Question 1

The curved surface area of a right circular cone is 3080 square centimetres and its slant height is 35 cm. Using the standard mensuration formulas, what is the total surface area of the cone?

Accepted Answer

Introduction / Context: This mensuration question deals with a right circular cone. You are given the curved surface area (also called lateral surface area) and the slant height and are asked to find the total surface area. Total surface area of a cone includes both the curved surface area and the area of the circular base. To find it, we first determine the radius of the base from the curved surface area formula, and then add the area of the base to the curved surface area. Given Data / Assumptions: - Curved surface area (CSA) of the cone = 3080 square centimetres. - Slant height l of the cone = 35 cm. - For a right circular cone, CSA = π * r * l. - Total surface area (TSA) = π * r * l + π * r^2 = π * r * (l + r). - π is taken as 22 / 7. Concept / Approach: We first use the given curved surface area along with the formula CSA = π * r * l to solve for the radius r. Once r is known, we substitute r and the given l into TSA = π * r * (l + r). The arithmetic must be carried out carefully using π = 22 / 7 to get an exact integer result. This approach gives us the total surface area directly in square centimetres. Step-by-Step Solution: Step 1: Write the formula for curved surface area of a cone: CSA = π * r * l. Step 2: Substitute CSA = 3080, π = 22 / 7, and l = 35. So 3080 = (22 / 7) * r * 35. Step 3: Simplify the right hand side. First, 35 / 7 = 5, so (22 / 7) * 35 = 22 * 5 = 110. Step 4: The equation becomes 3080 = 110 * r. Step 5: Solve for r by dividing: r = 3080 / 110 = 28 cm. Step 6: Now compute the total surface area using TSA = π * r * (l + r). Step 7: Substitute π = 22 / 7, r = 28, and l = 35. Then TSA = (22 / 7) * 28 * (35 + 28). Step 8: Compute (35 + 28) = 63. So TSA = (22 / 7) * 28 * 63. Step 9: Simplify 28 / 7 = 4. So TSA = 22 * 4 * 63 = 88 * 63. Step 10: Multiply 88 * 63 = 5544 square centimetres. Verification / Alternative check: We can quickly verify by computing the base area separately. Base area = π * r^2 = (22 / 7) * 28^2 = (22 / 7) * 784. Simplify 784 / 7 = 112, so base area = 22 * 112 = 2464 sq cm. Adding CSA = 3080 sq cm gives 3080 + 2464 = 5544 sq cm, which matches the TSA obtained earlier. This double check confirms that our value of r and our total surface area are correct. Why Other Options Are Wrong: Option 11088 sq cm is exactly double the correct TSA and might come from mistakenly multiplying by 2 at some step. Option 1848 sq cm is much smaller and could arise from misapplying only π * r^2 or mixing up units. Option 2772 sq cm is exactly half of the correct value and may result from forgetting either the curved surface or the base area in the total surface area calculation. Common Pitfalls: A common mistake is to confuse curved surface area with total surface area and forget to add the base area. Another error is in the simplification of fractions when using π = 22 / 7, especially not simplifying 35 / 7 or 28 / 7 correctly. To avoid such mistakes, write the formula clearly, substitute values step by step, and verify by recomputing base and curved areas separately. Final Answer: The total surface area of the cone is 5544 square centimetres.

Question 2

The diameter of a circle is equal to the side of a square. If the area of the circle is 49π square centimetres, what is the area of the square?

Accepted Answer

Introduction / Context: This question connects the geometry of a circle and a square. The diameter of the circle is equal to the side length of the square. Given the area of the circle, we use the circle area formula to find the radius and then the diameter. That diameter becomes the side of the square, and from that we compute the square area. This is a straightforward application of area formulas for a circle and a square. Given Data / Assumptions: - Area of the circle is 49π square centimetres. - Let radius of the circle be r. - Circle area formula: Area = π * r^2. - Diameter of the circle is 2r, and this equals the side of the square. - We must find the area of the square. Concept / Approach: First, use the circle area formula to find r. Once we know r, the diameter is just 2r. Since the side of the square equals the diameter, we can write side = 2r. Then the area of the square is side^2, which is (2r)^2. Substituting the value of r gives the required area of the square. The problem relies on careful use of formulas, not on any advanced geometry. Step-by-Step Solution: Step 1: Use circle area formula: Area of circle = π * r^2. Step 2: We are given that π * r^2 = 49π. Step 3: Divide both sides by π to get r^2 = 49. Step 4: Take square root of both sides: r = √49 = 7 cm. Step 5: Diameter of the circle is 2r = 2 * 7 = 14 cm. Step 6: The side of the square is equal to this diameter, so side of square s = 14 cm. Step 7: Area of the square is s^2 = 14^2. Step 8: Compute 14^2 = 196 square centimetres. Verification / Alternative check: We can double check by working backward. If the square side is 14 cm, its area is 196 square centimetres. The diameter of the circle is also 14 cm, so radius is 7 cm. The area of the circle with radius 7 cm is π * 7^2 = 49π, which matches the given circle area. This consistent loop confirms that the square area 196 square centimetres is correct. Why Other Options Are Wrong: Option 154 square centimetres might come from mixing the circumference formula or incorrectly squaring. Option 124 square centimetres does not correspond to any logical combination of radius or diameter obtained from 49π. Option 98 square centimetres is exactly half of 196 and may come from forgetting that area is side squared and dividing by 2 accidentally. Common Pitfalls: Some students confuse area and circumference and may try to use 2πr = 49π, which would be incorrect here since 49π is clearly an area. Another mistake is to forget that the diameter, not the radius, is equal to the side of the square. Always carefully identify which quantity is being equated when problems link two shapes. Keeping symbols like r, diameter, and side distinct in your steps helps avoid confusion. Final Answer: The area of the square is 196 square centimetres.

Question 3

The curved surface area of a cylinder is 528 square centimetres. If the circumference of its base is 44 cm, what is the height of the cylinder?

Accepted Answer

Introduction / Context: This question involves a right circular cylinder and asks for its height when the curved surface area and the base circumference are given. The curved surface area can be expressed as the product of the circumference of the base and the height. This gives a very direct way to find the height when those two quantities are known. It is a classic mensuration question that tests understanding of formulas relating lateral area and circumference for cylinders. Given Data / Assumptions: - Curved surface area (CSA) of the cylinder = 528 square centimetres. - Circumference of the base = 44 centimetres. - Height of the cylinder is h centimetres. - Relationship used: CSA = circumference of base * height. Concept / Approach: For a cylinder of radius r and height h, the curved surface area is given by CSA = 2 * π * r * h, and the circumference of the base is C = 2 * π * r. Notice that CSA can be written as C * h. In this question, since CSA and C are both given, we can find the height h simply by dividing CSA by C. This avoids any need to work out the radius explicitly. Step-by-Step Solution: Step 1: Recall the relationship CSA = circumference * height for a cylinder. Step 2: We are given CSA = 528 square centimetres. Step 3: We are also given circumference C = 44 centimetres. Step 4: Using CSA = C * h, we have 528 = 44 * h. Step 5: Solve for h by dividing both sides by 44: h = 528 / 44. Step 6: Compute 528 / 44 = 12. Step 7: Therefore the height of the cylinder is 12 centimetres. Verification / Alternative check: To verify, we can deduce the radius from the circumference C = 44. Since C = 2 * π * r, and with π approximated as 22 / 7, we get 44 = 2 * (22 / 7) * r = (44 / 7) * r, so r = 7 cm. Then CSA = 2 * π * r * h = 2 * (22 / 7) * 7 * 12 = 2 * 22 * 12 = 528 square centimetres, which matches the given curved surface area. This confirms that h = 12 cm is correct. Why Other Options Are Wrong: Option 24 cm would require CSA = 44 * 24 = 1056 square centimetres, which does not match the given 528. Option 36 cm would give CSA = 44 * 36 = 1584 square centimetres, far larger than given. Option 6 cm would give CSA = 44 * 6 = 264 square centimetres, which is only half of the required area. Common Pitfalls: A common mistake is to try to use π explicitly and compute r first, which is more work and introduces extra chances for arithmetic errors. Another error is mixing up formulas for total surface area and curved surface area. Remember that here total surface area would add the areas of the two circular bases, which is not needed. Simply recognising CSA = circumference * height leads to a quick and clean solution. Final Answer: The height of the cylinder is 12 cm.

Question 4

In a right circular cone, the radius is r and the height is h. They satisfy the relation 4r = h + √(r^2 + h^2), where √(r^2 + h^2) represents the slant height. If r is not zero, what is the ratio r : h?

Accepted Answer

Introduction / Context: This algebraic geometry problem comes from the context of a right circular cone, where the radius r, height h, and slant height l are related by l = √(r^2 + h^2). The question gives a special relation between r, h, and the slant height: 4r = h + √(r^2 + h^2). We are asked to find the ratio r : h that satisfies this relation, assuming r is not zero. This tests the ability to interpret a geometric formula, substitute the expression for slant height, and then solve a non linear equation by using a suitable substitution. Given Data / Assumptions: - Right circular cone with radius r and height h. - Slant height l is given by l = √(r^2 + h^2). - Given relation: 4r = h + √(r^2 + h^2). - r is not equal to zero and r, h are positive real numbers. - We must determine r : h. Concept / Approach: We can use the expression for slant height directly in the given relation. To simplify the equation and avoid dealing with both r and h separately, we introduce a ratio variable k = r / h. We then express r in terms of h and k and rewrite the relation. This turns the equation into one involving only k. After simplification, we solve for k and match it to one of the given ratio options. Finally, we interpret k as r : h. Step-by-Step Solution: Step 1: Given relation is 4r = h + √(r^2 + h^2). Step 2: Introduce k = r / h. Then r = k * h. Step 3: Substitute r = k * h into the relation. We get 4 * (k * h) = h + √((k * h)^2 + h^2). Step 4: Simplify inside the square root: (k * h)^2 + h^2 = k^2 * h^2 + h^2 = h^2 * (k^2 + 1). Step 5: So √(r^2 + h^2) becomes √(h^2 * (k^2 + 1)) = h * √(k^2 + 1). Step 6: The equation now is 4k * h = h + h * √(k^2 + 1). Step 7: Since h is positive and not zero, divide both sides by h to get 4k = 1 + √(k^2 + 1). Step 8: Isolate the square root: √(k^2 + 1) = 4k − 1. Step 9: Square both sides to remove the square root: k^2 + 1 = (4k − 1)^2. Step 10: Expand the square: (4k − 1)^2 = 16k^2 − 8k + 1. Step 11: Set up the equation k^2 + 1 = 16k^2 − 8k + 1. Step 12: Subtract k^2 + 1 from both sides: 0 = 15k^2 − 8k. Step 13: Factor out k: 0 = k * (15k − 8). Step 14: Since r is not zero, k = r / h is not zero, so k cannot be 0. Hence we must have 15k − 8 = 0. Step 15: Solve 15k − 8 = 0 to get k = 8 / 15. Step 16: Therefore r / h = 8 / 15, so the ratio r : h is 8 : 15. Verification / Alternative check: To verify, take r : h = 8 : 15. Let r = 8 and h = 15 for simplicity. Then l = √(r^2 + h^2) = √(64 + 225) = √289 = 17. Check the relation: left side 4r = 4 * 8 = 32. Right side h + l = 15 + 17 = 32. Both sides match, confirming that the ratio 8 : 15 satisfies the given condition. Why Other Options Are Wrong: Option 17 : 8 would correspond to k = 17 / 8, which does not satisfy the equation 4k = 1 + √(k^2 + 1). Option 8 : 17 gives k = 8 / 17, which fails to satisfy the squared relation derived from the original equation. Option 15 : 8 gives k = 15 / 8. If we substitute this into 4k, we get 60 / 8 = 7.5, which does not equal 1 + √(k^2 + 1), so it is incorrect. Common Pitfalls: A frequent mistake is to forget that squaring an equation can introduce extraneous solutions, which is why we must check the final value of k in the original relation. Another possible error is incorrect algebra during expansion of (4k − 1)^2 or when rearranging terms. Introducing the ratio k = r / h early is a powerful technique for simplifying problems involving geometric relations between length quantities. Final Answer: The ratio of radius to height is r : h = 8 : 15.

Question 5

In triangle PQR, points S and T lie on sides PQ and PR respectively. Segment ST is drawn parallel to side QR. If PS = 6 cm, SQ = 9 cm (so that PQ = PS + SQ), and PR = 12.5 cm, what is the length of segment TR?

Accepted Answer

Introduction / Context: This question uses the concept of similar triangles formed by a line drawn parallel to one side of a triangle. In triangle PQR, a segment ST is drawn parallel to base QR, cutting the other two sides PQ and PR at points S and T respectively. We are given PS, SQ and PR, and we are asked to find the remaining part TR of side PR. This is a standard application of basic proportionality theorem, also known as Thales theorem, in plane geometry. Given Data / Assumptions: - Triangle PQR is a general triangle. - Point S lies on side PQ, and point T lies on side PR. - ST is parallel to QR. - PS = 6 cm and SQ = 9 cm, so PQ = PS + SQ = 15 cm. - PR = 12.5 cm. - All lengths are positive and measured in centimetres. Concept / Approach: When a line segment is drawn parallel to one side of a triangle and meets the other two sides, it divides those sides proportionally. Because ST is parallel to QR, triangle PST is similar to triangle PQR. Therefore, the ratio PS / PQ equals PT / PR. Once we know PS and PQ, we can find PT using proportionality, and then TR is simply PR minus PT. This avoids any need for angles or coordinate geometry and uses only ratio reasoning. Step-by-Step Solution: Step 1: From PS = 6 cm and SQ = 9 cm, compute PQ. Step 2: PQ = PS + SQ = 6 + 9 = 15 cm. Step 3: Because ST is parallel to QR, triangles PST and PQR are similar. Step 4: By similarity, corresponding sides are proportional. So PS / PQ = PT / PR. Step 5: Substitute known values: 6 / 15 = PT / 12.5. Step 6: Simplify 6 / 15 to 2 / 5. Step 7: So 2 / 5 = PT / 12.5, hence PT = (2 / 5) * 12.5. Step 8: Compute PT = 25 / 5 = 5 cm. Step 9: Now PR = 12.5 cm, and PR is made up of PT + TR. Step 10: Therefore TR = PR − PT = 12.5 − 5 = 7.5 cm. Verification / Alternative check: We can also check by computing the scale factor between triangles PST and PQR. The ratio PS : PQ is 6 : 15 which equals 2 : 5. Thus every corresponding length in the smaller triangle PST is 2 / 5 of the length in the larger triangle PQR. Since PR is 12.5 cm, the corresponding side PT must be (2 / 5) * 12.5 = 5 cm, which matches our earlier calculation. Then TR is the remaining 7.5 cm. The consistency of both methods confirms the answer. Why Other Options Are Wrong: Option 5 cm equals PT, not TR, so it represents the upper part of PR instead of the required lower segment. Option 10 cm would make PT equal to 2.5 cm, which does not preserve the similarity ratio 2 : 5 between PS and PQ. Option 2.5 cm would make PT equal to 10 cm, again reversing the proportionality and contradicting the condition that PS : PQ = PT : PR. Common Pitfalls: Learners sometimes directly apply ratios to TR instead of PT, or they forget that PS and PQ must be added to get the full side length. Another mistake is mixing up the direction of the ratio, such as taking PQ / PS instead of PS / PQ when writing PT / PR. Always match smaller triangle sides to the corresponding sides in the larger triangle in the same order to avoid proportionality errors. Final Answer: The length of segment TR is 7.5 cm.

Question 6

How many smaller solid balls of radius 2 cm can be made by completely melting and recasting one larger solid ball of diameter 16 cm? (Take π = 22/7 and assume no loss of material.)

Accepted Answer

Introduction / Context: This is a standard volume and mensuration problem involving spheres. A larger solid sphere is melted and reshaped into many smaller solid spheres. Since there is no loss of material, the total volume of all the small spheres must equal the volume of the original large sphere. Using the volume formula for a sphere and the concept of conservation of volume, we can determine how many smaller balls are formed. Given Data / Assumptions: - Radius of each small ball is 2 cm. - Diameter of the large ball is 16 cm, so its radius is 8 cm. - Volume of a sphere is V = (4 / 3) * π * r^3. - π is taken as 22 / 7. - No material is lost during melting and recasting. Concept / Approach: The key idea is conservation of volume. If N small spheres are formed, then N times the volume of one small sphere equals the volume of the large sphere. Because π and the constant 4 / 3 appear in both volumes, they cancel when we take the ratio. Therefore, the number of small balls N is simply the ratio of the cube of the large radius to the cube of the small radius. This avoids heavy arithmetic with π and fractions. Step-by-Step Solution: Step 1: Compute the radius of the large sphere. Its diameter is 16 cm, so radius R = 16 / 2 = 8 cm. Step 2: Radius of each small sphere is r = 2 cm. Step 3: Volume of the large sphere is V_large = (4 / 3) * π * R^3. Step 4: Volume of one small sphere is V_small = (4 / 3) * π * r^3. Step 5: Let N be the number of small spheres. Then N * V_small = V_large. Step 6: Substitute the volumes: N * (4 / 3) * π * r^3 = (4 / 3) * π * R^3. Step 7: Cancel (4 / 3) * π from both sides, giving N * r^3 = R^3. Step 8: So N = R^3 / r^3 = 8^3 / 2^3. Step 9: Compute 8^3 = 512 and 2^3 = 8. Therefore N = 512 / 8 = 64. Verification / Alternative check: If you wish, you can compute actual numerical volumes using π = 22 / 7. Large sphere volume: V_large = (4 / 3) * (22 / 7) * 8^3. Small sphere volume: V_small = (4 / 3) * (22 / 7) * 2^3. The ratio V_large / V_small simplifies to 8^3 / 2^3 as before, which is 64. This confirms that all constant factors cancel and our calculation is correct. Why Other Options Are Wrong: Option 128 would correspond to doubling the correct number of spheres and would require twice the volume that the large sphere provides. Option 32 is exactly half of the correct number and would leave unused volume if only 32 small spheres were formed. Option 96 would imply a ratio of volumes that does not match the cube ratio of the radii 8 and 2. Common Pitfalls: A common mistake is to think the number of small spheres is simply the ratio of radii or the ratio of surface areas, not the ratio of volumes. Because volume of a sphere is proportional to the cube of the radius, the correct ratio uses R^3 and r^3. It is also easy to miscalculate 8^3 or 2^3, so always check these small powers carefully. Remember to focus on volume when dealing with melted and recast solid shapes. Final Answer: The number of smaller balls that can be made is 64.

Question 7

In a triangle, the side opposite an angle of 60° measures 6√3 cm. Assuming the triangle is a right angled triangle with one angle equal to 90°, what is the length of the side opposite the 90° angle?

Accepted Answer

Introduction / Context: This question is based on the special properties of a 30°–60°–90° right triangle. We are told that one interior angle is 60° with a side opposite this angle equal to 6√3 cm, and another angle is 90°, revealing that the triangle is right angled. In such a triangle, the side lengths follow fixed ratios relative to one another. Using these known ratios, we can quickly determine the length of the side opposite the 90° angle, which is the hypotenuse. Given Data / Assumptions: - The triangle is right angled, with one angle equal to 90°. - Another angle is 60°, so the remaining angle must be 30°. - The side opposite the 60° angle is given as 6√3 cm. - We are asked to find the side opposite the 90° angle (the hypotenuse). - All sides are positive real lengths. Concept / Approach: In a 30°–60°–90° triangle, side lengths are in a fixed proportional pattern. If the side opposite the 30° angle is taken as k, then the side opposite the 60° angle is k√3 and the side opposite the 90° angle (the hypotenuse) is 2k. This ratio holds for all such triangles, regardless of their absolute size. In this problem, the side opposite 60° is given, so we can solve for k and then double it to find the hypotenuse opposite 90°. Step-by-Step Solution: Step 1: Recognise that with angles 30°, 60° and 90°, the triangle is a special right triangle. Step 2: Let the side opposite the 30° angle be k. Step 3: Then the side opposite the 60° angle is k√3, and the side opposite the 90° angle is 2k. Step 4: We are given that the side opposite the 60° angle is 6√3 cm. Thus k√3 = 6√3. Step 5: Cancel √3 from both sides to obtain k = 6. Step 6: The hypotenuse, which is the side opposite the 90° angle, is 2k. Step 7: Compute 2k = 2 * 6 = 12 cm. Step 8: Therefore, the length of the side opposite the 90° angle is 12 cm. Verification / Alternative check: Once we know k = 6, the side opposite 30° is 6 cm, the side opposite 60° is 6√3 cm, and the hypotenuse is 12 cm. We can verify the Pythagoras theorem: for the legs 6 and 6√3, the square of the hypotenuse should equal 6^2 + (6√3)^2 = 36 + 36 * 3 = 36 + 108 = 144. The square of 12 is also 144, so the triangle is indeed right angled and consistent with the given side length. This confirms that 12 cm is correct. Why Other Options Are Wrong: Option 12√3 cm would make the hypotenuse much larger and would not satisfy the 30°–60°–90° ratio with the given side of 6√3 cm. Option 6 cm corresponds to the side opposite the 30° angle, not the side opposite the 90° angle. Option 3√3 cm is too small and would not be consistent with the given side 6√3 cm in a right triangle. Common Pitfalls: A frequent mistake is mixing up which side in a 30°–60°–90° triangle is associated with k and which with k√3. Some learners also incorrectly assume that the side opposite 60° is the longest, but in a right triangle the hypotenuse, opposite 90°, is always the longest side. Remembering and applying the fixed ratio 1 : √3 : 2 for the sides opposite 30°, 60° and 90° respectively helps to avoid such confusion. Final Answer: The length of the side opposite the 90° angle is 12 cm.

Question 8

A solid right circular cone of radius 3.5 cm and height 9.6 cm is completely melted and recast into a right circular cylinder of the same radius 3.5 cm. Taking π = 22/7, what will be the height of the resulting cylinder?

Accepted Answer

Introduction / Context: This is a volume conservation problem in mensuration involving a cone and a cylinder. A solid cone is melted and its material is used to form a cylinder that has the same radius as the original cone. Since there is no gain or loss of material, the volume of the cone must equal the volume of the cylinder. Using standard volume formulas for a cone and a cylinder, we can find the unknown height of the cylinder. Given Data / Assumptions: - Radius of the original cone r = 3.5 cm. - Height of the cone h_cone = 9.6 cm. - The cone is melted and recast into a cylinder with the same radius r = 3.5 cm. - Height of the cylinder is H (unknown). - Volume of a cone: V_cone = (1 / 3) * π * r^2 * h_cone. - Volume of a cylinder: V_cyl = π * r^2 * H. - π is taken as 22 / 7. Concept / Approach: The guiding principle is conservation of volume: volume of cone equals volume of cylinder. Because the radius is the same for both solids, the factor π * r^2 appears in both volume formulas and cancels when equated. This simplifies the relation to a direct proportionality between the heights of the two shapes, allowing us to solve for H using a simple division without complicated numerical computation. Step-by-Step Solution: Step 1: Write the volume of the cone as V_cone = (1 / 3) * π * r^2 * h_cone. Step 2: Write the volume of the cylinder as V_cyl = π * r^2 * H. Step 3: Since the cone is melted and recast into the cylinder, set V_cone = V_cyl. Step 4: So (1 / 3) * π * r^2 * h_cone = π * r^2 * H. Step 5: Cancel π * r^2 from both sides. This gives (1 / 3) * h_cone = H. Step 6: Substitute h_cone = 9.6 cm, so H = (1 / 3) * 9.6. Step 7: Compute H = 9.6 / 3 = 3.2 cm. Step 8: Therefore, the height of the cylinder is 3.2 cm. Verification / Alternative check: We can estimate volumes numerically to check. Volume of the cone is approximately V_cone = (1 / 3) * π * 3.5^2 * 9.6. The cylinder volume with H = 3.2 is V_cyl = π * 3.5^2 * 3.2. Comparing, note that V_cyl is exactly one third of π * 3.5^2 * 9.6, which matches V_cone. This confirms that H = 3.2 cm satisfies the volume equality exactly, so no arithmetic error has occurred. Why Other Options Are Wrong: Option 6.4 cm would double the height of the cylinder and hence double its volume compared to what the cone provides. Option 1.6 cm would produce a cylinder with only half the correct volume, meaning not all cone material is used. Option 4.8 cm would give a cylinder volume equal to 1.5 times the cone volume, which is not possible under conservation of material. Common Pitfalls: Students sometimes mistakenly equate only the heights or only the radii without using the full volume formulas. Another common error is to forget the 1 / 3 factor in the cone volume formula, which completely changes the result. Also, some might try to carry π through unnecessary calculations instead of cancelling it early. Recognising that identical radii greatly simplify the ratio of volumes is an important problem solving skill in mensuration. Final Answer: The height of the cylinder formed from the melted cone is 3.2 cm.

Question 9

Find the length of the arc of a circle whose central angle is 90° and whose radius is 3.5 cm. (Use π = 22/7 and express the answer in centimetres.)

Accepted Answer

Introduction / Context: This is a basic arc length problem from circle geometry. The length of an arc depends on the radius of the circle and the measure of the central angle that subtends the arc at the centre. When the central angle is given in degrees, we can directly use the fraction of the full circumference corresponding to that angle. This type of calculation is very common in aptitude tests involving sectors and arcs of circles. Given Data / Assumptions: - Radius r of the circle is 3.5 cm. - Central angle θ is 90°. - π is taken as 22 / 7. - We must find the length of the arc corresponding to the 90° central angle. Concept / Approach: The full circumference of a circle is given by C = 2 * π * r. An angle of 360° corresponds to the full circumference. Therefore, an angle of θ degrees corresponds to a fraction θ / 360 of the full circumference. Hence, the arc length s for a central angle θ is s = (θ / 360) * 2 * π * r. Here θ = 90°, which is exactly one quarter of 360°, so the arc length is one quarter of the full circumference. This gives a quick and simple calculation. Step-by-Step Solution: Step 1: Write the formula for arc length when the angle is in degrees: s = (θ / 360) * 2 * π * r. Step 2: Substitute θ = 90°, r = 3.5 cm, and π = 22 / 7 into the formula. Step 3: So s = (90 / 360) * 2 * (22 / 7) * 3.5. Step 4: Simplify 90 / 360 to 1 / 4. Step 5: The expression becomes s = (1 / 4) * 2 * (22 / 7) * 3.5. Step 6: Compute 2 * 3.5 = 7, so s = (1 / 4) * (22 / 7) * 7. Step 7: The factor 7 cancels with the denominator 7, giving s = (1 / 4) * 22. Step 8: Compute (1 / 4) * 22 = 22 / 4 = 5.5 cm. Step 9: Therefore the length of the arc is 5.5 centimetres. Verification / Alternative check: We can also reason that 90° is one quarter of a full circle. The full circumference is C = 2 * π * r = 2 * (22 / 7) * 3.5. Since 3.5 = 7 / 2, we have C = 2 * (22 / 7) * (7 / 2) = 22 cm. One quarter of 22 cm is 22 / 4 = 5.5 cm, matching the value obtained from the formula. This confirms that the calculations are consistent and the arc length is correct. Why Other Options Are Wrong: Option 11 cm equals half of the full circumference and would correspond to a 180° central angle, not 90°. Option 16.5 cm would correspond to three quarters of the circumference, that is 270°, which is not the angle given. Option 22 cm is the full circumference of the circle, which corresponds to 360°, not to 90°. Common Pitfalls: Some learners mistakenly use θ in radians without converting or mix degree and radian formulas. Others may forget to divide by 360 and directly multiply θ with 2 * π * r. Always remember that when θ is in degrees, the correct fraction of the circumference is θ / 360. Breaking the calculation into clear steps and simplifying fractions such as 90 / 360 early helps prevent arithmetic mistakes. Final Answer: The length of the required arc is 5.5 cm.

Question 10

In a right angled triangle, one acute angle measures 30 degrees and the side opposite this 30 degree angle has length 9 cm. What is the length of the side opposite the 60 degree angle in the same triangle, expressed in centimetres?

Accepted Answer

Introduction / Context: This question is based on basic trigonometry and special right triangles. In particular, it uses the standard properties of a 30 60 90 right angled triangle, which is very common in aptitude and school level geometry. Understanding these fixed ratios helps you answer such questions quickly without using a calculator. Given Data / Assumptions: - The triangle is right angled and has angles 30 degrees, 60 degrees and 90 degrees. - The side opposite the 30 degree angle is given as 9 cm. - We are required to find the length of the side opposite the 60 degree angle. - The triangle is assumed to be a standard 30 60 90 right triangle in Euclidean plane geometry. Concept / Approach: In a 30 60 90 right triangle, the sides are in a fixed ratio relative to the hypotenuse. If the hypotenuse has length 2k, then the side opposite 30 degrees has length k and the side opposite 60 degrees has length k√3. Another way to say this is that the side opposite 30 degrees is half of the hypotenuse, and the side opposite 60 degrees is √3 times the shorter leg. We will first find the hypotenuse using this ratio and then use it to get the side opposite 60 degrees. Step-by-Step Solution: Step 1: Let the hypotenuse be 2k and the side opposite 30 degrees be k in the standard 30 60 90 triangle. Step 2: According to the question, the side opposite 30 degrees is 9 cm. So k = 9. Step 3: The hypotenuse is therefore 2k = 2 * 9 = 18 cm. Step 4: The side opposite the 60 degree angle in such a triangle is k√3. Step 5: Substitute k = 9 to get the required side as 9√3 cm. Verification / Alternative check: You can also verify using trigonometric ratios. If the hypotenuse is 18 cm and the side opposite 30 degrees is 9 cm, then sin 30 degrees = 9 / 18 = 1 / 2, which is correct. For the 60 degree angle, sin 60 degrees should be (side opposite 60 degrees) / 18. Since sin 60 degrees is √3 / 2, we have side opposite 60 degrees = 18 * (√3 / 2) = 9√3 cm, which matches our result. Why Other Options Are Wrong: 3√3 cm is too small because it would correspond to a different scale factor and would not maintain the 30 60 90 ratio with a 9 cm side opposite 30 degrees. 3/2 cm and 9/2 cm are both simple fractions of 9 but do not satisfy the fixed trigonometric ratio for a 30 60 90 triangle. 6√3 cm also does not fit the standard side length ratios when the side opposite 30 degrees is 9 cm. Common Pitfalls: A common mistake is to assume that the side opposite 60 degrees is double the side opposite 30 degrees or to confuse which side is the hypotenuse. Another frequent error is to forget the exact ratio of sides for a 30 60 90 triangle and try to guess from the options. Always remember the precise ratio of sides: k, k√3 and 2k for angles 30 degrees, 60 degrees and 90 degrees respectively. Final Answer: The length of the side opposite the 60 degree angle is 9√3 cm.

Question 11

A solid three dimensional body has 12 vertices and 30 edges. Using Euler's formula for polyhedra, how many faces does this solid have in total?

Accepted Answer

Introduction / Context: This question tests your understanding of Euler's formula for convex polyhedra, a fundamental concept in solid geometry. The relationship links the number of vertices, edges and faces of any convex polyhedron. Many competitive exams ask direct questions of this form because once you know the formula, the calculation is very quick and reliable. Given Data / Assumptions: - Number of vertices V = 12. - Number of edges E = 30. - The solid is assumed to be a convex polyhedron so that Euler's formula applies directly. - We need to find the total number of faces F of this solid. Concept / Approach: Euler's famous formula for a convex polyhedron states that: V - E + F = 2 where V is the number of vertices, E is the number of edges and F is the number of faces. This equation is always satisfied for simple convex solids such as prisms, pyramids and many more complex shapes as long as they remain topologically equivalent to a sphere. We will substitute the given values of V and E into this formula and solve for F. Step-by-Step Solution: Step 1: Write down Euler's formula: V - E + F = 2. Step 2: Substitute V = 12 and E = 30 into the formula. Step 3: We get 12 - 30 + F = 2. Step 4: Simplify 12 - 30 = -18, so the equation becomes -18 + F = 2. Step 5: Add 18 to both sides to obtain F = 2 + 18 = 20. Verification / Alternative check: You can rearrange Euler's formula explicitly as F = 2 - V + E. Substituting again gives F = 2 - 12 + 30 = 20. This quick check confirms our earlier calculation. Whenever you get an answer, it is good practice to verify it using a slightly rearranged version of the same formula to avoid simple arithmetic mistakes. Why Other Options Are Wrong: 22, 24 and 26 do not satisfy Euler's relation when V = 12 and E = 30. For example, if F were 22 then V - E + F would be 12 - 30 + 22 = 4, which does not equal 2. Similar checks show that 24 and 26 also violate Euler's formula, so they cannot be correct. 18 would give 12 - 30 + 18 = 0, again not equal to 2, so it is also incorrect. Common Pitfalls: A frequent error is to mix up the signs when substituting into V - E + F = 2 or to use a wrong version of the formula. Some students also confuse polyhedra with planar graphs and try to apply graph theory ideas incorrectly. Always remember that for standard convex 3D solids, Euler's formula V - E + F = 2 is directly applicable and extremely reliable. Final Answer: The solid has 20 faces in total.

Question 12

The curved surface area of a right circular cylinder is 1232 square centimetres. If the circumference of its circular base is 154 centimetres, what is the height of the cylinder? (Take π = 22/7.)

Accepted Answer

Introduction / Context: This problem focuses on the relationship between the curved surface area of a right circular cylinder, the circumference of its base and its height. Cylinders occur frequently in practical applications, such as tanks, pipes and cans, so curved surface area and circumference formulas are widely used in aptitude and engineering questions. Given Data / Assumptions: - Curved surface area (CSA) of the cylinder = 1232 square centimetres. - Circumference of the base = 154 centimetres. - The cylinder is right circular, so the axis is perpendicular to the base. - Take π = 22 / 7 as directed in the question. - We must find the height h of the cylinder in centimetres. Concept / Approach: For a right circular cylinder, the curved surface area is given by: CSA = 2 * π * r * h The circumference of the base is given by: Circumference = 2 * π * r A very useful shortcut is that the curved surface area can also be written as: CSA = (circumference of base) * height In this question, we already know the curved surface area and the circumference. So we can simply divide CSA by the circumference to get the height directly. Step-by-Step Solution: Step 1: Use the relation CSA = circumference * height. Step 2: Substitute the given values: 1232 = 154 * h. Step 3: Solve for h by dividing both sides by 154. Step 4: h = 1232 / 154. Step 5: Compute 1232 / 154 = 8, so the height of the cylinder is 8 cm. Verification / Alternative check: We can double check by starting with the standard formulas. From the circumference 2 * π * r = 154, using π = 22 / 7, we get r = 154 / (2 * 22 / 7) = 154 * 7 / 44 = 24.5 cm. Then the curved surface area formula 2 * π * r * h should equal 1232. Substituting r = 24.5 and h = 8, we get 2 * 22 / 7 * 24.5 * 8 = 1232, confirming that the height is correct. Why Other Options Are Wrong: Heights 4 cm, 6 cm, 12 cm and 16 cm would all give either too small or too large a curved surface area when multiplied by the given circumference 154 cm. None of them satisfy the exact relation 154 * h = 1232 except h = 8 cm. Common Pitfalls: Many students first try to compute the radius from the circumference formula and then go back to the curved surface area formula. Although this works, it takes more time and increases the chance of arithmetic mistakes. Another common error is confusing total surface area with curved surface area. In this question, only the curved surface area is relevant, so the simple product circumference times height is enough. Final Answer: The height of the cylinder is 8 cm.

Question 13

The area of a rectangle is 120 square centimetres and its breadth is 8 centimetres. What must be the length of the diagonal of this rectangle, in centimetres?

Accepted Answer

Introduction / Context: This problem combines two basic properties of rectangles. First, the area of a rectangle is the product of its length and breadth. Second, the diagonal of a rectangle can be obtained by applying the Pythagoras theorem to the length and breadth. Such questions are standard in aptitude tests and help you practise linking area with side lengths and diagonals. Given Data / Assumptions: - Area of the rectangle = 120 square centimetres. - Breadth (one side) = 8 centimetres. - The rectangle has right angles at each corner by definition. - We need to find the length of the diagonal in centimetres. Concept / Approach: The area A of a rectangle is given by: A = length * breadth So we can find the length by dividing the area by the breadth. Once we know length and breadth, we use the Pythagoras theorem because the diagonal forms the hypotenuse of a right angled triangle with the length and breadth as legs. Thus: diagonal^2 = length^2 + breadth^2 and diagonal is the positive square root of this sum. Step-by-Step Solution: Step 1: Use the area formula A = length * breadth. Step 2: Substitute A = 120 and breadth = 8. So 120 = length * 8. Step 3: Solve for length: length = 120 / 8 = 15 cm. Step 4: Apply Pythagoras theorem to find the diagonal d: d^2 = length^2 + breadth^2. Step 5: Compute d^2 = 15^2 + 8^2 = 225 + 64 = 289, so d = √289 = 17 cm. Verification / Alternative check: As a quick check, note that the triple (8, 15, 17) is a well known Pythagorean triple where 8^2 + 15^2 = 17^2. Therefore, a rectangle with sides 8 cm and 15 cm will always have a diagonal of 17 cm. This gives additional confidence that the answer is correct. Why Other Options Are Wrong: 30 cm and 34 cm are much larger than the sum of the sides and would make the Pythagoras relation invalid because 15^2 + 8^2 does not equal 30^2 or 34^2. 15 cm is simply the length of the rectangle, not the diagonal, so it ignores the breadth. 25 cm also fails the Pythagoras check because 15^2 + 8^2 is not equal to 25^2. Common Pitfalls: Some learners forget to find the length first and try to guess or use incorrect formulas for the diagonal. Others mistakenly add the sides rather than adding the squares of the sides. Always remember that the diagonal of a rectangle is obtained using the Pythagoras theorem, not by simple addition of length and breadth. Final Answer: The length of the diagonal of the rectangle is 17 cm.

Question 14

Point P with coordinates (8, 5) is the midpoint of line segment AB. The coordinates of A are (5, y) and the coordinates of B are (x, -3). What is the value of x, the x coordinate of point B?

Accepted Answer

Introduction / Context: This question checks your understanding of the midpoint formula in coordinate geometry. Midpoints are very common in analytic geometry problems because they relate the endpoints of a segment to the point exactly in the middle. Knowing how to handle x and y coordinates separately is essential for solving such questions quickly and accurately. Given Data / Assumptions: - Midpoint P has coordinates (8, 5). - Endpoint A has coordinates (5, y). - Endpoint B has coordinates (x, -3). - The point P lies exactly halfway between A and B on the segment AB. - We need to find x, the x coordinate of B. The y coordinate y of A is not asked in this question. Concept / Approach: If a point P(xm, ym) is the midpoint of a segment joining A(x1, y1) and B(x2, y2), the midpoint formula is: xm = (x1 + x2) / 2 ym = (y1 + y2) / 2 In other words, the midpoint coordinates are simply the averages of the x coordinates and the y coordinates of the endpoints. Since we know P, A and the y coordinate of B, we can set up equations and solve for x and y independently. Step-by-Step Solution: Step 1: Apply the midpoint formula for the x coordinates: 8 = (5 + x) / 2. Step 2: Multiply both sides by 2 to remove the denominator: 16 = 5 + x. Step 3: Solve for x by subtracting 5 from both sides: x = 16 - 5 = 11. Step 4: For completeness, you can apply the midpoint formula for the y coordinates: 5 = (y + (-3)) / 2. Step 5: From this, y + (-3) = 10, which gives y = 13, but this value is not required for the final answer. Verification / Alternative check: Check the x coordinate: with A(5, 13) and B(11, -3), the average of the x coordinates is (5 + 11) / 2 = 16 / 2 = 8, which matches the x coordinate of P. For the y coordinates, (13 + (-3)) / 2 = 10 / 2 = 5, which matches the y coordinate of P. This confirms that P really is the midpoint and that x = 11 is consistent. Why Other Options Are Wrong: -11 and -7 would produce a midpoint x coordinate that is negative or too small and would not match xm = 8. 7 also gives (5 + 7) / 2 = 6, not 8, so it is incorrect. 5 would give (5 + 5) / 2 = 5, again not equal to 8. Common Pitfalls: A common mistake is to confuse midpoint with distance and try to use distance formulas instead of simple averages. Others forget to multiply by 2 or mishandle the algebraic signs when solving for x. Always treat x and y coordinates independently and carefully apply the midpoint formula component wise. Final Answer: The value of x, the x coordinate of point B, is 11.

Question 15

In triangle ABC, which is equilateral, points D and E are the midpoints of sides AB and BC respectively. What is the ratio of the area of triangle ABC to the area of the quadrilateral ADEC (called a trapezium in the statement)?

Accepted Answer

Introduction / Context: This is a geometry question about area ratios inside an equilateral triangle. When midpoints of sides are joined or used to form sub figures, the resulting smaller triangles and quadrilaterals often have areas in simple ratios relative to the original triangle. Recognising such patterns saves time in exams, especially in questions on equilateral triangles and mid segments. Given Data / Assumptions: - Triangle ABC is equilateral, so all sides and all angles are equal. - D is the midpoint of side AB. - E is the midpoint of side BC. - Quadrilateral ADEC is formed by points A, D, E and C inside the triangle. - We need the ratio area of triangle ABC : area of quadrilateral ADEC. Concept / Approach: An equilateral triangle has many symmetry properties. When midpoints of sides are used, line segments joining them create smaller congruent triangles. A helpful approach is to assign a side length to the equilateral triangle, use coordinate geometry or known area formulas to compute exact areas and then take the ratio. Another approach is to use similar triangles and proportionality, because segments joining midpoints often create triangles that are similar to the original triangle with predictable area ratios. Step-by-Step Solution: Step 1: Assume the side length of equilateral triangle ABC is s units for convenience. Step 2: The area of an equilateral triangle with side s is (√3 / 4) * s^2, so area of ABC is (√3 / 4) * s^2. Step 3: Place the triangle in a coordinate system. Let A be at (0, 0), B at (s, 0) and C at (s / 2, (√3 / 2) * s). Step 4: D, the midpoint of AB, has coordinates ((0 + s) / 2, 0) = (s / 2, 0). E, the midpoint of BC, has coordinates ((s + s / 2) / 2, (0 + (√3 / 2) * s) / 2) = (3s / 4, (√3 / 4) * s). Step 5: Use a standard polygon area method or known results to find that the area of quadrilateral ADEC is (3 / 4) of the area of triangle ABC. Hence area ABC : area ADEC = 4 : 3. Verification / Alternative check: An alternative more conceptual check is to observe that joining the midpoints in an equilateral triangle partitions it into four small congruent equilateral triangles, each of area (1 / 4) of the original. The quadrilateral ADEC covers three of these smaller parts when you visualise the figure, leaving one small triangle outside near vertex B. Thus the quadrilateral ADEC has an area equal to 3 / 4 of the original triangle, again giving the ratio 4 : 3. Why Other Options Are Wrong: 5 : 3, 4 : 1, 8 : 5 and 3 : 2 do not match the geometric partitioning implied by midpoints in an equilateral triangle. None of these ratios correspond to a pattern where one small triangular region is exactly one quarter of the whole, with the remaining region being three quarters. Common Pitfalls: A common mistake is to assume that each new segment always divides the triangle into halves or equal parts, which is not true in general. Another pitfall is misidentifying which small triangles are included inside the quadrilateral and which one remains outside. Drawing a clear diagram and, if needed, marking equal side segments can prevent such confusion. Final Answer: The required ratio of areas is 4 : 3.

Question 16

In an isosceles triangle ABC with AB = AC, a line segment XY is drawn parallel to the base BC such that X lies on AB and Y lies on AC. If angle A measures 30 degrees, then what is the measure of angle BXY?

Accepted Answer

Introduction / Context: This question tests your understanding of basic angle properties in isosceles triangles and the effect of a line drawn parallel to the base. Such constructions appear frequently in geometry because parallel lines preserve corresponding and alternate interior angles, making it easier to relate interior angles of smaller similar triangles to those of the original triangle. Given Data / Assumptions: - Triangle ABC is isosceles with AB = AC. - Therefore the base is BC and angles at B and C are equal. - Angle A measures 30 degrees. - XY is drawn parallel to BC with X on AB and Y on AC. - We need to find angle BXY at point X. Concept / Approach: In an isosceles triangle with AB = AC, the base angles at B and C are equal. The sum of angles in a triangle is 180 degrees, so once we know angle A, we can find the base angles. Drawing XY parallel to BC creates a smaller triangle AXY that is similar to triangle ABC by the angle angle similarity rule because corresponding angles are equal. Therefore, the angle at X corresponding to angle B will be equal to angle B itself. Step-by-Step Solution: Step 1: In triangle ABC, sum of angles is 180 degrees, so A + B + C = 180 degrees. Step 2: Given AB = AC, triangle ABC is isosceles with equal base angles, so B = C. Step 3: Angle A is 30 degrees, so B + C = 180 - 30 = 150 degrees. Step 4: Since B = C, each base angle is B = C = 150 / 2 = 75 degrees. Step 5: XY is parallel to BC, so angle BXY is corresponding to angle ABC. Hence angle BXY equals 75 degrees. Verification / Alternative check: If you draw a clear diagram, mark angle A as 30 degrees and the base angles as 75 degrees each, and then draw XY parallel to BC, you will see that triangle AXY is a smaller version of ABC. The angle at X between AX and XY is clearly parallel to the angle at B between AB and BC, confirming that angle BXY = angle ABC = 75 degrees. Why Other Options Are Wrong: 30 degrees is the angle at A, not at X or B, so it cannot equal angle BXY. 60 degrees and 105 degrees do not match any angle directly related to the given isosceles configuration in this setup. 150 degrees is the sum of the two base angles, not a single interior angle formed by the parallel line construction. Common Pitfalls: A typical mistake is to confuse the roles of the base angles and the apex angle or to misapply parallel line angle relationships. Some students also try to subtract angles in an incorrect order. Always start by finding all angles in the original triangle and then transfer the appropriate angle measures to the smaller triangle formed by the parallel line. Final Answer: The measure of angle BXY is 75 degrees.

Question 17

What are the coordinates of the reflection of the point (5, -1) in the horizontal line y = 2 in the coordinate plane?

Accepted Answer

Introduction / Context: This problem belongs to coordinate geometry and deals with reflection of a point in a horizontal line. Reflections across lines such as y = c or x = c occur frequently in analytical geometry and transformation questions. Understanding how the coordinates change under reflection lets you answer such questions quickly without drawing a detailed graph each time. Given Data / Assumptions: - Original point is P(5, -1). - The line of reflection is the horizontal line y = 2. - Reflection means that the line y = 2 will act like a mirror. - We need to find the coordinates of the reflected point P prime. Concept / Approach: Reflection of a point across a horizontal line y = c keeps the x coordinate the same, because the point moves straight up or down. The y coordinate changes such that the reflected point is the same perpendicular distance on the other side of the line. If the original y coordinate is y1 and the line is y = c, then the distance from the point to the line is d = c - y1 when the point is below the line. The reflected y coordinate will be c + d. In summary, the y values are symmetric with respect to the line y = c. Step-by-Step Solution: Step 1: Identify the x coordinate of the original point P. It is x = 5. Step 2: Identify the y coordinate of P. It is y = -1. Step 3: The line of reflection is y = 2. The vertical distance from P to the line is 2 - (-1) = 3 units. Step 4: The reflected point must lie the same distance above the line, so its y coordinate is 2 + 3 = 5. Step 5: The x coordinate remains unchanged, so the reflected point P prime is (5, 5). Verification / Alternative check: Plotting a quick sketch on graph paper or mentally visualising the axes, you see that point (5, -1) lies 3 units below the line y = 2. To reflect it, you move directly upward 3 units above the line, landing at y = 5. The x coordinate does not change, confirming that the reflection is at (5, 5). Why Other Options Are Wrong: (5, -5) and (-5, -5) both have y coordinates further below the line y = 2 instead of being above it. (-5, 5) has the correct y coordinate but the x coordinate has changed sign, which does not happen in reflection across a horizontal line. (5, 2) lies on the line of reflection itself and would correspond to a point already on the mirror, not the reflection of (5, -1). Common Pitfalls: One frequent mistake is to accidentally change both coordinates or only the y coordinate without preserving the distance from the mirror line. Another error is to confuse reflections in horizontal lines with reflections in vertical lines of the form x = c. Always remember that reflections in y = c keep x unchanged and adjust y symmetrically around c. Final Answer: The reflection of the point (5, -1) in the line y = 2 is (5, 5).

Question 18

In right angled triangle ABC, angle B is 90 degrees and angle A is 30 degrees. If the hypotenuse AC has length 8 centimetres, what is the length of side AB in centimetres?

Accepted Answer

Introduction / Context: This problem again uses the special properties of a 30 60 90 right angled triangle. When one of the acute angles in a right triangle is 30 degrees, the ratios of the side lengths become fixed. Remembering these ratios makes it much faster to solve questions without repeatedly using trigonometric functions. Given Data / Assumptions: - Triangle ABC is right angled at B, so angle B is 90 degrees. - Angle A is 30 degrees, so angle C must be 60 degrees. - The hypotenuse is AC with length 8 cm. - We need to find side AB, which is one of the legs of the triangle. Concept / Approach: In a 30 60 90 triangle, the side opposite 30 degrees is the smallest, the side opposite 60 degrees is medium and the side opposite 90 degrees is the hypotenuse. If the hypotenuse is 2k, then the side opposite 30 degrees is k and the side opposite 60 degrees is k√3. In this triangle, side BC is opposite angle A, which is 30 degrees, and side AB is opposite angle C, which is 60 degrees. Knowing the hypotenuse AC, we can first find k and then compute the required side AB. Step-by-Step Solution: Step 1: Let the hypotenuse AC be 2k in the standard 30 60 90 triangle. Step 2: Given AC = 8 cm, so 2k = 8, which gives k = 4. Step 3: The side opposite 60 degrees is k√3. Here angle C is 60 degrees, and its opposite side is AB. Step 4: Substitute k = 4 to get AB = 4√3 cm. Step 5: Hence the required side length AB is 4√3 cm. Verification / Alternative check: We can use a trigonometric ratio such as sine. Since angle A is 30 degrees, sin 30 degrees = opposite / hypotenuse = BC / AC. This gives BC = 8 * 1 / 2 = 4 cm. Then side AB, which is opposite the 60 degree angle at C, must satisfy Pythagoras theorem: AB^2 = AC^2 - BC^2 = 8^2 - 4^2 = 64 - 16 = 48, so AB = √48 = 4√3 cm. This is consistent with the special triangle ratio method. Why Other Options Are Wrong: 2√3 cm and 2 / √3 cm are too small and do not satisfy Pythagoras theorem with hypotenuse 8 cm. 4 / √3 cm is also too small and would give a hypotenuse less than 8 cm when combined with the other leg. 8√3 cm is far too large to be a leg when the hypotenuse is only 8 cm. Common Pitfalls: A common mistake is to take the side opposite 30 degrees as k√3 instead of k, thereby reversing the ratios. Another error is to confuse which side is opposite which angle when the triangle is named. Drawing a quick sketch and marking each angle and its opposite side can help avoid confusion. Final Answer: The length of side AB is 4√3 cm.

Question 19

The surface area of a sphere is 5544 square centimetres. Using π = 22/7, what is the diameter of the sphere in centimetres?

Accepted Answer

Introduction / Context: This question involves the formula for the surface area of a sphere and asks you to find the diameter. Spheres are common in geometry and physics, and knowing how to move between radius, diameter and surface area using the standard formulas is important for many quantitative exams. Given Data / Assumptions: - Surface area of the sphere S = 5544 square centimetres. - Use π = 22 / 7 as indicated by the problem style, which is common in aptitude questions. - We need to find the diameter of the sphere in centimetres. Concept / Approach: The surface area S of a sphere with radius r is given by: S = 4 * π * r^2 Once we know S and π, we can rearrange this formula to solve for r^2 and then r. The diameter D is simply 2r. So the main steps are to find r from the given area and then double r to get the diameter. Step-by-Step Solution: Step 1: Write the surface area formula S = 4 * π * r^2. Step 2: Substitute S = 5544 and π = 22 / 7 into the equation: 5544 = 4 * (22 / 7) * r^2. Step 3: Compute the constant part: 4 * (22 / 7) = 88 / 7. Step 4: So 5544 = (88 / 7) * r^2, which implies r^2 = 5544 * 7 / 88. Step 5: Simplify 5544 / 88 = 63, so r^2 = 63 * 7 = 441. Therefore r = √441 = 21 cm, and the diameter D = 2 * 21 = 42 cm. Verification / Alternative check: Substitute r = 21 cm back into the formula to verify. S = 4 * π * r^2 = 4 * 22 / 7 * 21^2. First compute 21^2 = 441. Then S = 4 * 22 / 7 * 441 = 4 * 22 * 63 = 88 * 63 = 5544, which exactly matches the given surface area. This confirms that r = 21 cm and D = 42 cm are correct. Why Other Options Are Wrong: A diameter of 21 cm would correspond to a radius of 10.5 cm and a much smaller surface area. 84 cm and 63 cm would correspond to radii that produce surface areas very different from 5544 square centimetres. 28 cm would give radius 14 cm and again an incorrect surface area when substituted into the formula. Common Pitfalls: Students sometimes confuse diameter and radius, using r instead of 2r when reading the answer choices. Another common error is not simplifying the fraction correctly when solving for r^2. Careful arithmetic and clear separation of each step help avoid these issues. Final Answer: The diameter of the sphere is 42 cm.

Question 20

Line segment AB intersects the straight line CD at point B. For which value of angle ABD (in degrees) will the length of AB be the smallest, that is, the distance from A to line CD is minimised?

Accepted Answer

Introduction / Context: This question is about the shortest distance from a point to a straight line in geometry. When a line segment from a point A meets a line CD at B, the distance AB is smallest when AB is perpendicular to CD. This is a classic property often used in geometric proofs and also appears in optimisation problems. Given Data / Assumptions: - CD is a straight line in the plane. - Point A lies off the line CD somewhere in the plane. - Line segment AB intersects line CD at point B. - Angle ABD is the angle at B between segment BA and the part of line CD that extends toward D. - We want the length AB to be as small as possible, that is, the shortest distance from A to line CD. Concept / Approach: In Euclidean geometry, the shortest distance from a point to a straight line is along the perpendicular from the point to the line. This means that the segment from the point to the line must form a 90 degree angle with the line. If the angle is anything other than 90 degrees, the segment from the point will be longer because it will have both a perpendicular component and a component along the line. Step-by-Step Solution: Step 1: Visualise or sketch a line CD and a point A not on that line. Step 2: Draw several segments from A to the line CD, meeting the line at different points such as B. Step 3: Observe that the length of AB is smallest when the segment AB is perpendicular to CD. Step 4: A perpendicular intersection corresponds to an angle of 90 degrees between AB and CD at B. Step 5: Therefore, AB is minimised when angle ABD is exactly 90 degrees. Verification / Alternative check: If you imagine rotating the point of intersection B along the line CD, any tilt away from the perpendicular makes AB the hypotenuse of a right triangle whose one leg is the shortest distance to the line. Since the hypotenuse of a right triangle is always longer than either leg, any non perpendicular segment will be longer than the perpendicular one. This reasoning confirms that the smallest possible length occurs at a 90 degree angle. Why Other Options Are Wrong: Angles 30, 45, 60 and 75 degrees correspond to oblique intersections between AB and CD. In each case, AB is the hypotenuse of a right triangle formed by the perpendicular distance and a segment along the line CD. The hypotenuse is always longer than the perpendicular distance, so these angles cannot give the minimal length. Common Pitfalls: A common misunderstanding is to think that any acute angle may minimise the distance, but only the right angle gives the true shortest distance. Some learners also confuse this with projection along a line. Always remember that the perpendicular from a point to a line represents the minimum distance between them. Final Answer: AB is smallest when angle ABD is 90 degrees.

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