Question 1

A can go round a circular path 8 times in 40 minutes. If the diameter of the circle is increased to 10 times the original diameter, then the time required by A to go round the new path once, travelling at the same speed as before, is

Accepted Answer

Concept & Distance, Speed, and Time: The distance around a circular path is its circumference, which is given by \pi × diameter (\pi d). Time required is proportional to the distance traveled if the speed remains constant: T = (D/S) Step-by-Step Solution: • Given: 8 rounds take 40 minutes. • Deduction: 1 round of the original path takes (40/8) = 5 minutes. • Calculation: The original distance for 1 round is \pi d. • Calculation: The new diameter is 10d, so the new distance for 1 round is \pi(10d) = 10(\pi d). • Deduction: The new path is 10 times as long as the original path. • Calculation: Since speed is constant, the time required will also be 10 times the original time. 10 × 5 minutes = 50 minutes. Exam Strategy & Shortcut: Realize that distance is directly proportional to diameter. If diameter becomes 10×, distance becomes 10×. Time for 1 old round is 5 min. Therefore, time for 1 new round is 10 × 5 = 50 min. Common Pitfall: A common error is confusing the number of rounds (8) and applying the multiplier to the total 40 minutes directly without finding the time for a single round first. Final Answer: Therefore, the correct answer is 50 min.

Question 2

If the radius of a circle is doubled, its area is increased by

Accepted Answer

Concept & Scaling Factors in Geometry: When the linear dimension (radius) of a 2D shape is scaled by a factor of k, its area is scaled by a factor of k^2. New Area = k^2 × Old Area Step-by-Step Solution: • Given: The radius is doubled. This means the scaling factor k = 2. • Calculation: Let the original radius be r. Original area A_1 = \pi r^2. • Calculation: New radius is 2r. New area A_2 = \pi (2r)^2 = 4\pi r^2. • Deduction: The new area is 4 times the original area. • Calculation: The *increase* in area is 4\pi r^2 - \pi r^2 = 3\pi r^2. • Calculation: Percentage increase = ((3\pi r^2/\pi r^2)) × 100 = 300%. Exam Strategy & Shortcut: "Doubled" means a multiplier of 2. Area multiplier is 2^2 = 4. If you have 4 times the amount you started with, you have gained 3 times your original amount. 3 × 100% = 300% increase. Common Pitfall: A classic trap is selecting 400%. The area *becomes* 400% of its original size, but the question specifically asks what it is *increased by*, which requires subtracting the original 100%. Final Answer: Therefore, the correct answer is 300%.

Question 3

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 4

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 5

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 6

Four horses are tethered at four corners of a square plot of side $63	ext{ metres}$ so that they just cannot reach one another. The area left ungrazed is

Accepted Answer

Concept & Area of Grazing Sectors: This is practically identical to the "four touching circles" problem. If the horses "just cannot reach one another", their grazing regions represent four quarter-circles that meet exactly at the midpoints of the square's sides. Ungrazed Area = Total Plot Area - Total Grazed Area Step-by-Step Solution: • Given: Side of the square plot s = 63 m. • Deduction: The radius r of each horse's grazing sector is exactly half the side length: r = (63/2) = 31.5 m. • Calculation: Area of the square plot = 63^2 = 3969 m^2. • Calculation: The four 90^\circ grazing sectors form one full circle of radius 31.5 m. Area grazed = \pi r^2. • Calculation: Grazed Area = (22/7) × (63/2) × (63/2) = 11 × 9 × (63/2) = 99 × 31.5 = 3118.5 m^2. • Calculation: Ungrazed Area = 3969 - 3118.5 = 850.5 m^2. Exam Strategy & Shortcut: Using the derived formula from the four touching circles concept, the ungrazed area is directly (6/7)r^2. (6/7) × ((63/2))^2 = (6/7) × (3969/4) = (3/7) × (3969/2) = 3 × (567/2) = (1701/2) = 850.5. This significantly reduces the arithmetic steps. Common Pitfall: A common misstep is calculating the area of four separate circles instead of realizing the grazing areas inside the plot only amount to four quarter-sectors (one single full circle equivalent). Final Answer: Therefore, the correct answer is 850.5 m^2.

Question 7

The difference between the areas of the circumcircle and the incircle of a regular polygon of $n$ sides with each side of length $2a$, is

Accepted Answer

Concept & Incircle and Circumcircle Geometry: For any regular polygon, the circumradius (R), the inradius (r), and half of the side length form a right-angled triangle. By using the Pythagorean theorem on this fundamental triangle, we can find the difference in their areas without needing the number of sides n. R^2 = r^2 + (Half-side)^2 Step-by-Step Solution: • Let the regular polygon have n sides, each of length 2a. • Let the center of the polygon be O. • The distance from O to a vertex is the circumradius, R. • The perpendicular distance from O to the midpoint of a side is the inradius, r. • This forms a right-angled triangle with the hypotenuse R, height r, and base equal to half the side length, which is (2a/2) = a. • By the Pythagorean theorem: R^2 = r^2 + a^2. • Rearranging gives: R^2 - r^2 = a^2. • The area of the circumcircle is \pi R^2. • The area of the incircle is \pi r^2. • The difference in their areas = \pi R^2 - \pi r^2 = \pi (R^2 - r^2). • Substitute R^2 - r^2 = a^2 into the equation: Area Difference = \pi a^2. Exam Strategy & Shortcut: Notice that the options are mostly independent of n (Option A). In geometry, if a relationship holds for an n-sided polygon, it must hold for the simplest one. Assume a square (n=4) with side 2a. Inradius r = a, circumradius R = a\sqrt{2}. Area difference = \pi(a\sqrt{2})^2 - \pi(a)^2 = 2\pi a^2 - \pi a^2 = \pi a^2. Common Pitfall: Students often assume the area difference depends on the number of sides n and mistakenly guess a complex option like (2n + 1)\pi a^2. Final Answer: Therefore, the correct answer is \pi a^2.

Question 8

Two equal circles are drawn in square in such a way that a side of the square forms diameter of each circle. If the remaining area of the square is 42 cm$^2$. How much will the diameter of the circle measures? [IBPS—RRB Officers Gr. 'B' Exam, 2015]

Accepted Answer

Concept & Area Subtraction: The problem refers to semicircles formed on the sides of a square that cut into its area. The remaining area is the area of the square minus the area of these two semicircles (which combine to form one full circle). Area_{remaining} = Area_{square} - Area_{circle} Step-by-Step Solution: • Let the side of the square be a. The total area of the square is a^2. • Two semicircles are drawn with a diameter equal to the side of the square (a). Their radius is r = (a/2). • The combined area of these two semicircles is equal to the area of one full circle: \pi × ((a/2))^2 = (\pi a^2/4). • Substitute \pi = (22/7): Area = (22/7) × (a^2/4) = (11a^2/14). • The remaining area of the square is a^2 - (11a^2/14) = (3a^2/14). • We are given that the remaining area is 42 cm^2. So, (3a^2/14) = 42. • Solving for a^2: a^2 = 42 × (14/3) = 14 × 14 = 196. • Therefore, a = 14 cm. Since the diameter of the circle is equal to the side of the square, the diameter is also 14 cm. Exam Strategy & Shortcut: Memorize that when a full circle (or two semicircles) of maximum diameter is inscribed in a square, the circle occupies (11/14) of the square's area, and the remaining space is (3/14) of the area. Setting (3/14) × a^2 = 42 quickly yields a = 14. Common Pitfall: Taking "two equal circles" literally as two independent full circles inside the square, which would cause an impossible negative remaining area. Contextually, this phrasing in competitive exams implies semicircles on opposite edges. Final Answer: Therefore, the correct answer is 14 cm.

Question 9

Area of a rectangle is 150 metre sq. When the breadth of the same rectangle is increased by 2 meter and the length decreased by 5 metre the area of the rectangle decreases by 30 metre square. What is the perimeter of the square whose side are equal …

Accepted Answer

Concept & Algebraic Formulation: This problem requires setting up a system of equations based on the area of a rectangle (Area = length × breadth). By manipulating the dimensions as described, we can solve for the original length and breadth. Area = l × b Perimeter_{square} = 4 × side Step-by-Step Solution: • Let the original length be l and original breadth be b. We are given l × b = 150. • The new breadth is b + 2 and the new length is l - 5. • The new area decreases by 30, making it 150 - 30 = 120 m^2. • Equation: (l - 5)(b + 2) = 120. • Expand the equation: lb + 2l - 5b - 10 = 120. • Substitute lb = 150: 150 + 2l - 5b - 10 = 120 \Rightarrow 2l - 5b = -20 \Rightarrow 5b - 2l = 20. • From the first equation, b = (150/l). Substitute this into the linear equation: 5((150/l)) - 2l = 20. • Multiply through by l: 750 - 2l^2 = 20l \Rightarrow 2l^2 + 20l - 750 = 0. • Divide by 2: l^2 + 10l - 375 = 0. • Factor the quadratic equation: (l + 25)(l - 15) = 0. Since length cannot be negative, l = 15 meters. • The question asks for the perimeter of a square with side equal to this length (15 m). • Perimeter = 4 × 15 = 60 meters. Exam Strategy & Shortcut: Instead of fully solving the quadratic, you can test factor pairs of 150 for l and b. Pairs of 150: (10, 15), (25, 6), (30, 5). If l=15 and b=10, new l=10 and new b=12. New area = 10 × 12 = 120. This matches the condition perfectly! Perimeter = 15 × 4 = 60. Common Pitfall: A common error is solving for the length (15) and selecting it as the final answer, neglecting the second part of the question which specifically asks for the *perimeter* of a square with that side length. Final Answer: Therefore, the correct answer is 60 m.

Question 10

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 11

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 12

The height of a triangle is equal to the perimeter of a square whose diagonal is $8\sqrt{2}$ metre and the base of the same triangle is equal to the side of a square whose area is 729sq. metre. What is the area of the triangle? (In sq. metre)  [Unit…

Accepted Answer

Concept & Multi-Shape Integration: This problem connects the properties of squares (diagonal, perimeter, area, side) to the dimensions of a triangle (base, height) to calculate the triangle's area. Diagonal_{square} = a\sqrt{2} Area_{square} = s^2 Area_{triangle} = (1/2) × base × height Step-by-Step Solution: • Step 1: Find the triangle's height. • The diagonal of the first square is 8\sqrt{2} m. • Formula for diagonal is side × \sqrt{2}. Thus, the side of the first square is 8 m. • The perimeter of this square is 4 × 8 = 32 m. • Given: Height of triangle = Perimeter of this square = 32 m. • Step 2: Find the triangle's base. • The area of the second square is 729 sq. m. • Formula for area is side^2 = 729. Taking the square root, the side is \sqrt{729} = 27 m. • Given: Base of triangle = Side of this square = 27 m. • Step 3: Calculate the area of the triangle. • Area = (1/2) × base × height. • Area = (1/2) × 27 × 32. • Area = 27 × 16. • 27 × 16 = 27 × (10 + 6) = 270 + 162 = 432 sq. m. Exam Strategy & Shortcut: Since you need to multiply (1/2) × 27 × 32, you can just calculate the unit digit. The unit digit of 27 × 16 is 7 × 6 = 42, which ends in 2. Looking at the options, both (b) 206 and (c) 472 and (d) 432 end in 2, but 27 × 16 is close to 25 × 16 = 400, making 432 the only logical choice. Common Pitfall: Confusing the area of the square with its side or perimeter, or forgetting the (1/2) multiplier in the triangle area formula. Final Answer: Therefore, the correct answer is 432.

Question 13

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.

Question 14

The base of triangle is 15 cm and height is 12 cm. the height of another triangle of double the area having base 20cm is

Accepted Answer

Concept & Formula: The area of a triangle is given by half the product of its base and height. Area = (1/2) × base × height Step-by-Step Solution: 1. Given: Base of the first triangle (b_1) = 15 cm, Height (h_1) = 12 cm. 2. Calculate Area of First Triangle: Area (A_1) = (1/2) × 15 × 12 = 90 cm^2. 3. Determine Area of Second Triangle: The problem states the area is double, so A_2 = 2 × 90 = 180 cm^2. 4. Given: Base of the second triangle (b_2) = 20 cm. Let its height be h_2. 5. Solve for Height: Apply the area formula for the second triangle: 180 = (1/2) × 20 × h_2 180 = 10 × h_2 h_2 = (180/10) = 18 cm. Exam Strategy & Shortcut: You can set up a direct ratio if you understand the area formulas. \frac{Area_1}{Area_2} = (b_1 h_1/b_2 h_2). Since Area_2 = 2 × Area_1, the ratio is (1/2) = (15 × 12/20 × h_2) \rightarrow (1/2) = (180/20 h_2) \rightarrow 20 h_2 = 360 \rightarrow h_2 = 18. Common Pitfall: A common mistake is forgetting the (1/2) in the area formula, which would lead to incorrect intermediate areas, though in a ratio comparison, it cancels out. Always double-check your initial area calculation. Final Answer: Therefore, the correct answer is 18 cm.

Question 15

The base of an isosceles is 14 cm and its perimeter is 36 cm. Find its area.

Accepted Answer

Concept & Formula: The perimeter of a triangle is the sum of all its sides. For an isosceles triangle with equal sides a and base b, the perimeter is 2a + b. Heron's formula can be used to find the area if all three sides are known: Area = \sqrt{s(s-a)(s-b)(s-c)} where s is the semi-perimeter. Alternatively, a specific formula for isosceles triangles is: Area = (b/4)\sqrt{4a^2 - b^2} Step-by-Step Solution: 1. Given: Base (b) = 14 cm, Perimeter = 36 cm. Let the two equal sides be a. 2. Find Equal Sides: 2a + 14 = 36 2a = 22 \Rightarrow a = 11 cm. The sides are 11, 11, and 14. 3. Calculate Semi-perimeter (s): s = (36/2) = 18 cm. 4. Calculate Area using Heron's Formula: Area = \sqrt{18 × (18-11) × (18-11) × (18-14)} Area = \sqrt{18 × 7 × 7 × 4} Area = \sqrt{9 × 2 × 49 × 4} Area = 3 × 7 × 2 × \sqrt{2} = 42\sqrt{2} sq. cm. Exam Strategy & Shortcut: Instead of Heron's formula, using the altitude is faster if you know Pythagorean triplets, but here (b/2) = 7 and hypotenuse is 11. Altitude h = \sqrt{11^2 - 7^2} = \sqrt{121 - 49} = \sqrt{72} = 6\sqrt{2}. Area = (1/2) × base × height = (1/2) × 14 × 6\sqrt{2} = 42\sqrt{2}. This requires less calculation. Common Pitfall: Misidentifying which side is the base versus the equal sides when applying the perimeter formula leads to the wrong lengths for the unknown sides. Final Answer: Therefore, the correct answer is 42\sqrt{2} sq. cm.

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