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- Khuram Ali Khan
- Results, J. Inequal. Appl., Vol. 2010, 16 pages, (Published on October 18, 2010), Impact Factor 0.879.... . Appl., Vol. 2011, Article ID 350973, 19 pages, (Published on March 14, 2011) Impact Factor 0.73 ([[... lities and Applications, 1 (2012), No. 1, 12-32, (Published on July 9, 2012). ([[http://scik.org/index... al, Vol. 2014 (2014), Article ID 283982, 9 pages (Published on May 15, 2014) ([[http://www.hindawi.co
- Question 1 and 2 Exercise 4.1 @math-11-kpk:sol:unit04
- unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB... Given: $$a_n=\dfrac{n(n+1)}{2}$$ For first term, put $n=1,$ $$a_1=\dfrac{1(1+1)}{2}=1$$ For second term, put $n=2$. $$a_2=\dfrac{2(2+1)}{2}=3$$ For third term, put $n=3,$ $$a_3=\dfrac{3(3+1)}{2}=6$$ For forth ter
- FSc/ICS Part 1 (Mathematics): PTB
- >Textbook of Algebra and Trigonometry Class XI is published by Punjab Textbook Board (PTB) Lahore, Pakistan. The book has total of 14 chapters.</lead> On... Intermediate and Secondary Educations (BISE)) of Punjab and Federal Board of Intermediate and Seconda... wnloaded from the download page of [[https://pctb.punjab.gov.pk/download_books | Punjab Curriculum and
- Multiple Choice Questions (BSc/BS/PPSC) by Akhtar Abbas @notes
- viding these notes and appreciates his efforts to publish these notes on MathCity.org. Multiple Choice... these notes, which might be helpful in BSc, BS or Punjab Public Service Commission (PPSC) exams. ^ Name |Multiple Choice Questions (BSc/BS/PPSC) | ^ Aut... al part are called: - imaginary numbers - pure non real numbers - pure imaginary numbers
- Topology and Functional Analysis Solved Paper by Noman Khalid @notes
- very thankful to him for providing these notes to publish on MathCity.org. Topology and Functional An... e said subject is given for the University of the Punjab (PU), Lahore. ^ Name |Topology and Functional Analysis Solved Papers | ^ Author |Noman Khalid ... *{{ notes:topology-and-functional-analysis-solved-pu-a2011.pdf |Download PDF ~ 7.54MB}}** %%|%% <btn t
- Question 4 Exercise 4.5 @math-11-kpk:sol:unit04
- unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB... 8}{1-0.1}=\dfrac{0.8}{0.9}=\dfrac{8}{9}$$\\ Hence putting $S_{\infty}$ in (i), we get $$0 . \overline{... ots \ldots \ldots . . . \text { (ii) }\end{align} Putting (ii) in (i), we get\\ $$1.63=1+\dfrac{7}{11}... s and given by $$S_{\infty}=\dfrac{a_1}{1-r}$$,\\ putting $a_1, r$ we get\\ $$S_{\infty}=\dfrac{0.15}{
- Question 1 and 2 Exercise 6.5 @math-11-kpk:sol:unit06
- unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB... now by complementary events $$P(B)=1-P(\bar{B})$$ Putting \begin{align}P(\bar{B})&=\dfrac{5}{8}\\ P(B)... frac{3}{8}\end{align} Also $$P(\bar{A})=1-P(A)$$ Putting $P(A)=\dfrac{1}{2}$, we get \begin{align}P(\... now by complementary events $$P(B)=1-P(\bar{B})$$ Putting $P(\bar{B})=\dfrac{5}{8}$, we get $$P(B)=1-\
- MCQs/Objective: HSSC-I @fsc
- : Text Book of Algebra and Trigonometry Class XI (Punjab Textbook Board, Lahore) </well><well> **[[FSc... \ Text Book of Algebra and Trigonometry Class XI (Punjab Textbook Board, Lahore). </well><well> **[[FS... : Text Book of Algebra and Trigonometry Class XI (Punjab Textbook Board, Lahore). </well><well> **[[FS... ll chapters of Algebra and Trigonometry Class XI (Punjab Textbook Board, Lahore) with answers. </well
- Question 1, Exercise 10.3 @fsc-part1-kpk:sol:unit10
- unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB... a =\cos (\alpha +\beta )-\cos (\alpha -\beta ).$$ Put $\alpha =6x$ and $\beta =x$ \begin{align}-\,2... a =\sin (\alpha +\beta )+\sin (\alpha -\beta )$$ Put $\alpha ={{55}^{\circ }}$ and $\beta ={{123}^... ta =\sin (\alpha +\beta )+\sin (\alpha -\beta )$$ Put $\alpha =\dfrac{A+B}{2}$ and $\beta =\dfrac{A
- Question 2, Exercise 10.3 @fsc-part1-kpk:sol:unit10
- unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB... )\cos \left( \dfrac{\alpha -\beta }{2} \right).$$ Put $\alpha ={{37}^{\circ }}$, $\beta ={{43}^{\circ ... t)\sin \left( \dfrac{\alpha -\beta }{2} \right)$$ Put $\alpha =36^\circ$, adn $\beta =82^\circ$ \begin... )\sin \left( \dfrac{\alpha -\beta }{2} \right).$$ Put $\alpha =\dfrac{P+Q}{2}$ and $\beta =\dfrac{P-Q}
- Question 5 Exercise 7.2 @math-11-kpk:sol:unit07
- unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB... }{x})^{8-r}(b x)^r$$ To get middle term $T_5$, we put $r=4$ \begin{align}T_5&=\dfrac{8 !}{(8-4) ! 4 !}... 9 !}{(9-r) ! r !}(3 x)^{9-r}(-\dfrac{x^2}{2})^r$$ Putting $r=4$ to get first middle term \begin{align}... x^{13}\\ & T_5=\dfrac{15309}{8} x^{13}\end{align} Putting $r=5$ to get the $2^{\text {nd }}$ middle te
- Question 1, Exercise 10.3 @math-11-kpk:sol:unit10
- unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB... a =\cos (\alpha +\beta )-\cos (\alpha -\beta ).$$ Put $\alpha =6x$ and $\beta =x$ \begin{align}-\,2... a =\sin (\alpha +\beta )+\sin (\alpha -\beta )$$ Put $\alpha ={{55}^{\circ }}$ and $\beta ={{123}^... ta =\sin (\alpha +\beta )+\sin (\alpha -\beta )$$ Put $\alpha =\dfrac{A+B}{2}$ and $\beta =\dfrac{A
- Question 2, Exercise 10.3 @math-11-kpk:sol:unit10
- unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB... )\cos \left( \dfrac{\alpha -\beta }{2} \right).$$ Put $\alpha ={{37}^{\circ }}$, $\beta ={{43}^{\circ ... t)\sin \left( \dfrac{\alpha -\beta }{2} \right)$$ Put $\alpha =36^\circ$, adn $\beta =82^\circ$ \begin... )\sin \left( \dfrac{\alpha -\beta }{2} \right).$$ Put $\alpha =\dfrac{P+Q}{2}$ and $\beta =\dfrac{P-Q}
- PPSC General Information, Syllabus, Paper Pattern
- syllabus and paper pattern of paper couducted by Punjab Public Service Commission (PPSC) for the post of Lecturer in Mathematics. This page might be helpful for other jobs as subject specialist or for public service commission of other provinces. =====... quations with Boundary Value Problems, Ed. 3. PWS Publishing Co., 1997. - Abell, Braselton, Modem Di
- Question 8, Exercise 1.2 @fsc-part1-kpk:sol:unit01
- unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB... how that $\overline{z}=-z$ if and only if $z$ is pure imaginary. ====Solution==== Suppose that $z=a+... it in (1), we get $$z=0+bi=bi,$$ that is, $z$ is pure imaginary. Conversly, suppose that $z$ is pure imaginary, then its real part will be zero, that is