Math - Arithmetic Progression: In an arithmetic progression, the first term is 5, and the common difference is 3. What is the sum of the first 10 terms of the progression?
Physics - Projectiles: A ball is thrown horizontally off a cliff with an initial speed of 10 m/s. If it takes 4 seconds for the ball to hit the ground, how far from the base of the cliff does the ball land? (Neglect air resistance and use g = 10 m/s² for the acceleration due to gravity)
Math - Set Theory: In a group of 30 people, 17 like chocolate, 13 like vanilla, and 8 like both chocolate and vanilla. How many people don’t like either flavor?
Math - Geometry: A circle has a diameter of 10 cm. What is the area of the circle? (Use π = 3.14)
Math - Recursion: A sequence is defined by the rule f(n) = f(n-1) + f(n-2), with f(1) = 1 and f(2) = 1. What is the value of f(7)?
Physics - Relative Motion: Two trains are traveling towards each other on parallel tracks. Train A is traveling at 60 km/h, and Train B is traveling at 90 km/h. If they are 300 km apart, how long will it take for them to meet?
Math - Algebra: A store sells 3 apples for $1.50. At the same rate, how much would 10 apples cost?
Math - Geometry: The sum of the interior angles of a polygon is 1,080 degrees. How many sides does the polygon have?
Math - Probability: A box contains 4 red balls, 6 blue balls, and 10 green balls. What is the probability of drawing a red ball, followed by a blue ball, without replacing the first ball?
Math - Logic: You have a balance scale and 9 identical-looking coins. One of the coins is slightly heavier than the others. Using the balance scale only 2 times, determine which coin is the heavier one.
Math - Arithmetic Progression: The sum of the first n terms of an arithmetic progression is given by the formula: S_n = n(a_1 + a_n)/2, where n is the number of terms, a_1 is the first term, and a_n is the nth term. In this case, n = 10, a_1 = 5, and a_n = a_1 + (n-1)d = 5 + (10-1)3 = 32. So, S_10 = 10(5 + 32)/2 = 10*37/2 = 185.
Physics - Projectiles: The horizontal distance is given by the formula: distance = horizontal_speed * time. In this case, horizontal_speed = 10 m/s and time = 4 seconds, so distance = 10*4 = 40 meters.
Math - Set Theory: Using the inclusion-exclusion principle, the number of people who don’t like either flavor is the total number of people minus those who like chocolate or vanilla. Total = 30, Chocolate_and_Vanilla = 17 + 13 - 8 = 22. So, 30 - 22 = 8 people don’t like either flavor.
Math - Geometry: The area of a circle is given by the formula: A = πr^2, where r is the radius. In this case, the diameter is 10 cm, so the radius is 5 cm. The area is A = 3.145^2 = 3.14*25 ≈ 78.5 cm².
Math - Recursion: f(1) = 1, f(2) = 1, f(3) = 2, f(4) = 3, f(5) = 5, f(6) = 8, and f(7) = f(6) + f(5) = 8 + 5 = 13.
Physics - Relative Motion: The relative speed of the two trains is the sum of their speeds: 60 + 90 = 150 km/h. Time = distance / relative speed = 300 / 150 = 2 hours.
Math - Algebra: Let x be the cost of 10 apples. Then, 3/1.50 = 10/x, which implies x = 1.50*10/3 = $5.00.
Math - Geometry: The sum of the interior angles of an n-sided polygon is given by the formula: (n-2)180°. So, (n-2)180° = 1080°, which implies n-2 = 1080°/180° = 6, and n = 8. The polygon has 8 sides.
Math - Probability: The probability of drawing a red ball is 4/(4+6+10) = 4/20 = 1/5. After drawing a red ball, there are 19 balls left in the box. The probability of drawing a blue ball is now 6/19. The combined probability of drawing a red ball followed by a blue ball is (1/5)*(6/19) = 6/95.
Math - Logic: First, divide the 9 coins into three groups of three coins each. Use the balance scale to compare the weight of two of these groups. If they balance, the heavier coin is in the third group