12th Public Examination Mathematics: Calculus, Probability, Vectors, Matrices, and Differential Equations

12th Public Examination Important Questions for Mathematics: Focus on Calculus, Probability, Vectors, Matrices, and Differential Equations

As we approach the 12th public examination, we're going to dive into some important questions from the realm of mathematics, covering calculus, probability, vectors, matrices, and differential equations. These subtopics are crucial for students who want to strengthen their mathematical foundations and prepare for the exams.

Calculus

1. Limits: Evaluate the limit of the sequence: lim_(n→∞) (n^2 - 5n + 6)/n^2.
2. Derivatives: Find the derivative of the function f(x) = (x^2 - 5x + 6)/x with respect to x.

Probability

3. Discrete Probability: In a class of 10 students, 4 are girls. A committee of 3 students is to be selected. Calculate the probability that 2 of them are girls.
4. Continuous Probability: A car's speedometer shows speeds uniformly from 0 to 120 mph. If a car's speed is 60 mph, find the probability that its speed is less than 80 mph.

Vectors

5. Vector Addition: Given the vectors A = 2i - 3j and B = 4i + j, find the vector C = A + B.
6. Scalar Multiplication: Find the vector D = 2A - 3B, where A and B are the vectors from question 5.

Matrices

7. Matrix Multiplication: Calculate the product AB, given the matrices A = | 1 2 | and B = | 3 2 |.
8. Matrix Transpose: Find the transpose of the matrix A from question 7.

Differential Equations

9. First-Order DE: Solve the first-order differential equation dy/dx + 2x*y = 3x^2, given that y(1) = 1.
10. Second-Order DE: Solve the homogeneous second-order differential equation y'' + y = 0 with initial conditions y(0) = 1, y'(0) = -2.

As a mathematically rich and educational resource, these questions are designed to strengthen your understanding of the subtopics and foster problem-solving skills. By working through these questions, you'll not only deepen your mathematical knowledge but also hone your ability to use mathematical concepts to solve real-world problems. With diligent practice and a solid foundation in these subtopics, you'll be well-prepared for the 12th public examination.