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- Definitions: FSc Part 1 (Mathematics): PTB
- ===== * **Rational number:** A number which can be written in the form of $\frac{p}{q}$, where $p,q ... **Irrational number:** A real number which cannot be written in the form of $\frac{p}{q}$, where $p,q... nating and recurring decimal is a rational number because it can be converted into common fraction. * **Non-terminating decimal or non-recurring decima
- Trigonometric Formulas
- d defined trigonometric functions. A PDF file can be downloaded for high quality printing and a word f... anel><grid><col sm="6"> * $\sin \left( \alpha +\beta \right)=\sin \alpha \cos \beta +\cos \alpha \sin \beta$ * $\sin \left( \alpha -\beta \right)=\sin \alpha \cos \beta -\cos \alph
- Definitions: FSc Part 1 (Mathematics): PTB by Aurang Zaib
- ===== ====Rational Number==== A number which can be expressed in the form \( \dfrac{p}{q} \), where \... ==Irrational Number==== A real number that cannot be represented as a fraction of two integers is call... her terminates nor repeats. These decimals cannot be expressed as a fraction with integer numerator an... another example, with its decimal representation being \( 1.41421356... \). ====Binary Operations===
- Ch 03: Matrices and Determinants @fsc-part1-ptb:important-questions
- == <list-group> * Fin $x$ and $y$ if $ \left[ {\begin{array}{c} x+3&1\\ -3& 3y-4 \end{array}} \right]= \left[ {\begin{array}{c} 2&1\\ -3&2 \end{array}} \right]$ -... ala(2015)// * Solve for matrix $A$ if $\left[ {\begin{array}{c}4&3\\ 2&2 \end{array}} \right]A-\left[ {\begin{array}{c} 2&3\\ -1&-2 \end{array}} \right]= \l
- Exercise 1.1 (Solutions) @fsc-part1-ptb:sol:ch01
- or '$\times$'. **Solutions** Addition Table \[\begin{array}{|c|c|} \hline + & 0 \\ \hline 0 & 0 ... ure property w.r.t. '+'. Multiplication Table \[\begin{array}{|c|c|} \hline \times & 0 \\ \hline 0... or '$\times$'. **Solutions** Addition Table \[\begin{array}{|c|c|} \hline + & 1 \\ \hline 1 & 2 ... ure property w.r.t. '+'. Multiplication Table \[\begin{array}{|c|c|} \hline \times & 1 \\ \hline 1
- Exercise 2.8 (Solutions) @fsc-part1-ptb:sol:ch02
- $G=\{0,1\}$ is shown in the adjoining table. \[ \begin{array}{|c|c|c|} \hline \oplus & 0 & 1 \\ \h... able, show that the set is an Abelian group? \[ \begin{array}{|c|c|c|c|c|} \hline \oplus & 0 & 1 &... Closure property holds in $\mathbb{Q}$ under $+$ because sum of two rational number is also rational.... is set? Show that this set is Abelian group. \[ \begin{array}{|c|c|c|} \hline \oplus & E & O \\ \
- Solution and Area of Oblique Triangle
- d defined trigonometric functions. A PDF file can be downloaded for high quality printing and a word f... b^2+c^2-2bc\cos \alpha$ * $b^2=c^2+a^2-2ca\cos \beta$ * $c^2=a^2+b^2-2ab\cos \gamma$ </col><col s... $\cos\alpha =\dfrac{b^2+c^2-a^2}{2bc}$ * $\cos\beta =\dfrac{c^2+a^2-b^2}{2ac}$ * $\cos\gamma =\df... ne"> * $\dfrac{a}{\sin \alpha }=\dfrac{b}{\sin \beta }=\dfrac{c}{\sin \gamma }$ </panel> <panel titl
- MCQs: Ch 04 Quadratic Equations @fsc-part1-ptb:mcq-bank
- -4ac}}{2a}$ - A quadratic equation which cannot be solved by factorization, that will be solved by - Comparing coefficients - Completing square ... on - None of these - A polynomial in$x$ can be considered as a - Non-linear equation - P... a+\sqrt{b}$ - $a+\sqrt{b}$ - If $\alpha$, $\beta$ are the roots of a quadratic equation then
- Ch 04: Quadratic Equations @fsc-part1-ptb:important-questions
- * If the roots of $px^2+qx+q=0$ are $\alpha$, $\beta$,then prove that $$\sqrt {\frac{\alpha}{\beta}}+\sqrt {\frac{\beta}{\alpha}}+\sqrt{\frac{p}{q}}=0$$ --- //BISE Gujrawala(2017),BISE Sagodha(2017... tem of equations --- //BISE Gujrawala(2017)// $${\begin{array}{c} x^2-5xy+6y^2=0\\x^2+y^2=45\end{array
- MCQs: Ch 02 Sets, Functions and Groups @fsc-part1-ptb:mcq-bank
- - Elements - None of these - A set can be describing in different no. of ways are - One... ne of these - A declarative statement which may be true or false but not both is called a - Indu... urdity - Quantifier - A statement which can be true or false depending upon the truth values of ... ymbol $\forall$ stand for - There exist - Belongs to - Such that - For all - The sym
- Ch 07: Permutation, Combination and Probability @fsc-part1-ptb:important-questions
- anwala(2015)// * How many $6-digit$ numbers can be formed from the digits $2,2,3,3,4,4$? How many of them with lie between $400,000$ and $430,000$? ---// BISE Gujrawala(2015)// * How many words can be formed from the letter $``PLANE"$ using all lette... // * In how many way can a hockey team of $11$ be selected out of $15$ players. ---// BISE Sargodha
- Exercise 1.2 (Solutions) @fsc-part1-ptb:sol:ch01
- lex numbers. **Solution** Let \(z,w,\) and \(v\) be complex numbers. Then the following properties ho... =z\nonumber \] - Existence of Additive Inverse \[\begin{array}{l} \mbox{For each} \; z\in \mathbb{C}, ... 4(iii)** Simplify: ${-i}^{19}$ **Solutions** \begin{align} {-i}^{19}& =[(-1)(i)] ^{19}=(-1)^{19}\... laystyle {{(-1)}^{-\frac{21}{2}}}$ **Solution** \begin{align} (-1)^{-\frac{21}{2}}&=\frac{1}{(-1)^\f
- Ch 06: Sequences and Series @fsc-part1-ptb:important-questions
- hat $\frac{a^n+b^n}{a^{n-1}+b^{n-1}}$ maybe $A.M$ between $a$ and $b$. --- //BISE Gujrawala(2015)// ... argodha(2015)// * If $15$ and $8$ are two A.Ms between $a$ and $b$, find $a$ and $b$. --- //BISE Sa... -- //BISE Sargodha(2015)// * Find three $A.Ms$ between $3$ and $11$ --- //BISE Sargodha(2016)// ... rgodha(2016),FBISE(2016)// * Insert two $G.Ms$ between $2$ and $16$ --- //BISE Sargodha(2017)//
- Ch 12: Applications of Trigonometry @fsc-part1-ptb:important-questions
- {\circ}$, $\alpha=62^{\circ}40'$, $b=796$, find $\beta$ anf $a$--- //BISE Gujrawala(2017)// * Find t... Gujrawala(2017)// * Show that $r_2=s tan\frac{\beta}{2}$--- //BISE Sargodha(2015)// * Show that $... FBISE(2017)// * Solve the triangle $ABC$, if $\beta=60^{\circ}$, $\gamma=15^{\circ}$, $b=\sqrt{6}$-... argodha(2015)// * Prove that $abc(sin\alpha+sin\beta+sin\gamma)=4\triangle s$--- //BISE Sargodha(201
- Definitions: Mathematics 11: PTB by Muzzammil Subhan
- or view PDF for all definitions. Samples is given below =====Sample===== * **Exponential Function:... 2=4$ * **Even Function:** A function is said to be an even function if $f(-x)=f(x)$ * **Odd Function:** A function is said to be an odd function if $f(-x)=-f(x)$ * **Circle:** ... ch that distance of each point from a fixed point bear a constant ratio less then one to the distance