Question 1

In a circle, a chord is drawn whose length is equal to the radius of the circle. A tangent is drawn at one end of this chord. What is the measure of the angle between the tangent and the chord?

Accepted Answer

Introduction / Context: This circle geometry problem involves a chord whose length equals the radius and a tangent drawn at one end of that chord. It tests knowledge of the tangent chord theorem, which connects the angle between a tangent and a chord with an angle inside the circle in the alternate segment. Given Data / Assumptions: A circle has radius r. A chord AB is drawn such that AB = r. A tangent is drawn at point A, one end of the chord AB. We must find the angle between this tangent and the chord AB. Concept / Approach: First, note that OA and OB are radii of the circle (O is the centre), and AB is a chord equal in length to the radius. Thus triangle AOB has all three sides equal to r, making it an equilateral triangle, so each angle in triangle AOB is 60°. The tangent at point A is perpendicular to radius OA, forming a 90° angle between OA and the tangent. The angle between the tangent at A and chord AB equals the difference between 90° and angle OAB in triangle AOB. Because triangle AOB is equilateral, angle OAB is 60°. Step-by-Step Solution: Step 1: Identify triangle AOB where OA and OB are radii and AB is a chord equal to the radius, so OA = OB = AB = r. Step 2: Conclude triangle AOB is equilateral, hence each internal angle is 60°. Step 3: The radius OA is perpendicular to the tangent at A, so the angle between OA and the tangent is 90°. Step 4: At point A, angle between the tangent and chord AB equals 90° minus angle OAB because OAB is inside the triangle between OA and AB. Step 5: Angle OAB = 60°, so required angle = 90° − 60° = 30°. Verification / Alternative Check: The tangent chord theorem states that the angle between a tangent and a chord through the point of contact equals the angle in the alternate segment of the circle that subtends the same chord. Here, the angle between the tangent at A and chord AB equals the angle in the circle at point C on the circumference such that chord AB subtends angle ACB. In equilateral triangle AOB, the corresponding inscribed angle at C is 30°, confirming the computed value. Why Other Options Are Wrong: 45°, 60°, and 75° do not arise from the perpendicular tangent and 60° vertex angle inside the equilateral triangle. In particular, 60° would incorrectly assume that the angle between tangent and chord equals the internal angle at the centre or at the vertex of the equilateral triangle, which is not true in this configuration. Common Pitfalls: Many learners forget that the tangent is perpendicular to the radius and either use the 60° angle directly or misapply the tangent chord theorem. Another frequent mistake is confusing the central angle AOB (60°) with the angle between the tangent and the chord. Always remember that the radius to the point of tangency forms a right angle with the tangent, and work from there. Final Answer: The angle between the tangent and the chord is 30°.

Question 2

In triangle ABC, if angle A + angle C = 140° and angle A + 3 × angle B = 180°, then what is the measure of angle A (in degrees)?

Accepted Answer

Introduction / Context: This algebraic geometry problem tests how to use angle relationships and the angle sum property of a triangle to solve for unknown angles. It involves forming and solving simple linear equations in terms of the triangle's angles. Given Data / Assumptions: Triangle ABC has interior angles A, B, and C. Angle A + angle C = 140°. Angle A + 3 × angle B = 180°. We must find angle A. Concept / Approach: The key is to use the angle sum property A + B + C = 180° together with the given equations. From A + C = 140°, we can express B in terms of given values. This allows us to find B first and then substitute back into the second equation A + 3B = 180° to find A. The approach consists of solving two linear equations with two unknowns (A and B) using substitution. Step-by-Step Solution: Step 1: Use the triangle angle sum property: A + B + C = 180°. Step 2: From A + C = 140°, subtract from the sum: (A + B + C) − (A + C) = 180° − 140°, which gives B = 40°. Step 3: Now use the second given relation: A + 3B = 180°. Step 4: Substitute B = 40°: A + 3 × 40° = 180°. Step 5: Simplify: A + 120° = 180°. Step 6: Subtract 120° from both sides: A = 60°. Verification / Alternative Check: With A = 60° and B = 40°, we can find C from A + C = 140°: 60° + C = 140°, so C = 80°. Check the angle sum: A + B + C = 60° + 40° + 80° = 180°, which is correct. Also check A + 3B: 60° + 3 × 40° = 60° + 120° = 180°, matching the second condition. All constraints are satisfied, so A = 60° is consistent. Why Other Options Are Wrong: If A were 80°, then from A + C = 140° we would get C = 60°, and from A + 3B = 180° we would get 3B = 100° so B would not be an integer and the angle sum would fail. Similar contradictions arise for A = 40° or A = 20°, where one or both of the given relationships would not hold once the remaining angles are computed. Only A = 60° satisfies all equations at once. Common Pitfalls: A frequent mistake is to try guessing angle values without systematically using the angle sum property and both given equations. Another pitfall is mismanaging algebra when subtracting or dividing, which can lead to incorrect values for B or C. Writing out each step carefully and checking all given conditions after finding an answer helps avoid such errors. Final Answer: The measure of angle A is 60°.

Question 3

From an external point P, two tangents PA and PB are drawn to a circle with centre O such that angle APB = 80°. What is the measure of angle AOP (in degrees)?

Accepted Answer

Introduction / Context: This circle geometry question deals with two tangents drawn from an external point to a circle and asks for a central angle involving the radius and line to the external point. It tests understanding of tangent properties, isosceles triangles formed by radii, and angle bisector symmetry in such configurations. Given Data / Assumptions: PA and PB are tangents drawn from external point P to a circle with centre O. Angle APB, the angle between the tangents, equals 80°. We must determine angle AOP. Concept / Approach: First, note that OA and OB are radii drawn to the points of tangency A and B, and each radius is perpendicular to the tangent at that point. Thus, angle OAP and angle OBP are right angles (90°). Because tangents from an external point are equal in length (PA = PB), triangle APB is isosceles. The configuration is symmetric about line OP, which acts as the angle bisector of angle APB and also the perpendicular bisector of chord AB. We use this symmetry to find angle APO, angle OAP, and finally angle AOP by applying the triangle angle sum in triangle AOP. Step-by-Step Solution: Step 1: Recognise that PA = PB, so triangle APB is isosceles with vertex at P. Step 2: The line OP, joining the centre and external point, is the symmetry axis and bisects angle APB. Step 3: Therefore angle APO = angle OPB = (1/2) × 80° = 40°. Step 4: Radius OA is perpendicular to tangent PA, so angle OAP = 90°. Step 5: Consider triangle AOP. The three angles are angle OAP = 90°, angle APO = 40°, and angle AOP (unknown). Step 6: Use triangle angle sum: angle OAP + angle APO + angle AOP = 180°. Step 7: Substitute known values: 90° + 40° + angle AOP = 180°. Step 8: Simplify: 130° + angle AOP = 180°, so angle AOP = 50°. Verification / Alternative Check: Another known relation is that the angle between two tangents (angle APB) is supplementary to the central angle AOB that subtends the chord AB. That is, angle APB = 180° − angle AOB. With angle APB = 80°, we get angle AOB = 100°. Since OP bisects angle AOB as well, angle AOP = 100° / 2 = 50°, confirming the earlier calculation. Why Other Options Are Wrong: Values like 40°, 60°, or 70° do not satisfy both the symmetry and the right angle conditions simultaneously. For example, 40° equals the half angle at P, not the central angle at O. A value of 60° or 70° would make the angles in triangle AOP sum to more than or less than 180°, contradicting basic triangle geometry. Common Pitfalls: A common mistake is to confuse angle APB with angle AOB or to forget that OP bisects both the angle between the tangents and the corresponding central angle. Students may also overlook the perpendicularity of radius and tangent, forgetting to use the 90° angle at A. Drawing a clear diagram and labelling all known angles helps to avoid these issues. Final Answer: The measure of angle AOP is 50°.

Question 4

Which of the following sets of three side lengths cannot form a triangle (all measurements in centimetres)?

Accepted Answer

Introduction / Context: This problem is another application of the triangle inequality, focusing on identifying which set of three positive numbers cannot be side lengths of a triangle. This concept is fundamental in geometry and often appears in objective tests to quickly eliminate impossible configurations. Given Data / Assumptions: Each option lists three positive numbers representing potential side lengths. All side lengths are in centimetres. A valid triangle must satisfy the triangle inequality: sum of any two sides must be strictly greater than the third side. Concept / Approach: For three lengths a, b, and c to form a non degenerate triangle, it must be true that a + b > c, b + c > a, and c + a > b. If even one of these is violated (less than or equal to), a triangle cannot be formed. We check each option to see where the inequality fails. The option where the sum of the two shorter sides does not exceed the longest side is the one that cannot form a triangle. Step-by-Step Solution: Step 1: Option a: 5, 6, 7. Check 5 + 6 = 11 > 7, 6 + 7 = 13 > 5, 7 + 5 = 12 > 6. All conditions are satisfied, so this set can form a triangle. Step 2: Option b: 5, 8, 15. Order them as 5, 8, 15 with 15 the largest. Check 5 + 8 = 13, which is less than 15. This fails the triangle inequality. Step 3: Because 5 + 8 is not greater than 15, these three lengths cannot form a triangle. Step 4: Option c: 8, 15, 18. Check 8 + 15 = 23 > 18, 15 + 18 = 33 > 8, 18 + 8 = 26 > 15. All good, so this set can form a triangle. Step 5: Option d: 6, 7, 11. Check 6 + 7 = 13 > 11, 7 + 11 = 18 > 6, 11 + 6 = 17 > 7. This set can also form a triangle. Verification / Alternative Check: Once you find a single failing case in a set (like 5 + 8 ≤ 15 in option b), you know that the set cannot form a triangle. For the other options, all three inequalities hold strictly, so there is no contradiction. Thus, option b is uniquely the impossible combination. Why Other Options Are Wrong: Option a corresponds to a typical acute or obtuse triangle configuration with lengths that clearly satisfy the inequalities. Options c and d also show sums of the two smaller sides that comfortably exceed the third side. Only option b fails the basic requirement that the two shorter sides together must exceed the longest side. Common Pitfalls: Sometimes students check only the smallest two sides against the largest and forget to verify the other two inequalities. While in many cases that is enough, relying on only one inequality can occasionally lead to mistakes. Another common error is misordering the sides, which can confuse which pair should be compared with the largest side. Sorting the three numbers first and then checking the sum of the two smaller ones against the largest is a safe and quick strategy. Final Answer: The set of side lengths that cannot form a triangle is 5 cm, 8 cm, 15 cm.

Question 5

A solid sphere of radius 9 cm is melted and recast to form a smaller solid sphere of radius 6 cm and a right circular cylinder with the same radius 6 cm. What is the height of the cylinder (in centimetres)?

Accepted Answer

Introduction / Context: This question combines volume formulas for a sphere and a right circular cylinder under the principle of conservation of volume. When a solid object is melted and recast into new shapes, the total volume remains the same (ignoring any loss), which allows us to set up an equation and solve for the missing dimension, here the cylinder height. Given Data / Assumptions: A solid sphere has radius R₁ = 9 cm. It is melted to form two solids: a sphere of radius R₂ = 6 cm and a right circular cylinder. The cylinder has radius r = 6 cm, equal to R₂. The total volume before and after melting is conserved. We need to find the height h of the cylinder. Concept / Approach: The volume of a sphere of radius R is V = (4/3) * π * R^3. The volume of a right circular cylinder with radius r and height h is V = π * r^2 * h. Conservation of volume gives: Volume of original sphere = volume of smaller sphere + volume of cylinder. Substituting the formulas and known radii allows us to solve for h. π appears in every term, so it can be cancelled out to simplify the algebra. Step-by-Step Solution: Step 1: Write volume of the original sphere: V₁ = (4/3) * π * 9^3. Step 2: Write volume of the smaller sphere: V₂ = (4/3) * π * 6^3. Step 3: Write volume of the cylinder: V_cyl = π * 6^2 * h. Step 4: Use conservation of volume: V₁ = V₂ + V_cyl. Step 5: Substitute and cancel π: (4/3) * 9^3 = (4/3) * 6^3 + 6^2 * h. Step 6: Compute powers: 9^3 = 729 and 6^3 = 216, 6^2 = 36. Step 7: Substitute: (4/3) * 729 = (4/3) * 216 + 36h. Step 8: Compute (4/3) * 729 = 4 * 243 = 972, and (4/3) * 216 = 4 * 72 = 288. Step 9: So 972 = 288 + 36h, giving 36h = 972 − 288 = 684. Step 10: Divide by 36: h = 684 / 36 = 19 cm. Verification / Alternative Check: We can confirm by recomputing approximate volumes with π factored out. The original sphere's volume factor is (4/3) * 729 = 972. The smaller sphere's factor is (4/3) * 216 = 288. With h = 19 and radius 6, the cylinder's factor is 6^2 * 19 = 36 * 19 = 684. Adding 288 and 684 gives 972, matching the original sphere volume factor, which verifies that h = 19 cm is consistent. Why Other Options Are Wrong: Heights 21 cm, 23 cm, or 25 cm would give cylinder volume factors 756, 828, or 900 respectively, which would make the total volume greater than or less than the original sphere's volume. They do not satisfy the equation V₁ = V₂ + V_cyl when the two spheres are included. Common Pitfalls: Students sometimes forget to cube the radii when computing sphere volumes or square the radius for the cylinder. Another common error is to forget that the cylinder has the same radius as the smaller sphere and mistakenly use 9 cm instead of 6 cm. Cancelling π early in the equation helps to simplify the calculations and reduce arithmetic mistakes. Final Answer: The height of the cylinder is 19 cm.

Question 6

In one circle, an arc subtends a central angle of 30°. The length of this arc is double the length of an arc in a second circle whose radius is three times the radius of the first circle. What is the measure of the central angle subtended by the arc…

Accepted Answer

Introduction / Context: This question tests the relationship between arc length, radius, and central angle in circles. It uses proportional reasoning across two different circles, which is a common theme in aptitude and geometry problems involving circles and arcs. Given Data / Assumptions: Circle 1 has radius r₁ and an arc that subtends a central angle of 30°. Circle 2 has radius r₂ = 3r₁. The length of the arc in circle 1 is twice the length of an arc in circle 2. The arcs are compared by length, not by angle. We must find the central angle θ₂ of the arc in circle 2. Concept / Approach: Arc length is directly proportional to both the radius and the central angle (in the same units, typically radians). For a given unit system, we can write arc length s = k * r * θ, where k is a constant depending on unit choice. Since we compare ratios, the constant cancels out. We set up a proportion comparing the two arc lengths, using r₂ = 3r₁ and the fact that the arc in circle 1 is twice the arc in circle 2, then solve for θ₂ (in degrees). Step-by-Step Solution: Step 1: Let s₁ be the arc length in circle 1 and s₂ be the arc length in circle 2. Step 2: For circle 1: s₁ ∝ r₁ * θ₁, with θ₁ = 30°. Step 3: For circle 2: s₂ ∝ r₂ * θ₂ = 3r₁ * θ₂. Step 4: We are told s₁ = 2 * s₂. Step 5: Replace s₁ and s₂ with their proportional forms: r₁ * 30° = 2 * (3r₁ * θ₂). Step 6: Cancel r₁ from both sides: 30° = 2 * 3 * θ₂ = 6θ₂. Step 7: Solve for θ₂: θ₂ = 30° / 6 = 5°. Verification / Alternative Check: Think in terms of ratios. If radius is tripled (from r₁ to 3r₁) but the arc in the larger circle is half the length of the arc in the smaller circle, then the product radius × angle for the second circle must be one sixth of the product radius × angle for the first circle. Because 30° divided by 6 is 5°, this is consistent with the proportional relationship and supports the result θ₂ = 5°. Why Other Options Are Wrong: Angles 3°, 4°, and 6° would not satisfy the required ratio. For example, with θ₂ = 6°, the product 3r₁ * 6° = 18r₁ would make s₂ too large, so it would not be half of s₁. Only θ₂ = 5° makes the proportional equality hold true. Common Pitfalls: Some students mistakenly think that if the radius is tripled and the arc is half as long, the angle must be divided by 3 or 2 separately, leading to incorrect values. Others forget that arc length is proportional to the product of radius and angle, not to each separately. Setting up a clear proportional equation helps avoid these errors. Final Answer: The central angle in the second circle is 5°.

Question 7

Two circles touch each other externally. The distance between their centres is 7 cm. If the radius of one circle is 4 cm, what is the radius of the other circle (in centimetres)?

Accepted Answer

Introduction / Context: This question tests the basic geometric property of two circles that touch each other externally. It is a straightforward application of the relation between the distance of the centres and the radii of the two circles. Given Data / Assumptions: Two circles touch each other externally. The distance between their centres is 7 cm. The radius of one circle is 4 cm. We must find the radius of the other circle. Concept / Approach: When two circles touch externally, the distance between their centres is equal to the sum of their radii. If the radii are r₁ and r₂, and d is the distance between the centres, then for external tangency we have d = r₁ + r₂. Here d is given as 7 cm, and one radius is known to be 4 cm. We simply rearrange the equation to solve for the unknown radius. Step-by-Step Solution: Step 1: Let the unknown radius be r cm. Step 2: Use the external tangency relation: distance between centres = sum of radii. Step 3: Substitute the known values: 7 = 4 + r. Step 4: Solve for r: r = 7 − 4 = 3. Step 5: Thus the radius of the other circle is 3 cm. Verification / Alternative Check: Check by recomputing the distance using the found radius. If one circle has radius 4 cm and the other has radius 3 cm, then r₁ + r₂ = 4 + 3 = 7 cm, which matches the given distance between centres. This confirms that the two circles would just touch externally with no overlap and no gap. Why Other Options Are Wrong: A radius of 4 cm for the other circle would give 4 + 4 = 8 cm between centres, not 7 cm. A radius of 5.5 cm or 3.5 cm would yield distances 9.5 cm or 7.5 cm, which also do not match the given distance. Only 3 cm satisfies the condition of external tangency with the given centre separation. Common Pitfalls: Sometimes learners mix up formulas for external and internal tangency. For internal tangency (one circle inside another), the distance between centres is the difference of the radii, not the sum. In this question, the circles touch externally, so it must be the sum. Carefully noting the word externally helps to apply the correct relation. Final Answer: The radius of the other circle is 3 cm.

Question 8

In a triangle, the lengths of the sides are a, b, and c respectively. If a^2 + b^2 + c^2 = ab + bc + ca, what type of triangle is it?

Accepted Answer

Introduction / Context: This algebraic geometry question asks you to classify a triangle based on a relationship between the squares and products of its side lengths. Recognising the pattern and manipulating the expression helps identify whether the triangle is equilateral, isosceles, or some other type. Given Data / Assumptions: A triangle has side lengths a, b, and c. They satisfy the equation a^2 + b^2 + c^2 = ab + bc + ca. We need to determine the type of triangle (isosceles, equilateral, scalene, or right angled). Concept / Approach: We rearrange the given equation to see if it can be factored into a sum of squares. A well known identity states that: (a − b)^2 + (b − c)^2 + (c − a)^2 = 2(a^2 + b^2 + c^2 − ab − bc − ca). This shows that if a^2 + b^2 + c^2 − ab − bc − ca = 0, then the sum of these squared differences must be zero, which in turn forces a = b = c. Thus the triangle must be equilateral, with all sides equal. Step-by-Step Solution: Step 1: Start with the given equation: a^2 + b^2 + c^2 = ab + bc + ca. Step 2: Subtract the right side from the left side: a^2 + b^2 + c^2 − ab − bc − ca = 0. Step 3: Use the identity: (a − b)^2 + (b − c)^2 + (c − a)^2 = 2(a^2 + b^2 + c^2 − ab − bc − ca). Step 4: Substitute the expression from Step 2 into the identity: (a − b)^2 + (b − c)^2 + (c − a)^2 = 2 × 0 = 0. Step 5: A sum of squares equals zero only if each squared term is zero. Thus (a − b)^2 = 0, (b − c)^2 = 0, and (c − a)^2 = 0. Step 6: Therefore a − b = 0, b − c = 0, and c − a = 0, which implies a = b = c. Step 7: A triangle with all three sides equal is an equilateral triangle. Verification / Alternative Check: We can check with a specific example. Suppose a = b = c = k. Then a^2 + b^2 + c^2 = 3k^2 and ab + bc + ca = 3k^2, so the equality holds. If we try a simple non equilateral triangle, for example a = 2, b = 3, c = 4, then a^2 + b^2 + c^2 = 4 + 9 + 16 = 29 and ab + bc + ca = 6 + 12 + 8 = 26. These are not equal, so the condition does not hold. This supports the conclusion that only equilateral triangles satisfy the given relation. Why Other Options Are Wrong: An isosceles triangle would have at least two equal sides but not necessarily all three, and the identity above shows that all differences must vanish, not just one. A scalene triangle has all sides different, which contradicts the result that a = b = c. A right angled triangle satisfies the Pythagorean relation a^2 + b^2 = c^2 for appropriate labelling, which is a different condition and does not force a^2 + b^2 + c^2 = ab + bc + ca. Common Pitfalls: Some learners try to interpret the relation as a variant of the Pythagorean theorem or attempt to plug in random values instead of working with identities. Others may not recall the sum of squares identity and thus miss the elegant path to showing that all sides must be equal. Remembering and applying algebraic identities can simplify many geometry problems involving side lengths and classification of triangles. Final Answer: The triangle is equilateral.

Question 9

A right circular conical structure stands on a circular base of diameter 21 m and has a vertical height of 14 m. If the cost of colour washing its curved surface is Rs. 6 per square metre (take π = 22/7), what is the total cost of the colour wash?

Accepted Answer

Introduction / Context: This problem involves the curved surface area of a right circular cone and a simple cost calculation based on area. It tests your ability to compute slant height from the radius and vertical height, apply the correct surface area formula, and then multiply by the given rate per square metre to get the total cost. Given Data / Assumptions: The conical structure has a circular base with diameter 21 m, so radius r = 21 / 2 = 10.5 m. The vertical height of the cone h = 14 m. Cost of colour washing the curved surface is Rs. 6 per square metre. Use π = 22/7. We need the total cost of colour washing the curved surface only. Concept / Approach: For a cone, the curved surface area (CSA) is given by CSA = π * r * l, where r is the radius and l is the slant height. The slant height is found from the radius and vertical height using the Pythagorean theorem: l = √(r^2 + h^2). Once we compute the curved surface area, we multiply it by the rate (Rs. 6 per square metre) to get the total cost. Step-by-Step Solution: Step 1: Compute the radius: r = 21 / 2 = 10.5 m. Step 2: Use the Pythagorean theorem to find slant height l: l = √(r^2 + h^2) = √(10.5^2 + 14^2). Step 3: Calculate 10.5^2 = 110.25 and 14^2 = 196. So r^2 + h^2 = 110.25 + 196 = 306.25. Step 4: Take square root: l = √306.25 = 17.5 m. Step 5: Compute curved surface area: CSA = π * r * l = (22/7) * 10.5 * 17.5. Step 6: Simplify r * l = 10.5 * 17.5 = (21/2) * (35/2) = 735 / 4 = 183.75. Step 7: Multiply: CSA = (22/7) * 183.75. First 183.75 / 7 = 26.25, then 26.25 * 22 = 577.5 square metres. Step 8: Multiply area by rate: total cost = 577.5 * 6 = 3465 rupees. Verification / Alternative Check: You can approximate π as 22/7 and verify the calculations stepwise or use fractional form throughout to avoid decimals. For example, CSA = (22/7) * (735/4) = (22 * 735) / 28 = (22 * 105) / 4 = 2310 / 4 = 577.5. Multiplying by 6 still gives 3465. This confirms that all intermediate steps are consistent and the final cost is correct. Why Other Options Are Wrong: Options Rs. 4365, Rs. 4465, and Rs. 3365 correspond to larger or smaller areas than the actual curved surface area when back calculated. They would require incorrect slant heights or misapplied formulas, such as using total surface area instead of curved surface area or miscomputing the radius or height. Common Pitfalls: Students sometimes mistakenly use the vertical height instead of the slant height in the curved surface area formula, or they confuse curved surface area with total surface area (which also includes the base). Another common error is using the diameter instead of the radius directly in the CSA formula. Always confirm that you have computed l correctly and that you are using r, not the diameter, inside π * r * l. Final Answer: The total cost of colour washing the curved surface is Rs. 3465.

Question 10

Points P, Q, and R lie on a circle such that angle PQR = 40° and angle QRP = 60°. What is the measure of the central angle subtended by arc QR (in degrees)?

Accepted Answer

Introduction / Context: This circle geometry problem connects angles at the circumference with the central angle subtending the same arc. It also uses the angle sum property of a triangle formed by three points on the circle. These relationships are core results in circle theorems and frequently appear in competitive exams. Given Data / Assumptions: P, Q, and R are points on the circumference of a circle. Angle PQR = 40°. Angle QRP = 60°. We must find the central angle (at the centre of the circle) subtended by arc QR. Concept / Approach: First, note that triangle PQR is an inscribed triangle in the circle, and its angles at Q and R are given. The third angle at P can be found using the triangle angle sum property. This angle at P is an inscribed angle that subtends arc QR. A basic circle theorem says that a central angle subtending a given arc is twice any inscribed angle that subtends the same arc. So, once we find angle QPR at P, we double it to get the central angle for arc QR. Step-by-Step Solution: Step 1: Use the triangle angle sum property for triangle PQR: angle PQR + angle QRP + angle QPR = 180°. Step 2: Substitute the given angles: 40° + 60° + angle QPR = 180°. Step 3: Add 40° and 60°: 100° + angle QPR = 180°. Step 4: Solve for angle QPR: angle QPR = 180° − 100° = 80°. Step 5: Angle QPR is an inscribed angle subtending arc QR. Step 6: The central angle subtending the same arc QR, call it angle QOR, satisfies angle QOR = 2 × angle QPR. Step 7: Therefore angle QOR = 2 × 80° = 160°. Verification / Alternative Check: Drawing a sketch of the circle with P, Q, and R on it and marking the angles helps visualise the situation. The large central angle QOR, opening over arc QR, must be more than 90° because angle QPR is already 80° at the circumference. Doubling 80° to 160° matches this intuition and fits neatly with the inscribed–central angle relationship, providing a strong visual check. Why Other Options Are Wrong: Option a, 80°, equals the inscribed angle at P and would only apply if P were at the centre, which it is not. Options b (120°) and c (140°) do not equal twice the inscribed angle 80° and therefore violate the central–inscribed angle theorem for the same arc. They cannot be correct in this configuration. Common Pitfalls: A common error is forgetting to compute the third angle in triangle PQR and using one of the given angles directly as the inscribed angle subtending arc QR. Another mistake is halving instead of doubling the inscribed angle when finding the central angle, mixing up which is larger. Always remember: central angle = 2 × inscribed angle for the same arc. Final Answer: The central angle subtended by arc QR is 160°.

Question 11

In an isosceles triangle ABC, sides AB and AC are equal and ∠A = 80°. The internal bisectors of angles ∠B and ∠C meet at point D inside the triangle. What is the measure (in degrees) of angle ∠BDC?

Accepted Answer

Introduction / Context: In this geometry question we work with an isosceles triangle and angle bisectors meeting at a special interior point. The goal is to find the angle formed at that point between the two bisectors, which is a standard property related to the incenter of a triangle. Given Data / Assumptions: Triangle ABC is isosceles with AB = AC. Angle ∠A = 80°. The internal bisectors of ∠B and ∠C meet at point D. We have to find the measure of angle ∠BDC. Concept / Approach: The intersection of the internal angle bisectors of a triangle is called the incenter. A key property is that the angle between the bisectors of ∠B and ∠C at the incenter is 90° + ∠A / 2. We use the fact that in an isosceles triangle with AB = AC, angles at B and C are equal. Step-by-Step Solution: Step 1: Since AB = AC, triangle ABC is isosceles with vertex at A, so ∠B = ∠C.Step 2: Sum of angles in a triangle is 180°, so ∠B + ∠C + ∠A = 180°.Step 3: With ∠A = 80° and ∠B = ∠C, we get 2∠B + 80° = 180°, so 2∠B = 100° and ∠B = ∠C = 50°.Step 4: The point where the internal bisectors of ∠B and ∠C meet is the incenter, which we call D.Step 5: A known result for the incenter is ∠BDC = 90° + ∠A / 2.Step 6: Substitute ∠A = 80° to get ∠BDC = 90° + 80° / 2 = 90° + 40° = 130°. Verification / Alternative check: Another way is to remember the general formulas: ∠BIC (incenter at I) equals 90° + ∠A / 2, and similarly for other vertex combinations. Substituting the given angle confirms that 130° is consistent with this standard incenter property. Why Other Options Are Wrong: 90°: This would be true only if ∠A were 0°, which is impossible in a real triangle. 100° and 80° do not follow from the formula 90° + ∠A / 2. 60° is too small and does not match any central or interior angle relationship here. Common Pitfalls: A common mistake is to assume that ∠BDC equals either ∠B or ∠C or to average them. Another error is to forget that the meeting point of angle bisectors is the incenter and to ignore the special formula 90° + ∠A / 2. Some learners also miscompute half of 80° or misapply the triangle angle sum rule. Final Answer: The angle between the bisectors of ∠B and ∠C at point D is 130°.

Question 12

A solid right circular cylindrical metal rod has a radius of 3.2 decimetres. It is completely melted and recast into 44 equal solid cubes, each having an edge length of 8 cm. What is the length (in cm) of the original cylindrical rod? (Take π = 22/7…

Accepted Answer

Introduction / Context: This problem tests your understanding of volume conservation when a solid is melted and recast into new shapes. The cylindrical rod and the cubes have the same total volume of material, so the volume of the cylinder must equal the sum of the volumes of all cubes. Given Data / Assumptions: Radius of the cylindrical rod r = 3.2 dm. 1 dm = 10 cm, so r = 32 cm. The rod is melted into 44 solid cubes. Edge of each cube = 8 cm. Take π = 22/7. Concept / Approach: We use the formula for volume of a cylinder, V = π * r^2 * h, and for a cube, V = a^3. Since the rod is melted and recast, total volume of the cylinder equals the sum of volumes of the 44 cubes. From this equality we solve for the unknown height h (length) of the rod in centimetres. Step-by-Step Solution: Step 1: Convert radius from decimetres to centimetres: r = 3.2 dm = 32 cm.Step 2: Volume of one cube = 8^3 = 512 cubic cm.Step 3: Total volume of 44 cubes = 44 * 512 = 22528 cubic cm.Step 4: Let h be the length of the cylindrical rod in cm. Volume of cylinder = π * r^2 * h = (22 / 7) * 32^2 * h.Step 5: Equate volumes: (22 / 7) * 32^2 * h = 22528.Step 6: Compute 32^2 = 1024, so (22 / 7) * 1024 * h = 22528.Step 7: Simplify: (22528 * 7) / (22 * 1024) = h, which evaluates to h = 7 cm. Verification / Alternative check: You can quickly check by substituting h = 7 cm back into the cylinder volume formula and confirming that the result is 22528 cubic cm, which matches the combined volume of the 44 cubes. This confirms the calculation is consistent. Why Other Options Are Wrong: 56 cm and 5.6 cm give cylinder volumes that are far too large or too small compared to 22528 cubic cm. The option 0.7 cm is off by a factor of ten and clearly unrealistic. Only 7 cm exactly matches the required equality of volumes. Common Pitfalls: Students often forget to convert decimetres to centimetres or misapply π. Another common mistake is to multiply instead of equating total volumes, or to forget that all 44 cubes together form the cylinder material. Carefully managing units prevents these errors. Final Answer: The length of the original cylindrical rod is 7 cm.

Question 13

The volumes of two right circular cones are in the ratio 1 : 4 and the diameters of their circular bases are in the ratio 4 : 5. What is the ratio of the heights of the two cones?

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Introduction / Context: This question checks your understanding of how the volume of a right circular cone depends on both its base radius and height. You are given the ratio of volumes and the ratio of diameters, and you must infer the resulting ratio of heights using proportional reasoning. Given Data / Assumptions: Volume ratio of cones V1 : V2 = 1 : 4. Ratio of diameters d1 : d2 = 4 : 5. Therefore radii also satisfy r1 : r2 = 4 : 5. We need h1 : h2, the ratio of the heights. Concept / Approach: The volume of a cone is given by V = (1 / 3) * π * r^2 * h. For two cones, the ratio of their volumes equals the product of the ratio of the squares of their radii and the ratio of their heights. We set up an equation using the given ratios and solve for the height ratio h1 : h2. Step-by-Step Solution: Step 1: Write the volume ratio: V1 / V2 = (1 / 3 * π * r1^2 * h1) / (1 / 3 * π * r2^2 * h2) = (r1^2 * h1) / (r2^2 * h2).Step 2: Substitute the given volume ratio: (r1^2 * h1) / (r2^2 * h2) = 1 / 4.Step 3: Use r1 : r2 = 4 : 5, so r1^2 : r2^2 = 16 : 25.Step 4: Therefore (16 * h1) / (25 * h2) = 1 / 4.Step 5: Cross multiply to get 64 * h1 = 25 * h2.Step 6: Hence h1 / h2 = 25 / 64, so the ratio of heights is 25 : 64. Verification / Alternative check: You can assume a convenient value such as h1 = 25 and h2 = 64 and compute the corresponding volumes numerically to see that their ratio indeed becomes 1 : 4, confirming the derived relationship. Why Other Options Are Wrong: The ratios 1 : 5, 4 : 25 and 16 : 25 do not satisfy the volume ratio when combined with the squared radius ratio 16 : 25. Only the height ratio 25 : 64 produces the required overall volume ratio of 1 : 4. Common Pitfalls: A frequent mistake is to directly divide the volume ratio by the radius ratio instead of using the squared radius. Another error is to confuse diameter ratio with radius ratio. Remember that volume depends on r^2, not r alone. Final Answer: The heights of the two cones are in the ratio 25 : 64.

Question 14

Consider the following statements about similar triangles and polygons. Which one of the following statements is true?

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Introduction / Context: This is a conceptual question from geometry about the definition and properties of similar triangles and polygons. It asks you to identify which statement correctly describes when triangles are similar and to carefully distinguish similarity from congruence and equal area. Given Data / Assumptions: We deal with triangles and polygons in Euclidean geometry. Four candidate statements about similarity are given. We must select the single correct statement. Concept / Approach: Two figures are similar if they have the same shape. For triangles, a key criterion is that corresponding sides are in proportion and corresponding angles are equal. In fact, for triangles, proportionality of corresponding sides alone is sufficient to guarantee similarity because angle equality follows automatically. Step-by-Step Solution: Step 1: Review option (a): Similar triangles have equal shape but can be of different size, so they are not always congruent. Thus (a) is false.Step 2: Option (b): Similar triangles can be scaled versions of each other, so their areas are generally not equal. This is false.Step 3: Option (c): For triangles, if the ratios of corresponding sides are equal, then the triangles are similar. This is a standard similarity criterion, so (c) is true.Step 4: Option (d): For general polygons, proportional sides alone do not guarantee similarity; corresponding angles must also be equal. Hence this statement is incomplete and false. Verification / Alternative check: You can recall standard triangle similarity tests such as side side side similarity, angle side angle similarity and angle angle angle similarity. The side side side similarity test directly states that proportionality of corresponding sides implies similarity, supporting option (c). Why Other Options Are Wrong: Options (a) and (e) incorrectly equate similarity with congruence. Option (b) incorrectly states equal areas, which is not required. Option (d) ignores the angle condition for polygons with more than three sides. Common Pitfalls: Many learners confuse similarity and congruence, thinking that same shape always implies same size. Others wrongly assume equal area means similarity. Remember that similar figures can be scaled versions of each other, preserving shape but not necessarily size or area. Final Answer: The correct statement is Two triangles are similar if their corresponding sides are proportional.

Question 15

In triangle ABC, OB and OC are the internal bisectors of angles ∠B and ∠C respectively, meeting at point O. If angle ∠BAC = 60°, what is the measure (in degrees) of angle ∠BOC?

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Introduction / Context: This geometry question deals with angle bisectors inside a triangle and the special point where they intersect. That point is the incenter. The question uses a well known relationship between the angle at the incenter and the opposite vertex angle of the triangle. Given Data / Assumptions: Triangle ABC is any triangle. OB is the internal bisector of angle ∠B. OC is the internal bisector of angle ∠C. These two bisectors meet at point O, the incenter. Angle ∠BAC = 60°. We must find angle ∠BOC. Concept / Approach: The incenter is the intersection point of the three internal angle bisectors of a triangle. A key property is that the angle between the bisectors from vertices B and C at the incenter equals 90° plus half of the third angle, that is ∠BOC = 90° + ∠A / 2 where ∠A is the angle at vertex A of the triangle. Step-by-Step Solution: Step 1: Recognize that OB and OC are internal angle bisectors, so their intersection O is the incenter of triangle ABC.Step 2: Recall the formula for the incenter angle between the bisectors from B and C: ∠BOC = 90° + ∠A / 2.Step 3: Here ∠A = ∠BAC = 60°.Step 4: Substitute into the formula: ∠BOC = 90° + 60° / 2.Step 5: Compute 60° / 2 = 30°, then 90° + 30° = 120°.Step 6: Therefore the required angle is ∠BOC = 120°. Verification / Alternative check: You can imagine a specific triangle, for example an equilateral triangle with all angles 60°, and check that the incenter equals the circumcenter. In that case, ∠BOC would subtend the same arc as ∠BAC and numerical calculations also confirm the value of 120° using central angle properties. Why Other Options Are Wrong: 150°, 100° and 90° do not satisfy the formula 90° + ∠A / 2 for ∠A = 60°. Option 60° simply repeats the vertex angle and ignores the incenter relationship. Common Pitfalls: Common mistakes include thinking that ∠BOC equals ∠A, or applying the formula with the wrong vertex angle. Some students also confuse the incenter with the circumcenter and misapply central angle results. Remember that the special formula 90° + ∠A / 2 applies to the incenter angle. Final Answer: The measure of angle ∠BOC is 120°.

Question 16

In a right-angled triangle, the difference between the measures of the two acute (non right) angles is 8°. What is the measure (in degrees) of the smallest angle of the triangle?

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Introduction / Context: This is a basic angle sum question for a right angled triangle. Every right angled triangle has one 90° angle and two acute angles. The problem gives the difference between those two acute angles and asks you to find the smaller one using simple algebra. Given Data / Assumptions: The triangle is right angled, so one angle is 90°. Let the acute angles be θ and φ. The difference between the two smaller angles is 8°. We need the smallest angle of the triangle. Concept / Approach: In any triangle, the sum of the three interior angles is 180°. In a right triangle, one angle is 90°, so the other two must add up to 90°. Given their difference, you can set up a pair of linear equations and solve for the individual angles using basic algebraic techniques. Step-by-Step Solution: Step 1: Let the smaller acute angle be x degrees.Step 2: Then the larger acute angle is x + 8 degrees because their difference is 8°.Step 3: In a right triangle, sum of all angles is 180°. One angle is 90°, so x + (x + 8) + 90 = 180.Step 4: Simplify: 2x + 8 + 90 = 180, so 2x + 98 = 180.Step 5: Solve: 2x = 180 - 98 = 82, hence x = 41.Step 6: Therefore the smallest angle measures 41°. Verification / Alternative check: Check by computing the other acute angle: 41° + 8° = 49°. Now sum of angles is 90° (right angle) + 41° + 49° = 180°, which confirms the result is consistent. Why Other Options Are Wrong: 37°, 42°, 49° and 35° do not satisfy both conditions of adding with another angle to 90° and having a difference of exactly 8°. Only 41° leads to a matching pair of 41° and 49° that works. Common Pitfalls: Students sometimes forget to include the right angle in the 180° sum or subtract the difference incorrectly. Carefully defining variables for the smaller and larger acute angles and writing the equation avoids confusion. Final Answer: The smallest angle of the triangle is 41°.

Question 17

What is the slope of the straight line represented by the equation 2x + 3y = 12?

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Introduction / Context: This coordinate geometry question checks whether you can convert a line from its standard form to slope intercept form. Once the line is written as y = mx + c, the slope m can be read directly. Understanding this skill is important for graphing and analysing linear equations. Given Data / Assumptions: Equation of the line: 2x + 3y = 12. We need to find the slope of this line. The coordinate system is the usual Cartesian plane. Concept / Approach: The slope intercept form of a line is y = mx + c, where m is the slope and c is the y intercept. To find the slope from an equation given in standard form Ax + By + C = 0 or Ax + By = C, we solve for y and compare with y = mx + c to identify m. Step-by-Step Solution: Step 1: Start from the given equation: 2x + 3y = 12.Step 2: Isolate the term involving y: 3y = -2x + 12.Step 3: Divide every term by 3 to solve for y: y = (-2 / 3)x + 4.Step 4: Now the equation is in the form y = mx + c with m = -2 / 3 and c = 4.Step 5: Therefore, the slope of the line is -2 / 3. Verification / Alternative check: You can find the slope by calculating the change in y over the change in x between any two points on the line. For example, when x = 0, y = 4; when x = 3, y = 2. Then slope = (2 - 4) / (3 - 0) = -2 / 3, which matches the result. Why Other Options Are Wrong: 2/3 and 3/2 have the wrong sign or magnitude. The option -3/2 is the reciprocal with negative sign, which is not correct here. The value 1/2 has no relation to the coefficients of x and y in the given equation. Common Pitfalls: Students sometimes forget to divide the constant term when isolating y or they misplace the negative sign during rearrangement. Always solve carefully for y and then read the slope from the coefficient of x. Final Answer: The slope of the line 2x + 3y = 12 is -2/3.

Question 18

The surface area of a sphere is 6.16 square centimetres. What is the diameter (in cm) of the sphere?

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Introduction / Context: This is a mensuration question involving a sphere. You are given the total surface area and asked to find the diameter. To answer, you must recall the formula for the surface area of a sphere and then work backward to find the radius and diameter. Given Data / Assumptions: Surface area of the sphere S = 6.16 square cm. Surface area formula: S = 4 * π * r^2. We assume π is approximately 3.14 for numerical evaluation. We need the diameter, which equals 2r. Concept / Approach: The relationship between surface area and radius for a sphere is S = 4 * π * r^2. Given S, we solve for r^2 = S / (4 * π). Taking the square root gives r, and then we multiply by 2 to get the diameter. Finally, we match the calculated value to the closest option. Step-by-Step Solution: Step 1: Use S = 4 * π * r^2.Step 2: Substitute S = 6.16 and π ≈ 3.14: 6.16 = 4 * 3.14 * r^2.Step 3: Compute the denominator: 4 * 3.14 = 12.56.Step 4: Solve for r^2: r^2 = 6.16 / 12.56 ≈ 0.49.Step 5: Take the square root: r ≈ √0.49 = 0.7 cm.Step 6: Diameter d = 2 * r = 2 * 0.7 = 1.4 cm. Verification / Alternative check: Substitute r = 0.7 back into the surface area formula: S = 4 * 3.14 * 0.7^2 = 4 * 3.14 * 0.49 ≈ 6.16 square cm, which matches the given surface area and confirms that the diameter is correct. Why Other Options Are Wrong: 0.7 cm is the radius, not the diameter. The options 2.1 cm, 2.8 cm and 3.5 cm all correspond to larger radii, which would produce much larger surface areas than 6.16 square cm. Common Pitfalls: Students often forget to double the radius to obtain the diameter, or they invert the formula incorrectly. Another common error is rounding too early instead of keeping sufficient precision until the final step. Final Answer: The diameter of the sphere is 1.4 cm.

Question 19

What is the equation of the straight line that is perpendicular to the line 2x + 3y = -6 and has a y-intercept equal to 3?

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Introduction / Context: This problem involves finding the equation of a line in the coordinate plane that is perpendicular to a given line and has a specified y-intercept. It tests your understanding of slopes of perpendicular lines and converting between slope intercept and standard forms of linear equations. Given Data / Assumptions: Original line: 2x + 3y = -6. The new line must be perpendicular to this line. The y-intercept of the new line is 3. Concept / Approach: First, we find the slope of the given line by rewriting it in the form y = mx + c. Then, use the fact that the slope of a line perpendicular to a line with slope m is the negative reciprocal, that is -1 / m. With the perpendicular slope and the given y-intercept, we write the new line as y = m2 * x + 3 and then convert to standard form for comparison with the options. Step-by-Step Solution: Step 1: Rewrite 2x + 3y = -6 in slope intercept form: 3y = -2x - 6, so y = (-2 / 3)x - 2.Step 2: The slope of this line is m1 = -2 / 3.Step 3: The slope of a line perpendicular to this is m2 = 3 / 2, the negative reciprocal of -2 / 3.Step 4: Use the y-intercept 3 to write the new line: y = (3 / 2)x + 3.Step 5: Multiply both sides by 2: 2y = 3x + 6, or rearranged as 3x - 2y + 6 = 0.Step 6: Hence the standard form is 3x - 2y = -6. Verification / Alternative check: We can verify the y-intercept by setting x = 0 in 3x - 2y = -6, which gives -2y = -6 and y = 3, matching the given intercept. The product of slopes m1 * m2 = (-2 / 3) * (3 / 2) = -1, confirming perpendicularity. Why Other Options Are Wrong: The equation 3x - 2y = 6 has the correct slope but gives y-intercept -3. The other equations correspond to slopes that are not perpendicular to -2 / 3 or do not have the correct intercept of 3. Common Pitfalls: Students sometimes take the reciprocal without changing the sign or forget to use the correct y-intercept when forming the new line. Another frequent error is rearranging incorrectly when converting between forms. Final Answer: The required line is given by 3x - 2y = -6.

Question 20

What are the coordinates of the centroid of a triangle whose vertices are A(2, 5), B(-4, 0) and C(5, 4)?

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Introduction / Context: This problem is from coordinate geometry and asks you to find the centroid of a triangle from the coordinates of its vertices. The centroid is the point where the three medians of the triangle intersect and has a simple formula in terms of the coordinates of the vertices. Given Data / Assumptions: Triangle vertices are A(2, 5), B(-4, 0) and C(5, 4). We need the centroid coordinates G(x, y). The centroid is the average of the coordinates of the three vertices. Concept / Approach: For any triangle with vertices (x1, y1), (x2, y2) and (x3, y3), the centroid G has coordinates G = ((x1 + x2 + x3) / 3, (y1 + y2 + y3) / 3). We simply plug in the given coordinates and simplify to get the required point. Step-by-Step Solution: Step 1: Identify the coordinates of the vertices: A(2, 5), B(-4, 0), C(5, 4).Step 2: Compute the x coordinate of the centroid: (2 + (-4) + 5) / 3 = (2 - 4 + 5) / 3 = 3 / 3 = 1.Step 3: Compute the y coordinate of the centroid: (5 + 0 + 4) / 3 = 9 / 3 = 3.Step 4: Therefore the centroid G is at (1, 3). Verification / Alternative check: You can check that the centroid lies roughly at the middle of the triangle by plotting the three vertices on a rough sketch and visually estimating the intersection point of medians, which should be close to (1, 3), supporting the calculation. Why Other Options Are Wrong: The points (-1, 3), (1, -3), (-1, -3) and (0, 3) do not satisfy the centroid formula for the given vertices. They usually arise from sign mistakes or averaging only two points instead of all three. Common Pitfalls: Typical errors include forgetting to divide by 3, mixing x and y coordinates, or incorrectly adding negative values. Carefully grouping the sums and dividing by 3 for each coordinate prevents these mistakes. Final Answer: The coordinates of the centroid are (1, 3).

Volume and Surface Area Questions