class 9th math 2.6 solution

Question#01: Identify the following statement as true or false. \[\begin{align} \left( \text{i} \right) \qquad&\sqrt{-3}\sqrt{-3}=3\quad\textbf{False} \\  \left( \text{ii} \right) \qquad&{{i}^{73}}=-i\quad \textbf{False} \\  \left( \text{iii} \right) \qquad&{{i}^{10}}=-1\quad \textbf{True} \\  \left( \text{iv} \right) \qquad&\text{Complex}\,\text{conjugate}\,\text{of}\,\left( -6i+{{i}^{2}} \right)\text{ is }\left( -1+6i \right) \quad \textbf{True} \\  \left( \text{v} \right) \qquad&\text{Difference of a complex number }z=a+bi\text{ and its conjugate}\\& \text{is a real number.}\quad … Read more

class 9th math 2.5 solution

Question#01: Evaluate \((i)\) \[{{i}^{7}}\] Solution: \[\begin{align}  & ={{i}^{7}} \\ & ={{i}^{6}}\times i \\ & ={{\left( {{i}^{2}} \right)}^{3}}\times i \\ & ={{\left( -1 \right)}^{3}}\times i \\ & =-1\times i \\ & =-i \\\end{align}\] \((ii)\) \[{{i}^{50}}\] Solution: \[\begin{align}  & ={{i}^{50}} \\ & ={{\left( {{i}^{2}} \right)}^{25}} \\ & ={{\left( -1 \right)}^{25}} \\ & =-1 \\  \end{align}\] \((iii)\) \[{{i}^{12}}\] Solution: \[\begin{align}  & ={{i}^{12}} \\ & ={{\left( {{i}^{2}} \right)}^{6}} … Read more

class 9th math 2.4 solution

Question#1: Use the law of exponents to simplify. \((i)\) \[\frac{{{\left( 243 \right)}^{-\frac{2}{3}}}{{\left( 32 \right)}^{\frac{1}{5}}}}{\sqrt{{{\left( 196 \right)}^{-1}}}}\] Solution: \[\begin{aligned}&=\frac{{{\left( 3^5 \right)}^{-\frac{2}{3}}}{{\left( 2^5 \right)}^{-\frac{1}{5}}}}{\sqrt{{{\left( 14^2 \right)}^{-1}}}}\\&=\frac{{{\left( 3 \right)}^{-\frac{5\times 2}{3}}}\times{{\left( 2^\cancel{5} \right)}^{-\frac{1}{\cancel{5}}}}}{\sqrt{{{\left( 14^{-1} \right)}^{2}}}}\\&=\frac{{{\left( 3 \right)}^{-\frac{10}{3}}}\times{{\left( 2 \right)}^{-1}}}{\sqrt{{{\left( 14^{-1} \right)}^{\cancel{2}}}}}\\&=\frac{{{\left( 3 \right)}^{-\frac{10}{3}}}\times{{ 2}^{-1}}}{{{ 14^{-1}}}}\\&=\frac{14}{{{3}^{\frac{10}{3}}}\times 2}\\&=\frac{\cancelto{7}{14}}{3^{\frac{9}{3}}\times {{3}^{\frac{1}{3}}}\times\cancel{2} }\\&=\frac{7}{3^{\frac{\cancelto{3}{9}}{\cancel{3}}}\times {{3}^{\frac{1}{3}}}}\\&=\frac{7}{{{3}^{3}}\times \sqrt[3]{3}}\\&=\frac{7}{27 \sqrt[3]{3}}\end{aligned}\] \((ii)\) \[\left( 2{{x}^{5}}{{y}^{-4}} \right)\left( -8{{x}^{-3}}{{y}^{2}} \right)\] Solution: \[\begin{aligned}&=\left( … Read more

9th class general math exercise 1.1 Urdu Q1 Solution

سوال نمبر۱- درج زیل فی صد کو کسروں کی آسان شکلوں میں واضح کیجئے۔ \((i)\: 95\%\) \[\frac{95}{100}\] سے کاٹیں گے۔\(5\) \[\frac{\cancelto{19}{95}}{\cancelto{20}{100}}\] \[\frac{19}{20}\] \((ii)\,65\%\) \[\frac{65}{100}\] سے کاٹیں گے۔\(5\) \[\frac{\cancelto{13}{65}}{\cancelto{20}{100}}\] \[\frac{13}{20}\] \((iii)\,75\%\) \[\frac{75}{100}\] سے دو بارکاٹیں گے۔\(5\) \[\frac{\cancelto{15}{75}}{\cancelto{20}{100}}\] \[\frac{\cancelto{3}{15}}{\cancelto{4}{20}}\] \[\frac{3}{4}\] \((iv)\,25\%\) \[\frac{25}{100}\] سے دو بارکاٹیں گے۔\(5\) \[\frac{\cancelto{5}{25}}{\cancelto{20}{100}}\] \[\frac{\cancelto{1}{5}}{\cancelto{4}{20}}\] \[\frac{1}{4}\] \((v)\,56\%\) \[\frac{56}{100}\] سے دو بارکاٹیں گے۔\(2\) \[\frac{\cancelto{28}{56}}{\cancelto{50}{100}}\] \[\frac{\cancelto{14}{28}}{\cancelto{25}{50}}\] … Read more

Gauss Jordan’s elimination In Sagemath

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

sagemath matrix

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

Sagemath Integral – A Complete Guide

Integration in sagemath

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