Math - Number Patterns: The sequence of numbers 1, 4, 9, 16, 25, … follows a pattern. Identify the pattern and find the next two numbers in the sequence.
Physics - Kinematics: A ball is thrown straight up into the air with an initial speed of 20 m/s. How long will it take for the ball to reach its highest point? (Neglect air resistance and use g = 10 m/s² for the acceleration due to gravity)
Math - Combinatorics: In how many different ways can 5 different books be arranged on a bookshelf?
Math - Geometry: A right-angled triangle has two sides measuring 5 cm and 12 cm. What is the length of the third side?
Math - Modular Arithmetic: Calculate the remainder when 2^100 is divided by 7 without calculating the entire value of 2^100.
Physics - Conservation of Energy: A 1 kg ball is dropped from a height of 10 meters. How fast is it moving just before it hits the ground? (Neglect air resistance and use g = 10 m/s² for the acceleration due to gravity)
Math - Algebra: A secret number is multiplied by 4 and then 7 is subtracted from the result, giving 29. What is the secret number?
Math - Pigeonhole Principle: There are 10 different colored socks in a drawer. It’s dark, and the colors can’t be seen. How many socks need to be pulled out of the drawer to guarantee that there is at least a matching pair?
Math - Probability: A fair six-sided die is rolled. What is the probability of rolling two sixes in a row?
Math - Logic: There are three light switches outside a room, and one of them controls a lightbulb inside the room. You can only enter the room once, and the door is closed, so you can’t see the lightbulb from outside. How can you determine which switch controls the lightbulb by only entering the room once?
Math - Number Patterns: The pattern is the square of consecutive integers (1^2, 2^2, 3^2, …). The next two numbers in the sequence are 6^2 = 36 and 7^2 = 49.
Physics - Kinematics: Using the equation v = u - gt, where v is the final velocity, u is the initial velocity, g is the acceleration due to gravity, and t is time, we have: 0 = 20 - 10t t = 2 seconds
Math - Combinatorics: There are 5! (5 factorial) ways to arrange the books, which is 54321 = 120 different arrangements.
Math - Geometry: By the Pythagorean theorem, the third side (hypotenuse) is sqrt(5^2 + 12^2) = sqrt(25 + 144) = sqrt(169) = 13 cm.
Math - Modular Arithmetic: 2^100 % 7 = ((2^3 % 7)^33 * 2) % 7 = (1^33 * 2) % 7 = 2.
Physics - Conservation of Energy: Using the equation v^2 = u^2 + 2as, where v is the final velocity, u is the initial velocity, a is the acceleration due to gravity, and s is the distance, we have: v^2 = 0^2 + 21010 v^2 = 200 v = sqrt(200) = 10*sqrt(2) m/s (approximately)
Math - Algebra: Let the secret number be x. Then, 4x - 7 = 29. 4x = 36 x = 9
Math - Pigeonhole Principle: You need to pull out 3 socks to guarantee a matching pair. If the first two socks are different colors, the third one will definitely match one of the first two.
Math - Probability: The probability of rolling a six on a fair six-sided die is 1/6. The probability of rolling two sixes in a row is (1/6) * (1/6) = 1/36.
Math - Logic: Turn on switch 1 and wait for a few minutes. Then turn off switch 1 and turn on switch 2. Enter the room. If the lightbulb is on, it’s controlled by switch 2. If it’s off and hot, it’s controlled by switch 1. If it’s off and cool, it’s controlled by switch 3.