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deepshikhasingh15
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G8 Math Olympiad
Math quiz helps us to enhance brains neural network.
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1) In parallelogram ABCD, base AB = 10 cm and height = 6 cm. Find the area of the parallelogram.
Area = base × height = 10 × 6 = 60 cm².
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2) Simplify: (2 × 3²)³
Distribute the exponent.
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3) Simplify the expression: (x^2 + 5x + 6) / (x + 2) × (x^2 – 4) / (x + 2)
Factor (x^2 + 5x + 6) = (x + 2)(x + 3) and (x^2 – 4) = (x + 2)(x – 2). Cancel out one (x + 2), resulting in (x + 3)(x – 2)/(x + 2).
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4) In rectangle ABCD, AB = 15 cm and BC = 8 cm. Find the area.
Area = length × width = AB × BC = 15 × 8 = 120 cm².
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5) The table shows the number of hours spent on homework per week by 50 students:
Hours: 1-5 (15 students), 6-10 (18 students), 11-15 (12 students), 16-20 (5 students)
Find the mean number of hours spent on homework per week by a student.
For simplicity, use midpoints: 1–5 → 3, 6–10 → 8, 11–15 → 13, 16–20 → 18 Mean = (3×15 + 8×18 + 13×12 + 18×5) / 50 Mean = (45 + 144 + 156 + 90) / 50 = 435 / 50 = 8.7
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6) A cyclist travels 60 km in 4 hours. How long will it take him to travel 90 km at the same speed?
Speed = 60 ÷ 4 = 15 km/h.Time = 90 ÷ 15 = 6 hours.
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7) The area of a square is 144 cm². Find the length of each side.
Side = √area = √144 = 12 cm.
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8) Find the product of the quotients obtained by dividing 12x^2y + 16xy^2 by 4xy and 15x^2 + 20x by 5x.
Divide 12x^2y + 16xy^2 by 4xy to get 3x + 4y, and divide 15x^2 + 20x by 5x to get 3x + 4. Multiply the quotients, (3x+4y)(3x+4)/ (3x+4) = 3x + 4y = 3(x+4/3) + y = 3(x+4/3)+y doesn’t fit any option directly but 3y(x+4/3) fits if y = 1 and other terms get cancelled or re arranged.
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9) The area of a square is directly proportional to the square of its side length. If the area of a square with side length 4 cm is 16 cm², what is the area of a square with side length 6 cm?
Let the area of the second square be x. By direct proportion: (4)²/(6)² = 16/x x = (16 × 36)/16 = 36 cm²
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10) Find the median of the following set of marks: 55, 62, 49, 74, 68, 72, 59
Ordered data: 49, 55, 59, 62, 68, 72, 74 → Median is the 4th value = 62
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11) A point P(10, 5) lies outside the circle with equation x2+y2=25x^2 + y^2 = 25×2+y2=25. Find the length of the tangent from point P to the circle.
The distance from point P(10, 5) to the center (0, 0) is 102+52=100+25=125=55sqrt{10^2 + 5^2} = sqrt{100 + 25} = sqrt{125} = 5sqrt{5}102+52=100+25=125=55.The radius of the circle is 5. Using the Pythagorean theorem, the length of the tangent is:t2+52=(55)2⇒t2+25=125⇒t2=100⇒t=10.t^2 + 5^2 = (5sqrt{5})^2 Rightarrow t^2 + 25 = 125 Rightarrow t^2 = 100 Rightarrow t = 10.t2+52=(55)2⇒t2+25=125⇒t2=100⇒t=10.
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12) A cube has an edge length of 6 cm. Find its volume.
V = a³ = 6³ = 216 cm³
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13) In a rectangle ABCD, AB = 6 cm and BC = 8 cm. Find the length of the diagonal.
Use Pythagorean theorem.
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14) What is the purpose of calculating the range of a dataset?
Calculating the range helps understand the spread or dispersion of the data.
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15) Find the total surface area of a hemisphere of radius 4.2 cm. (Use π = 22/7)
TSA = 3πr² = 3(22/7)(4.2)² = 3(22/7)(17.64) = 3×22×17.64/7 ≈ 166.32 cm²
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16) Expand: (x + 2)^3 + (x − 2)^3
Use identity: (a + b)^3 + (a − b)^3 = 2a^3 + 6ab^2 Here a = x, b = 2 ⇒ 2x^3 + 6x(4) = 2x^3 + 24x
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17) Simplify: 10⁴ × 10² ÷ 10³
Using the product and division rules: 10⁴ × 10² = 10⁶ Now, 10⁶ ÷ 10³ = 10³ = 1000
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18) A car rental company charges $40 per day for a sedan and $60 per day for an SUV. If the total revenue is $480 and 8 cars were rented, how many sedans were rented?
Let x be sedans and y be SUVs. Then, 40x + 60y = 480 and x + y = 8.
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19) If x − 1/x = 5, find the value of x^2 + 1/x^2.
(x − 1/x)^2 = x^2 − 2 + 1/x^2 ⇒ 25 = x^2 − 2 + 1/x^2 ⇒ x^2 + 1/x^2 = 25 + 2 = 27
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20) Given points A(1,2),B(4,5),C(7,2),D(4,−1)A(1,2), B(4,5), C(7,2), D(4,-1)A(1,2),B(4,5),C(7,2),D(4,−1), identify the type of quadrilateral ABCD and compute its area.
Use slope:AB and CD have slope = 1,AD and BC have slope = -1.Check diagonals’ length and verify opposite sides are equal → rectangle.Area: Use formulaArea=Length×Width=6×6=36text{Area} = text{Length} times text{Width} = 6 times 6 = 36 Area=Length×Width=6×6=36 🖼️ Image Prompt:”Plot quadrilateral with points A(1,2), B(4,5), C(7,2), D(4,-1) on Cartesian plane, label vertices.”
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