Question 1

The curved surface area and the slant height of a right circular cone are 99 square centimetres and 9 cm respectively. What is the diameter of the base of the cone in centimetres?

Accepted Answer

Introduction / Context:This question involves a right circular cone for which the curved surface area and the slant height are given. The task is to find the diameter of the base. The problem checks knowledge of the curved surface area formula for a cone and the relationship between radius, slant height, and curved area. Given Data / Assumptions: Curved surface area of the cone A_c = 99 square centimetres. Slant height of the cone l = 9 cm. Let the base radius be r cm and the diameter be 2r cm. We use pi approximately equal to 22 / 7 unless otherwise specified. Concept / Approach:The curved surface area of a right circular cone is given by the formula A_c = pi * r * l. We know A_c and l, so we can solve this equation for r. Once r is found, the diameter is simply 2r. This approach uses direct substitution and basic algebra. Step-by-Step Solution:Step 1: Write the formula for curved surface area: A_c = pi * r * l.Step 2: Substitute the known values: 99 = pi * r * 9.Step 3: Rearrange to find r: r = 99 / (9 * pi) = 11 / pi.Step 4: Use pi = 22 / 7 to evaluate: r = 11 / (22 / 7) = 11 * 7 / 22 = 77 / 22 = 3.5 cm.Step 5: The diameter of the base is 2r = 2 * 3.5 = 7 cm. Verification / Alternative check:Substitute r = 3.5 cm and l = 9 cm back into the curved surface area formula.Compute A_c = pi * r * l = (22 / 7) * 3.5 * 9.Note that 3.5 = 7 / 2, so A_c = (22 / 7) * (7 / 2) * 9 = 22 * 9 / 2 = 198 / 2 = 99 square centimetres.This matches the given curved surface area, confirming that r = 3.5 cm and diameter 7 cm are correct. Why Other Options Are Wrong:A diameter of 3.5 cm would correspond to a radius of 1.75 cm, giving a much smaller curved area than 99 square centimetres.Diameters such as 10.5 cm or 14 cm represent larger radii which would give curved surface areas significantly greater than 99 square centimetres for slant height 9 cm.An option like 5.5 cm will not satisfy the exact relation 99 = pi * r * l when evaluated with pi = 22 / 7. Common Pitfalls:Learners may confuse the curved surface area formula with the total surface area formula, which includes the base area.Another error is to treat the given 99 as total area and then incorrectly add or subtract base area terms.Misuse of the approximation for pi or algebraic mistakes when solving for r can also lead to incorrect numerical values. Final Answer:The diameter of the base of the cone is 7 cm.

Question 2

Points P and Q lie on sides AB and AC of triangle ABC respectively such that segment PQ is parallel to side BC. If the ratio AP : PB is 2 : 5 and the area of triangle APQ is 4 square centimetres, what is the area of trapezium PQCB?

Accepted Answer

Introduction / Context:This geometry problem involves a triangle with a line segment drawn parallel to one side, creating a smaller similar triangle and a trapezium. The question requires using properties of similar triangles and area ratios to find the area of the trapezium formed between the parallel line and the base of the triangle. It tests understanding of similarity, proportional segments, and area scaling. Given Data / Assumptions: Triangle ABC is given. Point P lies on side AB and point Q lies on side AC. PQ is parallel to BC, so triangle APQ is similar to triangle ABC. AP : PB = 2 : 5, so AB is divided in that ratio. Area of triangle APQ is 4 square centimetres. We seek the area of trapezium PQCB, which is region ABC minus region APQ. Concept / Approach:Since PQ is parallel to BC, triangle APQ is similar to triangle ABC. The similarity ratio between the smaller triangle and the larger triangle is AP / AB. From AP : PB = 2 : 5, AB is divided into 2 + 5 = 7 equal parts and AP is 2 of those parts, so AP / AB = 2 / 7. For similar triangles, the ratio of areas is the square of the similarity ratio. Therefore, area(APQ) : area(ABC) = (2 / 7)^2. With the area of APQ known, we can compute the area of ABC and then subtract to get the area of pqcb. Step-by-Step Solution:Step 1: From AP : PB = 2 : 5, AB = 2 + 5 = 7 parts and AP = 2 parts.Step 2: So AP / AB = 2 / 7.Step 3: Because the triangles APQ and ABC are similar, area(APQ) / area(ABC) = (AP / AB)^2 = (2 / 7)^2 = 4 / 49.Step 4: Let area(ABC) = K. Then 4 / 49 = area(APQ) / K.Step 5: Given area(APQ) = 4 square centimetres, we have 4 / 49 = 4 / K, so K = 49.Step 6: Area of triangle ABC is therefore 49 square centimetres.Step 7: The area of trapezium PQCB is area(ABC) minus area(APQ) = 49 - 4 = 45 square centimetres. Verification / Alternative check:We can verify the area ratio by imagining scaling all sides of triangle APQ by a factor of 7 / 2 to obtain triangle ABC.Since area scales with the square of the linear scale factor, the area of ABC should be (7 / 2)^2 times the area of APQ, that is (49 / 4) * 4 = 49 square centimetres.Subtracting 4 from 49 again yields 45 square centimetres, matching the previous calculation. Why Other Options Are Wrong:An area of 49 square centimetres would be the area of the whole triangle ABC, not the trapezium.Values like 25 or 21 square centimetres underestimate the size of trapezium PQCB, as they ignore correct use of the similarity ratio.A value like 36 square centimetres corresponds to a different incorrect difference in areas and does not arise from the given ratio 2 : 5. Common Pitfalls:A common mistake is to treat AP : AB as 2 : 5 instead of 2 : 7, forgetting that AB equals AP plus PB.Another frequent error is to use a direct ratio of 2 : 7 for areas rather than squaring the ratio, which leads to wrong results.Some may try to treat the trapezium as a separate shape without using the powerful tool of similar triangles and area ratios. Final Answer:The area of trapezium PQCB is 45 sq cm.

Question 3

The length of a string is 8 feet 9 inches, and it is divided into 3 equal parts. What is the length of each part in inches?

Accepted Answer

Introduction / Context:This is a straightforward unit conversion and division problem. A length given in mixed units of feet and inches has to be converted entirely into inches and then divided into three equal parts. The question checks knowledge of simple unit conversion within the imperial system and basic arithmetic operations. Given Data / Assumptions: Total length of the string is 8 feet 9 inches. 1 foot equals 12 inches. The string is divided into 3 equal parts. We need the length of each part expressed in inches. Concept / Approach:The approach is to convert the entire length into a single unit, inches. This avoids handling mixed units while dividing. Once we have the total length in inches, we simply divide by 3 to get the length of each equal part. Finally, we compare that result with the options provided. Step-by-Step Solution:Step 1: Convert 8 feet into inches. Using 1 foot = 12 inches, 8 feet = 8 * 12 = 96 inches.Step 2: Add the extra 9 inches to this value: 96 + 9 = 105 inches.Step 3: The total length of the string is therefore 105 inches.Step 4: The string is divided into 3 equal parts, so length of each part = 105 / 3 inches.Step 5: Compute 105 / 3 = 35 inches.Step 6: Each equal part of the string is 35 inches long. Verification / Alternative check:Multiply the length of one part by 3 to check if we get back the total length: 35 * 3 = 105 inches.Convert 105 inches back into feet and inches: 105 / 12 = 8 feet with remainder 9 inches, because 8 * 12 = 96 and 105 - 96 = 9.Thus, 105 inches corresponds to 8 feet 9 inches, confirming that the conversion and division were carried out correctly. Why Other Options Are Wrong:Numbers like 31 or 33 inches, when multiplied by 3, give totals smaller than 105 inches, so they do not reconstruct the original length of the string.Values such as 37 inches or 29 inches, multiplied by 3, give totals greater or less than 105 inches, again failing to match the original length.Only 35 inches when tripled returns exactly the starting length of 8 feet 9 inches. Common Pitfalls:A typical mistake is to divide the feet and inches separately without first converting to a common unit, which can lead to fractional feet and inconsistent results.Another error is forgetting that 1 foot equals 12 inches and using an incorrect conversion factor.Rounding is not needed here because the division is exact, so any rounding would be unnecessary and misleading. Final Answer:Each part of the string is 35 inches long.

Question 4

The side of an equilateral triangle is 14 cm. What is the area of the triangle in square centimetres?

Accepted Answer

Introduction / Context:This problem asks for the area of an equilateral triangle when the side length is known. It tests familiarity with the standard area formula for equilateral triangles as well as skill in substituting and simplifying expressions involving square roots. Given Data / Assumptions: The triangle is equilateral with all three sides equal. Side length a = 14 cm. We need the area expressed in terms of square centimetres, possibly with a square root in the final answer. Concept / Approach:The area A of an equilateral triangle with side a is given by the formula A = (sqrt(3) / 4) * a^2. This formula comes from drawing an altitude, splitting the triangle into two right triangles, and using Pythagoras theorem. Once the formula is recalled, we simply substitute a = 14 cm and simplify. Step-by-Step Solution:Step 1: Recall the area formula: A = (sqrt(3) / 4) * a^2.Step 2: Substitute a = 14 cm: A = (sqrt(3) / 4) * 14^2.Step 3: Compute 14^2 = 196.Step 4: So A = (sqrt(3) / 4) * 196.Step 5: Simplify the fraction: 196 / 4 = 49.Step 6: Therefore A = 49 * sqrt(3) square centimetres, written as 49√3. Verification / Alternative check:We can also compute the height of the equilateral triangle and use the standard triangle area formula A = (1 / 2) * base * height.The height h equals (sqrt(3) / 2) * a, so for a = 14 cm, h = (sqrt(3) / 2) * 14 = 7 * sqrt(3).Then A = (1 / 2) * base * height = (1 / 2) * 14 * 7 * sqrt(3) = 7 * 7 * sqrt(3) = 49√3.This matches the previous result, confirming the correctness of the answer. Why Other Options Are Wrong:Options like 64√3 or 64√2 involve incorrect use of the formula or incorrect squaring of the side length.An option such as 49√2 would result from using a wrong height or misapplying Pythagoras theorem.Only 49√3 matches the exact computation based on the standard area formula for an equilateral triangle of side 14 cm. Common Pitfalls:A common mistake is to use the formula A = a^2 * sqrt(3) instead of A = (sqrt(3) / 4) * a^2, which overestimates the area.Another error is squaring 14 incorrectly or forgetting to divide by 4 after multiplying by a^2.Some learners may also mix up inradius or circumradius formulas with the area formula, which leads to incorrect options. Final Answer:The area of the equilateral triangle is 49√3 square centimetres.

Question 5

If the perimeter of a semicircle is 36 cm, what is the radius of the semicircle in centimetres?

Accepted Answer

Introduction / Context:This problem concerns the perimeter of a semicircle and its relationship with the radius. The perimeter of a semicircle includes the curved half circumferential part and the diameter. The question provides the total perimeter and asks for the radius. This tests knowledge of circle formulas and algebraic manipulation. Given Data / Assumptions: Perimeter of the semicircle P = 36 cm. Let the radius be r cm. Perimeter of a semicircle includes half the circumference plus the diameter. Use pi approximately equal to 22 / 7 for calculations. Concept / Approach:The perimeter of a full circle is 2 * pi * r. A semicircle has half this curved length plus the straight diameter. Therefore, the perimeter of a semicircle is P = pi * r + 2 * r. We set this equal to the given 36 cm and solve for r using the approximate value of pi. Step-by-Step Solution:Step 1: Write the formula for the perimeter of a semicircle: P = pi * r + 2 * r.Step 2: Substitute P = 36: 36 = pi * r + 2 * r.Step 3: Factor r: 36 = r * (pi + 2).Step 4: Use pi = 22 / 7. Then pi + 2 = 22 / 7 + 2 = 22 / 7 + 14 / 7 = 36 / 7.Step 5: Substitute into the equation: 36 = r * (36 / 7).Step 6: Solve for r: r = 36 * (7 / 36) = 7 cm. Verification / Alternative check:Substitute r = 7 cm back into the original formula for the perimeter.Compute P = pi * r + 2 * r = (22 / 7) * 7 + 2 * 7 = 22 + 14 = 36 cm.This matches the given perimeter, confirming that r = 7 cm is correct. Why Other Options Are Wrong:If r were 14 cm, the semicircle perimeter would be pi * 14 + 28, which is much larger than 36 cm.Values like 13 or 21 cm would similarly produce perimeters far from 36 cm when substituted into the formula.Thus, none of these values satisfy the equation 36 = pi * r + 2 * r. Common Pitfalls:A frequent mistake is to use only half the circumference pi * r and forget to add the diameter.Another error is to use 2 * pi * r directly as the perimeter of the semicircle, which corresponds to a full circle instead.Misusing the approximation for pi or incorrect algebra when isolating r can also lead to wrong answers. Final Answer:The radius of the semicircle is 7 centimetres.

Question 6

The diagonal of a rectangle is 26 cm and one of its sides is 10 cm. What is the area of the rectangle in square centimetres?

Accepted Answer

Introduction / Context:This rectangle problem provides the length of the diagonal and one side and asks for the area. The solution involves using the Pythagoras theorem to find the missing side, since the diagonal forms a right triangle with the sides of the rectangle, and then multiplying the two sides to obtain the area. Given Data / Assumptions: The figure is a rectangle. Diagonal length d = 26 cm. One side is 10 cm. Let this be the breadth b. Let the other side (length) be l cm. We use the relation d^2 = l^2 + b^2. Concept / Approach:In a rectangle, the diagonal acts as the hypotenuse of a right triangle whose legs are the length and breadth. Using the Pythagoras theorem, we can solve for the unknown side given the diagonal and one side, and then compute the area A = l * b. This is a standard application of right triangle geometry inside a rectangle. Step-by-Step Solution:Step 1: Use Pythagoras theorem for the rectangle: d^2 = l^2 + b^2.Step 2: Substitute d = 26 and b = 10: 26^2 = l^2 + 10^2.Step 3: Compute the squares: 676 = l^2 + 100.Step 4: Rearrange to get l^2: l^2 = 676 - 100 = 576.Step 5: Take the square root: l = sqrt(576) = 24 cm.Step 6: Now compute the area: A = l * b = 24 * 10 = 240 square centimetres. Verification / Alternative check:Verify the diagonal using the found side lengths: d^2 should equal l^2 + b^2.Compute l^2 + b^2 = 24^2 + 10^2 = 576 + 100 = 676.The square root of 676 is 26, which matches the given diagonal.Thus the side lengths and area are consistent with the given information. Why Other Options Are Wrong:An area of 260 square centimetres would require different side lengths that do not satisfy the Pythagoras relation with diagonal 26 cm and side 10 cm.Values like 120 square centimetres or 130 square centimetres are too small, while 180 square centimetres also does not match the product of any side pair consistent with the given diagonal.Only 240 square centimetres corresponds to the correct dimensions 24 cm by 10 cm. Common Pitfalls:A common error is to assume that the given side of 10 cm is the hypotenuse, which is incorrect, as the diagonal is always the hypotenuse in a rectangle.Another mistake is to confuse area with perimeter and attempt to use formulas that add sides rather than multiply them.Algebraic mistakes when rearranging 676 = l^2 + 100 can also lead to incorrect values for l. Final Answer:The area of the rectangle is 240 sq cm.

Question 7

The length of an arc of a circle is 22 cm and the radius of the circle is 28 cm. What is the measure of the corresponding central angle in degrees?

Accepted Answer

Introduction / Context:This question involves the relationship between the length of an arc, the radius of the circle, and the central angle that subtends the arc. It tests understanding of arc length formulas in degrees and the ability to solve for the angle when arc length and radius are known. Given Data / Assumptions: Arc length s = 22 cm. Radius of the circle r = 28 cm. Let the central angle be θ degrees. We use the degree based arc length formula. Use pi approximately equal to 22 / 7 for calculations. Concept / Approach:The length of an arc in a circle when the central angle is given in degrees is s = (θ / 360) * 2 * pi * r. Given s and r, we can substitute and solve for θ. This involves straightforward algebra and substitution of the approximate value of pi. Step-by-Step Solution:Step 1: Write the arc length formula: s = (θ / 360) * 2 * pi * r.Step 2: Substitute s = 22, r = 28: 22 = (θ / 360) * 2 * pi * 28.Step 3: Simplify the constant part: 2 * 28 = 56, so 22 = (θ / 360) * 56 * pi.Step 4: Using pi = 22 / 7, we have 56 * pi = 56 * (22 / 7) = 8 * 22 = 176.Step 5: So the equation becomes 22 = (θ / 360) * 176.Step 6: Solve for θ: θ = 22 * 360 / 176.Step 7: Compute 22 * 360 = 7920 and then 7920 / 176 = 45 degrees. Verification / Alternative check:Verify by substituting θ = 45 degrees back into the formula.Compute s = (45 / 360) * 2 * pi * 28 = (1 / 8) * 56 * pi.Using pi = 22 / 7, s = (1 / 8) * 56 * 22 / 7 = (1 / 8) * 8 * 22 = 22 cm.This matches the given arc length, confirming θ = 45 degrees. Why Other Options Are Wrong:Angles like 60 degrees or 75 degrees would produce larger arcs than 22 cm for the same radius.A 90 degree angle would generate an arc equal to one quarter of the full circumference, which is clearly longer than 22 cm for radius 28 cm.An angle of 30 degrees would give an arc shorter than 22 cm.Only 45 degrees produces exactly the given arc length. Common Pitfalls:Students sometimes use the radian based arc length formula s = r * θ but incorrectly mix degrees and radians.Another common mistake is to forget the factor of 2 * pi * r in the full circumference, which leads to incorrect equations for arc length.Incorrect use of the approximation for pi or errors in arithmetic when simplifying 56 * pi can also lead to wrong answers. Final Answer:The measure of the central angle is 45°.

Question 8

The area of an equilateral triangle is 9√3 square centimetres. What is the perimeter of the triangle?

Accepted Answer

Introduction / Context:This problem gives the area of an equilateral triangle and asks for its perimeter. The key is to use the area formula for an equilateral triangle to recover the side length and then multiply by three to obtain the perimeter. This tests knowledge of standard formulas and algebraic manipulation with square roots. Given Data / Assumptions: The triangle is equilateral. Area A = 9√3 square centimetres. Let the side length be a cm. Perimeter P = 3a. Concept / Approach:The area of an equilateral triangle with side a is A = (sqrt(3) / 4) * a^2. Given the area, we can set this equal to 9√3 and solve for a. Once a is found, the perimeter is P = 3a. This requires careful handling of the square root and constants in the equation. Step-by-Step Solution:Step 1: Write the area formula: A = (sqrt(3) / 4) * a^2.Step 2: Set A equal to the given area: (sqrt(3) / 4) * a^2 = 9 * sqrt(3).Step 3: Divide both sides by sqrt(3): a^2 / 4 = 9.Step 4: Multiply both sides by 4: a^2 = 36.Step 5: Take the positive square root (since side length is positive): a = 6 cm.Step 6: Perimeter P = 3 * a = 3 * 6 = 18 cm. Verification / Alternative check:To verify, we can substitute a = 6 cm back into the area formula.Compute A = (sqrt(3) / 4) * a^2 = (sqrt(3) / 4) * 36 = 9 * sqrt(3) square centimetres, which matches the given area.Thus, the side length and perimeter are correctly determined. Why Other Options Are Wrong:If the perimeter were 9 cm, the side length would be 3 cm and the area would be much smaller than 9√3.If the perimeter were 9√3 cm, the side length would be 3√3 cm, leading to a different area.A perimeter of 6√3 cm similarly corresponds to a side length of 2√3 cm, giving an incorrect area.Only 18 cm corresponds to side length 6 cm, which yields the correct area. Common Pitfalls:Some learners forget to divide by sqrt(3) correctly when isolating a^2.Another error is to misremember the area formula and omit the factor 1 / 4, leading to an incorrect computation of a.Confusing perimeter with area and trying to directly manipulate 9√3 as if it were a length rather than an area can also cause mistakes. Final Answer:The perimeter of the equilateral triangle is 18 cm.

Question 9

ABCD is a kite in which angle A is 90 degrees and angle C is 60 degrees. If the length of side AB is 6 cm, what is the length of diagonal AC?

Accepted Answer

Introduction / Context:This problem concerns a kite, a special quadrilateral with two pairs of adjacent equal sides. Given two non adjacent angles and the length of one side, we are asked to find the length of a diagonal. This is a challenging geometry question that uses symmetry of kites, angle bisectors, and right triangle trigonometry or coordinate geometry reasoning. Given Data / Assumptions: ABCD is a kite with AB = AD and CB = CD. Angle A is 90 degrees. Angle C is 60 degrees. Side AB has length 6 cm. We are to find the length of diagonal AC. Diagonal AC is the axis of symmetry of the kite and bisects angles A and C. Concept / Approach:Because ABCD is a kite with AB = AD, diagonal AC is an axis of symmetry. Therefore AC bisects angle A into two angles of 45 degrees and bisects angle C into two angles of 30 degrees. A useful approach is to place A and C on a horizontal line and use vector or coordinate geometry. By expressing the coordinates of B and C in terms of the side length and angles, we form equations involving the unknown diagonal AC and then solve for its length. The final expression simplifies to a surd of the form 3(√2+√6). Step-by-Step Solution:Step 1: Place A at the origin with diagonal AC along the horizontal axis and let C have coordinate (L, 0), where L is the length of AC.Step 2: Since AB = 6 cm and angle between AB and AC at A is 45 degrees, the coordinates of B can be taken as (6 * cos 45°, 6 * sin 45°) = (3√2, 3√2).Step 3: Similarly, D is symmetric to B about AC and has coordinates (3√2, -3√2).Step 4: At C, diagonal AC bisects angle C, so the angle between AC and CB is 30 degrees.Step 5: Let C = (L, 0). Then the vectors CB and CD from C to B and D are (3√2 - L, 3√2) and (3√2 - L, -3√2).Step 6: The angle between CB and CD is given as 60 degrees. Using the dot product formula, cos 60° = 1 / 2 = (CB · CD) / (|CB| * |CD|).Step 7: Compute CB · CD and the squared magnitudes, set up the equation, and solve for L to obtain L = 3(√2 + √6). Verification / Alternative check:We can confirm that with L = 3(√2 + √6), the computed angle at C using the dot product of CB and CD indeed equals 60 degrees.The symmetry of the construction ensures that AB = AD and CB = CD, preserving the kite properties.Substituting the side length and this diagonal into right triangles formed by the diagonals further confirms consistency of the lengths and angles. Why Other Options Are Wrong:Options such as 6(√2+√3) cm or 2(√2+√3) cm arise from incorrect angle splitting or misinterpretation of the kite symmetry.Values involving √3 instead of √6 usually come from using 30 and 60 degree relationships incorrectly.Only 3(√2+√6) cm correctly satisfies both the given angle measures and the side length condition of the kite. Common Pitfalls:A common mistake is to assume that all sides of the kite are equal, confusing it with a rhombus.Another error is to forget that diagonal AC bisects angles A and C and that the diagonals of a kite are perpendicular only in a special case, not always needed here.Algebraic mistakes while handling surds in the dot product equation can also lead to incorrect expressions for the diagonal. Final Answer:The length of diagonal AC is 3(√2+√6) cm.

Question 10

A rectangular swimming pool of length 20 m and breadth 10 m has a uniformly sloping floor. The depth is 1 m at one end along the length and 3 m at the other end. How much water, in cubic metres, is needed to completely fill this pool?

Accepted Answer

Introduction / Context:This problem asks for the volume of water required to fill a swimming pool that has a rectangular top surface but a floor that slopes uniformly from one end to the other. Such pools are common in practical situations, and the question tests understanding of how to treat the varying depth as an average when computing the volume. Given Data / Assumptions: Top surface of the pool is a rectangle of length L = 20 m and breadth B = 10 m. Depth at one end along the length is 1 m. Depth at the other end is 3 m. The floor slopes uniformly, so depth varies linearly from 1 m to 3 m along the length. We assume vertical sides and no extra features such as steps that change volume. Concept / Approach:For a tank or pool that has a uniform slope in one direction, the depth at any point is a linear function between the shallow and deep ends. The volume can be found by taking the area of the horizontal top surface and multiplying by the average depth. The average depth is simply (shallow depth + deep depth) divided by 2, because of the linear variation. Step-by-Step Solution:Step 1: Compute the top surface area of the pool: A = L * B = 20 * 10 = 200 square metres.Step 2: Compute the shallow depth d1 = 1 m.Step 3: Compute the deep depth d2 = 3 m.Step 4: Find the average depth d_avg = (d1 + d2) / 2 = (1 + 3) / 2 = 4 / 2 = 2 m.Step 5: Volume of water required is V = A * d_avg = 200 * 2 = 400 cubic metres. Verification / Alternative check:Another way to see this is to imagine the pool volume as the average of two rectangular prisms: one with depth 1 m and one with depth 3 m.Volume with depth 1 m would be 200 * 1 = 200 cubic metres.Volume with depth 3 m would be 200 * 3 = 600 cubic metres.The average of 200 and 600 is (200 + 600) / 2 = 800 / 2 = 400 cubic metres, which matches the earlier computation. Why Other Options Are Wrong:An answer of 800 cubic metres corresponds to using the deepest depth 3 m for the entire pool, which overestimates the volume.An answer of 300 or 480 cubic metres arises from incorrect use of the average depth or miscalculations of the area.A value such as 600 cubic metres represents treating the entire pool as if it had the deep depth only.Only 400 cubic metres correctly reflects the average depth of 2 m. Common Pitfalls:Learners sometimes simply take the deeper depth and multiply by the area, ignoring the sloping nature of the floor.Another error is to average the lengths instead of the depths, which is not relevant here.Some may attempt unnecessarily complex integration when the linear variation allows a simple average depth method. Final Answer:The volume of water needed to fill the pool is 400 cubic metres.

Question 11

The area of a sector of a circle is 30.8 square centimetres and the corresponding central angle is 72 degrees. What is the radius of the circle in centimetres?

Accepted Answer

Introduction / Context:This problem connects the area of a sector of a circle with its central angle and radius. Given the area of the sector and the central angle in degrees, we are asked to find the radius of the circle. It tests the ability to apply the sector area formula and manipulate it to solve for the radius. Given Data / Assumptions: Area of the sector A = 30.8 square centimetres. Central angle θ = 72 degrees. Radius of the circle is r cm. Use pi approximately equal to 22 / 7. Concept / Approach:The area of a sector of a circle when the central angle is given in degrees is A = (θ / 360) * pi * r^2. With A and θ known, we can rearrange this formula to solve for r^2 and then take the square root to obtain r. Careful substitution and algebra will yield a nice value when using the given approximation for pi. Step-by-Step Solution:Step 1: Write the sector area formula: A = (θ / 360) * pi * r^2.Step 2: Substitute A = 30.8 and θ = 72 degrees: 30.8 = (72 / 360) * pi * r^2.Step 3: Simplify the fraction 72 / 360 = 1 / 5.Step 4: The equation becomes 30.8 = (1 / 5) * pi * r^2.Step 5: Multiply both sides by 5: 154 = pi * r^2.Step 6: Use pi = 22 / 7 to get 154 = (22 / 7) * r^2.Step 7: Multiply both sides by 7 / 22: r^2 = 154 * 7 / 22.Step 8: Compute 154 / 22 = 7, so r^2 = 7 * 7 = 49.Step 9: Take the square root: r = sqrt(49) = 7 cm. Verification / Alternative check:Substitute r = 7 cm back into the formula to verify the area.Compute A = (72 / 360) * pi * 7^2 = (1 / 5) * pi * 49.Using pi = 22 / 7, A = (1 / 5) * (22 / 7) * 49 = (1 / 5) * 22 * 7 = 154 / 5 = 30.8 square centimetres.This matches the given sector area, confirming the calculation. Why Other Options Are Wrong:If r were 3.5 cm, then r^2 would be 12.25 and the sector area would be far smaller than 30.8 square centimetres.If r were 14 cm, r^2 would be 196 and the area would be much larger than 30.8 square centimetres.Values like 10.5 cm or 21 cm also produce sector areas that do not match 30.8 when substituted into the formula.Only r = 7 cm satisfies the equation. Common Pitfalls:A common error is to forget to convert the central angle from degrees to a fraction of a full circle by dividing by 360.Another mistake is to use the full circle area formula pi * r^2 directly without applying the fraction θ / 360.Misplacing the factor of 1 / 5 when rearranging the equation can also lead to incorrect values of r^2. Final Answer:The radius of the circle is 7 cm.

Question 12

What is the measure of the central angle, in degrees, of a circular arc whose length is 11 cm, given that the radius of the circle of which it is a part is 14 cm?

Accepted Answer

Introduction / Context: In this problem we work with the relationship between the length of an arc and the central angle that subtends it. Such questions are very common in aptitude tests because they connect basic circle geometry with unit conversion between radians and degrees, and they test whether you know the standard formula for arc length as well as how to manipulate it to find the unknown angle. Given Data / Assumptions: The circle has radius r = 14 cm. The length of the arc is l = 11 cm. The central angle that subtends this arc at the centre of the circle is unknown and must be found in degrees. We assume a standard Euclidean circle and use the usual relationship between arc length, radius and angle in radians. Concept / Approach: The key formula for arc length of a circle is: l = r * θ where l is the arc length, r is the radius, and θ is the central angle in radians. Once we find θ in radians, we then convert it to degrees using the relation: degrees = θ * (180 / π) Because the options are all in degrees, our final step is this conversion, usually with π approximated as 22/7 in aptitude problems. Step-by-Step Solution: Step 1: Write the arc length formula: l = r * θ. Step 2: Substitute l = 11 cm and r = 14 cm. Step 3: 11 = 14 * θ. Step 4: Solve for θ in radians: θ = 11 / 14. Step 5: Convert θ to degrees using degrees = θ * (180 / π). Step 6: Use π = 22/7. Then degrees = (11 / 14) * (180 / (22 / 7)). Step 7: Simplify: degrees = (11 / 14) * 180 * 7 / 22. Step 8: Notice that 11 cancels with 22 and 7 cancels with 14, giving degrees = 45. Step 9: Therefore the measure of the central angle is 45°. Verification / Alternative check: We can quickly verify the arithmetic. Starting from (11 / 14) * 180 * 7 / 22, combine numerators and denominators. The factor 11 in the numerator cancels with 22 leaving 2 in the denominator. The factor 7 in the numerator cancels with 14 leaving 2 in the denominator. We then have 180 / (2 * 2) = 180 / 4 = 45 degrees. This check confirms that there was no mistake in simplification or cancellation and that the angle is exactly 45 degrees, which is a nice standard value. Why Other Options Are Wrong: 60° would correspond to a larger arc because for a fixed radius, a larger angle produces a longer arc than 11 cm. 75° and 90° are even larger, which would make the arc length significantly more than 11 cm for radius 14 cm. The value 135° is much more than 90°, giving an arc that is clearly too long. Only 45° is consistent with the formula and the given numbers. Common Pitfalls: A frequent mistake is to try to use degrees directly in the formula l = r * θ without converting to radians or without using a correct conversion. Another common error is to invert the relationship and write θ = r / l instead of θ = l / r. Some students also forget to use π = 22/7 when simplifying, which makes the arithmetic look messy and can lead to rounding errors. Finally, mixing up the roles of radius and diameter in the formula can lead to an answer that is off by a factor of two. Final Answer: Thus, the measure of the required central angle is 45°.

Question 13

A cylindrical vessel of height 5 cm and radius 4 cm is completely filled with sand, and the sand is then poured out to form a right circular cone of radius 6 cm. What will be the height of this cone, taking π = 22/7?

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Introduction / Context: This question examines your understanding of volume relationships between different solids, specifically a right circular cylinder and a right circular cone. In many practical situations, material is melted or transferred from one shape into another, and the volume must be conserved. Here, sand completely fills a cylinder and is then poured into a cone, and you must use volume conservation to determine the unknown height of the cone. Given Data / Assumptions: Height of the cylindrical vessel, h_cyl = 5 cm. Radius of the cylinder, r_cyl = 4 cm. Sand is poured to form a right circular cone. Radius of the cone, r_cone = 6 cm. Height of the cone, h_cone, is unknown. Take π = 22/7 and assume there is no loss or gain of sand, so volumes are equal. Concept / Approach: We use the standard volume formulas for a cylinder and a cone. For a cylinder: V_cylinder = π * r_cyl^2 * h_cyl. For a cone: V_cone = (1 / 3) * π * r_cone^2 * h_cone. Because all the sand in the cylinder is transferred to the cone, the volumes must be equal. So we set V_cylinder = V_cone and solve for the unknown height h_cone. Step-by-Step Solution: Step 1: Write the cylinder volume: V_cylinder = π * r_cyl^2 * h_cyl. Step 2: Substitute r_cyl = 4 cm and h_cyl = 5 cm: V_cylinder = π * 4^2 * 5 = π * 16 * 5 = 80π. Step 3: Write the cone volume: V_cone = (1 / 3) * π * r_cone^2 * h_cone. Step 4: Substitute r_cone = 6 cm: V_cone = (1 / 3) * π * 6^2 * h_cone = (1 / 3) * π * 36 * h_cone = 12π * h_cone. Step 5: Set the volumes equal: 80π = 12π * h_cone. Step 6: Cancel π on both sides to simplify: 80 = 12 * h_cone. Step 7: Solve for h_cone: h_cone = 80 / 12 = 20 / 3 ≈ 6.67 cm. Step 8: Match this with the closest option, which is 6.67 cm. Verification / Alternative check: To verify, compute both volumes numerically. The cylinder volume is 80π. The cone volume using h_cone = 20 / 3 is V_cone = 12π * (20 / 3) = (240 / 3)π = 80π, which matches the cylinder volume exactly. This confirms that our calculation is correct and that no arithmetic mistake was made when solving for the height. Why Other Options Are Wrong: A height of 2.22 cm or 3.33 cm would produce a cone with much smaller volume than the original cylinder, so all the sand could not fit into the cone. A height of 1.67 cm is even smaller and clearly inconsistent with the large cylinder volume. The value 5 cm might be tempting because it matches the cylinder height, but volume relationships depend on both radius and height, and the cone has a different radius, so the heights cannot simply be equal. Common Pitfalls: One common error is to forget the factor of 1 / 3 in the cone volume formula. Another is to confuse the radii of the cylinder and the cone or to assume they are the same when they are not. Some students also try to compare heights directly rather than equating volumes. Finally, careless cancellation of π or incorrect simplification of fractions can lead to wrong numerical answers, especially under exam pressure. Final Answer: Therefore, the height of the cone formed by the sand is 6.67 cm.

Question 14

Point P is the midpoint of segment AB. The coordinates of point P are (2, 1) and the coordinates of point A are (11, 5). What are the coordinates of point B?

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Introduction / Context: This is a straightforward coordinate geometry question that uses the midpoint formula for a line segment in the Cartesian plane. The problem asks you to find the coordinates of an unknown endpoint of a segment when the midpoint and the other endpoint are given. Such questions test your ability to manipulate basic formulas and to work with simple algebraic equations involving averages of coordinates. Given Data / Assumptions: Point P is the midpoint of segment AB. Coordinates of P are (2, 1). Coordinates of A are (11, 5). Coordinates of B are unknown and must be found. We work on the standard x-y coordinate plane using the usual midpoint formula. Concept / Approach: For a segment joining points A(x1, y1) and B(x2, y2), the coordinates of the midpoint P are given by: P_x = (x1 + x2) / 2 P_y = (y1 + y2) / 2 We are given P and A, and we need to reverse this relationship to solve for the unknown coordinates of B. This means we will set up equations based on these midpoint formulas and solve for x2 and y2. Step-by-Step Solution: Step 1: Let B have coordinates (x, y). Step 2: Use the midpoint formula for the x-coordinate: (11 + x) / 2 = 2. Step 3: Multiply both sides by 2: 11 + x = 4, so x = 4 - 11 = -7. Step 4: Use the midpoint formula for the y-coordinate: (5 + y) / 2 = 1. Step 5: Multiply both sides by 2: 5 + y = 2, so y = 2 - 5 = -3. Step 6: Therefore B = (-7, -3). Step 7: Compare this with the options, and we see that option (-7, -3) exactly matches our result. Verification / Alternative check: To verify, substitute A and B back into the midpoint formula. For x-coordinates: (11 + (-7)) / 2 = 4 / 2 = 2, which matches the given midpoint x-coordinate. For y-coordinates: (5 + (-3)) / 2 = 2 / 2 = 1, which matches the given midpoint y-coordinate as well. Both coordinates of the midpoint check out, confirming that B is indeed (-7, -3). Why Other Options Are Wrong: The point (6.5, 3) gives midpoint x-coordinate (11 + 6.5) / 2 = 8.75 which is not equal to 2. The point (7, 3) gives midpoint x-coordinate (11 + 7) / 2 = 9 which is incorrect. The point (-6.5, -3) does not produce integer midpoint coordinates and fails to give P = (2, 1). The point (9, -1) also produces a different midpoint. Only (-7, -3) satisfies both coordinates of the midpoint formula. Common Pitfalls: A common mistake is to forget that the midpoint takes the average of the coordinates, not the difference. Some students also write equations like (x - 11) / 2 instead of (x + 11) / 2. Another frequent error is to mix up the formulas for midpoint and distance, or to solve only one of the two coordinates and then guess the other. Careful use of both x and y midpoint formulas ensures the correct point is found. Final Answer: Thus, the coordinates of point B are (-7, -3).

Question 15

In triangle ABC, the median AD is drawn from vertex A to side BC. If AD is 6 cm and the length of side BC is 12 cm, then what is the measure of angle CAB?

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Introduction / Context: This geometry problem focuses on a special property of right triangles involving the median drawn to the hypotenuse. It checks whether you remember a key fact: in a right triangle, the median drawn from the right angle vertex to the hypotenuse has a very specific relationship with the hypotenuse. Recognizing this property allows you to conclude which angle is a right angle without lengthy calculations. Given Data / Assumptions: Triangle ABC is given. AD is a median drawn from vertex A to side BC, so D is the midpoint of BC. Length of median AD = 6 cm. Length of side BC = 12 cm. We are asked to find the measure of angle CAB. The triangle is assumed to lie in a standard Euclidean plane with usual triangle properties. Concept / Approach: In any triangle, a median joins a vertex to the midpoint of the opposite side. In a right triangle there is an important special result: the median drawn from the right angle to the hypotenuse is exactly half the hypotenuse. That means if a triangle has a hypotenuse of length h and a median from the right angle vertex to the hypotenuse of length h / 2, then that vertex must be the right angle. We will compare the given median length and half of BC to decide whether angle CAB is the right angle. Step-by-Step Solution: Step 1: D is the midpoint of BC since AD is a median. Step 2: The length of BC is given as 12 cm. Step 3: Compute half of BC: BC / 2 = 12 / 2 = 6 cm. Step 4: The given length of the median AD is also 6 cm. Step 5: Therefore AD = BC / 2. Step 6: In a triangle, if the median from a vertex to the opposite side equals half of that opposite side, then that opposite side is the hypotenuse and the vertex is a right angle. Step 7: Here, BC acts as the hypotenuse and A is the right angle vertex. Step 8: So angle CAB is the angle at A, and it must be 90°. Verification / Alternative check: Another way to think about this is to recall that in a right triangle, the midpoint of the hypotenuse is equidistant from all three vertices. Thus, if AD is the segment from the right angle vertex A to the midpoint D of BC, then AD equals the radius of the circumcircle, which is half of the hypotenuse. Since AD and BC / 2 are equal here, triangle ABC must be right angled at A. There is no need to find side lengths explicitly or to use trigonometry for this reasoning. Why Other Options Are Wrong: An angle of 30° or 60° is typical in special right triangles, but the question is asking which angle is the right angle, based on the median property. The values 30° and 60° do not correspond to the right angle. Similarly, 120° is an obtuse angle and cannot be part of a triangle that respects the condition about the median being half the opposite side. Only 90° is consistent with the known geometric property used in this problem. Common Pitfalls: Many students misinterpret the median property and think it always equals half the opposite side, which is not true. This happens only for the median drawn from the right angle to the hypotenuse in a right triangle. Others attempt to apply the Pythagoras theorem without enough side information, which leads nowhere. Some learners also confuse medians with altitudes or angle bisectors. Carefully remembering the special property about the median to the hypotenuse avoids these issues. Final Answer: Thus, angle CAB is a right angle and its measure is 90°.

Question 16

The sum of the measures of all the interior angles of a polygon is 2340°. How many sides does this polygon have?

Accepted Answer

Introduction / Context: This problem checks your knowledge of the formula for the sum of interior angles of a polygon. Aptitude tests frequently include such questions to see whether you can recall the correct expression, substitute the given total angle sum, and solve for the number of sides of the polygon using simple algebra. It also reinforces understanding of how polygons generalize from triangles and quadrilaterals. Given Data / Assumptions: The polygon is a simple convex polygon. The sum of all its interior angles is 2340°. We must find the number of sides n of the polygon. We will use the standard formula relating the number of sides and the sum of interior angles. Concept / Approach: For a polygon with n sides, the sum of the measures of all interior angles is given by: Sum of interior angles = (n - 2) * 180°. This formula comes from the fact that any polygon with n sides can be divided into (n - 2) triangles, each having a sum of interior angles equal to 180°. We are given the sum and must solve for n by setting (n - 2) * 180° equal to 2340°. Step-by-Step Solution: Step 1: Write the formula: (n - 2) * 180° = 2340°. Step 2: Divide both sides by 180° to isolate (n - 2). Step 3: (n - 2) = 2340 / 180. Step 4: Compute 2340 / 180 = 13. Step 5: So n - 2 = 13. Step 6: Add 2 to both sides: n = 13 + 2 = 15. Step 7: Therefore, the polygon has 15 sides. Verification / Alternative check: To verify, plug n = 15 back into the formula. The sum of interior angles should be (15 - 2) * 180° = 13 * 180°. Calculating, 13 * 180 = 2340°, which matches the given sum exactly. This confirms our value of n and shows that there is no arithmetic error in the division or addition steps. Why Other Options Are Wrong: If n = 13, the sum would be (13 - 2) * 180° = 11 * 180° = 1980°, which is less than 2340°. For n = 17, the sum would be (17 - 2) * 180° = 15 * 180° = 2700°, which is greater than 2340°. For n = 11, the sum is (11 - 2) * 180° = 9 * 180° = 1620°. A hypothetical n = 19 gives an even larger sum. Only n = 15 gives exactly 2340°. Common Pitfalls: Students sometimes mistakenly use 180n instead of (n - 2) * 180°, which overestimates the sum. Another frequent error is confusing interior and exterior angles. For exterior angles of a convex polygon, the sum is always 360°, which is a different concept. Also, forgetting to divide by 180° properly or mishandling the arithmetic division 2340 / 180 can lead to an incorrect number of sides. Final Answer: Hence, the polygon has 15 sides.

Question 17

A cylinder of radius 4.3 cm and height 4.8 cm is melted and recast into a right circular cone having the same radius as the cylinder. What will be the height of this cone? (Use π = 22/7.)

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Introduction / Context: This question involves conservation of volume when a solid is melted and recast into another shape. Here a cylinder is melted to form a cone, and you are required to find the new height. Such questions are common in quantitative aptitude, because they require you to apply volume formulas and to recognise that total volume of material remains constant during the transformation. Given Data / Assumptions: Radius of the cylinder, r_cyl = 4.3 cm. Height of the cylinder, h_cyl = 4.8 cm. The cylinder is melted and recast into a cone. The cone has the same radius as the cylinder, r_cone = 4.3 cm. Height of the cone, h_cone, is unknown. Use π = 22/7 and assume no loss of material during melting and recasting. Concept / Approach: The volumes of the original and the recast solid must be equal. The volume of a cylinder is: V_cylinder = π * r_cyl^2 * h_cyl. The volume of a cone is: V_cone = (1 / 3) * π * r_cone^2 * h_cone. Since r_cyl = r_cone, the radius cancels nicely when we equate the two volumes. This makes it straightforward to isolate and calculate h_cone. Step-by-Step Solution: Step 1: Write the cylinder volume: V_cylinder = π * r_cyl^2 * h_cyl. Step 2: Substitute r_cyl = 4.3 and h_cyl = 4.8: V_cylinder = π * (4.3)^2 * 4.8. Step 3: Write the cone volume: V_cone = (1 / 3) * π * r_cone^2 * h_cone. Step 4: Since r_cone = 4.3, V_cone = (1 / 3) * π * (4.3)^2 * h_cone. Step 5: Equate volumes: π * (4.3)^2 * 4.8 = (1 / 3) * π * (4.3)^2 * h_cone. Step 6: Cancel π and (4.3)^2 from both sides: 4.8 = (1 / 3) * h_cone. Step 7: Multiply both sides by 3: h_cone = 3 * 4.8 = 14.4 cm. Step 8: Match this with the options: 14.4 cm is available and is therefore the correct choice. Verification / Alternative check: We can verify by computing approximate numerical volumes. The cylinder volume is π * (4.3)^2 * 4.8 and the cone volume with h_cone = 14.4 is (1 / 3) * π * (4.3)^2 * 14.4. Since 14.4 / 3 = 4.8, the cone volume simplifies back to π * (4.3)^2 * 4.8, exactly matching the cylinder volume. This confirms that our reasoning and arithmetic are correct. Why Other Options Are Wrong: A height of 28.8 cm or 21.6 cm would produce a cone with volume larger than the original cylinder, which is impossible without adding material. Heights of 7.2 cm or 10.8 cm would yield smaller volumes than the original cylinder, meaning that some material would have disappeared. Only 14.4 cm gives a cone with the same volume as the cylinder, which is required by the melting and recasting process. Common Pitfalls: One typical mistake is to forget the factor of 1 / 3 in the cone volume formula, leading to incorrect proportionality between the heights. Another error is to square the radius incorrectly or to cancel the π factor prematurely without considering the rest of the expression. Some students also try to relate heights directly using simple ratios, but the correct relationship comes from equating full volume expressions, not from intuitive height comparisons. Final Answer: Therefore, the height of the recast cone is 14.4 cm.

Question 18

If the radius of a circle is increased by 25%, by what percentage does the area of the circle increase?

Accepted Answer

Introduction / Context: This is a classic percentage change question in geometry that examines your understanding of how the area of a circle depends on its radius. The radius is increased by a certain percentage, and you must determine the resulting percentage increase in the area. This kind of problem emphasizes that when a quantity depends on the square of a variable, small changes in the variable can produce larger changes in the dependent quantity. Given Data / Assumptions: Original radius of the circle is r. The radius is increased by 25% to become the new radius. We must find the percentage increase in the area of the circle. The relationship between area and radius for a circle is used in its standard form. Concept / Approach: The area of a circle is given by: A = π * r^2. If the radius is increased by 25%, the new radius r_new is: r_new = r * (1 + 25 / 100) = r * 1.25. The new area A_new becomes π * r_new^2. We compare A_new with the original area A and compute the percentage increase as: Percentage increase = [(A_new - A) / A] * 100 percent. Step-by-Step Solution: Step 1: Original area: A = π * r^2. Step 2: New radius: r_new = 1.25 * r. Step 3: New area: A_new = π * (r_new)^2 = π * (1.25r)^2. Step 4: Compute (1.25)^2: 1.25 * 1.25 = 1.5625. Step 5: So A_new = π * 1.5625 * r^2 = 1.5625 * A. Step 6: Increase in area = A_new - A = (1.5625A - A) = 0.5625A. Step 7: Percentage increase = (0.5625A / A) * 100 percent = 0.5625 * 100 percent = 56.25 percent. Step 8: So the area increases by 56.25 percent. Verification / Alternative check: For a quick check, assume a convenient radius. Let r = 4 units. Original area A = π * 4^2 = 16π. New radius is r_new = 1.25 * 4 = 5. New area A_new = π * 5^2 = 25π. The increase is 25π - 16π = 9π. Percentage increase is (9π / 16π) * 100 percent = (9 / 16) * 100 percent = 56.25 percent. This numerical example confirms the algebraic result. Why Other Options Are Wrong: An increase of 25 percent would be correct if the area depended linearly on the radius, but it depends on the square of the radius. Options 50 percent and 28.125 percent do not match the correct squared factor of 1.5625. The value 31.25 percent would correspond to a much smaller change and does not come from squaring a 25 percent increase in radius. Only 56.25 percent correctly reflects the transformed area. Common Pitfalls: A very common mistake is to assume that if the radius increases by 25 percent, the area also increases by 25 percent. This ignores the square relationship between area and radius. Another error is squaring 25 instead of 1.25, leading to 625 percent or other meaningless values. Some students also forget to subtract 1 when converting the factor 1.5625 into a percentage increase. Careful stepwise reasoning avoids these errors. Final Answer: Thus, when the radius is increased by 25%, the area of the circle increases by 56.25 percent.

Question 19

What is the length of the side of an equilateral triangle if its area is given to be 36√3 square centimetres?

Accepted Answer

Introduction / Context: This question tests your understanding of the formula for the area of an equilateral triangle and your ability to manipulate square roots and algebraic expressions. Equilateral triangles have equal sides and equal angles, which leads to a special and widely used area formula. Problems like this commonly appear in aptitude exams to check whether you can move smoothly from area back to side length. Given Data / Assumptions: The triangle is equilateral, so all sides are equal. Area of the triangle is A = 36√3 square centimetres. We must find the side length a of the triangle in centimetres. We will use the standard formula for the area of an equilateral triangle. Concept / Approach: For an equilateral triangle with side length a, the area is given by: A = (√3 / 4) * a^2. We know A and must solve this equation for a. This involves basic algebra: isolating a^2 and then taking the square root to find a. Careful handling of the square root factors ensures an exact and clean answer. Step-by-Step Solution: Step 1: Write the area formula: A = (√3 / 4) * a^2. Step 2: Substitute the given area: 36√3 = (√3 / 4) * a^2. Step 3: Multiply both sides by 4 to eliminate the denominator: 4 * 36√3 = √3 * a^2. Step 4: Compute 4 * 36√3 = 144√3. Step 5: The equation becomes 144√3 = √3 * a^2. Step 6: Divide both sides by √3: a^2 = 144. Step 7: Take the positive square root (side length is positive): a = √144 = 12. Step 8: Therefore, the side length of the equilateral triangle is 12 cm. Verification / Alternative check: Check by substituting a = 12 back into the area formula. Compute a^2 = 12^2 = 144. Then area A = (√3 / 4) * 144. Simplify: 144 / 4 = 36, so A = 36√3. This matches the area given in the problem exactly, confirming that the side length was calculated correctly and that no algebraic steps were missed or mishandled. Why Other Options Are Wrong: If a = 6 cm, the area would be (√3 / 4) * 36 = 9√3, which is too small. If a = 24 cm, the area would be (√3 / 4) * 576 = 144√3, which is much larger than 36√3. If a = 18 cm, the area becomes (√3 / 4) * 324 = 81√3, also too large. The value 9 cm gives an area (√3 / 4) * 81 = 20.25√3, which still does not match. Only a side length of 12 cm yields area 36√3. Common Pitfalls: A common problem is misremembering the formula and using (1 / 2) * base * height without computing the height correctly. Another error is incorrectly handling the factor √3 / 4 when solving for a^2, especially when multiplying both sides by 4 or dividing by √3. Some students also make arithmetic mistakes when squaring or taking square roots, such as forgetting that √144 is 12 and not another value. Writing each step clearly helps avoid these errors. Final Answer: Hence, the side length of the equilateral triangle is 12 cm.

Question 20

The area of a rectangle is 1200 square centimetres and its perimeter is 140 centimetres. What is the length of the diagonal of this rectangle?

Accepted Answer

Introduction / Context: This problem combines several key properties of rectangles: the relationships between area, perimeter, side lengths, and the diagonal. It requires algebraic manipulation of two equations to find the side lengths, and then application of the Pythagoras theorem to find the diagonal. Such multi-step questions are good practice for translating word statements into equations and connecting different geometric formulas. Given Data / Assumptions: Let the rectangle have length L and breadth B (both in cm). Area A = L * B = 1200 square centimetres. Perimeter P = 2(L + B) = 140 centimetres. We must find the length of the diagonal d of the rectangle. We will use the Pythagoras theorem: d^2 = L^2 + B^2. Concept / Approach: From the perimeter, we can express L + B. From the area, we know L * B. We thus have a system of two equations in L and B. Once we solve this system, we apply the Pythagoras theorem to find the diagonal d. Rectangles always have right angles, so the diagonal is the hypotenuse of a right triangle with legs L and B. Step-by-Step Solution: Step 1: Use the perimeter: 2(L + B) = 140 ⇒ L + B = 70. Step 2: Use the area: L * B = 1200. Step 3: Treat L and B as roots of a quadratic equation t^2 - (sum)t + product = 0. Step 4: So t^2 - 70t + 1200 = 0. Step 5: Factor or solve using the discriminant. Compute the discriminant: D = 70^2 - 4 * 1200 = 4900 - 4800 = 100. Step 6: Roots are: t = [70 ± √100] / 2 = [70 ± 10] / 2. Step 7: Thus L and B are 40 cm and 30 cm (order does not matter). Step 8: Now use the Pythagoras theorem for the diagonal: d^2 = 40^2 + 30^2 = 1600 + 900 = 2500. Step 9: Therefore d = √2500 = 50 cm. Verification / Alternative check: Verify by recomputing area and perimeter. With L = 40 cm and B = 30 cm, area A = 40 * 30 = 1200 square centimetres, which matches the given area. Perimeter P = 2(40 + 30) = 2 * 70 = 140 centimetres, also matching the problem statement. The diagonal of a 3-4-5 scaled triangle (here 30, 40, 50) is well known to be 50 cm, which further confirms our calculations. Why Other Options Are Wrong: If the diagonal were 30 cm or 40 cm, then at least one side would have to be shorter than is possible given the area and perimeter constraints. A diagonal of 60 cm would come from larger sides, producing a bigger area than 1200 square centimetres. A diagonal of 70 cm is even more extreme and would not be consistent with the given perimeter. Only 50 cm matches the side lengths determined from the simultaneous equations. Common Pitfalls: Mistakes often occur when setting up the quadratic equation, especially in writing the correct product term or computing the discriminant. Another common error is to guess side lengths without checking both area and perimeter simultaneously. Some students also misuse the Pythagoras theorem by adding perimeter values or mixing up which quantities are squared. Working systematically from equations and checking each step reduces these errors significantly. Final Answer: The length of the diagonal of the rectangle is 50 cm.

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