Question 1

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 2

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 3

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 4

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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