Q: In the given figure, ABCD is a rectangle with AD = 4 units and AE = EB. EF is perpendicular to DB and is half of DF. If the area of the triangle DEF is 5 sq. units, then what is the area of ABCD? <img src="/api/images/abcd-rectangle-ad-4-units-a…

Concept & Pythagorean Theorem in Area Calculations: By utilizing given area relations and right-angled triangles, we can deduce unknown side lengths step-by-step to find the dimensions of the larger rectangle. Step-by-Step Solution: • Given \triangle DEF is a right-angled triangle (since EF \perp DB) with area 5 sq. units. • Given EF = (1/2)DF. Let EF = x, then DF = 2x. • Area of \triangle DEF = (1/2) \cdot base \cdot height = (1/2) \cdot DF \cdot EF = (1/2) \cdot (2x) \cdot x = x^2. • Since Area = 5, x^2 = 5, meaning EF = \sqrt{5} and DF = 2\sqrt{5}. • In right \triangle DEF, applying Pythagoras theorem to find hypotenuse DE: DE^2 = EF^2 + DF^2 = (\sqrt{5})^2 + (2\sqrt{5})^2 = 5 + 20 = 25 • Therefore, DE = 5 units. • In right \triangle DAE (corner of rectangle ABCD), AD = 4 units and hypotenuse DE = 5 units. • Apply Pythagoras theorem to find AE: AE = \sqrt{DE^2 - AD^2} = \sqrt{25 - 16} = \sqrt{9} = 3 units. • Given AE = EB, so AB = 2 \cdot AE = 2 \cdot 3 = 6 units. • The area of rectangle ABCD = AB \cdot AD = 6 \cdot 4 = 24 sq. units. Exam Strategy & Shortcut: Recognize the classic 3-4-5 right triangle. Once you find DE = 5 and know AD = 4, AE must be 3. Since E is the midpoint, the full length AB is 6. Area is simply 6 × 4 = 24. Common Pitfall: Misinterpreting the relation between EF and DF, or failing to identify \triangle DAE as a right-angled triangle with sides 3-4-5. Final Answer: Therefore, the correct answer is 24 sq. units.

Q: Which two figure have an equal area? <img src="/api/images/which-two-figure-have-equal-area-circle-rectangle-square.png" alt="which two figure have equal area circle rectangle square" width="300" height="200" loading="lazy" decoding="a…

Concept & Identification of Typo: To find equal values, we must calculate the measurements of each figure. Note: This specific question contains a known historical typo in the source material. It asks for equal *area*, but mathematically, it is testing for equal *perimeter*. We will calculate both to demonstrate this. Step-by-Step Solution: • Calculate Areas: • Figure A (Circle): Area = \pi r^2 = (22/7) × 7^2 = 154 cm^2 • Figure B (Rectangle): Area = L × W = 12 × 10 = 120 cm^2 • Figure C (Square): Area = S^2 = 12^2 = 144 cm^2 • Figure D (Rectangle): Area = L × W = 14 × 12 = 168 cm^2 • *Result:* None of the areas are equal. • Calculate Perimeters: • Figure A (Circle): Perimeter = \pi d = (22/7) × 14 = 44 cm • Figure B (Rectangle): Perimeter = 2(L + W) = 2(12 + 10) = 44 cm • Figure C (Square): Perimeter = 4S = 4 × 12 = 48 cm • Figure D (Rectangle): Perimeter = 2(L + W) = 2(14 + 12) = 52 cm • Conclusion: The perimeters of Figure A and Figure B are exactly equal (44 cm). Given the options provided, the question intended to ask "Which two figures have an equal perimeter?". Exam Strategy & Shortcut: If you quickly estimate the areas (A \approx 150, B = 120, C = 144, D = 168) and see no matches, immediately pivot to checking perimeters. Exam boards occasionally print "area" when they mean "perimeter" or vice versa. Common Pitfall: Wasting time recalculating the areas multiple times assuming a math error was made, rather than identifying the question's textual flaw. Final Answer: Therefore, the correct answer is A and B.

Q: The area of a circular field is equal to the area of a rectangular field. The ratio of the length and the breadth of the rectangular field is 14 : 11 respectively and perimeter is 100 metres. What is the diameter of the circular field?

Concept & Area Equivalence: Use the given ratio and perimeter to find the exact dimensions of the rectangle. Then, equate its area to the circle's area formula to find the radius and diameter. Perimeter of Rectangle = 2(l + b) Area of Rectangle = l × b Area of Circle = \pi r^2 Step-by-Step Solution: • Given: Ratio of length to breadth l : b = 14 : 11. Let l = 14x and b = 11x. • Given: Perimeter = 100 m. • Set up equation: 2(14x + 11x) = 100 \Rightarrow 2(25x) = 100 \Rightarrow 50x = 100 \Rightarrow x = 2. • Find dimensions: Length l = 14 × 2 = 28 m, Breadth b = 11 × 2 = 22 m. • Calculate Rectangle Area: 28 × 22 = 616 m^2. • Equate to Circle Area: \pi r^2 = 616. • Solve for radius r: (22/7) × r^2 = 616 \Rightarrow r^2 = (616 × 7/22) = 28 × 7 = 196. • r = \sqrt{196} = 14 m. • Calculate Diameter: d = 2r = 2 × 14 = 28 m. Exam Strategy & Shortcut: Instead of multiplying 28 × 22 to get 616, leave it as 28 × 22. When equating to \pi r^2, write (22/7) × r^2 = 28 × 22. The 22 on both sides cancels out immediately, leaving r^2 = 28 × 7, which is faster to evaluate. Common Pitfall: Stopping at r = 14 m and selecting option (a), forgetting that the question specifically asks for the *diameter* of the circular field, not the radius. Final Answer: Therefore, the correct answer is 28 m.

Q: A circle and a square have the same area. The ratio of the side of the square and the radius of the circle is

Concept & Logic: Equating the area formulas for a circle and a square allows us to find the direct proportional relationship between the side of the square (a) and the radius of the circle (r). \pi r^2 = a^2 Step-by-Step Solution: 1. Given: Area of circle = Area of square. \pi r^2 = a^2 2. Find the ratio of side (a) to radius (r): Divide both sides by r^2: (a^2/r^2) = \pi 3. Take the square root of both sides: (a/r) = \sqrt{\pi} (a/r) = \frac{\sqrt{\pi}}{1} So, the ratio of the side of the square to the radius of the circle is \sqrt{\pi} : 1. Exam Strategy & Shortcut: Set area to a simple constant like \pi. If area = \pi, then a^2 = \pi \Rightarrow a = \sqrt{\pi}. For the circle, \pi r^2 = \pi \Rightarrow r = 1. The ratio a : r is immediately \sqrt{\pi} : 1. Common Pitfall: Reversing the requested ratio is a common trap. The question asks for "side of the square" compared to the "radius of the circle", so ensure a is the numerator/first term. Final Answer: Therefore, the correct answer is \sqrt{\pi} : 1.

Q: A circular grassy plot of land, 42 m in diameter, has a path 3.5 m wide running around it outside. The cost of gravelling the path at ₹ 4 per square metre is

Concept & Formula: Calculate the area of the circular path by finding the difference in areas between the outer circle and inner circle, then multiply by the cost rate. Area of path = \pi(R^2 - r^2) = \pi(R - r)(R + r) Total Cost = Area × Rate Step-by-Step Solution: • Given: • Diameter of plot = 42 m • Inner radius (r) = (42/2) = 21 m • Width of path = 3.5 m • Outer radius (R) = 21 + 3.5 = 24.5 m • Find the area of the path: Area = (22/7) × (24.5^2 - 21^2) Area = (22/7) × (24.5 - 21) × (24.5 + 21) Area = (22/7) × 3.5 × 45.5 Area = 22 × 0.5 × 45.5 Area = 11 × 45.5 = 500.5 sq m • Calculate total cost: Total Cost = 500.5 × 4 = ₹ 2002 Exam Strategy & Shortcut: Delay full multiplication until the end. Notice that 3.5 = (7/2). Area = (22/7) × (7/2) × 45.5 = 11 × 45.5. Then directly multiply by 4: 11 × 45.5 × 4 = 44 × 45.5 = 2002. Common Pitfall: Adding the path width (3.5 m) to the diameter (42 m) instead of the radius (21 m), which skews the outer boundary measurement entirely. Final Answer: Therefore, the correct answer is ₹ 2002.

Question 1

An error of 2% in excess is made while measuring the side of a square. The percentage of error in the calculated area of the square is

Accepted Answer

Concept & Error Propagation in Area: When there is a small percentage error in measuring the linear dimension of a square, the error propagates in two dimensions. We can use the successive percentage change formula to find the net error in area: Net Error = x + y + (xy/100) Here, x and y represent the percentage error in length and width, which are both +2%. Step-by-Step Solution: • Step 1: The percentage error in measuring the side is 2% in excess, meaning it is a +2% change. • Step 2: Let x = 2 and y = 2. • Step 3: Plug these values into the successive percentage formula: Error in Area = 2 + 2 + (2 × 2/100). • Step 4: Calculate the terms: 4 + (4/100) = 4 + 0.04. • Step 5: The total percentage error is 4.04%. Exam Strategy & Shortcut: Assume the true side is 100. The true area is 100 × 100 = 10,000. With a 2% excess error, the measured side is 102. The calculated area is 102^2 = 10,404. The excess area is 404. Percentage error = ((404/10000)) × 100 = 4.04%. Common Pitfall: A common mistake is simply adding the errors (2% + 2% = 4%) or assuming the area error is identical to the side error (2%). This ignores the compounding factor ((xy/100)). Final Answer: Therefore, the correct answer is 4.04%.

Question 2

If the diagonal of a square is made 1.5 times, then the ratio of the areas of two squares is

Accepted Answer

Concept & Formula: The area of a square can be calculated directly from its diagonal using the formula: Area = (d^2/2) where d is the length of the diagonal. Step-by-Step Solution: 1. Let the diagonal of the original square be d. 2. The area of the original square is A_1 = (d^2/2). 3. The diagonal of the new square is made 1.5 times, so it is 1.5d = (3/2)d. 4. The area of the new square is A_2 = ( (3d/2)^2 / 2 ) = ( \frac{9d^2/4) }{ 2 } = (9d^2/8). 5. The ratio of the areas of the original square to the new square is A_1 : A_2 = (d^2/2) : (9d^2/8). 6. Simplify the ratio by multiplying both sides by (8/d^2): 4 : 9. Exam Strategy & Shortcut: Since Area is proportional to the square of any linear dimension (like the diagonal), if the diagonal is multiplied by a factor k, the area is multiplied by k^2. Here, k = 1.5 = (3/2). Therefore, the area factor is ((3/2))^2 = (9/4). The ratio of original area to new area is 1 : (9/4), which simplifies to 4 : 9. Common Pitfall: Confusing the order of the ratio. The question asks for the ratio of the areas of the two squares, implying original to new. The inverse ratio 9 : 4 is not among the options, which helps prevent this error. Final Answer: Therefore, the correct answer is 4 : 9.

Question 3

The hypotenuse of a right-angled isosceles triangle is $5	ext{ cm}$. The area of the triangle is

Accepted Answer

Concept & Right-Angled Isosceles Triangle: A right-angled isosceles triangle has two equal perpendicular legs (let's call them a). By the Pythagorean theorem, the hypotenuse h is given by h = a\sqrt{2}. The area of this triangle is given by: Area = (1/2) × a × a = (1/2)a^2 Alternatively, area can be directly related to the hypotenuse: Area = (h^2/4) Step-by-Step Solution: 1. Let the equal legs of the right-angled isosceles triangle be a. 2. The given hypotenuse is 5 cm. 3. Using the relationship h = a\sqrt{2}, we get 5 = a\sqrt{2}, so a = (5/\sqrt{2)}. 4. Substitute a into the area formula: Area = (1/2) × ((5/\sqrt{2)})^2 Area = (1/2) × (25/2) Area = (25/4) = 6.25 cm^2. Exam Strategy & Shortcut: For a right-angled isosceles triangle, memorize the direct shortcut formula relating area and hypotenuse: Area = (h^2/4). Squaring the hypotenuse gives 25, and dividing by 4 gives 6.25 instantly. Common Pitfall: Forgetting to square the \sqrt{2} in the denominator when calculating the area is a common arithmetic error, leading to (25/2) = 12.5. Final Answer: Therefore, the correct answer is 6.25 cm^2.

Question 4

If the area of an equilateral triangle is $24\sqrt{3}$ sq. cm, then its perimeter is

Accepted Answer

Concept & Formula: For an equilateral triangle of side a: Area A = \frac{\sqrt{3}}{4}a^2 Perimeter P = 3a Step-by-Step Solution: 1. Given Area = 24\sqrt{3}. 2. \frac{\sqrt{3}}{4}a^2 = 24\sqrt{3}. 3. Divide both sides by \sqrt{3}: (a^2/4) = 24. 4. a^2 = 96 \Rightarrow a = \sqrt{96} = \sqrt{16 × 6} = 4\sqrt{6} cm. 5. Perimeter = 3a = 3(4\sqrt{6}) = 12\sqrt{6} cm. Exam Strategy & Shortcut: Write 24\sqrt{3} as \frac{\sqrt{3}}{4} × (96). The side squared is 96, so the side is \sqrt{96}. Multiply by 3 for perimeter: 3\sqrt{96} = 3 × 4\sqrt{6} = 12\sqrt{6}. Common Pitfall: Forgetting to take the square root of 96, or calculating the perimeter as just 4\sqrt{6} (which is the side, not the perimeter). Final Answer: Therefore, the correct answer is 12\sqrt{6} cm.

Question 5

The areas of two equilateral triangles are in the ratio 25 : 36. Their altitudes will be in the ratio

Accepted Answer

Concept & Formula: The area of similar figures is proportional to the square of their corresponding linear dimensions (sides, altitudes, medians). (A_1/A_2) = ((h_1/h_2))^2 Step-by-Step Solution: 1. We are given the ratio of the areas of two equilateral triangles: (A_1/A_2) = (25/36). 2. The ratio of their altitudes h_1 and h_2 is the square root of the ratio of their areas. 3. (h_1/h_2) = \sqrt{(A_1/A_2)} = \sqrt{(25/36)}. 4. (h_1/h_2) = (5/6). Exam Strategy & Shortcut: For any 2D similar shapes (including all equilateral triangles), Area Ratio = (Side Ratio)^2 = (Altitude Ratio)^2. Simply take the square root of 25:36 to get 5:6. Common Pitfall: Assuming altitude ratio is the same as area ratio, or squaring the area ratio instead of taking the square root. Final Answer: Therefore, the correct answer is 5 : 6.

Question 6

In the given figure, ABCD is a rectangle with AD = 4 units and AE = EB. EF is perpendicular to DB and is half of DF. If the area of the triangle DEF is 5 sq. units, then what is the area of ABCD?  <img    src="/api/images/abcd-rectangle-ad-4-units-a…

Accepted Answer

Concept & Pythagorean Theorem in Area Calculations: By utilizing given area relations and right-angled triangles, we can deduce unknown side lengths step-by-step to find the dimensions of the larger rectangle. Step-by-Step Solution: • Given 	riangle DEF is a right-angled triangle (since EF \perp DB) with area 5 sq. units. • Given EF = (1/2)DF. Let EF = x, then DF = 2x. • Area of 	riangle DEF = (1/2) \cdot base \cdot height = (1/2) \cdot DF \cdot EF = (1/2) \cdot (2x) \cdot x = x^2. • Since Area = 5, x^2 = 5, meaning EF = \sqrt{5} and DF = 2\sqrt{5}. • In right 	riangle DEF, applying Pythagoras theorem to find hypotenuse DE: DE^2 = EF^2 + DF^2 = (\sqrt{5})^2 + (2\sqrt{5})^2 = 5 + 20 = 25 • Therefore, DE = 5 units. • In right 	riangle DAE (corner of rectangle ABCD), AD = 4 units and hypotenuse DE = 5 units. • Apply Pythagoras theorem to find AE: AE = \sqrt{DE^2 - AD^2} = \sqrt{25 - 16} = \sqrt{9} = 3 units. • Given AE = EB, so AB = 2 \cdot AE = 2 \cdot 3 = 6 units. • The area of rectangle ABCD = AB \cdot AD = 6 \cdot 4 = 24 sq. units. Exam Strategy & Shortcut: Recognize the classic 3-4-5 right triangle. Once you find DE = 5 and know AD = 4, AE must be 3. Since E is the midpoint, the full length AB is 6. Area is simply 6 × 4 = 24. Common Pitfall: Misinterpreting the relation between EF and DF, or failing to identify 	riangle DAE as a right-angled triangle with sides 3-4-5. Final Answer: Therefore, the correct answer is 24 sq. units.

Question 7

Which two figure have an equal area?  <img    src="/api/images/which-two-figure-have-equal-area-circle-rectangle-square.png"   alt="which two figure have equal area circle rectangle square"   width="300"   height="200"   loading="lazy"   decoding="a…

Accepted Answer

Concept & Identification of Typo: To find equal values, we must calculate the measurements of each figure. Note: This specific question contains a known historical typo in the source material. It asks for equal *area*, but mathematically, it is testing for equal *perimeter*. We will calculate both to demonstrate this. Step-by-Step Solution: • Calculate Areas: • Figure A (Circle): Area = \pi r^2 = (22/7) × 7^2 = 154 cm^2 • Figure B (Rectangle): Area = L × W = 12 × 10 = 120 cm^2 • Figure C (Square): Area = S^2 = 12^2 = 144 cm^2 • Figure D (Rectangle): Area = L × W = 14 × 12 = 168 cm^2 • *Result:* None of the areas are equal. • Calculate Perimeters: • Figure A (Circle): Perimeter = \pi d = (22/7) × 14 = 44 cm • Figure B (Rectangle): Perimeter = 2(L + W) = 2(12 + 10) = 44 cm • Figure C (Square): Perimeter = 4S = 4 × 12 = 48 cm • Figure D (Rectangle): Perimeter = 2(L + W) = 2(14 + 12) = 52 cm • Conclusion: The perimeters of Figure A and Figure B are exactly equal (44 cm). Given the options provided, the question intended to ask "Which two figures have an equal perimeter?". Exam Strategy & Shortcut: If you quickly estimate the areas (A \approx 150, B = 120, C = 144, D = 168) and see no matches, immediately pivot to checking perimeters. Exam boards occasionally print "area" when they mean "perimeter" or vice versa. Common Pitfall: Wasting time recalculating the areas multiple times assuming a math error was made, rather than identifying the question's textual flaw. Final Answer: Therefore, the correct answer is A and B.

Question 8

The area of a circular field is equal to the area of a rectangular field. The ratio of the length and the breadth of the rectangular field is 14 : 11 respectively and perimeter is 100 metres. What is the diameter of the circular field?

Accepted Answer

Concept & Area Equivalence: Use the given ratio and perimeter to find the exact dimensions of the rectangle. Then, equate its area to the circle's area formula to find the radius and diameter. Perimeter of Rectangle = 2(l + b) Area of Rectangle = l × b Area of Circle = \pi r^2 Step-by-Step Solution: • Given: Ratio of length to breadth l : b = 14 : 11. Let l = 14x and b = 11x. • Given: Perimeter = 100 m. • Set up equation: 2(14x + 11x) = 100 \Rightarrow 2(25x) = 100 \Rightarrow 50x = 100 \Rightarrow x = 2. • Find dimensions: Length l = 14 × 2 = 28 m, Breadth b = 11 × 2 = 22 m. • Calculate Rectangle Area: 28 × 22 = 616 m^2. • Equate to Circle Area: \pi r^2 = 616. • Solve for radius r: (22/7) × r^2 = 616 \Rightarrow r^2 = (616 × 7/22) = 28 × 7 = 196. • r = \sqrt{196} = 14 m. • Calculate Diameter: d = 2r = 2 × 14 = 28 m. Exam Strategy & Shortcut: Instead of multiplying 28 × 22 to get 616, leave it as 28 × 22. When equating to \pi r^2, write (22/7) × r^2 = 28 × 22. The 22 on both sides cancels out immediately, leaving r^2 = 28 × 7, which is faster to evaluate. Common Pitfall: Stopping at r = 14 m and selecting option (a), forgetting that the question specifically asks for the *diameter* of the circular field, not the radius. Final Answer: Therefore, the correct answer is 28 m.

Question 9

A circle and a square have the same area. The ratio of the side of the square and the radius of the circle is

Accepted Answer

Concept & Logic: Equating the area formulas for a circle and a square allows us to find the direct proportional relationship between the side of the square (a) and the radius of the circle (r). \pi r^2 = a^2 Step-by-Step Solution: 1. Given: Area of circle = Area of square. \pi r^2 = a^2 2. Find the ratio of side (a) to radius (r): Divide both sides by r^2: (a^2/r^2) = \pi 3. Take the square root of both sides: (a/r) = \sqrt{\pi} (a/r) = \frac{\sqrt{\pi}}{1} So, the ratio of the side of the square to the radius of the circle is \sqrt{\pi} : 1. Exam Strategy & Shortcut: Set area to a simple constant like \pi. If area = \pi, then a^2 = \pi \Rightarrow a = \sqrt{\pi}. For the circle, \pi r^2 = \pi \Rightarrow r = 1. The ratio a : r is immediately \sqrt{\pi} : 1. Common Pitfall: Reversing the requested ratio is a common trap. The question asks for "side of the square" compared to the "radius of the circle", so ensure a is the numerator/first term. Final Answer: Therefore, the correct answer is \sqrt{\pi} : 1.

Question 10

A circular grassy plot of land, 42 m in diameter, has a path 3.5 m wide running around it outside. The cost of gravelling the path at ₹ 4 per square metre is

Accepted Answer

Concept & Formula: Calculate the area of the circular path by finding the difference in areas between the outer circle and inner circle, then multiply by the cost rate. Area of path = \pi(R^2 - r^2) = \pi(R - r)(R + r) Total Cost = Area × Rate Step-by-Step Solution: • Given: • Diameter of plot = 42 m • Inner radius (r) = (42/2) = 21 m • Width of path = 3.5 m • Outer radius (R) = 21 + 3.5 = 24.5 m • Find the area of the path: Area = (22/7) × (24.5^2 - 21^2) Area = (22/7) × (24.5 - 21) × (24.5 + 21) Area = (22/7) × 3.5 × 45.5 Area = 22 × 0.5 × 45.5 Area = 11 × 45.5 = 500.5 sq m • Calculate total cost: Total Cost = 500.5 × 4 = ₹ 2002 Exam Strategy & Shortcut: Delay full multiplication until the end. Notice that 3.5 = (7/2). Area = (22/7) × (7/2) × 45.5 = 11 × 45.5. Then directly multiply by 4: 11 × 45.5 × 4 = 44 × 45.5 = 2002. Common Pitfall: Adding the path width (3.5 m) to the diameter (42 m) instead of the radius (21 m), which skews the outer boundary measurement entirely. Final Answer: Therefore, the correct answer is ₹ 2002.

Question 11

The circumference of a circular ground is 88 metres. A strip of land, 3 metres wide, inside and along the circumference of the ground is to be levelled. What is the budgeted expenditure if the levelling costs ₹ 7 per square metre?

Accepted Answer

Concept & Area of a Circular Ring: The area of a circular path (or ring) located inside a circular boundary is the difference between the area of the outer circle and the area of the inner circle. Area = \pi R^2 - \pi r^2 = \pi(R^2 - r^2) Step-by-Step Solution: • Find the outer radius (R): The circumference is 2\pi R = 88 m. • 2 × (22/7) × R = 88 \Rightarrow R = (88 × 7/44) = 14 m. • Find the inner radius (r): The strip is 3 m wide and inside, so r = R - 3 = 14 - 3 = 11 m. • Calculate the area of the strip: Area = \pi(R^2 - r^2) = (22/7) × (14^2 - 11^2) Area = (22/7) × (196 - 121) = (22/7) × 75 sq m. • Calculate the cost: Total Cost = Area × Rate = ((22/7) × 75) × 7 Total Cost = 22 × 75 = 1650. Exam Strategy & Shortcut: Notice that the cost calculation involves multiplying by 7, which perfectly cancels out the denominator of \pi ((22/7)). Delaying the multiplication of (22/7) × 75 saves time since the 7s cancel out in the final step. Common Pitfall: A common mistake is adding the width to the radius instead of subtracting it, assuming the path is outside the ground when the problem explicitly states it is "inside". Final Answer: Therefore, the correct answer is ₹ 1650.

Question 12

In the given diagram, $ABCD$ is a square and semi-circular regions have been added to it by drawing two semi-circles with $AB$ and $CD$ as diameters. If the total area of the three regions is $350$ sq. cm, then the length of the side of the square i…

Accepted Answer

Concept & Area of Composite Figures: The figure is composed of a central square and two identical semi-circles extending outwards. Since the semi-circles share the side length of the square as their diameter, they can logically be combined into exactly one complete circle. Formulas: Area of square = side^2 Area of circle = \pi r^2 = (\pi d^2/4) Step-by-Step Solution: 1. Let the side of the square ABCD be x cm. Consequently, the diameter of both semi-circles is also x cm. 2. The area of the central square is x^2. 3. The two semi-circles combine to form a full circle with diameter x (meaning its radius is (x/2)). Area of these two semi-circles combined = \pi ((x/2))^2 = (\pi x^2/4). 4. Set up the equation for the total given area: x^2 + (\pi x^2/4) = 350 5. Substitute \pi = (22/7) to solve: x^2 (1 + (22/28)) = 350 x^2 (1 + (11/14)) = 350 x^2 ((25/14)) = 350 6. Solve for x: x^2 = 350 × (14/25) = 14 × 14 = 196 x = \sqrt{196} = 14 cm. Exam Strategy & Shortcut: Notice the total area equation: x^2(1 + \pi/4) = x^2(25/14). The fractional factor 25/14 tells us that x^2 must ideally be a multiple of 14 to yield a clean integer total like 350. Checking the options, 14 is an extremely strong candidate. Testing it: 14^2 × (25/14) = 14 × 25 = 350. It matches perfectly. Common Pitfall: A common mistake is treating the two semi-circles as having a *radius* of x instead of a *diameter* of x, which inflates the area of the circular components by a factor of 4. Final Answer: Therefore, the correct answer is 14 cm.

Question 13

The area of a circle inscribed in an equilateral triangle is $154 	ext{ cm}^2$. Find the perimeter of the triangle.

Accepted Answer

Concept & Reverse Engineering Perimeter: When given the area of an inscribed circle, we work backward to find the radius of the circle, then use the inradius relationship to find the side of the equilateral triangle, and finally calculate its perimeter. Area = \pi r^2 r = (a/2\sqrt{3)} Perimeter = 3a Step-by-Step Solution: • Given: Area of the inscribed circle = 154 cm^2. • Step 1: Find the inradius r. \pi r^2 = 154 \implies (22/7) r^2 = 154 \implies r^2 = 154 × (7/22) \implies r^2 = 7 × 7 \implies r = 7 cm. • Step 2: Find the side a of the equilateral triangle. We know r = (a/2\sqrt{3)}, so 7 = (a/2\sqrt{3)}. a = 14\sqrt{3} cm. • Step 3: Find the perimeter. Perimeter = 3a = 3 × 14\sqrt{3} = 42\sqrt{3} cm. • Step 4: Convert to a decimal value (using \sqrt{3} \approx 1.732). Perimeter \approx 42 × 1.732 = 72.744 cm. • Rounding to one decimal place gives 72.7 cm. Exam Strategy & Shortcut: Memorize that a circle with area 154 always has a radius of 7. It's one of the most common standard measurements in geometry problems. Once r=7 is known, a = 14\sqrt{3}, and Perimeter = 42\sqrt{3}. You can quickly estimate 42 × 1.73 = 72.66, pointing safely to 72.7. Common Pitfall: Forgetting to multiply the side by 3 to get the perimeter, or prematurely rounding \sqrt{3} to 1.7, which would give 42 × 1.7 = 71.4, an incorrect distractor value close to Option (A). Use at least three decimal places (1.732) for accuracy. Final Answer: Therefore, the correct answer is 72.7 cm.

Question 14

If the circumference of a circle increases form $4\pi$ to $8\pi$, what change occurs in its area?

Accepted Answer

Concept & Circumference and Area Relationship: The scale factor for the circumference of a circle is identical to the scale factor for its radius. Once the scale factor for the radius (k) is known, the area scales by a factor of k^2. k = (C_2/C_1) = (r_2/r_1) Step-by-Step Solution: • Given: Initial circumference C_1 = 4\pi. Final circumference C_2 = 8\pi. • Calculation: Determine the scale factor for circumference: k = (8\pi/4\pi) = 2. • Deduction: Since circumference doubles, the radius also doubles. • Calculation: Let original area be A_1 = \pi r^2. • Calculation: If radius doubles, new area A_2 = \pi(2r)^2 = 4\pi r^2. • Deduction: The new area is 4 times the original area, which means it quadruples. Exam Strategy & Shortcut: Skip finding the actual radius. Notice that 8\pi is twice 4\pi. The linear dimensions (circumference, radius, diameter) doubled (2×). Area is 2-dimensional, so it scales by 2^2 = 4×. Thus, it quadruples. Common Pitfall: A common mistake is assuming that area increases linearly with circumference and choosing "It doubles." Remember area is proportional to the square of linear dimensions. Final Answer: Therefore, the correct answer is It quadruples.

Question 15

If the circumference of a circle is increased by 20%, what will be the effect on the circle?

Accepted Answer

Concept & Circumference and Area Proportionality: An increase in the circumference by a certain percentage implies the exact same percentage increase in the radius, since C = 2\pi r. The effect on the area (which is implied by "effect on the circle" given the options) can be calculated using the successive percentage change formula: x + y + (xy/100) Step-by-Step Solution: • Given: Circumference increases by 20%. • Deduction: Because C \propto r, the radius also increases by 20%. So, x = 20. • Calculation: We need to find the change in area (A = \pi r^2). Apply successive change for two identical 20% increases: 20 + 20 + (20 × 20/100). • Calculation: 40 + (400/100). • Calculation: 40 + 4 = 44%. Exam Strategy & Shortcut: Any 20% linear increase always leads to a 44% area increase. Memorize standard values: 10% \rightarrow 21%, 20% \rightarrow 44%, 30% \rightarrow 69%. This saves calculation time entirely. Common Pitfall: Students often select "Cannot be determined" because the question vaguely asks for "effect on the circle" rather than specifically "effect on the area". However, analyzing the numerical options makes it clear area is being requested. Final Answer: Therefore, the correct answer is 44% increase.

Question 16

Three circles of radius $3.5	ext{ cm}$ are placed in such a way that each circle touches the other two. The area of the portion enclosed by the circles is

Accepted Answer

Concept & Area Enclosed by Touching Circles: When three equal circles touch each other, connecting their centers forms an equilateral triangle. The area enclosed between the circles is the area of this triangle minus the area of the three circular sectors within it. Enclosed Area = Area of Equilateral Triangle - Area of 3 Sectors Step-by-Step Solution: • Given: Radius r = 3.5 cm = (7/2) cm. • Deduction: The side of the equilateral triangle formed by the centers is a = 2r = 7 cm. • Calculation: Area of the equilateral triangle = \frac{\sqrt{3}}{4} a^2 = \frac{\sqrt{3}}{4} × 7^2 = \frac{49\sqrt{3}}{4}. • Calculation: Using \sqrt{3} \approx 1.732, Area = (49 × 1.732/4) \approx 21.217 cm^2. • Calculation: Each interior angle is 60^\circ. The three sectors form a semicircle (180^\circ). Area of sectors = (1/2)\pi r^2 = (1/2) × (22/7) × ((7/2))^2 = (11/7) × (49/4) = (77/4) = 19.25 cm^2. • Calculation: Enclosed Area = 21.217 - 19.25 = 1.967 cm^2. Exam Strategy & Shortcut: For three mutually touching circles of radius r, the standard formula for the enclosed area is r^2(\sqrt{3} - (\pi/2)). Plugging in r = 3.5 directly yields (3.5)^2(1.732 - 1.571) = 12.25 × 0.161 = 1.972. Due to approximation differences in \pi and \sqrt{3}, 1.967 is the intended precise answer using (22/7). Common Pitfall: A common mistake is failing to convert the three 60^\circ sectors into a single semi-circle for calculation, leading to overly complex and error-prone fraction math. Final Answer: Therefore, the correct answer is 1.967 cm^2.

Question 17

The ratio of circumference and diameter of a circle is 22 : 7. If the circumference be $1\frac{4}{7}$m. then the radius of the circle is

Accepted Answer

Concept & Circle Proportions: The ratio of the circumference (C) of a circle to its diameter (D) is a constant, mathematically defined as \pi (approximated as (22/7)). We can use this ratio directly to find the diameter and consequently the radius. (C/D) = \pi \approx (22/7) Radius = (Diameter/2) Step-by-Step Solution: • Given: The ratio (C/D) = (22/7). • The circumference (C) is given as a mixed fraction: 1(4/7) m. • Convert the mixed fraction to an improper fraction: C = (1 × 7 + 4/7) = (11/7) m. • Set up the proportion: (\frac{11/7)}{D} = (22/7). • Multiply both sides by D: (11/7) = (22/7) × D. • Solve for D: D = (11/7) × (7/22) = (11/22) = (1/2) m. • The radius (r) is half of the diameter: r = (D/2) = (\frac{1/2)}{2} = (1/4) m. Exam Strategy & Shortcut: Recognize that (11/7) is exactly half of (22/7). Since the circumference is halved, the diameter must also be halved (from a conceptual 1 unit to (1/2) unit). The radius is half of that, resulting in (1/4) instantly. Common Pitfall: A highly common mistake is stopping after calculating the diameter ((1/2) m) and selecting it as the answer, forgetting that the question specifically asked for the radius. Final Answer: Therefore, the correct answer is (1/4)m.

Question 18

A rectangular carpet has an area of 120 m$^2$ and a perimeter of 46 metre. The length of its diagonal is  [SSC—CHSL (10+2) Exam, 2015]

Accepted Answer

Concept & Algebraic Identity: To find the diagonal of a rectangle, we need the sum of the squares of its length (l) and breadth (b), which is \sqrt{l^2 + b^2}. We can derive l^2 + b^2 using the algebraic identity (l + b)^2 = l^2 + b^2 + 2lb, substituting the given perimeter and area values. Area = l × b Perimeter = 2(l + b) Diagonal = \sqrt{l^2 + b^2} Step-by-Step Solution: • Given: Area (l × b) = 120 m^2. • Given: Perimeter 2(l + b) = 46 m. Therefore, l + b = (46/2) = 23. • Use the algebraic identity: (l + b)^2 = l^2 + b^2 + 2lb. • Substitute the known values into the identity: (23)^2 = l^2 + b^2 + 2(120). • Calculate the squares and products: 529 = l^2 + b^2 + 240. • Solve for l^2 + b^2: l^2 + b^2 = 529 - 240 = 289. • The diagonal is \sqrt{l^2 + b^2}. Therefore, Diagonal = \sqrt{289} = 17 metres. Exam Strategy & Shortcut: If l × b = 120 and l + b = 23, think of factor pairs of 120: (10, 12) - sum is 22; (8, 15) - sum is 23. Thus, l = 15 and b = 8. The diagonal forms a right-angled triangle with sides 8 and 15. The Pythagorean triplet for 8, 15 is 17. The answer is instantly 17. Common Pitfall: Attempting to solve for l and b using complex quadratic equations instead of using the algebraic identity or Pythagorean triplets, which wastes valuable time. Final Answer: Therefore, the correct answer is 17 metre.

Question 19

A courtyard is 25m long and 16m broad is to be paved with bricks of dimensions 20cm by 10cm. What is the total number of bricks required?

Accepted Answer

Concept & Unit Conversion and Area: The number of bricks required to pave an area is calculated by dividing the total area to be paved by the surface area of a single brick. Crucially, all dimensions must be in the same units (either all meters or all centimeters). Number\ of\ Bricks = (Total\ Area/Area\ of\ One\ Brick) Step-by-Step Solution: • Given: Courtyard dimensions are 25 m by 16 m. Brick dimensions are 20 cm by 10 cm. • Convert courtyard dimensions to centimeters (1 m = 100 cm). • Length = 25 × 100 = 2500 cm. • Breadth = 16 × 100 = 1600 cm. • Calculate the total area of the courtyard: Area = 2500 × 1600 = 4,000,000 cm^2. • Calculate the area of a single brick: Area = 20 × 10 = 200 cm^2. • Calculate total bricks: Number of bricks = (4,000,000/200). • Divide to find the result: (40,000/2) = 20000. • A total of 20000 bricks are required. Exam Strategy & Shortcut: You can set up the calculation as a single fraction and cancel zeroes before multiplying to avoid dealing with massive numbers: (2500 × 1600/20 × 10) = (250 × 160/2 × 1) = 250 × 80 = 20000. Common Pitfall: Failing to convert the meters into centimeters before calculating the area. If you simply calculate (25 × 16/20 × 10), you will get 2, which is completely incorrect by a magnitude of 10,000 (since 1 m^2 = 10,000 cm^2). Final Answer: Therefore, the correct answer is 20000.

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