Umar Zaheer Question#1: Find the determinant of the following matrices. \(i\) \[A=\left[ \begin{matrix} -1 & 1 \\ 2 & 0 \\\end{matrix} \right]\] Solution: \[\begin{align} \left| A \right|&=\left| \begin{matrix} -1 & 1 \\ 2 & 0 \\\end{matrix} \right| \\ & =\left( -1 \right)\left( 0 \right)-\left(1 \right)\left( 2 \right) \\ & =0-2 \\ & =-2 \\ \end{align}\] \(ii\) \[B=\left[ \begin{matrix} 1 & 3 … Read more
Question#1: Identify which of the following are rational and irrational numbers. \[\begin{align} \left( \text{i} \right)\qquad &\sqrt{3}\qquad &&\text{Irrational number} \\ \left( \text{ii} \right) \qquad &\frac{1}{6}\qquad &&\text{Rational number} \\ \left( \text{iii} \right) \qquad &\pi \qquad &&\text{Irrational number} \\ \left( \text{iv} \right) \qquad &\frac{15}{2}\qquad &&\text{Rational number} \\ \left( \text{v} \right) \qquad &7.25\qquad &&\text{Rational number} \\ \left( \text{vi} \right) \qquad &\sqrt{29}\qquad &&\text{Irrational number} \\\end{align}\] Question#2: … Read more
Question#1: Identify the property used in the following. \[\begin{align} \left( \text{i} \right)\qquad&a+b=b+a\qquad &&\text{Commutative Property w}\text{.r}\text{.t}\text{. addition } \\ \left( \text{ii} \right) \qquad &\left( ab \right)c=a\left( bc \right) \qquad &&\text{Associatve Property w}\text{.r}\text{.t}\text{. multiplication} \\ \left( \text{iii} \right) \qquad &7\times 1=7\qquad &&\text{Multiplicative Identity} \\ \left( \text{iv} \right) \qquad &x>y\text{ or }x=y\text{ or }x<y\qquad &&\text{Trichotomy Property} \\ \left( \text{v} \right) \qquad &ab=ba\qquad … Read more
Question#1: Write each radical expression in exponential notation and each exponential expression in radical notation. Do not simplify. \[\begin{align} (\text{i})\qquad&\sqrt[3]{-64} \\ & ={{\left( -64 \right)}^{\frac{1}{3}}} \\ (\text{ii})\qquad&{{2}^{3/5}} \\ & =\sqrt[5]{{{2}^{3}}} \\ (\text{iii})\qquad&-{{7}^{1/3}} \\ & =\sqrt[3]{-7} \\ (\text{iv})\qquad&{{y}^{-2/3}} \\ & =\sqrt[3]{{{y}^{-2}}} \\\end{align}\] Question#2: Tell whether the following statements are true or false? \[\begin{align} \left( \text{i} \right) \qquad & {{5}^{1/5}}=\sqrt{5}\qquad & \textbf{False} \\ \left( \text{ii} \right)\qquad … Read more
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
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
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
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
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