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- Question 4 Exercise 8.2 @math-11-nbf:sol:unit08
- theta<\dfrac{\pi}{2}$, i.e. $\theta$ lies in QI. We have $$\sin\theta = \pm \sqrt{1-\cos^2}.$$ Since ... {7}}. \end{align*} (d) $\sin \dfrac{\theta}{2}$ We have $$\sin\left(\frac{\theta}{2} \right) = \pm \... {5}}} \end{align*} (e) $\cos \dfrac{\theta}{2}$ We have $$\cos\left(\frac{\theta}{2} \right) = \pm \... \frac{3\pi}{2}\), i.e., \(\theta\) lies in QIII. We have: \begin{align*} \sec \theta &= \pm \sqrt{1+
- Question 4, Exercise 1.3 @math-11-nbf:sol:unit01
- {7}{53}i\end{align} Put value of $\omega$ in (1), we have \begin{align} &(1-i) z+(1+i)\left(\dfrac{2}{... frac{155}{106}+\dfrac{145}{106}i\end{align} Thus, we have $$z=\dfrac{155}{106}+\dfrac{145}{106}i, \ome... ts(2) \end{align} Multiplying $(1)$ by $(1-2i)$, we get: \begin{align} &(1-2i)(2i z) + (1-2i)(3-2i) ... 3) \end{align} Multiplying equation (2) by $2i$, we get: \begin{align} &2i(1-2i) z + 2i(3+2i) \omega
- Question 9, Exercise 1.2 @math-11-nbf:sol:unit01
- **Solution.** Suppose $z=3 - \sqrt{-4}=3-2i$. We will use the following formulas: \[\text{Re}(z^{... \dfrac{7+2 i}{3-i}\right)^{-1}$. **Solution.** We use the following formulas: \[Re\left(\left(\fr... 2}.\] For \(z_1 = 7 + 2i\) and \(z_2 = 3 - i\), we have: \[x_1 = 7, \quad y_1 = 2, \quad x_2 = 3, ... frac{4+2 i}{2+5 i}\right)^{-2}$. **Solution.** We will use the following formulas: \begin{align}
- Question 5 Exercise 8.2 @math-11-nbf:sol:unit08
- $\sin 2\theta=\dfrac{24}{25}$, $2\theta$ in QII. We have $$\cos 2\theta = \pm \sqrt{1-\sin^2 2\theta}... frac{49}{625}} = -\frac{7}{25} \end{align*} Also we have $$\sin\theta = \pm \sqrt{\frac{1-\cos 2\thet... -\dfrac{7}{25}\) and \(2\theta\) lies in QIII. We have: \[\sin 2\theta = \pm \sqrt{1 - \cos^2 2\theta}\] Since \(2\theta\) lies in QIII, we know that \(\sin 2\theta < 0\). Therefore: \begin
- Notes of Mathematics
- notes are send by different students or teachers. We are very thankful to them for sending us these no... of Advanced Analysis provided by Mr. Anwar Khan. We are really very thankful to Mr. Anwar Khan for pr... are send and written by [[:people:asim-marwat]]. We are very thankful to him for sharing these notes.... rtial Differential Equations by Muzammil Tanveer. We are very thankful to him for sharing these notes.
- Mathematics 10 (Science Group) @matric
- re, Pakistan are available on this page. Whenever we found the notes we will update this page and will upload notes here. If you wish to contribute and se... k ====== These MCQs was sent by [[:people:amir]], we are very thankful for his contribution. * {{ :m... === These notes are send by Ms. [[:people:amir]], we are very thankful to him for sending these notes.
- Question 6, Exercise 1.3 @math-11-kpk:sol:unit01
- =0$$ $$(z^2+z+1 )=0$$ By using quadratic formula, we have $$z=\dfrac{-1\pm \sqrt{1-4}}{2}$$ $$z=\dfrac... 2}$$ $$(z^2-z+1 )=0$$ By using quadratic formula, we have $$z=\dfrac{1\pm \sqrt{1-4}}{2}$$ $$z=\dfrac{... t{3}i}{2}$$ The value of $z$ from both equations, we have $$z=\pm \dfrac{1}{2}\pm \dfrac{\sqrt{3}}{2}i... $z^2-2z+4=0$$ According to the quadratic formula, we have $a=1$, $b=-2$ and $c=4$ Thus, we have \begin
- Are the functions are same? @dyk
- usually know as //domain of the function//. When we are saying set of input values, it means every el... taken as input. <wrap em>So to define a function, we must first have set of input values.</wrap> Alternatively, we say a relation $f:A \to B$ is called function if ... athbb{N}$, $B=\mathbb{R}$ and $f(x)=x+1$. Clearly we see $1 \mapsto 2$, $\quad 2 \mapsto 3$, $\quad 3
- Question 2 Exercise 4.3 @math-11-kpk:sol:unit04
- OOD ====Solution==== Given: $a_1=2, n=17, d=3$ \\ We need to find $a_{17}$ and $S_{17}$. As we know $$a_{n}=a_1+(n-1)d.$$ Thus $$a_{17}=2+(17-1)(3)=50.$$ ... ution==== Given: $a_1=-40$ and $S_{21}=210$.\\ So we have $n=21$ and we have to find $a_{21}$ and $d$. As \begin{align}&S_{21}=\dfrac{21}{2}(a_1+a_{21}) \
- Question 10, Exercise 1.2 @math-11-nbf:sol:unit01
- \, -- (4) \end{align} From (1), (2), (3) and (4), we have: $$\left| z_1 \right| = \left| -z_1 \right| ... 10} + \frac{7}{10}i. \,\, -- (i) \end{align} Now, we have \begin{align} \overline{z_1} = -3 - 2i, \qua... + \frac{7}{10}i.\,\, -- (ii)$$ From (i) and (ii), we have \[ \overline{\left( \frac{z_1}{z_2} \right)}... n \[ z_1 = -3 + 2i, \quad z_2 = 1 - 3i. \] First we calculate \begin{align} z_1 z_2 &= (-3 + 2i)(1
- Question 1 and 2 Exercise 6.5 @math-11-kpk:sol:unit06
- ac{1}{2}$.\\ Find $P(A \cap B)$. ====Solution==== We know by addition law of probability \begin{align}... ign} Substituting $P(A), P(B)$ and $P(A \cup B)$, we get $$P(A \cap B)=\dfrac{2}{5}+\dfrac{2}{5}-\dfra... dfrac{3}{4}$. Find $P(A \cap B)$ ====Solution==== We are given: $$P(A)=\dfrac{1}{2}, P(\bar{B})=\dfrac{5}{8}, P(A \cup B)=\dfrac{3}{4}$$ We know by complementary events $$P(B)=1-P(\bar{B})$
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- MathCraft: PDF to LaTeX file: Sample-01 @mathcraft
- i% \fi } \begin{document} \maketitle ABSTRACT. We give a simple proof of the Stolarsky means inequa... h $r$ and $s$ i.e. for $r \leq u$ and $s \leq v$, we have \begin{equation*} E(x, y ; r, s) \leq E(x,... end{equation*} \vspace{2mm} In this paper, first we shall give a simple proof of inequality (1). Further we shall introduce two new classes of means of Stola
- Question 3, Exercise 2.1 @math-11-kpk:sol:unit02
- }$ and $C=\begin{bmatrix}x\\y\\z\end{bmatrix}$.\\ We have to prove that $$(AB)C=A(BC)$$ First, we take \begin{align} AB&=\begin{bmatrix}x & y & z\end{bmat... fyz+c{{z}^{2}} \right] \ldots (1) \end{align} Now we take \begin{align}BC&=\begin{bmatrix}a & h & g\\h... } \right] \ldots (2)\end{align} From (1) and (2), we have $$(AB)C=A(BC).$$ =====Question 3(ii)(a)===
- Question 1 Exercise 5.3 @math-11-kpk:sol:unit05
- onstants on the both sides of the above equation, we get $$A+B=0 \text{and} A=1$$ Putting $A=1$,then \... 1)}$$ Multiplying both sides by $(2 n-1)(2 n+1)$. we get, \begin{align} & \mathrm{I}=A(2 n+1)+B(2 n-1)... Solving the above two equations for $A$ and $B$, we get \begin{align}A&=\dfrac{1}{2}\\ \text{and} B&=... n+2}$$ Multiplying both sides by $(3 n-1)(3 n+2)$ we get, \begin{align} 1&=A(3 n+2)+B(3 n-1) \\ \Right
- How to prepare admission test (A short guide) @papers:old_admission_test_of_assms_for_ph.d._mathematics
- University of Gujrat, Gujrat, BSc Old Papers (Mathematics Only) @papers:old_papers_for_bsc_mathematics