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- 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 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
- Question 1, Exercise 1.3 @fsc-part1-kpk:sol:unit01
- unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB... &=\dfrac{11-11i}{11}\\ \implies w&=1-i\end{align} Put value of $w$ in (i). \begin{align}z&-4(1-i)=3i\\... =6i & -2\\ \end{array}$$ \[-w=6i-2\] \[w=2-6i\] Put value of $w$ in (i).\\ \begin{align}z&+(2-6i)=3i... hline 8z & 0 & =8\\ \end{array}\\ \implies z=1$$ Put value of $z$ in (i).\\ \begin{align} &3(1)+(2+i)
- Question 2, Exercise 1.3 @fsc-part1-kpk:sol:unit01
- unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB... orem, $(z-a)$ is a factor of $P(z)$ iff $P(a)=0$. Put $z=-2$ \begin{align} P(-2)&=(-2)^3+6(-2)+20\\ &... eorem $(z-a)$ is a factor of $()$ iff $P(a)=0$.\\ Put $z=2$\\ \begin{align}P(-2)&=(-2)^3+6(-2)+20\\ &
- Question 3 & 4, Exercise 1.3 @fsc-part1-kpk:sol:unit01
- unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB... tion==== Given: $$z^2+2z_1+2=0\quad \ldots (i)$$ Put the value of $z_1=-1+i$ in (i) \begin{align}L.H.... es $z_1=-1+i$ satisfied the given equation.\\ Now put $z_2=-1-i$ in (i) \begin{align} L.H.S&=(-1-i)^2+
- Definitions: FSc Part1 KPK
- ====== A Textbook of Mathematics for Class XI is published by Khybar Pakhtunkhwa Textbook Board (KPTB
- Unit 1: Complex Numbers (Solutions) @fsc-part1-kpk:sol
- This is a first unit of the book Mathematics 11 published by Khyber Pakhtunkhwa Textbook Board, Pesh
- Unit 10: Trigonometric Identities of Sum and Difference of Angles (Solutions) @fsc-part1-kpk:sol
- This is a tenth unit of the book Mathematics 11 published by Khyber Pakhtunkhwa Textbook Board, Pesh
- Question 1, Exercise 1.1 @fsc-part1-kpk:sol:unit01
- unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB
- Question 2 & 3, Exercise 1.1 @fsc-part1-kpk:sol:unit01
- unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB
- Question 4, Exercise 1.1 @fsc-part1-kpk:sol:unit01
- unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB
- Question 5, Exercise 1.1 @fsc-part1-kpk:sol:unit01
- unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB
- Question 6, Exercise 1.1 @fsc-part1-kpk:sol:unit01
- unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB
- Question 7, Exercise 1.1 @fsc-part1-kpk:sol:unit01
- unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB