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- One Day International Symposia on Pure and Applied Mathematics UoS Sargodha (January 27, 2014)
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- Ch 09: Fundamental of Trigonometry
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- Question 12 & 13 Exercise 4.2 @math-11-kpk:sol:unit04
- his salary during his twenty first year of work? GOOD ====Solution==== Suppose $a_1$ represents salar... n during his 21st year of work is dollars 18,500. GOOD =====Question 13(i)===== Find the arithmetic mean between $12$ and $18$. GOOD ====Solution==== Here $a=12, b=18$.\\ Let say $... 15.\end{align} Hence 15 is A.M between 12 and 18. GOOD =====Question 13(ii)===== Find the arithmetic
- Formatting Syntax @wiki
- al links are recognized automagically: http://www.google.com or simply www.google.com - You can set the link text as well: [[http://www.google.com|This Link points to google]]. Email addresses like this one: <[email protected]> are recogn
- Question 1, Exercise 1.1 @math-11-kpk:sol:unit01
- estion 1(i)===== Simplify ${{i}^{9}}+{{i}^{19}}$. GOOD ====Solution==== \begin{align}{{i}^{9}}+{{i}^{1... i\cdot\left( -1 \right)\\ &=i-i\\ &=0.\end{align} GOOD =====Question 1(ii)===== Simplify ${{\left( -i \right)}^{23}}$. GOOD ====Solution==== \begin{align}{{\left( -i \righ... 1}}\\ &=-i\cdot\left( -1 \right)\\ &=i\end{align} GOOD =====Question 1(iii)===== Simplify ${{\left( -1
- Question 2 Exercise 4.3 @math-11-kpk:sol:unit04
- Find the one that is missing: $a_1=2, n=17, d=3$. GOOD ====Solution==== Given: $a_1=2, n=17, d=3$ \\ W... 2.\end{align} Hence $a_{17}=50$ and $S_{17}=442$. GOOD =====Question 2(ii)===== Some of the component... d the one that are missing $a_1=-40, S_{21}=210$. GOOD ====Solution==== Given: $a_1=-40$ and $S_{21}=2... \end{align} Hence $a_{21}=60$, $d=5$ and $n=21$. GOOD =====Question 2(iii)===== Some of the component
- Question 1, Exercise 1.1 @math-11-nbf:sol:unit01
- because i^2=-1\\ &=i\cdot(-1)\\ &=-i.\end{align} GOOD ====Question 1(ii)==== Evaulate ${{\left( -i... )^{11} \cdot i \\ &=-(-1)\cdot i = i. \end{align} GOOD ====Question 1(iii)==== Evaluate ${{\left( -1 \... =\dfrac{i}{-\left( -1 \right)}\\ &=i\end{align} GOOD ====Question 1(iv)==== Evaluate $\dfrac{2}{(-1)^{\frac{3}{2}}}$. GOOD **Solution.** \begin{align}{{\left( -1 \right
- Question 2, Exercise 1.1 @math-11-nbf:sol:unit01
- +i2)+(2+i4)\\ =&(3+2)+(i2+i4)\\ =&5+i6\end{align} GOOD ====Question 2(ii)==== Write the following com... +3i)-(2+5i)\\ =&(4-2)+(3i-5i)\\ =&2-2i\end{align} GOOD ====Question 2(iii)==== Write the following com... i)+(4-7i)\\ =&(4+4)+(7i-7i)\\ =&8+0i. \end{align} GOOD ====Question 2(iv)==== Write the following com... )-(2-5i)\\ =&(2-2)+(5i+5i)\\ =&0+10i. \end{align} GOOD ====Question 2(v)==== Write the following comp
- Question 8, Exercise 1.2 @math-11-nbf:sol:unit01
- ies & 4x^2+4y^2-4y-15=0, \end{align} as required. GOOD ====Question 8(ii)==== Write $|z-1|=|\bar{z}+i|... \\ \implies & x - y =0, \end{align} as required. GOOD ====Question 8(iii)==== Write $|z-4 i|+|z+4 i|=... s & 25x^2 + 9y^2 = 225, \end{align} as required. GOOD ====Question 8(iv)==== Write $\frac{1}{2} \oper... } y=4 \\ \implies & y=8, \end{align} as required. GOOD ====Question 8(v)==== Write $lm\left(\dfrac{z-1
- Question 10, Exercise 1.2 @math-11-nbf:sol:unit01
- overline{z_1} \right| = \sqrt{13}.$$ As required. GOOD ====Question 10(ii)==== For $z_{1}=-3+2 i$ and ... ght)} = \frac{\overline{z_1}}{\overline{z_2}}. \] GOOD ====Question 10(iii)==== For $z_{1}=-3+2 i$ an... = \overline{z_1} \overline{z_2}, \] as required. GOOD ====Question 10(iv)==== For $z_{1}=-3+2 i$ and... e{z_1 + z_2} = \overline{z_1} + \overline{z_2}.\] GOOD ====Question 10(v)==== For $z_{1}=-3+2 i$ and $
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- Question 10 Exercise 6.5 @math-11-kpk:sol:unit06
- obability that either both are apples or both are good? ====Solution===== Total number of Apples $=20$... es $=5$ number of defective oranges $=3$. Totál good apples $=15$ Defective apples $=5$ Total good oranges $=10$ number of defective oranges $=3$ number of good fruits $=22$ Now two fruits are chosen at rand
- Question 3, Exercise 1.1 @math-11-nbf:sol:unit01
- -9i}{2}\\ =&\dfrac{7}{2}-\dfrac{9}{2}i\end{align} GOOD ====Question 3(ii)==== Simplify the following $... 25}\\ =&\dfrac{7}{25}-\dfrac{1}{25}i. \end{align} GOOD ====Question 3(iii)==== Simplify the following... \ =&-\dfrac{2}{10}i\\ =&-\dfrac{1}{5}i\end{align} GOOD ====Question 3(iv)==== Simplify the following $... \\ =&\dfrac{1}{2i}-\dfrac{1}{2i}\\ =&0\end{align} GOOD ====Question 3(v)==== Simplify: $(2+i)^2+\dfra
- Question 7, Exercise 1.1 @math-11-nbf:sol:unit01
- rt{265}\end{align} Hence $|11+12 i|=\sqrt{265}$. GOOD ====Question 7(ii)==== Find the magnitude of ... = 3. \end{align} Hence $$|(2+3 i)-(2+6 i)|=3.$$ GOOD ====Question 7(iii)==== Find the magnitude of t... t{5} = 15.\end{align} Hence $|(2-i)(6+3 i)|=15$. GOOD ====Question 7(iv)==== Find the magnitude of th... dfrac{3-2 i}{2+i} \right|=\sqrt{\dfrac{13}{5}}$. GOOD ====Question 7(v)==== Find the magnitude of the
- Question 3(vi, vii, viii, ix & x) Exercise 8.3 @math-11-nbf:sol:unit08
- = \sec y(\sin 4y - \sin 2y)\\ &= RHS \end{align*} GOOD =====Questio 3(vii)===== Prove the identity $\d... & = \tan 5\beta \cot \beta\\ &= RHS \end{align*} GOOD =====Questio 3(viii)===== Prove the identity $\... ta} \\ & = -2 \cos 2\theta \\ &= RHS \end{align*} GOOD :!: **Correction:**\\ The corrected version of... \ & = \cos 7x \sec 5x \cot x\\ &=RHS \end{align*} GOOD =====Questio 3(x)===== Prove the identity $\df
- Question 1, Exercise 9.1 @math-11-nbf:sol:unit09
- m value $(M) = 4$ \\ and minimum value $(m) = 0$. GOOD **Alternative Method:** Given: $$y=2-2 \opera... ximum value (M) = 4 \\ and mimimum value (m) = 0. GOOD =====Question 1(ii)===== Find the maximum an... 7}{6}$ \\ and minimum value $(m) = \dfrac{1}{6}$. GOOD **Alternative Method:** Given: $$y=\dfrac{2}{... 7}{6}$ \\ and mimimum value $(m) = \dfrac{1}{6}$. GOOD =====Question 1(iii)===== Find the maximum and
- How to prepare admission test (A short guide) @papers:old_admission_test_of_assms_for_ph.d._mathematics
- Recent Advances in Mathematical Methods, Models & Applications, LSC Lahore, Pakistan (April 13-14, 2019) @conferences
- 2nd International Conference on Pure and Applied Mathematics UoS Sargodha (November 26-27, 2016) @conferences