Math - Geometry: You have a rectangular piece of paper with a length of 10 cm and a width of 6 cm. You want to cut out the largest possible circle from the paper. What is the diameter and area of the circle?
Logic - Riddles: There are three boxes labeled “Apples,” “Oranges,” and “Apples and Oranges.” Each box is labeled incorrectly. You can only take out one fruit from one box (without looking inside). How can you correctly label all the boxes?
Math - Algebra: Solve for x: 5(x - 3) + 2x = 3(x + 4) + 4.
Physics - Mechanics: A car accelerates uniformly from rest to a speed of 20 m/s in 5 seconds. How far does the car travel during this time?
Math - Puzzles: You have a balance scale and 9 identical-looking coins. One coin is slightly heavier than the others. You can use the scale only twice. How can you find the heavier coin?
Chemistry - Reactions: When baking soda (sodium bicarbonate) is mixed with vinegar (acetic acid), a chemical reaction occurs that produces carbon dioxide gas. Write the balanced chemical equation for this reaction.
Math - Combinatorics: You have 5 books, and you want to arrange them on a shelf. In how many different ways can you arrange the books?
Physics - Optics: A convex lens has a focal length of 10 cm. An object is placed 30 cm away from the lens. Using the lens formula (1/f = 1/v - 1/u), where f is the focal length, v is the image distance, and u is the object distance, calculate the image distance and whether the image is real or virtual.
Math - Probability: You roll a fair six-sided die twice. What is the probability that the sum of the two rolls is equal to 7?
Biology - Genetics: In pea plants, the gene for purple flowers (P) is dominant over the gene for white flowers (p). If two heterozygous purple-flowered plants (Pp) are crossed, what are the genotypic and phenotypic ratios of the offspring?
Math - Geometry: The largest possible circle that can be cut out from the paper will have a diameter equal to the width of the paper, which is 6 cm. The radius of the circle is 3 cm. The area of the circle is π * r^2 = π * 3^2 = 9π square cm.
Logic - Riddles: Take out one fruit from the “Apples and Oranges” box. If it’s an apple, then that box is the “Apples” box. The box labeled “Apples” must be the “Oranges” box, and the box labeled “Oranges” must be the “Apples and Oranges” box. If the fruit taken out was an orange, the labels can be determined similarly.
Math - Algebra: 5(x - 3) + 2x = 3(x + 4) + 4 5x - 15 + 2x = 3x + 12 + 4 7x - 15 = 3x + 16 4x = 31 x = 31/4
Physics - Mechanics: Using the kinematic equation: s = ut + (1/2)at^2, where s is displacement, u is initial velocity, a is acceleration, and t is time. The initial velocity is 0, so s = (1/2)at^2. The acceleration is (20 - 0)/5 = 4 m/s^2. The distance traveled is s = (1/2) * 4 * 5^2 = 50 meters.
Math - Puzzles: Divide the coins into three groups: two groups of three coins and one group of three coins. Weigh the first two groups. If they balance, the heavier coin is in the third group. If not, the heavier coin is in the heavier group. Weigh two coins from the identified group. If they balance, the third coin is heavier. If not, the heavier coin is found.
Chemistry - Reactions: The balanced chemical equation for the reaction between sodium bicarbonate (NaHCO3) and acetic acid (CH3COOH) is: NaHCO3 + CH3COOH → CH3COONa + H2O + CO2.
Math - Combinatorics: The number of ways to arrange 5 books is 5! (5 factorial), which is 5 * 4 * 3 * 2 * 1 = 120 different ways.
Physics - Optics: Using the lens formula: 1/f = 1/v - 1/u, where f = 10 cm and u = -30 cm (object distance is negative in the lens formula). Solving for v: 1/10 = 1/v - 1/-30. Simplifying, 1/v = 1/10 + 1/30 = 3/30 + 1/30 = 4/30. So, v = 30/4 = 7.5 cm. The image is real (positive v) and located 7.5 cm from the lens.
Math - Probability: The possible rolls that sum to 7 are (1,6), (2,5), (3,4), (4,3), (5,2), (6,1). There are 6 favorable outcomes. Since each die has 6 sides, there are 6 * 6 = 36 total possible outcomes. The probability is 6/36 = 1/6.
10 Biology - Genetics: The Punnett square for the cross between two heterozygous purple-flowered plants (Pp) is as follows:
| P | p
--|----|---
P | PP | Pp
p | Pp | pp
The genotypic ratio is 1 PP : 2 Pp : 1 pp. The phenotypic ratio is 3 purple-flowered plants (PP or Pp) : 1 white-flowered plant (pp).