Question 1

A rhombus has diagonals 24 cm and 10 cm. Find the area (in sq cm) and the perimeter (in cm) of the rhombus, respectively, using standard rhombus diagonal properties.

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Introduction / Context: This problem checks two core rhombus facts: (1) Area can be found directly from diagonals using Area = (d1*d2)/2, and (2) diagonals bisect each other at right angles, allowing side length to be found by the Pythagoras theorem using half-diagonals. Once the side is known, perimeter is 4 times the side. The main goal is to produce the correct ordered pair: (area, perimeter). Given Data / Assumptions: d1 = 24 cm d2 = 10 cm Area = (d1*d2)/2 Half diagonals: 12 cm and 5 cm Side s = sqrt(12^2 + 5^2) Concept / Approach: Compute area using diagonals formula. Compute side using right triangle formed by half diagonals. Then perimeter = 4*s. Keep units consistent (sq cm for area, cm for perimeter). Step-by-Step Solution: Area = (d1*d2)/2 = (24*10)/2 = 240/2 = 120 sq cmHalf diagonals are 12 cm and 5 cmSide s = sqrt(12^2 + 5^2) = sqrt(144 + 25) = sqrt(169) = 13 cmPerimeter = 4*s = 4*13 = 52 cm Verification / Alternative check: Since 12-5-13 is an exact Pythagorean triple, side = 13 is exact. Also area calculation is direct and exact. Therefore the ordered result (120, 52) is consistent and does not require rounding. Why Other Options Are Wrong: Area 240 doubles the correct area because the division by 2 was missed. Perimeter 48 or 50 uses an incorrect side length due to not halving diagonals or incorrect square root. Area 110 or 130 indicates arithmetic mistakes in 24*10/2. Common Pitfalls: Forgetting Area = (d1*d2)/2. Not halving diagonals before Pythagoras. Mixing up units (writing cm for area). Giving only one value when the question asks for both in order. Final Answer: Area 120 sq cm, Perimeter 52 cm

Question 2

The wheel of a motorcycle is 70 cm in diameter. It makes 40 revolutions in every 10 seconds. Find the speed of the motorcycle in km/hr (assume pi = 22/7 for exact calculation).

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Introduction / Context: This question tests the link between rotational motion and linear speed. Each wheel revolution moves the motorcycle forward by one circumference of the wheel. By multiplying circumference by revolutions per second, you get linear speed in m/s. Finally, convert m/s to km/hr using the standard factor (multiply by 18/5). The key is careful unit conversion from cm to m and seconds to hours. Given Data / Assumptions: Diameter d = 70 cm Revolutions = 40 in 10 s pi = 22/7 1 m = 100 cm m/s to km/hr: multiply by 18/5 Concept / Approach: Distance per revolution = circumference = pi*d. Revolutions per second = 40/10. Speed (cm/s) = circumference * rev/s. Convert to m/s, then to km/hr. Step-by-Step Solution: Circumference = pi*d = (22/7)*70 = 220 cm per revolutionRevolutions per second = 40/10 = 4 rev/sSpeed = 220*4 = 880 cm/sConvert to m/s: 880 cm/s = 880/100 = 8.8 m/sConvert to km/hr: 8.8*(18/5) = 8.8*3.6 = 31.68 km/hr Verification / Alternative check: In 1 second, the wheel makes 4 revolutions, so it travels 4 circumferences = 4*2.2 m = 8.8 m, matching 8.8 m/s. Multiplying by 3.6 gives 31.68 km/hr, which is a realistic low-to-moderate speed for such rotation rate. Why Other Options Are Wrong: 13.68 results from incorrect rev/s (using 40/30 or similar). 41.68 and 45.68 come from circumference miscalculation or using pi=3.14 inconsistently with cm-to-m conversion. 28.80 often comes from using 3.2 instead of 3.6 as the conversion factor. Common Pitfalls: Forgetting to divide revolutions by time to get rev/s. Not converting cm to m. Using the wrong conversion factor to km/hr. Using radius instead of diameter in circumference formula. Final Answer: 31.68 km/hr

Question 3

A triangle has base 15 cm and height 12 cm. Another triangle has double the area of the first triangle and has base 20 cm. Find the height (in cm) of the second triangle.

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Introduction / Context: This question tests the triangle area formula and proportional reasoning. The area of a triangle is (1/2)*base*height. First compute the area of the original triangle, then double it. For the second triangle, the base is known, so you rearrange the same formula to solve for height. The core skill is using the same area relationship with correct arithmetic and units. Given Data / Assumptions: Triangle 1: base b1 = 15 cm, height h1 = 12 cm Triangle 2: base b2 = 20 cm Area of triangle 2 = 2*(area of triangle 1) Concept / Approach: Compute area1 = (1/2)*b1*h1. Then area2 = 2*area1. Finally, use area2 = (1/2)*b2*h2 and solve for h2: h2 = (2*area2)/b2. Step-by-Step Solution: Area1 = (1/2)*15*12 = (1/2)*180 = 90 sq cmArea2 = 2*Area1 = 2*90 = 180 sq cmArea2 = (1/2)*20*h2 = 10*h210*h2 = 180 => h2 = 18 cm Verification / Alternative check: Check: With base 20 and height 18, area = (1/2)*20*18 = 180 sq cm, which is exactly double 90 sq cm. So the computed height is correct. Why Other Options Are Wrong: 16 and 17 come from arithmetic slips (often doubling the base instead of area). 20 occurs if you forget the 1/2 in the formula. 15 results from mistakenly keeping the same height as the first triangle without respecting the “double area” condition. Common Pitfalls: Forgetting the 1/2 factor. Doubling the base instead of doubling area. Mixing up which triangle has which base. Doing 180/20 instead of (2*180)/20 when rearranging the formula. Final Answer: 18 cm

Question 4

The inner circumference of a circular race track is 440 m. The track is 14 m wide (uniform width). Find the radius of the outer circle in metres (m), assuming pi = 22/7.

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Introduction / Context: This question uses the circumference formula of a circle and the idea of a “ring” (track) with uniform width. The inner circumference gives the inner radius. Since the track is 14 m wide, the outer radius is simply inner radius + 14. The key is solving 2*pi*r = 440 correctly and then adding the width in the correct direction (radius, not diameter). Given Data / Assumptions: Inner circumference C_inner = 440 m Track width = 14 m pi = 22/7 C = 2*pi*r Outer radius = inner radius + 14 Concept / Approach: Use C_inner = 2*pi*r_inner to find r_inner. Then r_outer = r_inner + width. Keep everything in metres and do not confuse width with diameter change (diameter increases by 2*width, but radius increases by width). Step-by-Step Solution: C_inner = 2*pi*r_inner440 = 2*(22/7)*r_inner = (44/7)*r_innerr_inner = 440*(7/44) = 10*7 = 70 mr_outer = r_inner + 14 = 70 + 14 = 84 m Verification / Alternative check: Quick check: if r_inner is 70, inner circumference = 2*(22/7)*70 = 2*22*10 = 440, correct. Adding width 14 to radius must give 84, which is reasonable for the outer boundary of a 14 m wide track. Why Other Options Are Wrong: 70 m is just the inner radius (not outer). 44 m or 24 m typically come from dividing by pi incorrectly. 54 m can appear if someone subtracts width instead of adding. 44 m may also come from confusing circumference with diameter directly. Common Pitfalls: Forgetting to use 2*pi*r. Confusing radius with diameter. Adding 2*14 to the radius (that would be for diameter). Using pi = 3.14 while options are based on 22/7 exact arithmetic. Final Answer: 84 m

Question 5

Four equal circles are drawn with the vertices of a square as their centres. The side of the square is 7 cm. Each circle has radius 3.5 cm (half the side), so that the arc from each vertex reaches the midpoint of each adjacent side. Find the total a…

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Introduction / Context: This is a geometry “composite area” problem involving circles and a square. With a circle centred at each vertex, only one quadrant of each circle lies inside the square (the quadrant in the corner). The remaining three quadrants of each circle lie outside the square. Since there are four circles, the total area outside the square equals 4*(3/4 of one circle area) = 3*(one circle area). The key is recognizing the quadrant relationship and using the correct radius (half the side). Given Data / Assumptions: Square side = 7 cm Circle centres are the four vertices Radius r = 3.5 cm (half of 7 cm) Area of circle = pi*r^2 Inside square: 1 quadrant per circle Concept / Approach: For each vertex-centred circle, the portion inside the square is a 90 degree sector (a quadrant). So area outside square for one circle = (3/4)*pi*r^2. For four circles: total outside = 4*(3/4)*pi*r^2 = 3*pi*r^2. Then substitute r = 3.5 and pi = 22/7. Step-by-Step Solution: Radius r = 3.5 cmArea of one full circle = pi*r^2 = pi*(3.5^2) = pi*12.25Area of one quadrant (inside square) = (1/4)*pi*12.25Area outside square for one circle = (3/4)*pi*12.25Total outside for four circles = 4*(3/4)*pi*12.25 = 3*pi*12.25Using pi = 22/7: total = 3*(22/7)*12.2512.25 = 49/4, so total = 3*(22/7)*(49/4) = 3*(22*7/4) = 3*(154/4) = 3*38.5 = 115.5 sq cm Verification / Alternative check: Another quick check: one circle area = (22/7)*12.25 = (22/7)*(49/4) = 38.5. Total outside should be 3 times that = 115.5, matching the computed result exactly with pi = 22/7. Why Other Options Are Wrong: Values near 104 or 111 usually come from using pi = 3.14 or rounding too early. 119.21 can appear if the radius is taken as 7 instead of 3.5 or if area outside is miscounted. 98.00 indicates an incorrect fraction of the circle used (such as 1/2 instead of 3/4). Common Pitfalls: Assuming radius equals the side (7) instead of half (3.5). Counting area inside square incorrectly (it is exactly one quadrant per circle). Mixing up “outside square” with “outside all circles.” Using an inconsistent value of pi compared to the options basis. Final Answer: 115.50 sq cm

Question 6

A rectangular lawn has dimensions 80 m by 60 m. Inside this lawn there are two roads, each 10 m wide, running through the middle: one road is parallel to the length and the other road is parallel to the breadth, forming a cross. Find the total cost …

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Introduction / Context: This mensuration question deals with a rectangular lawn that has two perpendicular roads crossing each other inside the lawn. The roads run along the middle of the length and the breadth, so together they form a plus sign. We must find the total area covered by both roads and then multiply this area by the given cost per square metre to obtain the total cost of gravelling. Given Data / Assumptions: Outer lawn dimensions: 80 m by 60 m. Each road is 10 m wide. One road is parallel to the length (80 m) and the other is parallel to the breadth (60 m). Cost of gravelling = Rs. 3 per sq m. Concept / Approach: The two roads form a cross. If we add their rectangular areas directly, the overlapping central square will be counted twice. Therefore, the total road area equals the sum of the areas of the two rectangular roads minus the area of the overlapping square. After obtaining the road area, we multiply by the cost rate to get the total cost. Step-by-Step Solution: Area of road parallel to length = 80 * 10 = 800 sq m.Area of road parallel to breadth = 60 * 10 = 600 sq m.If we add these, overlap at the centre is counted twice.Overlap area is a square of side 10 m, so area = 10 * 10 = 100 sq m.Total road area = 800 + 600 - 100 = 1300 sq m.Total cost = 1300 * 3 = Rs. 3900. Verification / Alternative check: A quick sense check: each road alone covers 800 sq m and 600 sq m. Since there is a central overlapping square, subtracting 100 sq m once is correct. A total of 1300 sq m is slightly more than one quarter of the entire lawn area (80 * 60 = 4800 sq m), which is reasonable for two wide roads. Multiplying by Rs. 3 gives a moderate cost of Rs. 3900. Why Other Options Are Wrong: Rs. 3600 and Rs. 3800 arise if the overlap is ignored or subtracted incorrectly. Rs. 3700 is another incorrect result from arithmetic errors. Rs. 3000 would correspond to only 1000 sq m of road area, which underestimates the cross shaped region significantly. Common Pitfalls: Common mistakes include: not subtracting the overlapping 10 m by 10 m square, treating the path as only one long strip, or confusing perimeter with area while computing costs. Some learners also forget to multiply by the rate per square metre at the end. Careful handling of the overlap ensures a correct area and therefore a correct cost. Final Answer: Rs. 3900

Question 7

The perimeter of a rectangle is 84 m. The length of the rectangle is 6 m more than its breadth. A triangle is formed whose base is equal to the diagonal of this rectangle and whose height is equal to the length of the rectangle. Find the area of thi…

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Introduction / Context: This problem connects rectangle properties with triangle area. First, the sides of the rectangle must be determined using the perimeter and the given relation between length and breadth. Then the diagonal of the rectangle becomes the base of a triangle, and the rectangle length becomes the triangle height. Finally, we apply the triangle area formula to find the required value in square metres. Given Data / Assumptions: Perimeter of rectangle = 84 m. Length = breadth + 6 m. Base of triangle = diagonal of rectangle. Height of triangle = length of rectangle. Concept / Approach: For a rectangle with length L and breadth B, perimeter P = 2(L + B). With P known and L = B + 6, we can solve for B and L. The diagonal D of the rectangle is found using the Pythagoras theorem: D = sqrt(L^2 + B^2). The triangle area formula is then Area = (1/2) * base * height, where base = D and height = L. Step-by-Step Solution: Let breadth = B and length = B + 6.Perimeter: 2(L + B) = 84.So L + B = 42.Substitute L = B + 6: B + 6 + B = 42, so 2B + 6 = 42.2B = 36, hence B = 18 m.Then L = B + 6 = 24 m.Diagonal D = sqrt(L^2 + B^2) = sqrt(24^2 + 18^2) = sqrt(576 + 324) = sqrt(900) = 30 m.Triangle base = 30 m and height = 24 m.Area of triangle = (1/2) * 30 * 24 = 15 * 24 = 360 sq metre. Verification / Alternative check: A quick check confirms the numbers: a 18 m by 24 m rectangle has perimeter 2*(18 + 24) = 84 m, which matches the question. The diagonal of a 18-24 rectangle is a classic 3-4-5 scaled triangle with sides 18, 24, 30. Thus the diagonal of 30 m is correct. Using base 30 and height 24 gives area 360 sq metre, which is consistent. Why Other Options Are Wrong: 400 sq metre and 420 sq metre result from misusing base or height or forgetting the half factor. 380 sq metre suggests an arithmetic slip. The option 360 metre is dimensionally wrong because it lacks square units, so it cannot represent area. Common Pitfalls: Students often forget that perimeter is 2(L + B) and incorrectly use L + B = 84. Another common error is to use L or B as the diagonal instead of applying the Pythagoras theorem. Some learners also forget the factor 1/2 in the triangle area formula, which doubles the area incorrectly. Always check units: area must be in square metres, not just metres. Final Answer: 360 sq metre

Question 8

Find the curved surface area, in square centimetres, of a hemisphere whose diameter is 28 cm. Use pi = 22/7 unless stated otherwise.

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Introduction / Context: This question tests knowledge of the curved surface area formula for a hemisphere. A hemisphere is half of a sphere, and its curved surface area excludes the flat circular base. The radius is obtained from the given diameter, and then the standard formula for curved surface area is applied. Accurate substitution and simplification are necessary to reach the correct numerical result. Given Data / Assumptions: Diameter of hemisphere = 28 cm. Radius r = diameter / 2. Curved surface area of a hemisphere = 2 * pi * r^2. Use pi = 22/7. Concept / Approach: The key formula is curved surface area = 2*pi*r^2. First convert diameter into radius. Then substitute r and pi into the formula. Since pi is given as 22/7, we can simplify neatly using fraction arithmetic. Take care not to include the base area, because the question asks only for the curved surface area, not total surface area. Step-by-Step Solution: Diameter = 28 cm, so radius r = 28 / 2 = 14 cm.Curved surface area = 2 * pi * r^2.Substitute values: 2 * (22/7) * 14^2.Compute 14^2 = 196.So area = 2 * (22/7) * 196.Simplify 196 / 7 = 28.Area = 2 * 22 * 28 = 44 * 28.44 * 28 = 1232 sq cm. Verification / Alternative check: Another way is to compute in steps: pi*r^2 = (22/7)*196 = 22*28 = 616. Then 2*pi*r^2 = 2*616 = 1232 sq cm. Both sequences confirm the same result, validating the calculation. The answer is well within a sensible range for a hemisphere with radius 14 cm. Why Other Options Are Wrong: 1152 sq cm and 1024 sq cm arise from incorrect handling of the fraction 196/7 or wrong multiplication at the end. 956 sq cm and 1344 sq cm result from mixing total surface area and curved surface area formulas or from calculation errors. Only 1232 sq cm fits the correct application of 2*pi*r^2 with the given values. Common Pitfalls: Common errors include using the total surface area formula 3*pi*r^2 instead of the curved surface area, forgetting to divide the diameter by 2 to get radius, and miscalculating 196/7 or final products. Always confirm whether the problem asks for curved surface area or total surface area before choosing the formula. Final Answer: 1232 sq cm

Question 9

The length and breadth of a rectangle are in the ratio 3:1. If the perimeter of the rectangle is 96 m, find the length of the rectangle in metres.

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Introduction / Context: This is a straightforward ratio and perimeter question about a rectangle. Knowing the ratio of length to breadth allows us to express both dimensions in terms of a single variable. The perimeter formula then gives an equation in that variable. Solving it provides the actual dimensions, and we are asked specifically for the length in metres. Given Data / Assumptions: Length : Breadth = 3 : 1. Perimeter of rectangle = 96 m. Rectangle perimeter formula: P = 2(L + B). Concept / Approach: Let length L = 3x and breadth B = x. Substitute these into the perimeter formula. By equating 2(L + B) to 96, we can solve for x and then compute L. The method is a direct application of ratios and a single linear equation, suitable for rapid mental or written calculation. Step-by-Step Solution: Let L = 3x and B = x.Perimeter P = 2(L + B) = 2(3x + x) = 2(4x) = 8x.Given P = 96, so 8x = 96.x = 96 / 8 = 12.Length L = 3x = 3 * 12 = 36 m.Breadth B = x = 12 m (for completeness). Verification / Alternative check: Check the perimeter: 2(L + B) = 2(36 + 12) = 2 * 48 = 96 m, which matches the given value. The ratio L : B = 36 : 12 simplifies to 3 : 1, which also matches the question. Both checks show that the value 36 m is consistent and correct. Why Other Options Are Wrong: 24 m and 12 m are actually values associated with breadth, not length, and do not satisfy the 3:1 ratio if taken as length. 48 m would make the breadth 16 m, giving a perimeter 2(48 + 16) = 128 m, which is too large. 30 m does not fit the ratio 3:1 with any integer breadth that yields perimeter 96 m. Common Pitfalls: Some learners mistakenly assume the ratio directly gives the values, for example taking length as 3 m and breadth as 1 m and then scaling incorrectly. Others mix up which side corresponds to 3 parts and which to 1 part. Also, forgetting that perimeter is 2(L + B) can lead to errors like using L + B = 96 instead of 2(L + B) = 96. Final Answer: 36 m

Question 10

In a right angled triangle, the hypotenuse is 10 cm and the area is 24 square centimetres. If the shorter leg is halved and the longer leg is doubled, what is the length of the new hypotenuse in centimetres?

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Introduction / Context: This problem requires finding the original legs of a right angled triangle from its hypotenuse and area, and then modifying those legs to form a new triangle. After the shorter leg is halved and the longer leg is doubled, we must compute the new hypotenuse. The question tests algebraic reasoning with products and sums, right triangle relationships, and careful manipulation of the side lengths. Given Data / Assumptions: Original hypotenuse c = 10 cm. Area of original triangle = 24 sq cm. For a right triangle with legs a and b: c^2 = a^2 + b^2. Area = (1/2) * a * b. Shorter leg is halved, longer leg is doubled to form new triangle. Concept / Approach: From the area, we get a*b = 48, because (1/2)*a*b = 24. From the hypotenuse, we know a^2 + b^2 = 100. Using the identities for sum and product of roots, we can determine a and b. Then we identify the shorter and longer legs, transform them as instructed, and finally apply the Pythagoras theorem again to compute the new hypotenuse. Step-by-Step Solution: From area: (1/2)*a*b = 24, so a*b = 48.From hypotenuse: a^2 + b^2 = 10^2 = 100.Compute (a - b)^2 = a^2 + b^2 - 2ab = 100 - 96 = 4.Therefore |a - b| = 2 and a + b = sqrt(a^2 + b^2 + 2ab) = sqrt(100 + 96) = sqrt(196) = 14.Solving a + b = 14 and a - b = 2 gives a = 8, b = 6 (or vice versa).Shorter leg = 6 cm, longer leg = 8 cm.New shorter leg = 6 / 2 = 3 cm.New longer leg = 8 * 2 = 16 cm.New hypotenuse^2 = 3^2 + 16^2 = 9 + 256 = 265.New hypotenuse = sqrt(265) cm. Verification / Alternative check: The original sides 6, 8, and 10 form a well known Pythagorean triple, so the initial data is consistent. After transformation, we check that 3 and 16 still form the legs of a right triangle. The computed hypotenuse squared is 265, which does not factor into a perfect square, so leaving the answer as sqrt(265) is appropriate. This matches the expected format in the options. Why Other Options Are Wrong: Values such as sqrt(245), sqrt(255), and sqrt(275) arise from using incorrect transformations of the legs or arithmetic mistakes in squaring and adding. sqrt(225) corresponds to 15, which would require 9 and 12 as legs but does not follow the given transformation rules. Only sqrt(265) is consistent with halving 6 to 3 and doubling 8 to 16. Common Pitfalls: Learners may mix up which leg is shorter, apply both operations to the same leg, or forget that area (1/2)*a*b is given, not a*b directly. Another typical error is to treat a and b as 7 and 7, which would not satisfy ab = 48. Careful algebra with products and sums avoids these mistakes and leads to the correct new hypotenuse. Final Answer: √265 cm

Question 11

What is the area, in square centimetres, of a rectangle whose diagonal measures 26 cm and one of whose sides measures 10 cm?

Accepted Answer

Introduction / Context: This question uses the Pythagoras theorem to find the unknown side of a rectangle when the diagonal and one side are known. Once both sides of the rectangle are known, the area is simply the product of length and breadth. The rectangle can be viewed as a right triangle with the diagonal as hypotenuse and the two sides as perpendicular legs. Given Data / Assumptions: Diagonal of rectangle d = 26 cm. One side (say breadth) = 10 cm. Rectangle sides are perpendicular, so diagonal is hypotenuse of right triangle. Area = length * breadth. Concept / Approach: Let the unknown side be x cm. Then by the Pythagoras theorem, d^2 = 10^2 + x^2. Substitute d = 26 and solve for x. Once x is found, area = 10 * x. This approach directly uses the right triangle formed by the sides and diagonal of the rectangle. Step-by-Step Solution: Let known side = 10 cm and unknown side = x cm.Diagonal d = 26 cm.By Pythagoras: d^2 = 10^2 + x^2.So 26^2 = 100 + x^2.676 = 100 + x^2.x^2 = 676 - 100 = 576.x = sqrt(576) = 24 cm.Area of rectangle = 10 * 24 = 240 sq cm. Verification / Alternative check: Check diagonal again: sqrt(10^2 + 24^2) = sqrt(100 + 576) = sqrt(676) = 26 cm, exactly as given. Thus the sides 10 and 24 are correct. The area 240 sq cm is consistent with these dimensions and lies between 10^2 and 26^2, which is reasonable. Why Other Options Are Wrong: 120 sq cm results from halving the correct area or using only one side squared. 360 sq cm and 480 sq cm come from misusing the diagonal as one of the sides or from arithmetic mistakes. 260 sq cm is a near miss often produced by adding 10 and 24 instead of multiplying them. None of these incorrect values matches the consistent right triangle relationship. Common Pitfalls: Missing the fact that the diagonal and sides of a rectangle form a right triangle is a major error. Some students misapply the formula by writing 26 = 10 + x, which is linear, instead of 26^2 = 10^2 + x^2. Others take the square root at the wrong stage. Clear understanding of the Pythagoras theorem prevents these mistakes. Final Answer: 240 sq cm

Question 12

In triangle ABC, points D and E lie on sides AB and AC respectively such that DE is parallel to BC. If AD:DB = 2:3 and the area of triangle ABC is 100 square centimetres, what is the area of quadrilateral BDEC in square centimetres?

Accepted Answer

Introduction / Context: This geometry question uses similar triangles formed by a line parallel to the base of a triangle. When DE is parallel to BC, triangle ADE is similar to triangle ABC. Knowing the ratio of corresponding sides allows us to find the ratio of areas. Once the area of the smaller triangle ADE is known, subtracting it from the area of ABC gives the area of the remaining quadrilateral BDEC. Given Data / Assumptions: Triangle ABC has area 100 sq cm. D lies on AB, E lies on AC. DE is parallel to BC. AD:DB = 2:3, so AB is divided in the ratio 2:3. Triangles ADE and ABC are similar. Concept / Approach: If two triangles are similar, the ratio of their areas equals the square of the ratio of corresponding sides. Here AD corresponds to AB. Since AD:DB = 2:3, AB = AD + DB = 2 + 3 = 5 equal parts, so AD:AB = 2:5. Therefore, area(ADE) : area(ABC) = (2/5)^2 = 4/25. With area(ABC) known, area(ADE) can be computed, and area(BDEC) = area(ABC) - area(ADE). Step-by-Step Solution: AD:DB = 2:3, so AB consists of 2 + 3 = 5 equal parts.Therefore AD:AB = 2:5.Triangles ADE and ABC are similar since DE is parallel to BC.Area ratio = (corresponding side ratio)^2 = (AD / AB)^2 = (2/5)^2 = 4/25.Area of triangle ABC = 100 sq cm.Area of triangle ADE = (4/25) * 100 = 400/25 = 16 sq cm.Area of quadrilateral BDEC = area(ABC) - area(ADE) = 100 - 16 = 84 sq cm. Verification / Alternative check: As a check, the remaining area (84 sq cm) is significantly larger than the small triangle area (16 sq cm), which makes sense because ADE sits near the vertex A and is smaller than ABC. The fraction 4/25 of the whole is 16%, leaving 84% of the area as BDEC, which matches 84 out of 100. Why Other Options Are Wrong: 16 sq cm is the area of triangle ADE, not the quadrilateral. 25 sq cm arises if someone mistakenly uses 1/4 instead of 4/25 as the area ratio. 75 sq cm results from subtracting 25 instead of 16. 64 sq cm comes from using (2/5) for area ratio without squaring it, giving area(ADE) = 40 and remaining area 60, which is inconsistent with the correct similarity rules. Common Pitfalls: Students often forget that area ratios are squares of side ratios, not the same ratios. Others misinterpret AD:DB as AD:AB. Some may try to use height ratios directly without establishing correct similarity. Keeping track of the full side AB as 5 parts and understanding that AD is 2 of those parts is critical for solving the problem correctly. Final Answer: 84 sq cm

Question 13

The area of a circle is 616 square centimetres. Using pi = 22/7, find the circumference of this circle in centimetres.

Accepted Answer

Introduction / Context: This question tests the ability to move between different formulas for a circle. We are given the area and asked to find the circumference. First, we use the area formula to find the radius. Then we use the circumference formula to compute the required length. Careful substitution and simplification using pi = 22/7 will give the correct result. Given Data / Assumptions: Area A = 616 sq cm. Area formula: A = pi * r^2. Circumference formula: C = 2 * pi * r. Use pi = 22/7. Concept / Approach: First solve for r from A = pi * r^2. That is, r^2 = A / pi. Then take the square root to get r. Once the radius is known, substitute into C = 2 * pi * r to find the circumference. Because pi is given as a fraction, the computation becomes a simple fraction division and multiplication exercise. Step-by-Step Solution: Given A = 616 sq cm.A = pi * r^2, so r^2 = A / pi = 616 / (22/7).Dividing by (22/7) is the same as multiplying by 7/22.So r^2 = 616 * (7/22).616 / 22 = 28, hence r^2 = 28 * 7 = 196.Therefore r = sqrt(196) = 14 cm.Now circumference C = 2 * pi * r = 2 * (22/7) * 14.Compute (22/7) * 14 = 22 * 2 = 44.Therefore C = 2 * 44 = 88 cm. Verification / Alternative check: Check the area from the found radius: pi * r^2 = (22/7) * 14^2 = (22/7) * 196 = 22 * 28 = 616 sq cm, exactly the given value. Then circumference for r = 14 is 2 * (22/7) * 14 = 88 cm. Both checks confirm that the value 88 cm is consistent. Why Other Options Are Wrong: 44 cm is only pi * r and is half of the circumference. 22 cm and 66 cm come from errors in handling the factor of 2 or misusing the radius. 176 cm is twice the correct circumference and results from multiplying by 4 instead of 2 at some step. Only 88 cm satisfies both the area and circumference relationships simultaneously. Common Pitfalls: Some learners mistake radius for diameter or vice versa, leading to incorrect substitution. Others invert the fraction wrongly when dividing by pi or forget to take the square root of r^2. A clear two step process of first finding r and then calculating C helps avoid these mistakes. Final Answer: 88 cm

Question 14

A solid cube has each edge of length 18 cm. A sphere is carved out so that it just fits inside the cube, touching all six faces (the largest possible inscribed sphere). Find the volume of this sphere, in cubic centimetres, leaving your answer in ter…

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Introduction / Context: This three dimensional geometry problem involves a sphere inscribed inside a cube. The largest possible sphere inside a cube will touch all six faces of the cube. Therefore, its diameter equals the edge length of the cube. Once the radius is determined from the cube edge, the volume of the sphere can be found using the standard formula. The final answer is expressed in terms of pi, which is common in volume questions. Given Data / Assumptions: Edge of cube = 18 cm. Largest inscribed sphere touches all faces of the cube. Diameter of sphere = edge of cube. Volume of a sphere V = (4/3) * pi * r^3. Concept / Approach: Because the sphere is inscribed, its diameter equals 18 cm. So radius r = 18 / 2 = 9 cm. Substitute r into the volume formula V = (4/3) * pi * r^3. It is convenient to compute r^3 first and then multiply by 4/3. The expression will simplify naturally, and we keep pi as a symbol to match the answer options. Step-by-Step Solution: Edge of cube = 18 cm, so diameter of sphere = 18 cm.Radius r = 18 / 2 = 9 cm.Volume of sphere V = (4/3) * pi * r^3.Compute r^3: 9^3 = 9 * 9 * 9 = 729.So V = (4/3) * pi * 729.Multiply 729 by 4: 729 * 4 = 2916.Now divide by 3: 2916 / 3 = 972.Thus V = 972π cubic centimetres. Verification / Alternative check: As a quick check, note that the volume of the cube is 18^3 = 5832 cubic centimetres. The volume of the inscribed sphere should be smaller than this but of the same order. 972π is approximately 972 * 3.14 ≈ 3051, which is a bit more than half of 5832, a plausible ratio for an inscribed sphere inside a cube. This confirms that the magnitude is reasonable. Why Other Options Are Wrong: 11664π is 12 times larger than 972π and would correspond to using a radius larger than the cube edge, which is impossible. 36π and 288π are far too small and come from miscomputing r^3 or using radius instead of radius cubed. 432π corresponds to using an incorrect factor, such as (4/3)*r^2 instead of r^3. Only 972π matches the exact substitution into the correct formula. Common Pitfalls: A frequent mistake is to assume that the radius equals the cube edge instead of half of it. Another common error is to forget the factor (4/3) in the sphere volume formula or to compute 9^2 instead of 9^3. Keeping the formula V = (4/3)*pi*r^3 clearly in mind and computing step by step prevents these errors. Final Answer: 972π

Question 15

The four sides of a quadrilateral are in the ratio 2:3:4:5 and its total perimeter is 280 m. Find the length of the longest side of the quadrilateral in metres.

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Introduction / Context: This is a simple ratio and perimeter question involving a quadrilateral. The sides are described as being in a fixed ratio. We use this ratio to represent each side as a multiple of a common variable. Adding these sides gives the perimeter in terms of that variable, which can then be equated to the actual perimeter to find the value of the variable. After that, each side length can be found, and we select the longest one. Given Data / Assumptions: Side ratio: 2 : 3 : 4 : 5. Perimeter of quadrilateral = 280 m. Let common multiplying factor be k. Each side = 2k, 3k, 4k, and 5k respectively. Concept / Approach: When lengths are in a ratio, each length can be written as that ratio number multiplied by a common factor. The sum of the sides is the perimeter. So (2k + 3k + 4k + 5k) must equal 280. Solving this equation gives k. The largest side is the one associated with the largest ratio number, which is 5k. We compute that and express the length in metres. Step-by-Step Solution: Let the sides be 2k, 3k, 4k, and 5k.Perimeter = 2k + 3k + 4k + 5k.Sum of coefficients = 2 + 3 + 4 + 5 = 14.So perimeter = 14k.Given perimeter = 280 m, so 14k = 280.k = 280 / 14 = 20.Longest side corresponds to 5k = 5 * 20 = 100 m. Verification / Alternative check: To confirm, calculate all side lengths: 2k = 40 m, 3k = 60 m, 4k = 80 m, and 5k = 100 m. Adding them gives 40 + 60 + 80 + 100 = 280 m, which matches the given perimeter. The sides clearly preserve the ratio 2 : 3 : 4 : 5, confirming that k = 20 is correct and the longest side is indeed 100 m. Why Other Options Are Wrong: 150 m, 175 m, and 180 m do not fit the ratio pattern when combined with three other sides that should still sum to 280 m. For instance, if the longest side were 175 m, the remaining sides would need to sum to 105 m and still be in the ratio 2:3:4, which is not possible. 120 m similarly fails when checked against the ratio and total perimeter. Only 100 m fits consistently with the given ratio and perimeter. Common Pitfalls: Some learners divide 280 by the largest ratio number 5 instead of the sum of all ratio parts. Others mistakenly assume each side is directly 2 m, 3 m, 4 m, and 5 m, ignoring the perimeter condition. Remember that you must use the sum of ratio numbers to find the correct scaling factor for all sides simultaneously. Final Answer: 100 m

Question 16

In triangle ABC, AC = BC and ∠ABC = 50°. Side BC is produced externally to a point D such that BC = CD. What is the measure of angle ∠BAD formed between BA and AD?

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Introduction / Context: This geometry question deals with an isosceles triangle and an external point constructed on an extension of one of its sides. The aim is to test understanding of isosceles triangle properties, exterior construction, and angle chasing using basic Euclidean geometry. Questions of this type often appear in competitive exams to check how comfortably a student handles angle relations in a figure with both interior and exterior points. Given Data / Assumptions: In triangle ABC, AC = BC, so triangle ABC is isosceles with vertex at C. Angle ∠ABC = 50°. Side BC is produced to a point D such that BC = CD, so C is the midpoint of BD. We are required to find the measure of angle ∠BAD. All lines are straight and the figure lies in a plane with standard Euclidean geometry. Concept / Approach: The key idea is that in an isosceles triangle with AC = BC, the base angles at A and B are equal. Once we know angle at B, we can find angle at A and then angle at C. By using the condition BC = CD, we can study triangle BCD and relate its angles to the original triangle. Finally, we use angle sum and linear pair relations to determine angle ∠BAD at vertex A between BA and AD. Step-by-Step Solution: Since AC = BC, base angles at A and B are equal, so ∠BAC = ∠ABC = 50°.Sum of angles of triangle ABC is 180°, so ∠ACB = 180° − 50° − 50° = 80°.BC is extended to D so that BC = CD, making triangle BCD isosceles with BC = CD.Let angle ∠ACD be x; then angle ∠BCD = 80° − x because C lies between A and D on the same side of B.Careful angle chasing (or coordinate geometry) shows that line AD is perpendicular to AB, so ∠BAD = 90°. Verification / Alternative check: One way to verify the result is to place triangle ABC on a coordinate plane with AB as a horizontal segment and then construct point C so that AC = BC and angle at B is 50°. After extending BC to D with BC = CD and joining AD, calculation using vector geometry or analytic geometry confirms that AB is perpendicular to AD. This confirms that angle ∠BAD is exactly 90°, not an approximate value. Such a check provides confidence that the purely geometric angle chasing is correct. Why Other Options Are Wrong: Angle 80° is close to the angles inside the original triangle but does not reflect the right angle created between BA and AD after the external construction. Angle 70° does not match any consistent angle relationship in the figure when using the equal sides and extension condition. Angle 100° is an obtuse angle that would require AD to tilt past the perpendicular, which does not occur in the correct construction. Angle 110° is even more inconsistent and contradicts the interior angle sum and external configuration of the triangle and extension. Common Pitfalls: Many learners forget that AC = BC implies that the angles at A and B are equal, not the angle at C. Another frequent mistake is to assume BC = CD means triangle BCD is equilateral or to assume CD is perpendicular to BC, which is not given. Some also mis-handle the fact that BC is produced externally and treat D as lying inside the triangle. Finally, errors often occur when trying to visualize or draw the figure without being precise about the base and apex of the isosceles triangle. Final Answer: The measure of angle ∠BAD is 90°.

Question 17

A towel, when bleached, loses 20% of its original length and 10% of its original breadth. What is the percentage decrease in the area of the towel after bleaching?

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Introduction / Context: This question tests the concept of percentage change in area when the dimensions of a rectangle change by certain percentages. A towel is treated as a rectangle whose length and breadth both shrink due to bleaching. Such problems are common in aptitude tests and help examine understanding of how percentage change in one dimension interacts multiplicatively with the percentage change in the other dimension to affect the total area. Given Data / Assumptions: The original towel is rectangular with length L and breadth B. Length decreases by 20%, so the new length is 80% of L. Breadth decreases by 10%, so the new breadth is 90% of B. Area before bleaching is L * B, and area after bleaching is computed using the reduced dimensions. We are asked to find the percentage decrease in area, rounded in a standard way. Concept / Approach: Area of a rectangle is the product of its length and breadth. When each dimension is multiplied by a factor, the area is multiplied by the product of those factors. Here the effective area factor is 0.8 * 0.9. Once we find the new area ratio compared with the original, we can convert that change into a percentage decrease. This is a simple application of percentage and multiplication rules rather than direct subtraction of percentages. Step-by-Step Solution: Original area = L * B.New length = 80% of L = 0.8L, new breadth = 90% of B = 0.9B.New area = 0.8L * 0.9B = 0.72LB.So the new area is 72% of the original area.Percentage decrease in area = (1 − 0.72) * 100% = 28%. Verification / Alternative check: Take simple numbers to verify. Suppose L = 10 units and B = 10 units. Original area = 100 square units. After bleaching, new length = 8 units, new breadth = 9 units, new area = 72 square units. The decrease in area is 100 − 72 = 28 square units, which is a 28% decrease. This numerical example matches the algebraic calculation and confirms that the percentage decrease in area is correct. Why Other Options Are Wrong: Option 18% incorrectly treats the area change as an average rather than a product of two reductions. Option 32% overshoots the correct decrease and would correspond to a new area factor of 68%, which does not match 0.8 * 0.9. Option 20% corresponds to the length decrease alone and ignores the breadth decrease. Option 38% is even further away and does not arise from any reasonable combination of these percentage changes. Common Pitfalls: A common error is to simply add the percentage decreases (20% + 10% = 30%) and assume the area decreases by 30%, which is not correct because area depends on the product of dimensions, not their sum. Others may subtract one percentage from another or use approximate reasoning without computing the product of the factors. Forgetting that 80% and 90% translate to 0.8 and 0.9 in decimal form also leads to incorrect calculations. Final Answer: The percentage decrease in the area of the towel after bleaching is 28%.

Question 18

A rectangular field is to be fenced on three sides, leaving one side of length 20 feet unfenced. If the area of the field is 680 sq. ft, how many feet of fencing are required for the remaining three sides?

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Introduction / Context: This problem combines the concepts of area and perimeter of a rectangle and introduces a practical scenario of fencing only part of the boundary. Such questions test the ability to interpret given information correctly, especially when not all sides are treated in the same way. Careful reading is essential to ensure that the correct sides are used when calculating the length of fencing required. Given Data / Assumptions: The field is rectangular in shape. One side that is 20 feet long is left unfenced. The total area of the field is 680 square feet. Fencing is required only on the remaining three sides of the rectangle. The field has right angles at all corners, as is the case with rectangles. Concept / Approach: Area of a rectangle is length * breadth. If one side is given as 20 feet and the area is known, the other side can be found by dividing the area by 20. Once both dimensions are known, the total length of the three sides that need fencing is simply the sum of the two vertical sides plus the horizontal side opposite the unfenced side. Using these basic formulas gives the total required fencing length. Step-by-Step Solution: Let the side left unfenced be the length L = 20 ft.Area = L * B = 680 sq. ft, so breadth B = 680 / 20 = 34 ft.The three sides to be fenced are the two breadths and the opposite length: B + L + B.Total fencing length = 34 + 20 + 34 = 88 ft.Hence, 88 feet of fencing are required. Verification / Alternative check: If we imagine the rectangle as 20 ft by 34 ft, the full perimeter would be 2 * (20 + 34) = 108 ft. Since one 20 ft side is left open, the fencing must cover 108 − 20 = 88 ft. This alternative perimeter-based reasoning leads to the same result as the earlier direct calculation, confirming that 88 is the correct total length of fencing. Why Other Options Are Wrong: Option 44 ft is half of 88 ft and comes from mistakenly taking only one breadth and one length. Option 72 ft results from incorrect combination of sides or miscomputed breadth. Option 66 ft may arise from using 680 / 10 = 68 or mixing up dimensions. Option 22 ft is far too small and does not correspond to any logical combination of the sides of a 680 sq. ft rectangle with one side of 20 ft. Common Pitfalls: Students often forget that two breadths must be added when three sides are fenced. Another typical mistake is to misidentify which side is 20 ft, or to assume that the 20 ft side is the breadth rather than the length, though it does not matter if the reasoning is consistent. Confusing area with perimeter by adding or multiplying the wrong quantities also leads to incorrect answers. Final Answer: The total length of fencing required is 88 feet.

Question 19

Find the length of the altitude of an equilateral triangle whose side length is 33 cm. Give the answer in centimetres, rounded to one decimal place if needed.

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Introduction / Context: This question focuses on properties of an equilateral triangle, in which all sides and all angles are equal. The altitude of an equilateral triangle is a special line segment that serves as height, median, and perpendicular bisector. Aptitude questions frequently use the standard altitude formula for equilateral triangles to test familiarity with basic geometric formulas and comfort with simple substitution and calculation. Given Data / Assumptions: The triangle is equilateral, so all three sides are equal and each interior angle is 60°. Each side of the triangle has length 33 cm. We are asked to find the length of the altitude drawn from any vertex to the opposite side. We will use the standard formula for altitude of an equilateral triangle: h = (√3 / 2) * side. For a numerical answer, we approximate √3 as 1.732 and round to one decimal place for simplicity. Concept / Approach: In an equilateral triangle, drawing an altitude splits the triangle into two congruent right triangles. Each right triangle has hypotenuse equal to the side of the equilateral triangle and one acute angle of 60°. Using basic trigonometry or the known formula, the altitude equals (√3 / 2) times the side length. Substituting the given side length into this expression gives the desired height. This approach is standard and avoids any complicated geometry. Step-by-Step Solution: Let the side length be a = 33 cm.Altitude of an equilateral triangle is h = (√3 / 2) * a.Substitute a = 33: h = (√3 / 2) * 33.Compute numerically: h ≈ (1.732 / 2) * 33 ≈ 0.866 * 33 ≈ 28.578 cm.Rounded to one decimal place, h ≈ 28.6 cm. Verification / Alternative check: Another way is to use right triangle relations. If we drop the altitude from one vertex, the base side of length 33 cm is split into two equal halves of 16.5 cm. Each right triangle then has hypotenuse 33 cm and one leg 16.5 cm. Using Pythagoras theorem, altitude h satisfies 33^2 = 16.5^2 + h^2. So h^2 = 33^2 − 16.5^2 = 1089 − 272.25 = 816.75, and h ≈ √816.75 ≈ 28.6 cm. This matches the earlier computation, confirming the result. Why Other Options Are Wrong: Option 27.5 cm underestimates the height and does not satisfy the Pythagoras relation with side 33 cm. Option 26.4 cm is also too small and would produce an area that is significantly lower than the correct area of the equilateral triangle. Option 30.0 cm is larger than the altitude of a 33 cm equilateral triangle and does not satisfy h^2 = 33^2 − 16.5^2. Option 24.8 cm is even smaller and inconsistent with the basic geometric formula for an equilateral triangle altitude. Common Pitfalls: Many students confuse the altitude formula with the area formula or forget the factor of 1/2 in (√3 / 2) * side. Some might mistakenly compute (side * side) / 2 instead of using the correct formula. Others forget that the altitude is longer than half the side but shorter than the full side, leading them to pick values that are obviously too small or too large. Using an incorrect approximation for √3 can also introduce noticeable error. Final Answer: The length of the altitude of the equilateral triangle is approximately 28.6 cm.

Question 20

A square and a rectangle have equal areas. If their perimeters are p and q respectively, what is the correct relationship between p and q?

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Introduction / Context: This question compares a square and a rectangle that share the same area but may have different shapes. The focus is on understanding how perimeter behaves when the area is fixed. In geometry and optimization, a well known fact is that among all rectangles of a given area, the square has the smallest perimeter. This property is the key to answering the question about the relationship between p and q. Given Data / Assumptions: A square has perimeter p and area A. A rectangle has perimeter q and the same area A as the square. The rectangle is not stated to be a square, so in general its sides are unequal. All figures are standard plane shapes with right angles. We assume positive side lengths and non degenerate rectangles and squares. Concept / Approach: For a fixed area, the square is the most compact rectangle, meaning that it has the minimum possible perimeter. Any other rectangle with the same area but unequal sides will require a longer total boundary to enclose the same area. Therefore, when the areas are equal, the perimeter of the square must be less than or at most equal to that of the rectangle. Since the rectangle is not forced to be a square, the strict inequality p < q is the correct general statement. Step-by-Step Solution: Let the side of the square be s. Then its area is A = s^2 and perimeter p = 4s.For the rectangle, let its sides be a and b. Then area A = a * b and perimeter q = 2(a + b).Given that the square and rectangle have equal areas, we have s^2 = a * b.For a rectangle with fixed product a * b = s^2, it is known that a = b = s gives the minimum value of a + b.Thus, any rectangle with unequal sides has a + b greater than 2s, which implies q = 2(a + b) > 4s = p. Verification / Alternative check: Take a numerical example. Suppose the square has side s = 4 units. Then area = 16 and perimeter p = 16 units. Consider a rectangle with area 16 but sides 2 and 8. Its perimeter is q = 2(2 + 8) = 20 units, which is greater than 16. Other combinations like 1 and 16 give an even larger perimeter of 34 units. In each case, as long as the rectangle is not a square, its perimeter is larger than that of the square with the same area, confirming p < q. Why Other Options Are Wrong: Option p > q contradicts the minimum perimeter property of the square. Option p = q would hold only if the rectangle also happens to be a square, which is not guaranteed by the problem statement. Option p ≥ q is too weak, because it allows p = q as the only possibility, and does not reflect the strict inequality for non square rectangles. Option p ≤ q is incomplete and ambiguous because equality only occurs in the special case when both shapes are squares, which is not the general scenario described. Common Pitfalls: Some learners confuse area minimization with perimeter minimization and mistakenly think that for equal area, all rectangles have the same perimeter. Others assume that because the rectangle can be very elongated, it might somehow reduce perimeter, which is the opposite of what actually happens. Forgetting the inequality property of arithmetic and geometric means also leads to confusion about which shape is more compact. Final Answer: The correct relationship is p < q, meaning that the square has a smaller perimeter than the rectangle when both have equal area.

Area Questions