Question 1

if the side of a square be increased by 4 cms. The area increased by 60 sq. cms . The side of the square is?

Accepted Answer

Let each side = x cm Then, (x + 4 )2 - x2 = 60 ⇒ x 2 + 8x + 16 - x2 = 60 ∴ x = 5.5 cm

Question 2

A veranda 40 meters long 15 meters broad is to paved with stones each measuring 6 dm by 5 dm. the number of stones required is?

Accepted Answer

Length = (40 x 10 ) dm = 400 dm. Breadth = (15 x 10 ) dm = 150 dm.Area of veranda = (400 x 150 ) dm2 Area of one stone = (6 x 5 ) dm2∴ Required number of stones = (400 x 150) /(6 x 5) = 2000

Question 3

The length of a rectangular plot is twice of its width. If the length of a diagonal is 9√5 meters, the perimeter of the rectangular is?

Accepted Answer

Let breadth = y meters, Then, length = 2y meters∴ Diagonal = √y2 + (2y)2 = √5y2 metersSo, √5y2 = 9 √5 ∴ y= 9Thus, breadth = 9 m and length = 18 m ∴ Perimeter = 2 (18 + 9) m = 54m.

Question 4

The length and breadth of a playground are 36 m and 21 m respectively. Flagstaffs are required to be fixed on all along the boundary at a distance of 3 m apart. The number of flagstaffs will be?

Accepted Answer

Perimeter = 2 x (36 + 21 ) m = 144 m ∴ Number of flagstaffs = 144 / 3 = 38

Question 5

Square from Diagonal — Compute Area: If the diagonal AC of square ABCD is 5.2 cm, find the area of the square ABCD (in square centimetres).

Accepted Answer

Introduction / Context:For any square, the diagonal relates to the side by d = s * √2. Once the side is known, area follows directly from s^2. This question checks your ability to manipulate the diagonal–side relationship and maintain consistent units (centimetres). Given Data / Assumptions: Diagonal d = 5.2 cm Square property: d = s * √2 Area of square: A = s^2 All measures in centimetres Concept / Approach:The diagonal of a square is the hypotenuse of an isosceles right triangle with legs equal to the side length s. Therefore, s = d / √2. After computing s, square it to obtain the area in square centimetres. Only simple arithmetic and the √2 relation are required—no advanced trigonometry. Step-by-Step Solution: Compute side: s = d / √2 = 5.2 / √2.Since √2 ≈ 1.41421356, s ≈ 5.2 / 1.41421356 ≈ 3.676955 cm.Area: A = s^2 ≈ (3.676955)^2 ≈ 13.52 sq.cm (rounded to two decimals). Verification / Alternative check: Reverse check: If A = 13.52 ⇒ s ≈ √13.52 ≈ 3.6769 cm; diagonal = s * √2 ≈ 3.6769 * 1.4142 ≈ 5.20 cm, which matches the given 5.2 cm within rounding. Why Other Options Are Wrong: 15.12 and 12.62 result from rounding √2 or squaring errors. 10 sq.cm underestimates, implying a side near 3.16 cm, which would give a diagonal far below 5.2 cm. 14.04 sq.cm is a near miss from using √2 ≈ 1.45 or other approximations. Common Pitfalls: Using area = (diagonal)^2 instead of dividing by 2; remember A = d^2 / 2 for a square. Mixing units or rounding too early; keep sufficient precision until the final step. Final Answer:13.52 sq.cm.

Question 6

Rectangle with Length Twice Breadth — Area Change Scenario: The length of a rectangle is twice its breadth. If the length is decreased by 5 cm and the breadth is increased by 5 cm, the area increases by 75 cm^2. Find the original length of the recta…

Accepted Answer

Introduction / Context:This is a classic algebraic geometry problem linking dimensional changes to area differences. The relationship among original dimensions, their changes, and the resulting area increase yields a linear equation for the unknown breadth, and hence the original length (given length is twice breadth). Given Data / Assumptions: Original length L = 2B (B = original breadth) Modified dimensions: (L − 5) by (B + 5) Area increase = 75 cm^2 All dimensions in centimetres Concept / Approach:Compute the new area and subtract the old area. The difference equals 75. Substitute L = 2B to reduce to a single variable equation in B. Solve B, then obtain L = 2B. This approach avoids guessing and ensures consistency with the percentage-free, exact arithmetic setup. Step-by-Step Solution: Original area A = L * B = 2B^2.New area A' = (L − 5)(B + 5) = (2B − 5)(B + 5) = 2B^2 + 5B − 25.Increase: A' − A = (2B^2 + 5B − 25) − 2B^2 = 5B − 25 = 75.Solve 5B − 25 = 75 ⇒ 5B = 100 ⇒ B = 20 cm.Original length L = 2B = 40 cm. Verification / Alternative check: Old area: 40 * 20 = 800 cm^2; new area: 35 * 25 = 875 cm^2; increase = 75 cm^2 (matches). Why Other Options Are Wrong: 20 cm is the breadth, not the length. 30 cm and 50 cm do not satisfy the derived relationship. 25 cm would imply non-integer breadth inconsistent with the solved equation. Common Pitfalls: Expanding (2B − 5)(B + 5) incorrectly; watch signs in the cross term. Forgetting L = 2B when setting up the equation for B. Final Answer:40 cm.

Question 7

Longest Rod Across a Rectangular Park — Diagonal Length: A park measures 10 m by 8 m. What is the length of the longest pole (rod) that can be placed flat inside the park?

Accepted Answer

Introduction / Context:The longest straight object that can fit inside a rectangle lies along its diagonal. Therefore, the maximum pole length equals the rectangle's diagonal obtained by the Pythagorean theorem using the side lengths as perpendicular legs. Given Data / Assumptions: Length = 10 m Breadth = 8 m Right-angle corner implies Pythagoras applies Concept / Approach:For a rectangle with sides a and b, the diagonal d satisfies d^2 = a^2 + b^2. Compute the square root of the sum of squares to find d. This geometric maximum is independent of object orientation as any longer orientation would exceed the rectangle's bounds. Step-by-Step Solution: Compute squares: 10^2 = 100; 8^2 = 64.Sum: 100 + 64 = 164.Diagonal: d = √164 ≈ 12.806… m ≈ 12.8 m to one decimal place. Verification / Alternative check: Compare alternatives: 12 m is too short (12^2 = 144 < 164); 13.4 m is too long (13.4^2 = 179.56 > 164). Why Other Options Are Wrong: 10 m equals one side, not the diagonal. 18 m far exceeds the rectangular dimension constraints. 12 m underestimates the true diagonal length. Common Pitfalls: Forgetting that the diagonal is longer than either side but not longer than the hypotenuse computed via Pythagoras. Rounding early; carry enough precision before final rounding. Final Answer:12.8 metres.

Question 8

Squares with Diagonals in Ratio — Compare Areas: The ratio concerns two squares where the diagonal of one is double the diagonal of the other. What is the ratio of their areas (larger : smaller)?

Accepted Answer

Introduction / Context:For squares, both side and diagonal are linear measures. Because area scales with the square of a linear measure, doubling a linear dimension (like the diagonal) multiplies area by the square of that factor. This is a quick test of proportional reasoning rather than computation-heavy arithmetic. Given Data / Assumptions: Square 1 diagonal d1 Square 2 diagonal d2 = 2 * d1 Area proportionality: A ∝ (diagonal)^2 for squares Concept / Approach:Since A ∝ d^2 for a square (because s = d/√2 and A = s^2 = d^2/2), the ratio of areas equals the ratio of squared diagonals. With d2 = 2d1, A2/A1 = (2d1)^2 / d1^2 = 4. Therefore, the larger-to-smaller area ratio is 4:1. Step-by-Step Solution: Express A1 = k * d1^2 and A2 = k * d2^2 where k = 1/2.Substitute d2 = 2d1 ⇒ A2/A1 = (k * 4d1^2) / (k * d1^2) = 4.Hence the ratio (larger : smaller) = 4 : 1. Verification / Alternative check: Numeric example: Let d1 = 10 ⇒ A1 = 10^2/2 = 50. With d2 = 20 ⇒ A2 = 20^2/2 = 200 ⇒ ratio = 200:50 = 4:1. Why Other Options Are Wrong: 2:1 assumes linear, not quadratic, scaling. 3:1 and 3:2 arise from ad-hoc, unsupported factors. 5:2 ≈ 2.5:1 is also inconsistent with quadratic scaling. Common Pitfalls: Confusing side or diagonal doubling with doubling area; area quadruples when a linear dimension doubles. Final Answer:4:1.

Question 9

Effect of Doubling Diameter on Circle Area — Percent Increase: If the diameter of a circle is increased by 100% (i.e., doubled), by what percent does its area increase?

Accepted Answer

Introduction / Context:Circle area depends on the square of a linear dimension (radius or diameter). Doubling the diameter doubles the radius as well, so the area multiplies by 2^2 = 4. Converting that factor to a percentage increase distinguishes absolute area from relative change. Given Data / Assumptions: Original diameter D; new diameter = 2D Original radius r = D/2; new radius r' = D Area formula: A = πr^2 Concept / Approach:Compute the factor by which area changes: A'/A = (r'^2)/(r^2) = (D^2)/(D^2/4) = 4. A factor of 4 corresponds to a 300% increase because Increase% = (new − old)/old * 100% = (4A − A)/A * 100% = 300%. Step-by-Step Solution: Original area A = π(D/2)^2 = πD^2/4.New area A' = π(D)^2 = πD^2.Ratio A'/A = (πD^2) / (πD^2/4) = 4 ⇒ Increase% = (4 − 1)*100% = 300%. Verification / Alternative check: Numeric example: Let D = 10 ⇒ A = 78.54. New D = 20 ⇒ A' = 314.16. Increase = 235.62 ≈ 300% of 78.54. Why Other Options Are Wrong: 100% implies doubling area, not quadrupling. 200% implies tripling total, still short of 4A. 400% confuses new area with the percent increase. 150% is arbitrary and unsupported. Common Pitfalls: Reporting the new area as 400% instead of the increase of 300%. Final Answer:300% increase.

Question 10

Compare Areas — Square of side a vs Equilateral Triangle of side a: What is the ratio of the area of a square (side a) to the area of an equilateral triangle (side a)?

Accepted Answer

Introduction / Context:This comparison leverages two standard area formulas at the same scale (side length a). It checks familiarity with special-triangle areas and the ability to express a clean ratio without units, since the common factor a^2 cancels out. Given Data / Assumptions: Square of side a ⇒ Area A_s = a^2 Equilateral triangle of side a ⇒ Area A_t = (√3/4) * a^2 We seek A_s : A_t Concept / Approach:Form the ratio and cancel a^2. Express the result in simplest radical form. Since both shapes share the same side length, unit-free comparison is straightforward and does not require numeric substitution unless verifying. Step-by-Step Solution: A_s : A_t = a^2 : (√3/4 a^2).Cancel a^2 ⇒ 1 : (√3/4).Multiply both terms by 4 ⇒ 4 : √3. Verification / Alternative check: Let a = 4 ⇒ A_s = 16; A_t = (√3/4)*16 = 4√3 ⇒ ratio = 16 : 4√3 = 4 : √3. Why Other Options Are Wrong: 2:1 and 4:3 are integer ratios unrelated to √3/4. 2:√3 halves the correct first term. √3:4 inverts the desired ratio. Common Pitfalls: Using the triangle area as (1/2)ab without recognizing 60°; the special equilateral result is (√3/4)a^2. Final Answer:4 : √3.

Question 11

Recover Side Length from Area — Equilateral Triangle: The area of an equilateral triangle is 4√3 cm^2. Find the length of each side (in cm).

Accepted Answer

Introduction / Context:Equilateral triangle area depends solely on side length via a constant factor. Inverting the standard formula lets us find side length directly from a given area, reinforcing comfort with algebraic manipulation and radicals. Given Data / Assumptions: Area A = 4√3 cm^2 Formula: A = (√3/4) * a^2 a is the side length in cm Concept / Approach:Set 4√3 = (√3/4) * a^2 and solve for a^2 by isolating it. Cancel √3 on both sides to simplify early and avoid handling nested radicals later. Take the positive square root because side length is positive. Step-by-Step Solution: 4√3 = (√3/4) * a^2.Multiply both sides by 4/√3 ⇒ a^2 = 4√3 * (4/√3) = 16.Therefore, a = √16 = 4 cm. Verification / Alternative check: Plug back: (√3/4)*4^2 = (√3/4)*16 = 4√3 cm^2, matching the given area. Why Other Options Are Wrong: 4/√3 cm and √3/4 cm misapply inversion steps. 3 cm does not satisfy (√3/4)*9 = 2.25√3. 2√3 cm gives area (√3/4)*12 ≈ 3√3, not 4√3. Common Pitfalls: Forgetting to multiply by 4/√3 to isolate a^2. Taking the square root prematurely before simplifying constants. Final Answer:4 cm.

Question 12

Area of an Equilateral Triangle (side = 8 cm): Find the area (in cm^2) of an equilateral triangle whose side length is 8 cm.

Accepted Answer

Introduction / Context:The area of an equilateral triangle can be computed directly from side length using a fixed constant multiplier. This avoids trigonometric detours and emphasizes formula recall and clean substitution with units carried consistently. Given Data / Assumptions: Side a = 8 cm Area formula: A = (√3/4) * a^2 All lengths in centimetres Concept / Approach:Substitute a = 8 into A = (√3/4)a^2. Since a^2 = 64, the product streamlines neatly. Reporting the exact radical form (rather than decimals) keeps precision and matches typical aptitude answer formats. Step-by-Step Solution: Compute a^2: 8^2 = 64.Apply formula: A = (√3/4) * 64 = 16√3 cm^2.Approximate (optional): √3 ≈ 1.732 ⇒ A ≈ 27.712 cm^2. Verification / Alternative check: Using height h = (√3/2)a = 4√3 ⇒ A = (1/2)*base*height = (1/2)*8*(4√3) = 16√3 cm^2. Why Other Options Are Wrong: 64 cm^2 is the square area, not the triangle's. 4√3 cm^2 corresponds to side 4 cm. 21.3 cm^2 is an inaccurate decimal; exact is 27.7… 32√3 cm^2 doubles the correct value. Common Pitfalls: Using (1/2)ab with a wrong height; ensure equilateral height is (√3/2)a. Final Answer:16√3 cm2.

Question 13

Triangle Area from Three Sides — Heron’s Formula: Find the area of a triangle whose sides are 8 cm, 10 cm, and 12 cm. Express the answer in simplified radical form.

Accepted Answer

Introduction / Context:Heron’s formula computes the area of any triangle from its three side lengths, avoiding angle or height calculations. The key steps are finding the semiperimeter and then evaluating the square root expression carefully to keep radicals simplified. Given Data / Assumptions: Sides: a = 8 cm, b = 10 cm, c = 12 cm All sides satisfy triangle inequality (valid triangle) Semiperimeter s = (a + b + c)/2 Concept / Approach:Heron’s formula: Area = √[s(s − a)(s − b)(s − c)]. Compute s, the three differences, and multiply; then take the square root. Simplify the radical by extracting perfect squares to match common answer forms. Step-by-Step Solution: s = (8 + 10 + 12)/2 = 15.Compute factors: s − a = 7, s − b = 5, s − c = 3.Product: s(s − a)(s − b)(s − c) = 15 * 7 * 5 * 3 = 1575.Simplify: 1575 = 9 * 175 = 9 * 25 * 7 ⇒ √1575 = √(9*25*7) = 3*5*√7 = 15√7.Note that 15√7 = 5√(9*7) = 5√63, matching the given option form. Verification / Alternative check: Approximate value: √7 ≈ 2.6458 ⇒ area ≈ 15 * 2.6458 ≈ 39.69 sq cm. Why Other Options Are Wrong: 8√63 and 7√93 do not equal 15√7. 6√53 is unrelated to the factorization of 1575. 15√7 is equivalent but not listed as the primary option; 5√63 is the same value in the provided form. Common Pitfalls: Losing factors when simplifying √1575; factor completely to extract 9 and 25. Final Answer:5√63 sq cm (which equals 15√7 sq cm).

Question 14

Triangle Perimeter from Side Lengths — Direct Sum: Find the perimeter of a triangle with sides 6 cm, 4 cm, and 5 cm.

Accepted Answer

Introduction / Context:Perimeter is the simplest aggregate measure of a polygon—just the sum of side lengths. This item checks careful reading and arithmetic rather than geometric reasoning. Even so, confirming triangle validity is good practice to avoid degenerate cases. Given Data / Assumptions: Sides: 6 cm, 4 cm, 5 cm Units are all in centimetres Triangle inequality holds (6 < 4 + 5, etc.) Concept / Approach:No formulas beyond addition are required. Perimeter P equals the sum of the three sides in consistent units. Because the values are small integers, the result should be an integer in centimetres with no need for rounding. Step-by-Step Solution: Compute sum: 6 + 4 + 5 = 15.Therefore, P = 15 cm. Verification / Alternative check: Triangle inequality: 6 < 9, 4 < 11, 5 < 10 ⇒ valid triangle. Units consistent. Why Other Options Are Wrong: 14 cm and 16 cm are off by −1 and +1 due to common mental-sum slips. 18 cm and 20 cm ignore one side or add extra length. Common Pitfalls: Mistyping integer sums; always re-check totals when only simple arithmetic is involved. Final Answer:15 cm.

Question 15

Right-angled triangle — The area is 40 sq cm. If the base is 28 cm, find the corresponding height (use area = 1/2 * base * height).

Accepted Answer

Introduction / Context:This is a direct application of the right-triangle area relation. Knowing area and one leg (the base), we can back-compute the other leg (the height) without using Pythagoras because no hypotenuse is involved. Given Data / Assumptions: Area = 40 sq cm Base = 28 cm Right-angled triangle: area formula holds as 1/2 * base * height Concept / Approach:Area of a right triangle equals 1/2 * base * height. Rearranging gives height = (2 * Area) / base. Substitute the given values to get an exact fractional height. Step-by-Step Solution:height = (2 * 40) / 28height = 80 / 28 = 20 / 7 cmThus the required height is 20/7 cm ≈ 2.857 cm Verification / Alternative check:Check by plugging back: area = 1/2 * 28 * (20/7) = 14 * (20/7) = 40 ✔ Why Other Options Are Wrong:26/7, 36/7, 46/7, 56/7 give areas other than 40 when used in 1/2 * 28 * height; they are distractors based on arithmetic slips. Common Pitfalls:Forgetting the factor 1/2 in the area formula or attempting to use Pythagoras when only area and one leg are needed. Final Answer:20/7 cm

Question 16

Right-angled triangle — The area is 10 sq cm. If the perpendicular (height) is 20 cm, find the base.

Accepted Answer

Introduction / Context:For a right triangle, area = 1/2 * base * height. When area and one leg are known, the other leg can be computed directly. Units are in centimeters. Given Data / Assumptions: Area = 10 sq cm Perpendicular (height) = 20 cm Concept / Approach:Use height * base with the 1/2 factor. Solve base = (2 * Area) / height. Step-by-Step Solution:base = (2 * 10) / 20 = 20 / 20 = 1 cm Verification / Alternative check:Plug back: area = 1/2 * 1 * 20 = 10 sq cm ✔ Why Other Options Are Wrong:2, 3, 4, 5 cm all produce areas 20, 30, 40, 50 divided by 2, not equal to 10 with the given height. Common Pitfalls:Missing the factor 1/2 or confusing base and height values. Final Answer:1 cm

Question 17

Parallelogram — One side is 8.06 cm and the perpendicular distance to the opposite side is 2.08 cm. Find the approximate area.

Accepted Answer

Introduction / Context:The area of a parallelogram equals base * corresponding altitude (perpendicular distance). This avoids any need for angles or other sides when one side and its altitude are known. Given Data / Assumptions: Side (base) = 8.06 cm Altitude = 2.08 cm Area formula: A = base * altitude Concept / Approach:Multiply the given base by the perpendicular distance to get area; then round to two decimal places as the options are approximate. Step-by-Step Solution:A = 8.06 * 2.08 = 16.7648 cm2 ≈ 16.76 cm2 Verification / Alternative check:Units check: cm * cm = cm2; the magnitude fits between 8*2 = 16 and 8.5*2.1 ≈ 17.85, so 16.76 is sensible. Why Other Options Are Wrong:12.56, 14.56, 18.06, 22.56 cm2 stem from multiplying by the wrong dimension or misplacing decimals. Common Pitfalls:Using slanted height instead of perpendicular altitude; forgetting to align side with its corresponding altitude. Final Answer:16.76 cm2

Question 18

Rectangle — Find the area of a rectangle with length 15 m and breadth 8 m.

Accepted Answer

Introduction / Context:Area of a rectangle is the simplest product of its length and breadth. No diagonals or angles are needed when both sides are given. Given Data / Assumptions: Length = 15 m Breadth = 8 m Concept / Approach:A = L * B. Step-by-Step Solution:A = 15 * 8 = 120 sq m Verification / Alternative check:Units multiply correctly (m * m = m2) and the value is straightforward. Why Other Options Are Wrong:All other values differ from 120 due to arithmetic errors or guesses. Common Pitfalls:None significant; just ensure correct multiplication. Final Answer:120 sq m

Question 19

Rectangle — A field has length 100 m and breadth 40 m. Compute its area (in square meters).

Accepted Answer

Introduction / Context:Convert the product of length and breadth directly into square meters to avoid ambiguous symbolic notations. This removes confusion from misprinted expressions like 4 × 10^x. Given Data / Assumptions: L = 100 m B = 40 m Concept / Approach:Area of a rectangle A = L * B. Step-by-Step Solution:A = 100 * 40 = 4,000 sq m Verification / Alternative check:Magnitude check: 100 by 40 clearly covers 4,000 unit squares of 1 m by 1 m. Why Other Options Are Wrong:1,000, 1,400, 3,600, 4,400 sq m are arithmetic distractors and do not equal L * B. Common Pitfalls:Misreading symbolic options; using clear numerals prevents misinterpretation. Final Answer:4,000 sq m

Question 20

Rectangle — If both sides are increased by 5%, by what percentage does the diagonal increase?

Accepted Answer

Introduction / Context:Scaling every linear dimension of a rectangle by the same factor scales the diagonal by the same factor because the shape is similar. The diagonal is a linear measure, not an area. Given Data / Assumptions: Sides scale by 1.05 each. Diagonal d = sqrt(L^2 + B^2). Concept / Approach:New diagonal d' = sqrt((1.05L)^2 + (1.05B)^2) = 1.05 * sqrt(L^2 + B^2) = 1.05 * d. Hence percentage increase equals 5%. Step-by-Step Solution:Scale factor for diagonal = 1.05 ⇒ increase = (1.05 − 1) * 100% = 5% Verification / Alternative check:Pick L = 3, B = 4 ⇒ d = 5. After scaling by 1.05: d' = 5 * 1.05 = 5.25, which is 5% more. Why Other Options Are Wrong:4% or 6% come from confusing area vs length scaling; 9% would be the area-like change; 10% is arbitrary. Common Pitfalls:Applying area-percentage logic to a length measure; area would increase by about 10.25% (1.05^2 − 1). Final Answer:5%

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