High School Mathematics Practice

The sum of the first n positive integers is given by the formula n(n+1)/2. Therefore, the sum of the first 100 positive integers is 100(101)/2 = 5050.

The area of a circle is given by the formula πr^2. Therefore, the area of a circle with radius 5 cm is π(5)^2 = 25π cm^2.

The volume of a cube is given by the formula s^3. Therefore, the volume of a cube with side length 4 cm is 4^3 = 64 cm^3.

The equation of a line that passes through two points (x1, y1) and (x2, y2) is given by the formula y - y1 = (y2 - y1)/(x2 - x1) * (x - x1). Substituting the values of the two points, we get y - 3 = (7 - 3)/(5 - 2) * (x - 2). Simplifying this equation, we get y = 2x + 1.

We can solve this equation using the quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a. Substituting the values of a, b, and c, we get x = (-(-4) ± √((-4)^2 - 4(1)(3))) / 2(1). Simplifying this equation, we get x = (4 ± √(16 - 12)) / 2. Further simplifying, we get x = (4 ± √4) / 2. Therefore, the solutions to the equation are x = 1 and x = 3.

The GCD of two numbers is the largest positive integer that divides both numbers without leaving a remainder. We can find the GCD of 12 and 18 by using the Euclidean algorithm: 18 = 12 * 1 + 6, 12 = 6 * 2 + 0. Therefore, the GCD of 12 and 18 is 6.

The LCM of two numbers is the smallest positive integer that is divisible by both numbers. We can find the LCM of 12 and 18 by using the formula LCM = (a * b) / GCD. Substituting the values of a, b, and GCD, we get LCM = (12 * 18) / 6 = 36.

We can evaluate the expression by following the order of operations: 2^3 = 8, 3^2 = 9, 4^1 = 4. Therefore, the value of the expression is 8 + 9 - 4 = 15.

We can evaluate the expression by using the distributive property: (2 + 3i)(4 - 5i) = 2(4 - 5i) + 3i(4 - 5i). Simplifying this expression, we get 8 - 10i + 12i - 15i^2. Since i^2 = -1, we can simplify further to get 11 - 22i.

When flipping a coin, there are two possible outcomes: head or tail. Since both outcomes are equally likely, the probability of getting a head is 1/2.

When rolling two dice, there are 36 possible outcomes. The outcomes that result in a sum of 7 are (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), and (6, 1). Therefore, the probability of getting a sum of 7 is 6/36 = 1/6.

The sequence is a sequence of squares of natural numbers. The next term in the sequence is the square of the next natural number, which is 5. Therefore, the next term in the sequence is 5^2 = 25.

The sum of the first 100 prime numbers is 24133.

The largest prime number less than 100 is 97.

The smallest positive integer that is divisible by 2, 3, and 5 is 30.