Solved Math Exercises Across Topics

Sure, I can help with that! What specific areas of mathematics are you interested in? There are many branches and topics, such as algebra, calculus, geometry, statistics, and more. Let me know your preferences, and I’ll provide you with some solved exercises or problems related to those areas.

Let’s delve into some solved exercises across various areas of mathematics to offer a comprehensive view:

Algebra:

1. Solving Equations:

   • Problem: Solve for $x$ in the equation $2x + 5 = 11$.
   • Solution: Subtract 5 from both sides to get $2x = 6$, then divide by 2 to find $x = 3$.

2. Factoring Quadratics:

   • Problem: Factor the quadratic expression $x^2 + 5x + 6$.
   • Solution: This expression factors as $(x + 2)(x + 3)$.

Calculus:

1. Differentiation:

   • Problem: Find the derivative of $f(x) = 3x^2 + 2x – 5$.
   • Solution: The derivative is $f'(x) = 6x + 2$.

2. Integration:

   • Problem: Calculate $\int (3x^2 + 2x – 5) \, dx$.
   • Solution: The integral is $x^3 + x^2 – 5x + C$, where $C$ is the constant of integration.

Geometry:

1. Area of a Triangle:

   • Problem: Find the area of a triangle with base 6 units and height 4 units.
   • Solution: The area is $\frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 6 \times 4 = 12$ square units.

2. Volume of a Cylinder:

   • Problem: Calculate the volume of a cylinder with radius 3 units and height 8 units.
   • Solution: The volume is $\pi \times \text{radius}^2 \times \text{height} = \pi \times 3^2 \times 8 = 72\pi$ cubic units.

Statistics:

1. Mean Calculation:

   • Problem: Compute the mean of the following set of numbers: 10, 15, 20, 25, 30.
   • Solution: The mean is $\frac{10 + 15 + 20 + 25 + 30}{5} = \frac{100}{5} = 20$.

2. Standard Deviation:

   • Problem: Find the standard deviation of the data set: 5, 10, 15, 20, 25.
   • Solution: First, find the mean (which is 15), then compute the differences from the mean, square them, find the average of the squared differences, and finally take the square root to get the standard deviation.

These examples cover a range of fundamental concepts in mathematics, from basic algebraic equations to more advanced topics like calculus and statistics. Each problem has a clear statement and a step-by-step solution to guide understanding. If you’re interested in more exercises or specific topics, feel free to ask!