orthocentre and an angle at a vertex of the triangle without directly constructing all altitudes, which is very useful in time bound aptitude exams. Given Data / Assumptions: • Triangle ABC is a non degenerate triangle with orthocentre O. • Angle ∠BOC is given as 110 degrees. • O is the intersection point of the three altitudes of triangle ABC. • The triangle is assumed to be acute so that the usual interior angle relationships with the orthocentre apply directly. • We need to find angle ∠BAC, which is the angle at vertex A of the triangle. Concept / Approach: A key property of the orthocentre states that the angle at the orthocentre formed by joining it to any two vertices is supplementary to the angle at the remaining vertex. More precisely, in an acute triangle, the angle ∠BOC at the orthocentre is related to the angle at vertex A by the formula ∠BOC = 180° - ∠A. This is analogous to a similar relation for the circumcentre, where angles involving the circumcentre are often twice the vertex angles, but here for the orthocentre the relationship involves a supplement. Using this property, you can directly obtain the required angle at A from the given angle at O. Step-by-Step Solution: Step 1: Recall the orthocentre angle property: in triangle ABC with orthocentre O, angle ∠BOC = 180° - ∠A, where ∠A = ∠BAC. Step 2: Substitute the given value ∠BOC = 110° into the relation: 110° = 180° - ∠BAC. Step 3: Rearrange the equation to solve for ∠BAC: ∠BAC = 180° - 110°. Step 4: Compute the difference: 180° - 110° = 70°. Step 5: Therefore, the measure of angle ∠BAC is 70 degrees. Verification / Alternative check: If angle A is 70°, then angle ∠BOC for an acute triangle should be 180° - 70° = 110°, which matches the given information. This confirms that our use of the property is correct. Additionally, because 70° is an acute angle, it is consistent with the assumption that the triangle is acute, where the orthocentre lies inside the triangle and the standard relationships for orthocentre angles hold. There is no contradiction, so the answer is consistent and verified. Why Other Options Are Wrong: If ∠BAC were 110°, then ∠BOC would be 180° - 110° = 70°, which is the reverse of what is given. If ∠BAC were 100°, then ∠BOC should be 80°, not 110°. If ∠BAC were 90°, then ∠BOC should be 90°, which again does not match the given 110°. Thus all other options are inconsistent with the fundamental relation ∠BOC = 180° - ∠BAC, leaving 70° as the only correct choice. Common Pitfalls: A frequent mistake is to confuse the orthocentre with the circumcentre and incorrectly use a relation like ∠BOC = 2∠A, which actually belongs to the circumcentre in certain contexts, not to the orthocentre. Another common error is to think in terms of angle sums in the triangle alone and ignore the specific properties of the special point described in the question. Carefully noting that O is the orthocentre and recalling the correct identity ∠BOC = 180° - ∠A is crucial. Students should also be careful with subtraction, as reversing 180° - 110° can lead to incorrect values if done hastily. Final Answer: The measure of angle ∠BAC is 70°.

Question 19

In triangle ABC, BE and CF are two altitudes drawn from vertices B and C to the opposite sides. If AB = 6 cm, AC = 5 cm and altitude CF = 4 cm, then what is the length of altitude BE (in cm)?

Accepted Answer

Introduction / Context: This problem uses properties of altitudes in a triangle and the fact that the area of a triangle can be expressed in multiple ways depending on which side is chosen as the base. By equating these different expressions for the same area, you can relate different altitudes to each other. Such questions check your understanding of basic triangle area formulas and your ability to connect them in a simple algebraic way, which is very common in quantitative aptitude and geometry sections of competitive exams. Given Data / Assumptions: • Triangle ABC is a non degenerate triangle. • BE is an altitude from vertex B to side AC, so BE is perpendicular to AC. • CF is an altitude from vertex C to side AB, so CF is perpendicular to AB. • Side AB = 6 cm. • Side AC = 5 cm. • Altitude CF = 4 cm. • We are required to find the length of altitude BE in centimetres. Concept / Approach: The area of triangle ABC can be written using any side as the base with the corresponding altitude. If we choose AB as the base, the area is (1 / 2) * AB * CF, because CF is perpendicular to AB. If we choose AC as the base, the area is (1 / 2) * AC * BE, because BE is perpendicular to AC. Both expressions represent the same geometric area, so they must be equal. Equating these two formulas allows us to solve directly for BE. This technique is a straightforward but powerful application of the area formula for triangles. Step-by-Step Solution: Step 1: Express the area of triangle ABC using base AB and altitude CF: Area = (1 / 2) * AB * CF. Step 2: Substitute AB = 6 cm and CF = 4 cm to get Area = (1 / 2) * 6 * 4. Step 3: Compute this value: Area = (1 / 2) * 24 = 12 square centimetres. Step 4: Now express the same area using base AC and altitude BE: Area = (1 / 2) * AC * BE. Step 5: Substitute AC = 5 cm and Area = 12 square centimetres into this expression: 12 = (1 / 2) * 5 * BE. Step 6: Solve for BE: (1 / 2) * 5 * BE = 12 implies 5 * BE = 24, so BE = 24 / 5 = 4.8 cm. Verification / Alternative check: You can quickly check the consistency by recomputing the area with the found value of BE. Using base AC = 5 cm and altitude BE = 4.8 cm, Area = (1 / 2) * 5 * 4.8 = (1 / 2) * 24 = 12 square centimetres, which matches the area calculated earlier using base AB and altitude CF. This confirms that BE = 4.8 cm is consistent and correct. The dimensions are also reasonable since altitudes in triangles with moderate side lengths often fall in similar numeric ranges. Why Other Options Are Wrong: Option 7.5 cm would produce an area of (1 / 2) * 5 * 7.5 = 18.75 square centimetres, which does not match 12 square centimetres. Option 3.33 cm is an approximate value and would give area about (1 / 2) * 5 * 3.33 which is close to 8.325 square centimetres, again not equal to 12. Option 5.5 cm leads to area (1 / 2) * 5 * 5.5 = 13.75 square centimetres, which is also inconsistent. Hence, only 4.8 cm preserves the equality of the two area expressions. Common Pitfalls: Learners sometimes confuse which altitude corresponds to which base, or they may mistakenly try to use Pythagoras theorem without having enough information. Another common error is forgetting to multiply by 1 / 2 or losing track of units. It is important to remember that for any triangle, the area formula (1 / 2) * base * height works for any side chosen as base, as long as the corresponding altitude is used. Equating two such expressions is a simple and reliable method for problems involving multiple altitudes. Final Answer: The length of altitude BE is 4.8 cm.

Question 20

In triangle ABC, side BC is produced to a point D so that CD lies on the extension of BC. If the exterior angle ∠ACD = 120° and angle ∠B is equal to half of angle ∠A (that is, ∠B = (1/2)∠A), then what is the measure of angle ∠A (in degrees)?

Accepted Answer

Introduction / Context: This question combines the exterior angle property of triangles with a simple algebraic relationship between two interior angles. In many aptitude and geometry problems, you are given one exterior angle and some relationship between the interior angles, and you must apply the basic angle sum and exterior angle rules to find the required vertex angle. Understanding that an exterior angle of a triangle equals the sum of the two opposite interior angles is crucial for solving such problems efficiently. Given Data / Assumptions: • Triangle ABC is given, and side BC is extended to point D to form an exterior angle at C. • The exterior angle at C is ∠ACD = 120°. • Angle at vertex B satisfies ∠B = (1 / 2) ∠A, where ∠A is ∠BAC. • We need to determine the measure of angle ∠A in degrees. • The triangle is assumed to be a standard Euclidean triangle with positive angles summing to 180°. Concept / Approach: The main geometric fact used here is the exterior angle theorem: in any triangle, an exterior angle equals the sum of the two interior opposite angles. When BC is extended to D, ∠ACD is the exterior angle at C, and its two remote interior angles are ∠A and ∠B. Therefore, we have ∠ACD = ∠A + ∠B. We are also given a direct algebraic relationship between ∠A and ∠B, namely ∠B = (1 / 2) ∠A. Substituting this relationship into the exterior angle equation produces a simple linear equation in ∠A, which we can solve to get the required angle. Step-by-Step Solution: Step 1: Use the exterior angle property for triangle ABC with BC produced to D: ∠ACD = ∠A + ∠B. Step 2: Substitute the given values into the relationship: 120° = ∠A + ∠B. Step 3: Use the given condition ∠B = (1 / 2) ∠A. Step 4: Replace ∠B in the equation: 120° = ∠A + (1 / 2) ∠A. Step 5: Combine like terms: 120° = (3 / 2) ∠A. Step 6: Solve for ∠A: ∠A = (2 / 3) * 120° = 80°. Verification / Alternative check: Once we find ∠A = 80°, we can calculate ∠B from the given relationship ∠B = (1 / 2) ∠A, so ∠B = 40°. The exterior angle at C should equal the sum of these two interior angles, so ∠ACD = ∠A + ∠B = 80° + 40° = 120°, which matches the given value. Furthermore, the interior angles of triangle ABC should sum to 180°, so ∠C = 180° - (80° + 40°) = 60°, which is a valid positive angle. Everything is consistent, so ∠A = 80° is confirmed. Why Other Options Are Wrong: If ∠A were 60°, then ∠B would be 30°, and their sum would be 90°, not 120°. If ∠A were 75°, then ∠B would be 37.5°, and their sum would be 112.5°, again not 120°. If ∠A were 90°, then ∠B would be 45°, and the sum would be 135°. None of these combinations satisfy the exterior angle equation ∠ACD = 120°. Only ∠A = 80° leads to ∠A + ∠B = 120° and a valid triangle angle set. Common Pitfalls: One frequent mistake is to assume that the exterior angle is directly supplementary to one interior angle only, and then attempt to solve using 180° - 120° = 60° without using the given relationship with the second interior angle. Another error is misinterpreting the statement ∠B = (1 / 2) ∠A, either by inverting it or by using a wrong factor in the equation. To avoid these pitfalls, always write down the correct exterior angle relation and then carefully substitute the algebraic relationship between angles. Checking the sum of all interior angles at the end is a good way to diagnose mistakes. Final Answer: The measure of angle ∠A is 80°.

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