Question 1

The perimeter of a rectangle is 44 centimetres and its breadth is 10 centimetres. Find the area of the rectangle in square centimetres.

Accepted Answer

Introduction / Context: This problem checks basic understanding of rectangle properties and the use of perimeter and area formulas. Such questions are fundamental in quantitative aptitude and help you practise extracting unknown dimensions from given information and then using them to compute area. Given Data / Assumptions: The figure is a rectangle. Perimeter of the rectangle = 44 cm. Breadth (one side) = 10 cm. We are asked to find the area in square centimetres. All angles in a rectangle are right angles and opposite sides are equal. Concept / Approach: For a rectangle: Perimeter = 2 * (length + breadth). Area = length * breadth. We first use the perimeter to find the missing side, which is the length, and then substitute into the area formula. This is a straightforward two step application of standard formulas. Step-by-Step Solution: Step 1: Let length = L cm and breadth = B = 10 cm. Step 2: Perimeter = 2 * (L + B) = 44. Step 3: So, 2 * (L + 10) = 44. Step 4: Divide both sides by 2: L + 10 = 22. Step 5: Hence, L = 22 - 10 = 12 cm. Step 6: Area of rectangle = length * breadth = 12 * 10. Step 7: Area = 120 square centimetres. Verification / Alternative check: Check the perimeter using the found dimensions: 2 * (12 + 10) = 2 * 22 = 44 cm, which matches the given perimeter. This confirms that length = 12 cm and the area value 120 square centimetres is consistent. Why Other Options Are Wrong: 60 square centimetres: This could occur if a side was miscalculated or halved incorrectly. 180 square centimetres: This would imply dimensions like 18 by 10, giving a larger perimeter than 44 cm. 200 square centimetres: Requires a different length, inconsistent with the given perimeter. 240 square centimetres: Would correspond to 24 by 10 and a perimeter of 68 cm, which is not correct. Common Pitfalls: Learners often confuse perimeter and area formulas or forget to divide by 2 when using the perimeter equation. Another common mistake is using 44 directly as one of the sides. Always remember that perimeter for a rectangle involves the sum of all sides, not the product, and that area is obtained only after correctly finding both length and breadth. Final Answer: The area of the rectangle is 120 square centimetres.

Question 2

Triangles ABC and PQR are similar. The length of side AB is 18 centimetres and the length of the corresponding side PQ is 12 centimetres. If the area of triangle ABC is 324 square centimetres, find the area of triangle PQR.

Accepted Answer

Introduction / Context: This question uses the concept of similar triangles and the relationship between the ratio of their corresponding sides and the ratio of their areas. It is a classic geometry topic that appears frequently in aptitude tests and school level examinations. Given Data / Assumptions: Triangle ABC is similar to triangle PQR. Side AB in triangle ABC corresponds to side PQ in triangle PQR. AB = 18 cm. PQ = 12 cm. Area of triangle ABC = 324 square centimetres. We must find the area of triangle PQR. Concept / Approach: For two similar triangles: Ratio of corresponding sides = k. Ratio of their areas = k^2. Here, k is the scale factor from one triangle to the other. We first find the linear ratio using AB and PQ, then square it to get the area ratio, and finally apply it to find the unknown area. Step-by-Step Solution: Step 1: Identify the corresponding sides: AB corresponds to PQ. Step 2: Compute side ratio for triangle PQR relative to ABC: k = PQ / AB = 12 / 18. Step 3: Simplify: 12 / 18 = 2 / 3. Step 4: Area ratio of PQR to ABC = k^2 = (2 / 3)^2 = 4 / 9. Step 5: Area of triangle ABC = 324 square centimetres. Step 6: Area of triangle PQR = (4 / 9) * 324. Step 7: Compute: 324 / 9 = 36, then 36 * 4 = 144 square centimetres. Verification / Alternative check: We can reverse the logic: If area of PQR is 144 square centimetres and area of ABC is 324 square centimetres, ratio is 144 / 324 = 4 / 9. Taking square root gives side ratio 2 / 3, which matches PQ / AB. This confirms our calculations and preserves the similarity relation correctly. Why Other Options Are Wrong: 72 square centimetres: This corresponds to an area ratio of 2 / 9 rather than 4 / 9, which is too small. 108 square centimetres: Gives area ratio 108 / 324 = 1 / 3, not equal to k^2 where k is 2 / 3. 36 square centimetres: Far too small; would imply a much smaller side ratio inconsistent with 12 and 18. 487.5 square centimetres: Larger than the area of ABC, which contradicts the fact that PQ is smaller than AB. Common Pitfalls: Many students incorrectly assume that the area changes in the same ratio as the sides, forgetting to square the ratio. Another error is inverting the ratio and applying AB / PQ instead of PQ / AB for the target triangle. Always remember that if side ratio is a / b, then area ratio is (a / b)^2, and choose the direction of the ratio based on which triangle area you are trying to find. Final Answer: The area of triangle PQR is 144 square centimetres.

Question 3

The radius of a solid hemisphere is 7 centimetres. Find the volume of the hemisphere in cubic centimetres.

Accepted Answer

Introduction / Context: This numerical problem involves the volume of a solid hemisphere. Hemispheres are half spheres and appear in questions related to containers, domes, and various real life shapes. Knowing the formula for the volume of a hemisphere is important for geometry and mensuration. Given Data / Assumptions: The solid is a hemisphere. Radius r = 7 cm. We are asked to find the volume in cubic centimetres. Use the standard value of pi as 22 / 7 for calculation. Concept / Approach: Volume of a sphere of radius r is: V_sphere = (4 / 3) * pi * r^3. A hemisphere is half of a sphere, so: V_hemisphere = (1 / 2) * V_sphere = (2 / 3) * pi * r^3. We substitute r = 7 cm and pi = 22 / 7 into this formula to find the answer. Step-by-Step Solution: Step 1: Write the volume formula for a hemisphere: V = (2 / 3) * pi * r^3. Step 2: Substitute r = 7 cm and pi = 22 / 7. Step 3: Compute r^3: 7^3 = 7 * 7 * 7 = 343. Step 4: Substitute values: V = (2 / 3) * (22 / 7) * 343. Step 5: Simplify (22 / 7) * 343 = 22 * 49 = 1078. Step 6: Now V = (2 / 3) * 1078. Step 7: Compute 2 * 1078 = 2156, then divide by 3 to get approximately 718.67 cubic centimetres. Verification / Alternative check: Another way is to first compute the volume of the full sphere with radius 7 cm, then divide by 2. Sphere volume = (4 / 3) * pi * 7^3. Using pi = 22 / 7 gives approximately 1437.34 cubic centimetres. Half of this is 1437.34 / 2 = 718.67 cubic centimetres, which matches the previous result. Why Other Options Are Wrong: 1437.34 cubic centimetres: This is the volume of the full sphere, not the hemisphere. 239.56 cubic centimetres and 359.34 cubic centimetres: These are much smaller than half the sphere volume and result from incorrect application of the formula. 500 cubic centimetres: A rounded guess, not supported by proper computation and significantly off from the correct value. Common Pitfalls: A common mistake is to forget to take half of the sphere volume for a hemisphere, or to use the formula for a sphere directly. Some also incorrectly cube the radius or simplify fractions wrongly, especially when pi is taken as 22 / 7. Carefully substituting and simplifying step by step avoids these errors. Final Answer: The volume of the hemisphere is approximately 718.67 cubic centimetres.

Question 4

In triangle DEF, points G and H lie on sides DE and DF respectively, and segment GH is parallel to side EF. Point G divides side DE in the ratio 1 : 3 and segment HF is 7.2 centimetres long. Find the length of side DF in centimetres.

Accepted Answer

Introduction / Context: This question uses the concept of similar triangles formed when a line segment inside a triangle is drawn parallel to one of its sides. It combines ratio, similarity, and length calculations, which are common themes in geometry based aptitude problems. Given Data / Assumptions: Triangle DEF is given. Point G lies on side DE and point H lies on side DF. Segment GH is parallel to side EF. G divides DE in the ratio 1 : 3, meaning DG : GE = 1 : 3. Length of HF = 7.2 cm. We must find the total length of DF. Concept / Approach: Since GH is parallel to EF, triangles DGH and DEF are similar. This implies that corresponding sides of these triangles are in the same ratio. If DG : DE = 1 : 4 (because 1 + 3 = 4), then DH : DF also equals 1 : 4. Using this, we can express DF in terms of DH, and then relate DH and HF because DF = DH + HF. Step-by-Step Solution: Step 1: Ratio DG : GE = 1 : 3, hence DG : DE = 1 : (1 + 3) = 1 : 4. Step 2: Because triangles DGH and DEF are similar, the ratio DH : DF is the same as DG : DE. Step 3: Therefore, DH : DF = 1 : 4, which implies DH = DF / 4. Step 4: The full side DF is made of DH and HF: DF = DH + HF. Step 5: Substitute DH = DF / 4 into this: DF = DF / 4 + HF. Step 6: Rearrange: DF - DF / 4 = HF, so (3 / 4) * DF = HF. Step 7: HF is given as 7.2 cm, hence (3 / 4) * DF = 7.2. Step 8: Solve for DF: DF = 7.2 * (4 / 3) = 9.6 cm. Verification / Alternative check: Check DH: DH = DF / 4 = 9.6 / 4 = 2.4 cm. Then DF = DH + HF = 2.4 + 7.2 = 9.6 cm, which is consistent. Also, the ratio DH : DF = 2.4 : 9.6 = 1 : 4, matching the ratio obtained from the division of DE. This confirms the correctness of the answer. Why Other Options Are Wrong: 2.4 cm: This is only DH, not the full side DF. 3.6 cm and 4.8 cm: These values are too small to accommodate HF = 7.2 cm when DF must equal DH + HF. 12 cm: This makes DH = 3 cm and HF = 7.2 cm, which does not satisfy the 1 : 4 ratio between DH and DF. Common Pitfalls: A frequent mistake is to assume the ratio 1 : 3 directly compares DE and DG rather than DG and GE. Another error is forgetting that the full side DF includes both DH and HF. Some learners also misapply the concept of similar triangles and invert the ratios. Always carefully define which segments correspond and verify proportional relationships. Final Answer: The length of side DF is 9.6 cm.

Question 5

What is the circumference, in centimetres, of a circle whose area is 616 square centimetres?

Accepted Answer

Introduction / Context: This problem connects two key formulas for a circle: the formula for area and the formula for circumference. You are given the area and asked to find the circumference, which requires determining the radius first. Such multi step geometry questions are very common in competitive exams. Given Data / Assumptions: Area of the circle = 616 square centimetres. We need to find the circumference in centimetres. Use pi = 22 / 7 for calculations unless specified otherwise. Concept / Approach: The formulas involved are: Area A = pi * r^2. Circumference C = 2 * pi * r. We first use the area to compute r^2 and then r. After finding the radius, we plug it into the circumference formula to obtain the final answer. Step-by-Step Solution: Step 1: Write the area formula: A = pi * r^2. Step 2: Substitute A = 616 and pi = 22 / 7. Step 3: So, 616 = (22 / 7) * r^2. Step 4: Multiply both sides by 7 to remove the denominator: 616 * 7 = 22 * r^2. Step 5: Compute 616 * 7 = 4312, so 4312 = 22 * r^2. Step 6: Divide both sides by 22: r^2 = 4312 / 22 = 196. Step 7: Take square root: r = sqrt(196) = 14 cm. Step 8: Now use circumference formula: C = 2 * pi * r = 2 * (22 / 7) * 14. Step 9: Simplify: 14 / 7 = 2, so C = 2 * 22 * 2 = 88 cm. Verification / Alternative check: We can check by reversing the process. If r = 14 cm, area A = pi * r^2 = (22 / 7) * 196 = 22 * 28 = 616 square centimetres, confirming the given area. The circumference with r = 14 cm is again 88 cm, matching our computed result. Why Other Options Are Wrong: 44 cm: This is half of the correct circumference and would correspond to a radius of 7 cm, not 14 cm. 66 cm: This would correspond to a radius of approximately 10.5 cm and not give an area of 616 square centimetres. 132 cm: This is larger than required and corresponds to a radius greater than 14 cm. 22 cm: Much too small and clearly inconsistent with such a large area. Common Pitfalls: Students often mix up formulas for area and circumference or forget to take the square root when solving for the radius. Another issue is careless simplification of fractions when using pi = 22 / 7. Always proceed systematically: find r^2, find r, then compute the circumference carefully. Final Answer: The circumference of the circle is 88 cm.

Question 6

If the diameter of a circle is 35 centimetres, what is its circumference in centimetres?

Accepted Answer

Introduction / Context: This is a direct application of the circumference formula for a circle. The question gives the diameter and asks for the circumference, which is an essential and very frequently tested relationship in basic geometry and aptitude. Given Data / Assumptions: Diameter of the circle, d = 35 cm. We need to find the circumference in centimetres. Use pi = 22 / 7 for calculation. Concept / Approach: Circumference C of a circle can be written in two equivalent forms: C = pi * d, or C = 2 * pi * r. Here we are directly given the diameter, so we use C = pi * d. We substitute d = 35 cm and pi = 22 / 7 to compute C. Step-by-Step Solution: Step 1: Use the formula C = pi * d. Step 2: Substitute pi = 22 / 7 and d = 35 cm. Step 3: C = (22 / 7) * 35. Step 4: Simplify 35 / 7 = 5. Step 5: So C = 22 * 5 = 110 cm. Verification / Alternative check: We can also find the radius and then use the formula 2 * pi * r. Radius r = d / 2 = 35 / 2 = 17.5 cm. Then: C = 2 * pi * r = 2 * (22 / 7) * 17.5. 17.5 / 7 = 2.5, so C = 2 * 22 * 2.5 = 44 * 2.5 = 110 cm. This confirms the same result using an alternate method. Why Other Options Are Wrong: 70 cm: This would apply if pi was approximated as 2, which is incorrect. 77 cm: Equals pi * radius (22 / 7 * 17.5) only if one misapplies the formula; it does not match the correct circumference here. 140 cm: This value equals 4 * pi * r with r = 17.5 and is not the correct formula for circumference. 220 cm: Exactly double the correct answer and corresponds to using 2 * pi * d, which is wrong. Common Pitfalls: Learners sometimes confuse diameter with radius, mistakenly substituting 35 as the radius and using 2 * pi * r, leading to double the correct answer. Another common issue is forgetting that C = pi * d is already fully sufficient when diameter is given. Always identify whether you are given diameter or radius before picking the formula. Final Answer: The circumference of the circle is 110 cm.

Question 7

Find the volume, in cubic centimetres, of a solid sphere whose diameter is 7 centimetres.

Accepted Answer

Introduction / Context: This question tests your knowledge of the volume formula for a sphere and the ability to use the given diameter to first find the radius. Spherical shapes are common in mensuration problems related to geometry, physics, and everyday objects. Given Data / Assumptions: The solid is a sphere. Diameter d = 7 cm. Radius r = d / 2. We need the volume in cubic centimetres. Take pi = 22 / 7 for calculation. Concept / Approach: The volume of a sphere is given by: V = (4 / 3) * pi * r^3. We must first compute the radius as half the diameter, then cube the radius, and substitute into the formula with pi = 22 / 7. Step-by-Step Solution: Step 1: Diameter d = 7 cm, so radius r = d / 2 = 7 / 2 = 3.5 cm. Step 2: Use the formula V = (4 / 3) * pi * r^3. Step 3: Substitute r = 3.5 and pi = 22 / 7. Step 4: Compute r^3: 3.5 * 3.5 * 3.5 = 42.875. Step 5: So V = (4 / 3) * (22 / 7) * 42.875. Step 6: Combine constants: (4 / 3) * (22 / 7) * 42.875. Step 7: Using an exact fraction approach, 3.5^3 = 42.875, and simplifying gives approximately 179.67 cubic centimetres. Verification / Alternative check: We can approximate using 3.14 for pi to confirm. Then: V ≈ (4 / 3) * 3.14 * 3.5^3 ≈ 1.333 * 3.14 * 42.875. 1.333 * 3.14 ≈ 4.185, and 4.185 * 42.875 ≈ 179.5 cubic centimetres, close to 179.67 cubic centimetres. Both approximations are consistent and validate the answer. Why Other Options Are Wrong: 140.25 cubic centimetres: This is significantly lower than the correct volume and results from incorrect multiplication or using radius directly without cubing. 213.74 cubic centimetres and 337.16 cubic centimetres: These are larger than the correct volume and come from overestimation or misusing the formula. 250 cubic centimetres: A rounded guess that does not match exact calculations. Common Pitfalls: Common errors include using the diameter instead of the radius in the formula, failing to cube the radius, or using an incorrect formula such as 4 * pi * r^2, which is for surface area, not volume. Careful attention to the formula and proper identification of radius versus diameter are essential. Final Answer: The volume of the sphere is approximately 179.67 cubic centimetres.

Question 8

From a point outside a circle, a tangent is drawn. The distance from the point to the centre of the circle is 13 centimetres and the largest chord of the circle (its diameter) is 10 centimetres. Calculate the length of the tangent in centimetres.

Accepted Answer

Introduction / Context: This question explores the geometry of tangents to a circle from an external point. Tangents are important in many geometric constructions, and the right triangle formed by the radius and the tangent allows us to apply the Pythagoras theorem to find unknown lengths. Given Data / Assumptions: A circle has diameter 10 cm, so radius r = 5 cm. Point P is outside the circle. Distance OP (from the centre O to point P) = 13 cm. PT is the tangent from P to the circle. We must find the length PT in centimetres. Concept / Approach: A tangent to a circle is perpendicular to the radius at the point of contact. Therefore triangle OPT, where O is the centre and T is the point of tangency, is a right angled triangle with: OP as the hypotenuse, OT as the radius, PT as the tangent length. We can apply Pythagoras theorem: OP^2 = OT^2 + PT^2. Step-by-Step Solution: Step 1: Compute the radius: diameter = 10 cm, so radius OT = 10 / 2 = 5 cm. Step 2: OP, the distance from the external point to the centre, is given as 13 cm. Step 3: In right triangle OPT, OP^2 = OT^2 + PT^2. Step 4: Substitute values: 13^2 = 5^2 + PT^2. Step 5: Calculate squares: 169 = 25 + PT^2. Step 6: Rearrange: PT^2 = 169 - 25 = 144. Step 7: Take square root: PT = sqrt(144) = 12 cm. Verification / Alternative check: We can quickly check if 5, 12, 13 forms a Pythagorean triple. Indeed, 5^2 + 12^2 = 25 + 144 = 169 = 13^2. This confirms that the tangent length 12 cm fits perfectly in the right triangle OPT. Why Other Options Are Wrong: 8 cm and 10 cm: These do not satisfy the equation 5^2 + PT^2 = 13^2, since 5^2 + 8^2 = 25 + 64 = 89 and 5^2 + 10^2 = 25 + 100 = 125, both not equal to 169. 14 cm and 16 cm: These are too large. For example, 5^2 + 14^2 = 25 + 196 = 221, which is greater than 169, so they cannot be tangent lengths for this configuration. Common Pitfalls: Some students mistakenly treat the diameter as the hypotenuse or forget that the tangent is perpendicular to the radius at the point of contact. Others misapply Pythagoras theorem or confuse which side is the hypotenuse. Carefully identify the right angle at the point of tangency and always set the hypotenuse as the side opposite that right angle. Final Answer: The length of the tangent is 12 cm.

Question 9

The area of a rhombus is 54 square centimetres and the length of one diagonal is 9 centimetres. Find the length of its other diagonal in centimetres.

Accepted Answer

Introduction / Context: This question is about a rhombus, a special type of quadrilateral where all sides are equal. The diagonals of a rhombus have special properties and are often used to find its area or missing dimensions. Here you must use the relationship between diagonals and area. Given Data / Assumptions: The figure is a rhombus. Area of the rhombus = 54 square centimetres. One diagonal, say d1, is 9 cm. We must find the other diagonal, d2, in centimetres. Concept / Approach: For a rhombus with diagonals d1 and d2: Area = (1 / 2) * d1 * d2. This formula arises because the diagonals of a rhombus are perpendicular and bisect each other, effectively splitting it into four right angled triangles. We substitute the known area and one diagonal to solve for the second diagonal. Step-by-Step Solution: Step 1: Use the formula Area = (1 / 2) * d1 * d2. Step 2: Let d1 = 9 cm and d2 be the unknown diagonal. Step 3: Substitute values: 54 = (1 / 2) * 9 * d2. Step 4: Simplify (1 / 2) * 9 = 4.5, so 54 = 4.5 * d2. Step 5: Solve for d2: d2 = 54 / 4.5. Step 6: Compute 54 / 4.5 = 12. Step 7: Therefore, the other diagonal is 12 cm long. Verification / Alternative check: We can substitute back to confirm: Area = (1 / 2) * 9 * 12 = (1 / 2) * 108 = 54 square centimetres, which matches the given area. This confirms the correctness of d2 = 12 cm. Why Other Options Are Wrong: 6 cm: Would give area (1 / 2) * 9 * 6 = 27 square centimetres, which is too small. 9 cm: Would give area (1 / 2) * 9 * 9 = 40.5 square centimetres, still not 54. 18 cm: Produces area (1 / 2) * 9 * 18 = 81 square centimetres, which is larger than given. 24 cm: Produces area (1 / 2) * 9 * 24 = 108 square centimetres, double the required area. Common Pitfalls: Students sometimes confuse the rhombus area formula with that of a rectangle or parallelogram (base * height). They may also forget the factor of 1 / 2 or misidentify the given length as a side rather than a diagonal. Always confirm that you are dealing with diagonals and use the proper formula. Final Answer: The length of the other diagonal is 12 cm.

Question 10

Find the total surface area, in square centimetres, of a solid hemisphere with diameter 42 centimetres.

Accepted Answer

Introduction / Context: This question focuses on the total surface area of a solid hemisphere, which includes both the curved surface area and the area of the circular base. Such calculations are frequently used in mensuration problems about bowls, domes, and other half spherical objects. Given Data / Assumptions: The solid is a hemisphere. Diameter = 42 cm, so radius r = 21 cm. We must find the total surface area in square centimetres. Use pi = 22 / 7. Concept / Approach: For a hemisphere: Curved surface area (CSA) = 2 * pi * r^2. Area of the circular base = pi * r^2. Total surface area (TSA) = CSA + base area = 3 * pi * r^2. We simply substitute r = 21 cm and pi = 22 / 7 into TSA = 3 * pi * r^2. Step-by-Step Solution: Step 1: Compute radius r from the diameter: r = 42 / 2 = 21 cm. Step 2: Use formula TSA = 3 * pi * r^2. Step 3: Substitute pi = 22 / 7 and r = 21 cm. Step 4: Compute r^2: 21^2 = 441. Step 5: So TSA = 3 * (22 / 7) * 441. Step 6: Simplify 441 / 7 = 63. Step 7: TSA = 3 * 22 * 63. Step 8: First compute 22 * 63 = 1386. Step 9: Then TSA = 3 * 1386 = 4158 square centimetres. Verification / Alternative check: Alternatively, find curved surface area and base area separately. CSA = 2 * pi * r^2 = 2 * (22 / 7) * 441 = 2 * 22 * 63 = 2772 square centimetres. Base area = pi * r^2 = (22 / 7) * 441 = 22 * 63 = 1386 square centimetres. Summing: 2772 + 1386 = 4158 square centimetres, matching the previous calculation. Why Other Options Are Wrong: 3500 square centimetres: Underestimates the total area; likely due to dropping part of CSA or base area. 5782 and 6321 square centimetres: These overestimate the area and do not match any correct combination of CSA and base area with radius 21 cm. 7782 square centimetres: Far too large and inconsistent with the radius given. Common Pitfalls: A very common mistake is to use only the curved surface area formula 2 * pi * r^2 and forget to add the area of the base. Another error is to incorrectly halve or double the sphere surface area instead of applying the specific hemisphere formulas. Always remember that for a solid hemisphere you must include the base to get the total surface area. Final Answer: The total surface area of the hemisphere is 4158 square centimetres.

Question 11

In right angled triangle XYZ, angle Y is 90 degrees. If sin X = 4 / 5 and side XY is 6 centimetres, find the length of side YZ in centimetres.

Accepted Answer

Introduction / Context: This trigonometry question involves a right angled triangle and the sine of one of its acute angles. It shows how trigonometric ratios relate the sides of a right triangle and how we can use a known side and a given ratio to find another side. Given Data / Assumptions: Triangle XYZ is right angled at Y, so angle Y = 90 degrees. sin X = 4 / 5. Side XY = 6 cm. We must find the length of side YZ in centimetres. Concept / Approach: In a right triangle, sine of an angle is defined as: sin X = opposite side to angle X / hypotenuse. Here, with angle Y = 90 degrees, the hypotenuse is side XZ. For angle X, side opposite is YZ and side adjacent is XY. Given sin X = 4 / 5, the ratio of YZ to XZ is 4 : 5. We can scale this ratio using the given XY, which must match the adjacent side ratio corresponding to cos X. Step-by-Step Solution: Step 1: For angle X, opposite side = YZ and hypotenuse = XZ. So sin X = YZ / XZ = 4 / 5. Step 2: This tells us the sides are in the ratio YZ : XZ = 4 : 5. Step 3: In a 3, 4, 5 type right triangle, if sin X = 4 / 5, then cos X = 3 / 5 and the sides follow XY : YZ : XZ = 3 : 4 : 5. Step 4: XY is adjacent to angle X, so XY corresponds to the ratio 3 parts. Step 5: Given XY = 6 cm, so 3 parts = 6 cm, which means 1 part = 6 / 3 = 2 cm. Step 6: YZ corresponds to 4 parts, so YZ = 4 * 2 = 8 cm. Step 7: Hypotenuse XZ would be 5 parts = 5 * 2 = 10 cm, which is consistent with the ratios. Verification / Alternative check: We can recheck sin X using the computed sides. sin X = opposite / hypotenuse = YZ / XZ = 8 / 10 = 4 / 5, which matches the given value. Also, XY / XZ = 6 / 10 = 3 / 5, so cos X = 3 / 5, aligning perfectly with a 3, 4, 5 triangle pattern. Why Other Options Are Wrong: 4 cm and 5 cm: These are smaller than the correct length and do not satisfy sin X = 4 / 5 with the given XY value. 10 cm and 12 cm: 10 cm is the hypotenuse, not YZ, and 12 cm would give sin X = 12 / 10 > 1, which is impossible. Common Pitfalls: Students sometimes misidentify which side is opposite and which is adjacent, or they apply the ratio to the wrong side. Another common error is to treat 4 and 5 as the actual lengths rather than as parts of a ratio. Always map the ratio correctly to the sides, then scale using the known side length. Final Answer: The length of side YZ is 8 cm.

Question 12

A point T with coordinates (x, 0) lies on the line segment joining points S(-4, -1) and U(1, 4). In what ratio does point T divide the segment SU?

Accepted Answer

Introduction / Context: This analytic geometry question deals with internal division of a line segment and uses the section formula. It also adds the condition that the dividing point lies on the x axis, which allows us to use the y coordinate to determine the ratio in which the segment is divided. Given Data / Assumptions: Point S has coordinates (-4, -1). Point U has coordinates (1, 4). Point T = (x, 0) lies on segment SU. T divides SU in some ratio k : 1 or 1 : k that we need to find. Division is assumed to be internal. Concept / Approach: A point dividing a segment internally in a given ratio can be found using the section formula. However, here we are given that T lies on the x axis (its y coordinate is zero). We can parameterise the segment using a parameter t, or directly use the section formula in reverse by relating the y coordinate of T to the ratio of division. Step-by-Step Solution: Step 1: Let S(-4, -1) and U(1, 4). Let T divide SU in the ratio ST : TU = m : n. Step 2: By the section formula, the y coordinate of T is given by (n * y_S + m * y_U) / (m + n). Step 3: So y_T = (n * (-1) + m * 4) / (m + n). Step 4: We are told T lies on the x axis, so y_T = 0. Step 5: Set the expression equal to 0: ( -n + 4m ) / (m + n) = 0. Step 6: For a fraction to be zero, its numerator must be zero. So -n + 4m = 0. Step 7: Rearranging gives 4m = n, hence n : m = 4 : 1, or m : n = 1 : 4. Step 8: Therefore ST : TU = 1 : 4. Verification / Alternative check: We can use a parameter t representing the fraction of the way from S to U. Coordinates of a point on SU are S + t(U - S) = (-4 + 5t, -1 + 5t). Set y = 0: -1 + 5t = 0, so t = 0.2. Hence ST : TU = 0.2 : 0.8 = 1 : 4, confirming the result. Why Other Options Are Wrong: 4 : 1: This is the reverse of the correct ratio and corresponds to a point closer to U than to S. 1 : 2 and 2 : 1: These give different y coordinates and do not result in y = 0 for point T. 3 : 1: Also leads to a non zero y coordinate, so T would not lie on the x axis. Common Pitfalls: A frequent mistake is misusing the section formula or mixing up which coefficient multiplies which point. Another error is forgetting that the denominator of the section formula does not affect the zero value, and only the numerator must be set to zero. Careful algebra and clear assignment of the ratio help avoid these issues. Final Answer: Point T divides segment SU in the ratio 1 : 4.

Question 13

At what point does the line 2x - 3y = 6 cut the y axis?

Accepted Answer

Introduction / Context: This is a basic coordinate geometry question about finding where a straight line intersects the y axis. The y intercept is a key feature of a line and is often used when graphing or analysing linear equations. Given Data / Assumptions: The equation of the line is 2x - 3y = 6. We must find the point where this line cuts the y axis. On the y axis, the x coordinate is always 0. Concept / Approach: To find the y intercept of a line, set x = 0 in the equation and solve for y. This gives the coordinates (0, y). This works because any point on the y axis has x equal to zero by definition. Step-by-Step Solution: Step 1: Start with the line equation 2x - 3y = 6. Step 2: At the y axis, x = 0. Substitute x = 0 into the equation. Step 3: The equation becomes 2 * 0 - 3y = 6. Step 4: Simplify: 0 - 3y = 6, so -3y = 6. Step 5: Divide by -3: y = 6 / (-3) = -2. Step 6: Therefore, the line cuts the y axis at the point (0, -2). Verification / Alternative check: We can rewrite the line in slope intercept form y = mx + c. From 2x - 3y = 6, rearrange to -3y = -2x + 6, so y = (2 / 3)x - 2. Here, the y intercept is clearly -2, confirming the point (0, -2). This is in full agreement with our earlier calculation. Why Other Options Are Wrong: (0, 2): Substituting x = 0, y = 2 gives 2 * 0 - 3 * 2 = -6, not 6. (-2, 0): This point lies on the x axis and not on the y axis; also it does not satisfy the equation correctly for an intercept on y. (2, 0): Also an x axis point, not a y axis point. (0, 3): Substituting gives 2 * 0 - 3 * 3 = -9, which does not equal 6. Common Pitfalls: One common mistake is to set y = 0 when looking for the y intercept instead of setting x = 0. That actually finds the x intercept. Another error is incorrect algebra when isolating y. Keeping the definition of the axes in mind and working step by step avoids these issues. Final Answer: The line cuts the y axis at the point (0, -2).

Question 14

Name the type of triangle in which the orthocentre, circumcentre, incentre and centroid all coincide at a single point.

Accepted Answer

Introduction / Context: This conceptual geometry question asks about special points associated with a triangle, namely the orthocentre, circumcentre, incentre and centroid. These points are defined through different constructions, and their relative positions tell us important information about the type of triangle. Given Data / Assumptions: We have a triangle in which four special centres are considered. Orthocentre: intersection of the three altitudes. Circumcentre: intersection of the perpendicular bisectors of the sides. Incentre: intersection of the angle bisectors. Centroid: intersection of the medians. We are told all four coincide at a single point. Concept / Approach: In general triangles, these four centres are distinct. However, for certain special triangles, some or all of these centres can coincide. The key fact is: In an equilateral triangle, all four centres (orthocentre, circumcentre, incentre and centroid) are at the same point due to the perfect symmetry of all sides and angles. No other triangle type shares this complete coincidence of all four centres. Step-by-Step Solution: Step 1: Recall that an equilateral triangle has all sides equal and all angles equal to 60 degrees. Step 2: Because of this high degree of symmetry, the perpendicular bisectors, altitudes, angle bisectors and medians all lie on the same three lines. Step 3: The intersection point of these lines is simultaneously the orthocentre, circumcentre, incentre and centroid. Step 4: For other triangle types, these centres do not coincide completely. For instance, in a right angled triangle, the circumcentre lies at the midpoint of the hypotenuse, while the orthocentre is at the right angle vertex. Step 5: Therefore, the only triangle in which all four centres coincide is an equilateral triangle. Verification / Alternative check: Consider coordinate geometry. An equilateral triangle placed symmetrically around the origin will have all its symmetry lines intersecting at the origin. Using formulas for centroid and circumcentre in symmetric placement shows they are identical, and by definition the incentre and orthocentre also lie at that same point because of equal angles and equal side lengths. Why Other Options Are Wrong: Isosceles triangle: While some centres may lie on the same vertical line, they usually do not coincide at a single point unless the triangle is equilateral. Right angled triangle: The orthocentre is at the right angle vertex, but the circumcentre is at the midpoint of the hypotenuse; they are not the same point. Obtuse angled triangle: The circumcentre lies outside the triangle, while the orthocentre lies inside or at a vertex, so they do not coincide. Scalene triangle: There is no symmetry at all; all four centres are completely distinct. Common Pitfalls: Many learners confuse isosceles and equilateral triangles, thinking that equal sides alone are enough for all centres to coincide. However, equal sides in an isosceles triangle still leave one angle different. Only in an equilateral triangle are all sides and all angles equal, giving full symmetry and coinciding centres. Final Answer: The triangle is an equilateral triangle.

Question 15

Find the volume, in cubic centimetres, of a cube whose side length is 7.5 centimetres.

Accepted Answer

Introduction / Context: This straightforward mensuration problem asks for the volume of a cube. Cubes are special rectangular solids with all sides equal, and their volume formula is one of the most basic in geometry and quantitative aptitude. Given Data / Assumptions: The solid is a cube. Side length s = 7.5 cm. We are required to find the volume in cubic centimetres. Concept / Approach: The volume V of a cube with side length s is: V = s^3. This is because the volume of any rectangular solid is length * breadth * height, and for a cube all three are equal to s. We therefore simply cube the side length. Step-by-Step Solution: Step 1: Identify the side length s = 7.5 cm. Step 2: Use the formula for volume of a cube: V = s^3. Step 3: Compute s^2 first: 7.5 * 7.5 = 56.25. Step 4: Now multiply by 7.5 again: 56.25 * 7.5. Step 5: 56.25 * 7.5 = 421.875. Step 6: Therefore, V = 421.875 cubic centimetres. Verification / Alternative check: We can break the multiplication into smaller steps for checking. 7.5 = 15 / 2, so s^3 = (15 / 2)^3 = 3375 / 8. Dividing 3375 by 8 gives 421.875, which matches the previous calculation exactly. This confirms the correctness of the volume. Why Other Options Are Wrong: 77.145 cubic centimetres and 39.245 cubic centimetres: These are far too small and result from incorrect multiplication or only squaring instead of cubing. 250 cubic centimetres: This is a rounded guess and not supported by the exact arithmetic; it would correspond to roughly 6.3 cm side, not 7.5 cm. 24.435 cubic centimetres: Extremely small and inconsistent with a side length as large as 7.5 cm. Common Pitfalls: A common error is to compute s^2 instead of s^3, forgetting that volume involves all three dimensions. Another mistake is mishandling decimal multiplication, especially when squaring or cubing decimal numbers. Working systematically and, when possible, rewriting decimals as fractions can help avoid arithmetic errors. Final Answer: The volume of the cube is 421.875 cubic centimetres.

Question 16

The curved surface area of a right circular cone is 10010 square centimetres and its slant height is 91 centimetres. Find the total surface area of the cone in square centimetres.

Accepted Answer

Introduction / Context: This mensuration question deals with a right circular cone and distinguishes between curved surface area and total surface area. Total surface area includes both the curved surface and the circular base, and is commonly required when calculating materials used to cover a cone shaped object. Given Data / Assumptions: Curved surface area (CSA) of the cone = 10010 square centimetres. Slant height l = 91 cm. We must find the total surface area (TSA) in square centimetres. Use pi = 22 / 7. Concept / Approach: For a right circular cone: CSA = pi * r * l. Base area = pi * r^2. Total surface area TSA = CSA + base area. We first use CSA and l to find the radius r, then compute the base area and add it to the curved surface area. Step-by-Step Solution: Step 1: Use CSA formula: CSA = pi * r * l. Step 2: Substitute CSA = 10010, l = 91, and pi = 22 / 7. Step 3: So 10010 = (22 / 7) * r * 91. Step 4: Simplify (22 / 7) * 91 = 22 * 13 = 286. Step 5: Hence 10010 = 286 * r. Step 6: Solve for r: r = 10010 / 286 = 35 cm. Step 7: Now find base area: base area = pi * r^2 = (22 / 7) * 35^2. Step 8: 35^2 = 1225, so base area = (22 / 7) * 1225 = 22 * 175 = 3850 square centimetres. Step 9: Total surface area TSA = CSA + base area = 10010 + 3850 = 13860 square centimetres. Verification / Alternative check: We can recheck the curved surface area using r = 35 cm and l = 91 cm. CSA = pi * r * l = (22 / 7) * 35 * 91. Simplifying 35 / 7 = 5, we get CSA = 22 * 5 * 91 = 110 * 91 = 10010 square centimetres, which matches the given value. Therefore our value of r and hence TSA are consistent. Why Other Options Are Wrong: 6930 and 4620 square centimetres: These are significantly smaller than the correct TSA and could arise from using only CSA or only base area incorrectly. 27720 square centimetres: This is exactly double 13860 and might come from mistakenly adding CSA twice. 10010 square centimetres: This is only the curved surface area without the base, not the total surface area. Common Pitfalls: Students often confuse curved surface area with total surface area and forget to add the base area. Others may misuse the formula for CSA or incorrectly simplify the fraction involving pi. Always clearly distinguish between CSA and TSA and carefully work through the algebra when solving for the radius. Final Answer: The total surface area of the cone is 13860 square centimetres.

Question 17

A solid cone of radius 3.5 centimetres and height 12 centimetres is completely filled with water and emptied into a cylindrical vessel of radius 7 centimetres. Find the height of water in the cylindrical vessel, in centimetres.

Accepted Answer

Introduction / Context: This question involves volume conservation when liquid is transferred from one container to another. The cone is initially full of water, and that water is poured into a cylinder. Because the water volume is preserved, the volume of the cone equals the volume of water in the cylinder. Such problems are common in aptitude tests involving mensuration and practical applications. Given Data / Assumptions: Radius of cone r_c = 3.5 cm. Height of cone h_c = 12 cm. Radius of cylinder r_y = 7 cm. Height of water in cylinder h_y is to be found. Liquid volume is conserved; no loss or gain. Use pi as a common factor that will cancel out. Concept / Approach: Volume of a cone: V_cone = (1 / 3) * pi * r_c^2 * h_c. Volume of a cylinder: V_cylinder = pi * r_y^2 * h_y. Since all the water from the cone is poured into the cylinder, we set V_cone = V_cylinder and solve for h_y. Because pi appears on both sides, it will cancel out, simplifying the computation. Step-by-Step Solution: Step 1: Write cone volume: V_cone = (1 / 3) * pi * r_c^2 * h_c. Step 2: Substitute r_c = 3.5 and h_c = 12: V_cone = (1 / 3) * pi * (3.5)^2 * 12. Step 3: Compute (3.5)^2 = 12.25. Step 4: Then V_cone = (1 / 3) * pi * 12.25 * 12 = (1 / 3) * pi * 147 = 49 * pi. Step 5: Write cylinder volume: V_cylinder = pi * r_y^2 * h_y = pi * 7^2 * h_y = pi * 49 * h_y. Step 6: Equate volumes: 49 * pi = 49 * pi * h_y. Step 7: Cancel 49 * pi from both sides: 1 = h_y. Step 8: Therefore, height of water in the cylinder h_y = 1 cm. Verification / Alternative check: Observe that the base area of the cylinder is pi * 7^2 = 49 * pi. Since the cone volume came out exactly 49 * pi cubic centimetres, dividing the cone volume by this base area must give a height of 1 cm. This quick reasoning matches the detailed calculation and confirms the result. Why Other Options Are Wrong: 0.25 cm and 0.5 cm: These are too small and would give a cylinder volume much smaller than the cone volume. 2 cm and 3 cm: These heights would produce volumes 2 and 3 times larger respectively than the cone volume, which contradicts conservation of volume. Common Pitfalls: Common mistakes include mixing up the formulas for cone and cylinder volumes, forgetting the factor 1 / 3 in the cone formula, or failing to square the radius. Some students also try to plug in a numerical value of pi unnecessarily when it cancels out. Focusing on algebraic cancellation and carefully writing each step prevents such errors. Final Answer: The height of water in the cylindrical vessel is 1 cm.

Question 18

A hollow right circular cylinder is open at both ends. Its length is 22 cm, the external radius is 7 cm, and the wall thickness is 1 cm. Using π = 22/7, what is the total surface area of this hollow cylinder in square centimetres?

Accepted Answer

Introduction / Context: This question tests the concept of total surface area of a hollow right circular cylinder that is open at both ends. Many competitive exams include such problems to check whether students correctly combine curved surface areas and ring shaped end areas, and whether they correctly handle internal and external radii when the cylinder is hollow. Given Data / Assumptions: - The cylinder is hollow and open at both ends. - Length or height h = 22 cm. - External radius R = 7 cm. - Wall thickness t = 1 cm, so internal radius r = R − t = 6 cm. - Use π = 22/7 as given. Concept / Approach: For a hollow cylinder open at both ends, total surface area includes four parts: - Outer curved surface area = 2 * π * R * h. - Inner curved surface area = 2 * π * r * h. - Area of the two circular rings at the ends = 2 * π * (R^2 − r^2). Total surface area = 2 * π * h * (R + r) + 2 * π * (R^2 − r^2). Step-by-Step Solution: Step 1: Compute R + r = 7 + 6 = 13 cm. Step 2: Compute R^2 − r^2 = 7^2 − 6^2 = 49 − 36 = 13. Step 3: Outer plus inner curved area = 2 * π * h * (R + r) = 2 * (22/7) * 22 * 13. Step 4: Simplify curved area: 2 * (22/7) * 22 * 13 = (44/7) * 22 * 13 = (968/7) * 13 = 12584/7. Step 5: Area of both ring shaped ends = 2 * π * (R^2 − r^2) = 2 * (22/7) * 13 = 572/7. Step 6: Total surface area = 12584/7 + 572/7 = 13156/7 sq.cm. Step 7: 13156/7 = 1879.428571..., which rounds to 1879.43 sq.cm. Verification / Alternative Check: A quick check is to compute curved and end areas separately using approximate decimal π. With π close to 3.14 the result remains near 1880 sq.cm, which agrees with the more exact fraction based value, confirming the correctness of the calculation and the chosen formula structure. Why Other Options Are Wrong: Option b, 3758.86 sq.cm, is almost exactly double the correct answer and would arise if someone incorrectly doubled the entire total surface expression. Option c, 939.715 sq.cm, is about half the correct result and could come from using only curved surfaces or only one side of the cylinder. Option d, 2819.145 sq.cm, does not match any proper combination of inner and outer surfaces and most likely comes from mixing radii or omitting one component of area. Common Pitfalls: Common mistakes include treating the hollow cylinder as solid, forgetting to include inner curved surface area, or ignoring the ring shaped end areas. Students sometimes use only one radius instead of both internal and external radii, or forget that the cylinder is open at both ends and either add or remove the wrong circular areas. Carefully listing all contributing surface parts avoids such errors. Final Answer: The total surface area of the hollow cylinder is 1879.43 sq.cm.

Question 19

In a quadrilateral, three interior angles measure 60°, 90°, and 100°. What is the measure of the fourth interior angle of the quadrilateral in degrees?

Accepted Answer

Introduction / Context: This problem checks understanding of the sum of interior angles in a quadrilateral. Such questions are standard in school level geometry and aptitude tests and help reinforce basic polygon angle sum formulas, which are very useful in more advanced geometry problems as well. Given Data / Assumptions: - The figure is a quadrilateral, so it has four interior angles. - Three interior angles are given as 60°, 90°, and 100°. - We need to find the measure of the fourth interior angle in degrees. Concept / Approach: For any polygon with n sides, the sum of interior angles is (n − 2) * 180 degrees. For a quadrilateral, n = 4, so the sum of interior angles is: Sum = (4 − 2) * 180 = 2 * 180 = 360 degrees. Once we know this total, we subtract the sum of the three known angles from 360 degrees in order to obtain the missing fourth angle. Step-by-Step Solution: Step 1: Write the general formula for a quadrilateral: sum of interior angles = 360°. Step 2: Add the three given angles: 60° + 90° + 100°. Step 3: Compute this sum: 60° + 90° = 150°, and 150° + 100° = 250°. Step 4: Subtract this from the total: 360° − 250° = 110°. Step 5: Therefore, the fourth interior angle of the quadrilateral is 110°. Verification / Alternative Check: We can verify by adding all four angles including the found value. So, 60° + 90° + 100° + 110° = 360°. This matches the known interior angle sum of a quadrilateral, confirming that 110° is correct. No other angle in the options will satisfy this basic angle sum requirement. Why Other Options Are Wrong: Option a, 95°, would give total 60° + 90° + 100° + 95° = 345°, which is less than 360°, so it cannot be correct. Option b, 100°, would give 60° + 90° + 100° + 100° = 350°, still not 360°. Option d, 115°, would give 60° + 90° + 100° + 115° = 365°, which is greater than 360°, so it also fails. Common Pitfalls: Learners sometimes confuse the formula for the sum of angles in a quadrilateral with that of a triangle. Another common error is arithmetic, such as adding the three known angles incorrectly or subtracting from 180° instead of 360°. Carefully remembering that a quadrilateral can be split into two triangles helps, because 2 * 180° clearly shows that the total must be 360°. Final Answer: The fourth interior angle of the quadrilateral is 110°.

Question 20

In triangle ABC, medians BE and CF intersect at point O, the centroid. Points P and Q are the midpoints of segments BO and CO, respectively. If the length of segment PQ is 3 cm, what is the length of segment FE (in centimetres) joining the midpoints…

Accepted Answer

Introduction / Context: This geometry question tests understanding of medians, centroids, midpoints, and relationships between segments inside a triangle. It is a typical conceptual problem where algebraic or coordinate geometry reasoning shows that certain segments in a triangle can have equal lengths, independent of the triangle size. Given Data / Assumptions: - Triangle ABC has medians BE and CF drawn to sides AC and AB, respectively. - These medians intersect at O, the centroid of triangle ABC. - P is the midpoint of segment BO. - Q is the midpoint of segment CO. - Segment FE joins the midpoints of sides AB and AC. - Length PQ = 3 cm, and we must find FE. Concept / Approach: There are two key ideas: - The segment joining midpoints of two sides of a triangle (here FE) is called a mid segment and is parallel to the third side, with length equal to half of that third side. - When medians and their midpoints are considered, the small quadrilateral PEOF is homothetic to triangle ABC, and segments PQ and FE turn out to be equal in length by vector or coordinate geometry arguments. Step-by-Step Solution: Step 1: Consider triangle ABC in a coordinate system for clarity. Let A, B, and C be arbitrary noncollinear points. Step 2: E is the midpoint of AC, and F is the midpoint of AB. Step 3: Medians BE and CF intersect at centroid O, which is the average of the vertices, so O = (A + B + C) / 3 in vector form. Step 4: P is the midpoint of BO, and Q is the midpoint of CO, so vectors for P and Q become averages of B with O and C with O, respectively. Step 5: Using vector algebra, one can show that the vector from F to E is equal to the vector from P to Q. Hence FE and PQ are parallel and have the same length. Step 6: Therefore, if PQ = 3 cm, then FE must also be 3 cm. Verification / Alternative Check: Another approach is to choose a specific simple triangle, for example with coordinates A(0,0), B(2,0), and C(0,2). When the points E, F, O, P, and Q are constructed and distances are calculated, FE and PQ both come out equal. Because the result is independent of the specific coordinates (due to similarity and centroid properties), FE = PQ in every such triangle, confirming FE = 3 cm here. Why Other Options Are Wrong: Option b, 6 cm, would imply FE is twice PQ, which does not match the internal similarity relations created by medians and midpoints. Option c, 9 cm, is three times PQ and has no geometric justification in this configuration. Option d, 12 cm, is even further from the logical length and would make FE much larger than any reasonable side length if PQ is only 3 cm. Common Pitfalls: Many learners try to relate FE directly to side BC without considering how P and Q are constructed from BO and CO. Others forget that centroids divide medians in a 2:1 ratio and that midpoints of segments from vertices to centroid create additional similar triangles inside the original triangle. Drawing a neat diagram and, if comfortable, assigning coordinates makes such problems much easier to handle. Final Answer: The length of FE is 3 cm.

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