Umar Zaheer Note: This is the Solution of exercise 1.1 from newly published book by PCTB (Punjab Curriculum and Textbook Board, Pakistan) for new 9th session 2025 Onward. Exercise 1.1 Question # 01: Identify each of the following as rational or irrational number. \((i)\) \(\quad 2.353535 \) \((ii)\) \(\quad 0.\overline{6} \) \((iii)\) \(\quad 2.236067… \) \((iv)\) \(\quad … Read more
\(Q1.\:\) If \(x+3y=c\) and \(\left(1,2\right)\) is a solution of the equation, then the value of \(c\) is: \( a) \quad \) \(0\) \( b) \quad \) \(7\) \(c) \quad \) \(5\) \( d) \quad \) \(6\) \( \text{Answer/Explanation}\) \(Q2.\:\) \(2x+5=3x+4\) is equivalent to \( a) \quad \) \(x=1\) \( b) \quad \) \(2x+1=3\) \(c) \quad … Read more
Questions related to basic arithmetic involving absolute values. \(Q1.\quad |-2|=\_\_\_\_ \) \( a) \qquad 2 \) \( b) \quad -2\) \(c) \qquad 0 \) \( d) \qquad \text{Both a and b} \) \( \text{Answer/Explanation}\) \(Q2.\quad -|-3|=\_\_\_\_ \) \( a) \qquad 3 \) \( b) \quad -3\) \(c) \qquad 0 \) \( d) \qquad \text{Both a … Read more
\(Q1.\:\) Solve for \( x: 5x-1 \le 3 \) \( a) \quad x \le \frac{5}{4} \) \( b) \quad x \le \frac{4}{5}\) \(c) \quad x \le \frac{2}{5} \) \( d) \quad x \le \frac{5}{2} \) \( \text{Answer/Explanation}\) \(Q2.\: \) if \( \frac{1}{3}x \le 1\) then \( a) \quad x \in (3, \infty) \) \( b) … Read more
These are math definition notes of english medium of class 9th which is being taught in punjab, pakistan.
These are definition notes for 10th grade mathematics taught to science students in Punjab, Pakistan.
In sagemath, a system of linear equations can be solved with built-in functions matrix(), augment(), solve_right(), solve_left(), rref(), and by row, and column operations. Gauss Jordan elimination method is to solve \(n\) linear system of equations in \(n\) variables. Gauss Jordon method is similar to the Gauss elimination method, except entries above and below pivot … Read more
Matrix plays an important role in mathematics, and we can create, perform different operations, and also solve linear systems of equations via matrices in sagemath. In Sagemath the matrix() command is used to create matrices and perform different operations such as transpose, determinant, adjoint, inverse, etc. Sagemath can also solve a system of linear equations … Read more
Computing integration of complicated functions by hand is a challenging task, and sometimes, it almost becomes impossible, but with computational software, it is very easy to do. The command “integrate()” and its alias “integral()” are used in sagemath for computing definite, indefinite, improper, single, double or any multiple integrals. “numerical_integral()” is used for the numerical … Read more
Installation of Sagemath on Windows is not as easy as other software but here in this article, we will make it easy by providing an illustrated image of each step. Sagemath could not be operated directly on Windows, and to run it on Windows first we need to install a window sub-system for Linux and … Read more