Q: A rectangle has diagonal 25 cm and breadth (width) 7 cm. Find its perimeter in centimetres.

Introduction / Context:For a rectangle with length L and breadth W, the diagonal d satisfies d^2 = L^2 + W^2. From L, W we obtain the perimeter P = 2(L + W). Here d and W are given, so L follows from Pythagoras. Given Data / Assumptions: d = 25 cm, W = 7 cm. Concept / Approach:Compute L = √(d^2 − W^2) and then P = 2(L + W). Step-by-Step Solution: L = √(25^2 − 7^2) = √(625 − 49) = √576 = 24 cmPerimeter P = 2(L + W) = 2(24 + 7) = 62 cm Verification / Alternative check:Check the triple: 7, 24, 25 is a classic Pythagorean triple. Hence computations are consistent. Why Other Options Are Wrong:124 and 82 are double-counting or mixing values; 41 is half of 82 and not a valid perimeter given L and W. Final Answer:62

Q: For a regular hexagon (6 sides), by how many degrees is the measure of each interior angle greater than the corresponding exterior angle?

Introduction / Context:In any regular polygon with n sides, each exterior angle measures 360°/n, and each interior angle measures 180° − 360°/n. Their difference is straightforward to compute. Here n = 6 (regular hexagon). Given Data / Assumptions: n = 6. Concept / Approach:Compute exterior and interior angles explicitly and subtract: interior − exterior. Step-by-Step Solution: Exterior angle = 360°/6 = 60°Interior angle = 180° − 60° = 120°Difference = 120° − 60° = 60° Verification / Alternative check:The sum of one interior and its exterior is always 180°, so difference = 180° − 2*exterior = 180° − 120° = 60°, same result. Why Other Options Are Wrong:90°, 100°, 108° do not match the regular hexagon values; 108° is often confused with the interior angle of a regular pentagon. Common Pitfalls:Using 360°/n for the interior angle by mistake, or mixing n = 5 vs n = 6 values. Final Answer:60°

Q: A right circular cone has curved surface area (lateral area) 99 cm² and slant height 9 cm. Find its diameter in centimetres.

Introduction / Context:The curved surface area (CSA) of a right circular cone is CSA = π * r * l, where r is the base radius and l is the slant height. Given CSA and l, we can recover r and then the diameter 2r. Given Data / Assumptions: CSA = 99 cm² Slant height l = 9 cm Concept / Approach:Compute r = CSA / (π * l). Using π = 22/7 yields an exact value here. Then diameter D = 2r. Step-by-Step Solution: r = 99 / (π * 9) = 99 / (9 * 22/7) = 99 * 7 / 198 = 3.5 cmDiameter D = 2 * 3.5 = 7 cm Verification / Alternative check:CSA back-check: π * r * l = (22/7) * 3.5 * 9 = (22/7) * 31.5 = 99 cm², consistent. Why Other Options Are Wrong:3.5 is the radius (not diameter); 10.5 and 14 assume incorrect r; 7 is the correct diameter. Final Answer:7

Q: What is the measure, in degrees, of each exterior angle of a regular polygon that has 9 equal sides (a regular nonagon)?

Introduction: This geometry question tests your understanding of exterior angles of a regular polygon. For a regular polygon, all exterior angles are equal, and there is a simple relationship between the number of sides and each exterior angle. Given Data / Assumptions: The polygon is regular with 9 sides (a regular nonagon). Each exterior angle is equal in measure. We work with degrees. Concept / Approach: The sum of the measures of the exterior angles of any convex polygon, taking one exterior angle at each vertex, is always 360 degrees. For a regular polygon with n sides, each exterior angle is: Exterior angle = 360 / n degrees. We substitute n = 9 and compute this value. Step-by-Step Solution: Step 1: Identify the number of sides: n = 9. Step 2: Use the formula for one exterior angle of a regular polygon: Exterior angle = 360 / n. Step 3: Substitute n = 9: Exterior angle = 360 / 9 degrees. Step 4: Compute 360 / 9 = 40 degrees. Verification / Alternative check: As a check, you can find the interior angle first. Interior angle of a regular n sided polygon is: ( (n − 2) * 180 ) / n. For n = 9: Interior angle = (7 * 180) / 9 = 1260 / 9 = 140 degrees. Since each interior and exterior angle are supplementary for a regular polygon, interior + exterior = 180, so exterior = 180 − 140 = 40 degrees, which matches our earlier result. Why Other Options Are Wrong: 36, 30, 24, and 45 degrees do not equal 360 / 9. Choosing any of these would give an incorrect sum of exterior angles, not equal to 360 degrees. Common Pitfalls: Students sometimes confuse formulas for interior and exterior angles and use (n − 2) * 180 / n incorrectly. Others forget that the sum of all exterior angles is always 360 degrees, regardless of the number of sides. Remember that for a regular polygon, finding each exterior angle is as simple as dividing 360 degrees by the number of sides. Final Answer: Each exterior angle of the regular 9 sided polygon measures 40 degrees.

Question 1

Equal areas: circle vs square — compare diameter and diagonal: If a circle and a square have equal areas, what is (diameter of circle / diagonal of square)^2?

Accepted Answer

Introduction / Context:Equate the areas to relate the circle radius to the square side. Then express the requested squared ratio in those terms. Using π = 22/7 gives a clean rational result matching options. Given Data / Assumptions: Area(circle) = Area(square). Let circle radius = r, square side = s. Use π = 22/7 for final numeric comparison to options. Concept / Approach:πr^2 = s^2 ⇒ (diameter/diagonal)^2 = (2r / (s√2))^2 = (4r^2)/(2s^2) = 2r^2/s^2 = 2/π. Step-by-Step Solution: From equality: s^2 = πr^2.(diameter/diagonal)^2 = 2r^2 / s^2 = 2 / π.With π = 22/7, 2/π = 2 / (22/7) = 14/22 = 7/11. Verification / Alternative check:Direct substitution with any r (e.g., r = 1) produces the same 2/π expression; rationalizing with 22/7 yields 7/11. Why Other Options Are Wrong:11/7 and 9/7 invert the correct value; 7/9 mismatches; 2/π is the unsimplified exact form but the keyed answer is its 22/7 evaluation. Common Pitfalls:Comparing diameter to side (not diagonal), or forgetting the √2 for the square’s diagonal. Final Answer:7/11

Question 2

Sector angle from area and radius: The area of a circular sector is 462 sq cm and the circle’s radius is 21 cm. Find the sector’s central angle (in degrees).

Accepted Answer

Introduction / Context:Sector area relates to the full circle area by the angle fraction θ/360. With tidy numbers, the computation becomes clean using π = 22/7. Given Data / Assumptions: Sector area A_sector = 462 cm^2. Radius r = 21 cm ⇒ full circle area A_circle = πr^2. Concept / Approach:A_sector = (θ/360) * πr^2 ⇒ θ = 360 * A_sector / (πr^2). Step-by-Step Solution: πr^2 = (22/7)*441 = 22*63 = 1386.θ = 360 * 462 / 1386 = 360 / 3 = 120°. Verification / Alternative check:Plug back: (120/360)*1386 = 462, consistent. Why Other Options Are Wrong:90°, 60°, 30°, 150° do not satisfy the exact proportion with area 462. Common Pitfalls:Forgetting to divide by 360 or using diameter instead of radius in the area. Final Answer:120°

Question 3

Capacity of a canal with isosceles trapezium cross-section: Cross-section is an isosceles trapezium, 3 m wide at bottom and 5 m at top, depth 2 m, length 110 m. Find the maximum capacity (in cubic metres).

Accepted Answer

Introduction / Context:The canal’s capacity equals cross-sectional area times length. The cross-section is a trapezium with known parallel sides and height (depth). Given Data / Assumptions: Bottom width b1 = 3 m; top width b2 = 5 m. Depth h = 2 m. Length L = 110 m. Concept / Approach:Area(trapezium) = (1/2) * (b1 + b2) * h. Capacity = Area * L. Step-by-Step Solution: Area = (1/2) * (3 + 5) * 2 = 8 m^2.Capacity = 8 * 110 = 880 m^3. Verification / Alternative check:Units: (m^2)*m → m^3, correct. Why Other Options Are Wrong:1760, 1650, 1056, 990 do not equal 8*110. Common Pitfalls:Using the average width incorrectly or forgetting to multiply by the length. Final Answer:880 cubic metres

Question 4

In triangle PQR, the perpendicular bisectors OX, OY, and OZ meet at O (the circumcenter). Given ∠QPR = 65° and ∠PQR = 60°, find the value (in degrees) of ∠QOR + ∠POR.

Accepted Answer

Introduction / Context:This problem uses the circumcenter O of triangle PQR. The perpendicular bisectors of the three sides meet at O. For any arc in a circle, the central angle at O is twice the inscribed angle that subtends the same chord. We will convert given vertex angles into corresponding central angles and then add them appropriately. Given Data / Assumptions: ∠QPR = 65° (angle at P) ∠PQR = 60° (angle at Q) O is the circumcenter; ∠QOR subtends arc QR; ∠POR subtends arc PR. Concept / Approach:In a circumcircle, a central angle equals twice the inscribed angle that subtends the same arc. Therefore, ∠QOR = 2 * ∠QPR and ∠POR = 2 * ∠PQR. Sum them to obtain the target value. Step-by-Step Solution: ∠QOR = 2 * ∠QPR = 2 * 65° = 130°∠POR = 2 * ∠PQR = 2 * 60° = 120°∠QOR + ∠POR = 130° + 120° = 250° Verification / Alternative check:Using triangle angle sum, ∠PRQ = 180° − (65° + 60°) = 55°. The three corresponding central angles would be 130°, 120°, and 110°, which sum to 360°, consistent with a full circle. Our selected pair (130° and 120°) adds to 250° as computed. Why Other Options Are Wrong: 180: Would require ∠QPR + ∠PQR = 90°, which is not the case. 210 or 125: Do not match 2*(65°) + 2*(60°). Common Pitfalls: Confusing the inscribed angle with the central angle (forgetting the factor 2). Using the wrong vertex angle for a given arc. Final Answer:250

Question 5

In right triangle PQR with ∠PQR = 90°, the leg PQ = 10 cm and the hypotenuse PR = 26 cm. Find the inradius r (radius of the incircle) in centimetres.

Accepted Answer

Introduction / Context:The inradius r of a right triangle has a convenient formula in terms of side lengths: if the legs are a and b and the hypotenuse is c, then r = (a + b − c) / 2. We will first find the missing leg using the Pythagorean theorem, then apply the inradius formula. Given Data / Assumptions: Right angle at Q, so PR is the hypotenuse. PQ = 10 cm, PR = 26 cm. Concept / Approach:Use Pythagoras to get the other leg QR. For a right triangle, r = (a + b − c)/2, where c is the hypotenuse. Step-by-Step Solution: QR = √(PR^2 − PQ^2) = √(26^2 − 10^2) = √(676 − 100) = √576 = 24 cmLet a = 10, b = 24, c = 26.r = (a + b − c) / 2 = (10 + 24 − 26) / 2 = 8 / 2 = 4 cm Verification / Alternative check:Area K = (1/2) * a * b = (1/2) * 10 * 24 = 120. Also, K = r * s with semiperimeter s = (a + b + c)/2 = (10 + 24 + 26)/2 = 30. Then r = K/s = 120/30 = 4, consistent. Why Other Options Are Wrong: 9, 8, 6: Do not satisfy r = (a + b − c)/2 for the given triple 10–24–26. Common Pitfalls:Using (a + b + c)/2 instead of (a + b − c)/2; mixing up which side is the hypotenuse. Final Answer:4

Question 6

A single straight line cuts two concentric circles (same center). The chord lengths formed on the inner and outer circles are 4 cm and 16 cm, respectively. Find the difference between the squares of their radii (in cm²).

Accepted Answer

Introduction / Context:When a line at distance d from the center cuts a circle of radius r, the chord length L satisfies L = 2 * √(r^2 − d^2). With two concentric circles cut by the same line, d is common to both, enabling an easy elimination approach to find r2^2 − r1^2. Given Data / Assumptions: Inner circle chord length L1 = 4 cm. Outer circle chord length L2 = 16 cm. Both chords arise from the same line, so the perpendicular distance d from center to the line is the same. Concept / Approach:Use the relation (L/2)^2 = r^2 − d^2 for each circle, then subtract to eliminate d^2 and directly obtain the difference of squared radii. Step-by-Step Solution: For inner circle: (4/2)^2 = r1^2 − d^2 ⇒ 2^2 = r1^2 − d^2 ⇒ r1^2 − d^2 = 4For outer circle: (16/2)^2 = r2^2 − d^2 ⇒ 8^2 = r2^2 − d^2 ⇒ r2^2 − d^2 = 64Subtract: (r2^2 − d^2) − (r1^2 − d^2) = 64 − 4 ⇒ r2^2 − r1^2 = 60 Verification / Alternative check:Pick any d with 0 < d < r1; for example d = 1 gives r1^2 = 5, r2^2 = 65, difference = 60, confirming independence of d. Why Other Options Are Wrong: 240, 120, 90: Each assumes incorrect squaring or arithmetic; only 60 satisfies the chord–radius relation. Common Pitfalls:Forgetting to halve the chord length before squaring; mixing up r^2 − (L/2)^2 with (r − L/2)^2. Final Answer:60

Question 7

A rectangle has diagonal 25 cm and breadth (width) 7 cm. Find its perimeter in centimetres.

Accepted Answer

Introduction / Context:For a rectangle with length L and breadth W, the diagonal d satisfies d^2 = L^2 + W^2. From L, W we obtain the perimeter P = 2(L + W). Here d and W are given, so L follows from Pythagoras. Given Data / Assumptions: d = 25 cm, W = 7 cm. Concept / Approach:Compute L = √(d^2 − W^2) and then P = 2(L + W). Step-by-Step Solution: L = √(25^2 − 7^2) = √(625 − 49) = √576 = 24 cmPerimeter P = 2(L + W) = 2(24 + 7) = 62 cm Verification / Alternative check:Check the triple: 7, 24, 25 is a classic Pythagorean triple. Hence computations are consistent. Why Other Options Are Wrong:124 and 82 are double-counting or mixing values; 41 is half of 82 and not a valid perimeter given L and W. Final Answer:62

Question 8

For a regular hexagon (6 sides), by how many degrees is the measure of each interior angle greater than the corresponding exterior angle?

Accepted Answer

Introduction / Context:In any regular polygon with n sides, each exterior angle measures 360°/n, and each interior angle measures 180° − 360°/n. Their difference is straightforward to compute. Here n = 6 (regular hexagon). Given Data / Assumptions: n = 6. Concept / Approach:Compute exterior and interior angles explicitly and subtract: interior − exterior. Step-by-Step Solution: Exterior angle = 360°/6 = 60°Interior angle = 180° − 60° = 120°Difference = 120° − 60° = 60° Verification / Alternative check:The sum of one interior and its exterior is always 180°, so difference = 180° − 2*exterior = 180° − 120° = 60°, same result. Why Other Options Are Wrong:90°, 100°, 108° do not match the regular hexagon values; 108° is often confused with the interior angle of a regular pentagon. Common Pitfalls:Using 360°/n for the interior angle by mistake, or mixing n = 5 vs n = 6 values. Final Answer:60°

Question 9

A right circular cone has curved surface area (lateral area) 99 cm² and slant height 9 cm. Find its diameter in centimetres.

Accepted Answer

Introduction / Context:The curved surface area (CSA) of a right circular cone is CSA = π * r * l, where r is the base radius and l is the slant height. Given CSA and l, we can recover r and then the diameter 2r. Given Data / Assumptions: CSA = 99 cm² Slant height l = 9 cm Concept / Approach:Compute r = CSA / (π * l). Using π = 22/7 yields an exact value here. Then diameter D = 2r. Step-by-Step Solution: r = 99 / (π * 9) = 99 / (9 * 22/7) = 99 * 7 / 198 = 3.5 cmDiameter D = 2 * 3.5 = 7 cm Verification / Alternative check:CSA back-check: π * r * l = (22/7) * 3.5 * 9 = (22/7) * 31.5 = 99 cm², consistent. Why Other Options Are Wrong:3.5 is the radius (not diameter); 10.5 and 14 assume incorrect r; 7 is the correct diameter. Final Answer:7

Question 10

In a regular decagon (a polygon with 10 equal sides), what is the measure of each interior angle in degrees?

Accepted Answer

Introduction: This geometry problem checks your knowledge of interior angles of regular polygons. A regular decagon has 10 equal sides and 10 equal interior angles. You must apply the standard formula for interior angles of a polygon and then find the measure of each angle. Given Data / Assumptions: The polygon is a regular decagon. Number of sides n = 10. All interior angles are equal in a regular polygon. We work in degrees. Concept / Approach: The sum of interior angles of an n sided polygon is given by: Sum of interior angles = (n − 2) * 180 degrees. For a regular polygon, each interior angle is this sum divided by n. We substitute n = 10 for a decagon and simplify to get the measure of one interior angle. Step-by-Step Solution: Step 1: Number of sides n = 10. Step 2: Use the formula for sum of interior angles: Sum = (n − 2) * 180 degrees. Step 3: Substitute n = 10: Sum = (10 − 2) * 180 = 8 * 180 = 1440 degrees. Step 4: In a regular decagon, each interior angle is equal, so: Each interior angle = Sum / n = 1440 / 10. Step 5: Compute 1440 / 10 = 144 degrees. Verification / Alternative check: As a rough check, note that for large n, interior angles approach 180 degrees. For n = 10, we expect a value somewhat less than 180 degrees but greater than 120 degrees (which would correspond to a hexagon). The result of 144 degrees fits neatly between these limits and matches typical values known for regular polygons. Why Other Options Are Wrong: 120° and 140°: These are too small for a regular decagon, which has more sides than a hexagon or octagon and therefore larger interior angles. 150° and 160°: These are too large. They would correspond to polygons with more than 10 sides. Common Pitfalls: Students sometimes confuse the formula for exterior angles with that for interior angles, or forget to divide the total sum by the number of sides. Another frequent mistake is using (n * 180) instead of (n − 2) * 180. Always remember that a triangle (n = 3) has 180 degrees in total, which leads to the general formula (n − 2) * 180. Final Answer: Each interior angle of a regular decagon measures 144°.

Question 11

The circumference of a circle is 88 cm. Using the standard value of π = 22/7, what is the area of the circle in square centimetres?

Accepted Answer

Introduction: This problem combines properties of circles and basic algebra. You are given the circumference of a circle and asked to find its area. To solve it, you first determine the radius using the circumference formula, then use the area formula for a circle. Given Data / Assumptions: Circumference of the circle (C) = 88 cm. Use π = 22/7 for calculations. We must find the area A in square centimetres. Standard formulas for circumference and area of a circle apply. Concept / Approach: The circumference of a circle is given by: C = 2 * π * r, where r is the radius. Once r is found, we use the area formula: A = π * r^2. Both formulas must use the same value of π and consistent units (centimetres here). Step-by-Step Solution: Step 1: Use the circumference formula C = 2 * π * r. Step 2: Substitute C = 88 and π = 22/7: 88 = 2 * (22/7) * r. Step 3: Simplify the right side: 2 * (22/7) = 44/7, so 88 = (44/7) * r. Step 4: Multiply both sides by 7: 88 * 7 = 44 * r, so 616 = 44r. Step 5: Solve for r: r = 616 / 44 = 14 cm. Step 6: Now compute the area using A = π * r^2: A = (22/7) * 14^2 = (22/7) * 196. Step 7: Calculate: 196 / 7 = 28, so A = 22 * 28 = 616 sq cm. Verification / Alternative check: If r = 14 cm, the circumference should be C = 2 * π * r = 2 * (22/7) * 14 = (44/7) * 14 = 44 * 2 = 88 cm, which matches the given value. This confirms that r and therefore the area calculation are correct. Why Other Options Are Wrong: 308 sq cm, 154 sq cm, and 77 sq cm do not correspond to the area of a circle with radius 14 cm. 484 sq cm would be the area of a circle with radius sqrt(484/π), which does not match r = 14 cm and does not satisfy the given circumference. Common Pitfalls: Some students mistakenly use diameter instead of radius or forget to square the radius in the area formula. Others mis-handle the fraction π = 22/7, especially when simplifying expressions like (44/7) * r. Working step by step and simplifying carefully reduces such errors. Final Answer: The area of the circle is 616 sq cm.

Question 12

In an equilateral triangle, the radius of the circumcircle (the circle passing through all three vertices) is 14 cm. What is the radius, in centimetres, of the incircle (the circle inscribed inside the triangle and tangent to all three sides)?

Accepted Answer

Introduction: This question assesses your understanding of special properties of an equilateral triangle, particularly the relationship between its circumradius (radius of the circumscribed circle) and inradius (radius of the inscribed circle). Knowing these standard geometric relationships allows you to solve the problem quickly without extensive trigonometry. Given Data / Assumptions: The triangle is equilateral, meaning all sides and all angles are equal. Circumradius R = 14 cm. We must find the inradius r in centimetres. Standard geometric relations for equilateral triangles apply. Concept / Approach: For an equilateral triangle with side length a, the circumradius R and inradius r have the following relationship: R = 2r. This is a well known result derived from the geometry of equilateral triangles and the position of the centroid, incenter, and circumcenter, which all coincide in such a triangle. Therefore, if we know R, we can get r directly as r = R / 2. Step-by-Step Solution: Step 1: Use the relation between circumradius and inradius for an equilateral triangle: R = 2r. Step 2: The given circumradius is R = 14 cm. Step 3: Rearrange the formula to find r: r = R / 2. Step 4: Substitute R = 14: r = 14 / 2 = 7 cm. Verification / Alternative check: Another approach is to express R and r in terms of the side a. For an equilateral triangle: R = a / (sqrt(3)) and r = a * sqrt(3) / 6. Dividing R by r gives: R / r = [a / sqrt(3)] / [a * sqrt(3) / 6] = 6 / 3 = 2, so R = 2r again. With R = 14, we get r = 7, which confirms our earlier result. Why Other Options Are Wrong: 14: This would imply r = R, which contradicts the known relation R = 2r. 28 and 35: These values are even larger than R, which is impossible because the incircle must fit inside the triangle. 21: Also larger than R, so it cannot represent the inradius. Common Pitfalls: Some learners confuse formulas of inradius and circumradius or misremember the relation. Others attempt to derive everything from scratch using height and side relationships, which is more time consuming. Memorising and correctly applying R = 2r for equilateral triangles is very helpful in competitive exams. Final Answer: The radius of the inscribed circle is 7 cm.

Question 13

What is the measure, in degrees, of each exterior angle of a regular polygon that has 9 equal sides (a regular nonagon)?

Accepted Answer

Introduction: This geometry question tests your understanding of exterior angles of a regular polygon. For a regular polygon, all exterior angles are equal, and there is a simple relationship between the number of sides and each exterior angle. Given Data / Assumptions: The polygon is regular with 9 sides (a regular nonagon). Each exterior angle is equal in measure. We work with degrees. Concept / Approach: The sum of the measures of the exterior angles of any convex polygon, taking one exterior angle at each vertex, is always 360 degrees. For a regular polygon with n sides, each exterior angle is: Exterior angle = 360 / n degrees. We substitute n = 9 and compute this value. Step-by-Step Solution: Step 1: Identify the number of sides: n = 9. Step 2: Use the formula for one exterior angle of a regular polygon: Exterior angle = 360 / n. Step 3: Substitute n = 9: Exterior angle = 360 / 9 degrees. Step 4: Compute 360 / 9 = 40 degrees. Verification / Alternative check: As a check, you can find the interior angle first. Interior angle of a regular n sided polygon is: ( (n − 2) * 180 ) / n. For n = 9: Interior angle = (7 * 180) / 9 = 1260 / 9 = 140 degrees. Since each interior and exterior angle are supplementary for a regular polygon, interior + exterior = 180, so exterior = 180 − 140 = 40 degrees, which matches our earlier result. Why Other Options Are Wrong: 36, 30, 24, and 45 degrees do not equal 360 / 9. Choosing any of these would give an incorrect sum of exterior angles, not equal to 360 degrees. Common Pitfalls: Students sometimes confuse formulas for interior and exterior angles and use (n − 2) * 180 / n incorrectly. Others forget that the sum of all exterior angles is always 360 degrees, regardless of the number of sides. Remember that for a regular polygon, finding each exterior angle is as simple as dividing 360 degrees by the number of sides. Final Answer: Each exterior angle of the regular 9 sided polygon measures 40 degrees.

Question 14

In right triangle ΔDEF, the right angle is at E. If ∠D = 30° and EF = 6√3 cm, what is the length of side DE in centimetres?

Accepted Answer

Introduction: This trigonometry based geometry question involves a right angled triangle with one acute angle of 30 degrees. It uses the special properties of 30°–60°–90° triangles, where side lengths maintain a fixed ratio. You are given one side and must find another using these relationships. Given Data / Assumptions: Triangle ΔDEF is right angled at E, so ∠E = 90°. ∠D = 30°; hence ∠F = 60° (since the sum of angles in a triangle is 180°). Side EF = 6√3 cm. We need to find the length of side DE in centimetres. Concept / Approach: In a right triangle with angles 30°, 60°, and 90°, the side opposite 30° is the shortest, the side opposite 60° is √3 times the shortest side, and the hypotenuse is twice the shortest side. The ratio of sides opposite 30°, 60°, and 90° is: 1 : √3 : 2. We first identify which sides correspond to these angles, then use the ratio to compute DE. Step-by-Step Solution: Step 1: ∠D = 30°, ∠E = 90°, and ∠F = 60°. Step 2: Side EF is opposite ∠D (which is 30°), so EF is the side opposite 30°. Step 3: In the 1 : √3 : 2 ratio, let the side opposite 30° be x. Then: side opposite 60° = x√3 and hypotenuse = 2x. Step 4: Here, EF = x = 6√3? Actually, from the ratio, EF should equal x. But EF is given as 6√3, so x = 6√3. Step 5: Side DE is opposite ∠F (60°), so DE = x√3. Step 6: Substitute x = 6√3: DE = 6√3 * √3 = 6 * 3 = 18 cm. Verification / Alternative check: We can compute the hypotenuse DF = 2x = 2 * 6√3 = 12√3. Then check the Pythagorean theorem: DE^2 + EF^2 = 18^2 + (6√3)^2 = 324 + 36 * 3 = 324 + 108 = 432. Hypotenuse DF squared is (12√3)^2 = 144 * 3 = 432, so the values satisfy the Pythagorean theorem and are consistent. Why Other Options Are Wrong: 12, 9: These are too small and do not satisfy the 1 : √3 : 2 ratio for the given EF. 12√3 and 18√3: These are larger than required and would not maintain the correct trigonometric relationships with EF = 6√3. Common Pitfalls: A frequent mistake is mixing up which side is opposite which angle. Remember that the side opposite 30° is the smallest, and the side opposite 60° is √3 times that side. Another error is miscomputing √3 * √3 as 3, so care must be taken when simplifying surds. Final Answer: The length of DE is 18 cm.

Question 15

The length and breadth of a rectangle are 24 cm and 7 cm respectively. Using the standard perimeter formula, what is the perimeter of this rectangle in centimetres?

Accepted Answer

Introduction: This is a basic geometry question about rectangles. It tests whether you know and can apply the formula for the perimeter of a rectangle using its length and breadth. Such questions are common in aptitude tests to check speed and accuracy in applying simple formulas. Given Data / Assumptions: Length of the rectangle (L) = 24 cm. Breadth of the rectangle (B) = 7 cm. We are asked to find the perimeter P in centimetres. All sides of a rectangle are straight line segments and opposite sides are equal. Concept / Approach: The perimeter of a rectangle is the total distance around the figure. Since a rectangle has two equal lengths and two equal breadths, its perimeter is given by: P = 2 * (L + B). We simply substitute the given values of L and B into this formula and compute the result. Step-by-Step Solution: Step 1: Write the perimeter formula: P = 2 * (L + B). Step 2: Substitute the given length and breadth: L = 24 cm, B = 7 cm. Step 3: Compute L + B: L + B = 24 + 7 = 31. Step 4: Multiply by 2: P = 2 * 31 = 62 cm. Verification / Alternative check: You can also think of the perimeter as adding all four sides explicitly: P = L + B + L + B = 24 + 7 + 24 + 7 = 62 cm. This matches the formula based calculation and confirms that the perimeter is correct. Why Other Options Are Wrong: 48: Could result from 2 * 24, ignoring the breadth. 124: Looks like 4 * 31 or miscalculation; it is exactly double the correct perimeter. 96: Could come from 4 * 24, incorrectly assuming a square with side 24 cm. 31: This is just L + B, not 2(L + B), so it is half the true perimeter. Common Pitfalls: Some learners forget to multiply by 2 and stop at L + B. Others wrongly apply formulas from area (L * B) instead of perimeter. To avoid confusion, remember that perimeter is the boundary length around a shape, while area is the region inside it. Final Answer: The perimeter of the rectangle is 62 cm.

Question 16

What are the coordinates of the reflection of the point (−0.5, 6) in the x-axis on the Cartesian plane?

Accepted Answer

Introduction: This coordinate geometry question tests your understanding of reflections in the Cartesian plane, specifically about reflecting a point across the x-axis. It is a common concept used in graphs, transformations, and analytic geometry. Given Data / Assumptions: Original point: (−0.5, 6). We must reflect this point across the x-axis. The x-axis is the horizontal axis where y = 0. Concept / Approach: Reflection in the x-axis changes the sign of the y-coordinate, while the x-coordinate remains unchanged. In other words, the reflection of a point (x, y) in the x-axis is (x, −y). This arises because the x-axis is the line of symmetry and vertical distances are mirrored across it. Step-by-Step Solution: Step 1: Identify the coordinates of the given point: x = −0.5, y = 6. Step 2: Use the rule for reflection in the x-axis: (x, y) → (x, −y). Step 3: Apply this rule to the given point: (−0.5, 6) → (−0.5, −6). Step 4: Therefore, the reflected point is (−0.5, −6). Verification / Alternative check: Graphically, if you draw the point (−0.5, 6) on the plane, it lies 6 units above the x-axis. Its reflection must lie 6 units below the x-axis directly underneath, which has the same x-coordinate but a y-coordinate of −6. This confirms that (−0.5, −6) is indeed the correct reflected point. Why Other Options Are Wrong: (0.5, −6): Changes both the x and y signs; this reflects across the origin, not merely the x-axis. (−6, −0.5) and (6, −0.5): These swap coordinates and adjust signs; they do not follow the rule of reflection in the x-axis. (−0.5, 6): This is the original point and shows no reflection at all. Common Pitfalls: Students sometimes confuse reflections in the x-axis and y-axis. For reflection in the x-axis, only the y-coordinate changes sign. For reflection in the y-axis, only the x-coordinate changes sign. Mixing these up leads to the wrong image point. Final Answer: The reflection of the point (−0.5, 6) in the x-axis is (−0.5, −6).

Question 17

Two triangles ΔABC and ΔPQR are similar. The area of ΔABC is 36 sq cm and the area of ΔPQR is 9 sq cm. If side PQ = 4 cm corresponds to side AB, what is the length of AB in centimetres?

Accepted Answer

Introduction: This problem uses properties of similar triangles. When two triangles are similar, the ratio of their corresponding sides is related to the square root of the ratio of their areas. This question tests your understanding of this connection between side lengths and areas. Given Data / Assumptions: Triangles ΔABC and ΔPQR are similar. Area of ΔABC = 36 sq cm. Area of ΔPQR = 9 sq cm. Side PQ corresponds to side AB. PQ = 4 cm. We must find AB in centimetres. Concept / Approach: For similar triangles, the ratio of their areas equals the square of the ratio of corresponding sides. If k is the ratio of a side of ΔABC to the corresponding side of ΔPQR, then: (Area of ABC) / (Area of PQR) = k^2. We first compute the area ratio, then find k, and finally compute AB using the given length of PQ. Step-by-Step Solution: Step 1: Compute the ratio of areas: Area(ABC) / Area(PQR) = 36 / 9 = 4. Step 2: Let k be the ratio AB / PQ. Then: k^2 = 4. Step 3: Take the positive square root (since lengths are positive): k = sqrt(4) = 2. Step 4: Now AB / PQ = 2, so AB = 2 * PQ. Step 5: Substitute PQ = 4 cm: AB = 2 * 4 = 8 cm. Verification / Alternative check: If AB = 8 cm and PQ = 4 cm, then the ratio of corresponding sides is 8 : 4 = 2 : 1. The ratio of areas should then be the square of this, that is 2^2 = 4. Indeed, 36 / 9 = 4, which matches, so the result is consistent with similarity rules. Why Other Options Are Wrong: 16 and 12: These would give side ratios 16 : 4 = 4 and 12 : 4 = 3 respectively, leading to area ratios 16 and 9, which do not match 36 : 9. 6: This gives a side ratio of 6 : 4 = 1.5, area ratio 2.25, which is incorrect. 10: Also leads to an inconsistent area ratio when compared to 36 : 9. Common Pitfalls: Some learners directly compare side ratios with area ratios without squaring or taking square roots. Remember that area scales with the square of the linear scale factor in similar figures. Always relate side ratios and area ratios through the square relationship k^2. Final Answer: The length of AB is 8 cm.

Question 18

The perimeter of a semicircle (including its diameter) is 108 cm. Using π = 22/7, what is the radius of the semicircle in centimetres?

Accepted Answer

Introduction: This question examines your ability to apply the formula for the perimeter of a semicircle and to rearrange it to find the radius. A semicircle consists of half the circumference of a full circle plus the diameter as the straight edge. You are given the total perimeter and must work backwards to find the radius. Given Data / Assumptions: Perimeter of the semicircle (including the diameter) = 108 cm. Use π = 22/7. We must find the radius r in centimetres. The perimeter formula for a semicircle is known and includes both the curved part and the diameter. Concept / Approach: The perimeter P of a semicircle is given by: P = π * r + 2 * r, where π * r is half the circumference of the corresponding full circle and 2r is the diameter. We substitute P = 108 and π = 22/7, then solve the resulting linear equation for r. Step-by-Step Solution: Step 1: Write the formula for the perimeter of a semicircle: P = π * r + 2r. Step 2: Substitute P = 108 and π = 22/7: 108 = (22/7) * r + 2r. Step 3: Factor out r: 108 = r * (22/7 + 2). Step 4: Express 2 with denominator 7: 2 = 14/7, so (22/7 + 14/7) = 36/7. Step 5: Thus: 108 = r * (36/7). Step 6: Multiply both sides by 7: 108 * 7 = 36 * r, so 756 = 36r. Step 7: Solve for r: r = 756 / 36 = 21 cm. Verification / Alternative check: If r = 21 cm, then semicircular perimeter is: π * r + 2r = (22/7) * 21 + 2 * 21 = 22 * 3 + 42 = 66 + 42 = 108 cm, which matches the given perimeter. Therefore, the radius calculation is correct. Why Other Options Are Wrong: 28 and 42: These values are larger than 21 and would give a perimeter greater than 108 cm. 14: This would produce a perimeter π * 14 + 28 which is (22/7) * 14 + 28 = 44 + 28 = 72 cm, not 108 cm. 56: This is far too large and would lead to a much bigger perimeter than 108 cm. Common Pitfalls: A typical mistake is to use the full circumference formula 2 * π * r instead of π * r for the semicircular arc, or to forget to include the diameter 2r in the total perimeter. Another common error is mishandling the fraction 22/7 while simplifying. Carefully keeping track of terms and denominators avoids these errors. Final Answer: The radius of the semicircle is 21 cm.

Question 19

In plane geometry, if the length of each side of an equilateral triangle is 8 cm, what is the area of this equilateral triangle (in square centimetres)?

Accepted Answer

Introduction / Context: This is a standard mensuration question about the area of an equilateral triangle. Such problems are common in aptitude and competitive exams and test whether you remember and can correctly apply the formula for the area of an equilateral triangle in terms of its side length. Given Data / Assumptions: The triangle is equilateral (all sides and all angles equal). Each side has length 8 cm. We are asked to find the area in square centimetres. We assume Euclidean plane geometry and exact values with √3 left in surd form. Concept / Approach: For an equilateral triangle with side length a, the area formula is: Area = (√3 / 4) * a^2 This comes from drawing a height in the equilateral triangle, forming two 30°–60°–90° right triangles. One can also derive it using the general triangle area formula (1/2 * base * height) together with trigonometry for the height. Step-by-Step Solution: Let a = 8 cm. Use the formula Area = (√3 / 4) * a^2. Compute a^2: 8^2 = 64. Substitute: Area = (√3 / 4) * 64. Simplify 64 / 4 = 16, so Area = 16√3 sq cm. Verification / Alternative check: We know that an equilateral triangle with side 2 has area √3 sq units (since (√3 / 4) * 2^2 = √3). If the side is scaled up by a factor of 4 (from 2 to 8), area scales by the square of that factor, that is 4^2 = 16. Therefore the new area should be 16·√3, which matches our result of 16√3 sq cm. Why Other Options Are Wrong: 32√3 sq cm: This is double the correct area and would correspond to a larger side length. 16 sq cm and 32 sq cm ignore the √3 factor, so they do not respect the exact geometry of an equilateral triangle. Common Pitfalls: Students often confuse the formula for the area of an equilateral triangle with that of a general triangle or forget the √3 factor, writing (1/2)*a^2 instead. Another mistake is squaring only part of the expression. Always remember that for equilateral triangles, √3 is always present in the exact area expression. Final Answer: The area of the equilateral triangle is 16√3 sq cm.

Question 20

The lengths of the diagonals of a rhombus are 8 cm and 6 cm. Using the standard diagonal-based formula, what is the area of this rhombus in square centimetres?

Accepted Answer

Introduction / Context: Rhombus area problems test whether you know the important formula that links the area of a rhombus to the lengths of its diagonals. This formula is especially useful when side length is not given but diagonals are provided, as in many aptitude and geometry questions. Given Data / Assumptions: The figure is a rhombus. Length of the first diagonal d1 = 8 cm. Length of the second diagonal d2 = 6 cm. We are asked for the area in cm². Diagonals of a rhombus intersect at right angles and bisect each other. Concept / Approach: The area A of a rhombus in terms of its diagonals d1 and d2 is given by: A = (1/2) * d1 * d2 This comes from the fact that the diagonals divide the rhombus into four congruent right triangles, each with legs equal to half of the diagonals. Step-by-Step Solution: Given d1 = 8 cm and d2 = 6 cm. Use the formula A = (1/2) * d1 * d2. Substitute: A = (1/2) * 8 * 6. Compute 8 * 6 = 48. Then A = (1/2) * 48 = 24 cm2. Verification / Alternative check: Each right triangle formed by the diagonals has legs 4 cm and 3 cm (half of 8 and 6). The area of one right triangle is (1/2) * 4 * 3 = 6 cm2. There are four such triangles, so total area = 4 * 6 = 24 cm2, which matches our previous calculation. Why Other Options Are Wrong: 48 cm2 is the product d1 * d2 without multiplying by 1/2, so it is exactly double the correct area. 60 cm2 and 96 cm2 are even larger and have no basis in the standard rhombus area formula for these diagonal lengths. Common Pitfalls: A classic error is forgetting the factor of 1/2 and writing A = d1 * d2. Another pitfall is confusing side length with diagonal length; the usual parallelogram formula base * height is not directly usable without extra information, so the diagonal-based formula is the most efficient approach here. Final Answer: The area of the rhombus is 24 cm2.

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