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- Unit 1: Complex Numbers (Solutions)
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- Question 1, Exercise 1.1
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Fulltext results:
- Unit 1: Complex Numbers (Solutions)
- ===== Unit 1: Complex Numbers (Solutions) ===== This is a first unit of the book Mathematics 11 publis... awar, Pakistan. On this page we have provided the solutions of the questions. After reading this unit ... ib$. * Define $|z| = \sqrt{a^2+b^2}$ as the absolute value or modulus of a complex number $z=a+ib$ ... eal and imaginary parts of complex numbers. * Solve simultaneous linear equations with complex coef
- Unit 10: Trigonometric Identities of Sum and Difference of Angles (Solutions)
- etric Identities of Sum and Difference of Angles (Solutions) ===== This is a tenth unit of the book Ma... awar, Pakistan. On this page we have provided the solutions of the questions. After reading this unit ... rdion><panel type="default" title="Exercise 10.1 (Solutions)"> * [[fsc-part1-kpk:sol:unit10:ex10-1-p1|Question 1]] * [[fsc-part1-kpk:sol:unit10:ex10-1-
- Question 7, Exercise 10.2 @fsc-part1-kpk:sol:unit10
- ====== Question 7, Exercise 10.2 ====== Solutions of Question 7 of Exercise 10.2 of Unit 10: Trigonomet... \sin }^{4}}\theta =\dfrac{1}{\sec 2\theta }$. ====Solution==== \begin{align}L.H.S&={{\cos }^{4}}\theta ... }\dfrac{\theta }{2}=\dfrac{2}{\sin \theta }$. ====Solution==== \begin{align}L.H.S&=\tan \dfrac{\theta }... eta }{1-\cos 2\theta }={{\cot }^{2}}\theta $. ====Solution==== \begin{align}L.H.S&=\dfrac{1+\cos 2\thet
- Question 5, Exercise 1.3 @fsc-part1-kpk:sol:unit01
- ====== Question 5, Exercise 1.3 ====== Solutions of Question 5 of Exercise 1.3 of Unit 01: Complex Numb... awar, Pakistan. =====Question 5(i)===== Find the solutions of the equation ${{z}^{2}}+z+3=0$\\ ====Solution==== ${{z}^{2}}+z+3=0$\\ According to the quadr... 2}i\end{align} =====Question 5(ii)===== Find the solutions of the equation ${{z}^{2}}-1=z$.\\ ====Solu
- Question 6, Exercise 1.3 @fsc-part1-kpk:sol:unit01
- ====== Question 6, Exercise 1.3 ====== Solutions of Question 6 of Exercise 1.3 of Unit 01: Complex Numb... awar, Pakistan. =====Question 6(i)===== Find the solutions of the equation ${{z}^{4}}+{{z}^{2}}+1=0$\\ ====Solution==== \begin{align}{{z}^{4}}+{{z}^{2}}+1&=0\\... }}}\end{align} =====Question 6(ii)===== Find the solutions of the equation ${{z}^{3}}=-8$\\ ====Soluti
- Question 7, Exercise 1.2 @fsc-part1-kpk:sol:unit01
- ====== Question 7, Exercise 1.2 ====== Solutions of Question 7 of Exercise 1.2 of Unit 01: Complex Numb... eal and imaginary parts $\dfrac{2+3i}{5-2i}$. ====Solution==== \begin{align}&\dfrac{2+3i}{5-2i} \\ =&\... $\dfrac{{{\left( 1+2i \right)}^{2}}}{1-3i}$. ====Solution==== \begin{align}&\dfrac{(1+2i)^2}{1-3i}\\ ... ts $\dfrac{1-i}{{{\left( 1+i \right)}^{2}}}$. ====Solution==== \begin{align}&\dfrac{1-i}{{{\left( 1+i
- Question 2, Exercise 10.1 @fsc-part1-kpk:sol:unit10
- ====== Question 2, Exercise 10.1 ====== Solutions of Question 2 of Exercise 10.1 of Unit 10: Trigonome... === Evaluate exactly: $\sin \dfrac{\pi }{12}$ ===Solution=== We rewrite $\dfrac{\pi }{12}$ as $\dfrac{... ii)=== Evaluate exactly:$\tan {{75}^{\circ }}$ ==Solution== We rewrite ${{75}^{\circ }}$ as ${{45}^{\c... i)=== Evaluate exactly:$\tan {{105}^{\circ }}$ ==Solution== We rewrite ${{105}^{{}^\circ }}$ as ${{60}
- Question 5, Exercise 10.1 @fsc-part1-kpk:sol:unit10
- ====== Question 5, Exercise 10.1 ====== Solutions of Question 5 of Exercise 10.1 of Unit 10: Trigonome... find: $\sin \left( \alpha +\beta \right)$. ====Solution==== Given: $\tan\alpha =\dfrac{3}{4}$. As ... 8}{65}\end{align} $$\implies \bbox[4px,border:2px solid black]{\sin(\alpha +\beta)=\dfrac{33}{65}.}$$ ... find: $\cos \left( \alpha +\beta \right)$. ====Solution==== Given: $\tan\alpha =\dfrac{3}{4}$. As
- Question 2, Exercise 10.2 @fsc-part1-kpk:sol:unit10
- ====== Question 2, Exercise 10.2 ====== Solutions of Question 2 of Exercise 10.2 of Unit 10: Trigonomet... e second quadrant, then find $\sin 2\theta $. ====Solution==== Given: $\sin \theta =\dfrac{5}{13}$. U... right)\end{align} $$\implies \bbox[4px,border:2px solid black]{\sin 2\theta=-\dfrac{120}{169}.}$$ ===... e second quadrant, then find $\cos 2\theta $. ====Solution==== Given: $\sin \theta =\dfrac{5}{13}$. U
- Question 3, Exercise 10.2 @fsc-part1-kpk:sol:unit10
- ====== Question 3, Exercise 10.2 ====== Solutions of Question 3 of Exercise 10.2 of Unit 10: Trigonomet... the second quadrant, then find $\sin2\theta$. ====Solution==== Given: $\sin \theta =\dfrac{4}{5}$ Term... e reference triangle as shown: {{ :fsc-part1-kpk:sol:unit10:fsc-part1-kpk-ex10-2-q3.png?nolink |Refere... ight)\end{align} $$\implies \bbox[4px,border:2px solid black]{\sin 2\theta=-\dfrac{24}{25}.}$$ =====Q
- Question 4 and 5, Exercise 10.2 @fsc-part1-kpk:sol:unit10
- ====== Question 4 and 5, Exercise 10.2 ====== Solutions of Question 4 and 5 of Exercise 10.2 of Unit 10... uadrant, then find $\sin \dfrac{\theta }{2}$. ====Solution==== Given: $\cos \theta =-\dfrac{3}{7}$ and ... }{2}}\end{align} $$\implies \bbox[4px,border:2px solid black]{\sin\dfrac{\theta}{2}=-\sqrt{\dfrac{5}{7... to evaluate exactly $\sin \dfrac{2\pi }{3}$. ====Solution==== Given: $\sin \dfrac{2\pi }{3}$.\\ By usi
- Question 6, Exercise 10.2 @fsc-part1-kpk:sol:unit10
- ====== Question 6, Exercise 10.2 ====== Solutions of Question 6 of Exercise 10.2 of Unit 10: Trigonomet... s to evaluate exactly $\cos {{15}^{\circ }}$. ====Solution==== Because ${{15}^{\circ }}=\dfrac{{{30}^{\... to evaluate exactly $\tan {{67.5}^{\circ }}$. ====Solution==== Because ${{67.5}^{\circ }}=\dfrac{{{135}... to evaluate exactly $sin{{112.5}^{\circ }}$. ====Solution==== Because ${{112.5}^{\circ }}=\dfrac{{{225
- Question 8, Exercise 1.2 @fsc-part1-kpk:sol:unit01
- ====== Question 8, Exercise 1.2 ====== Solutions of Question 8 of Exercise 1.2 of Unit 01: Complex Numb... rline{z}=2\operatorname{Re}\left( z \right)$. ====Solution==== Assume $z=a+ib$, then $\overline{z}=a-i... line{z}=2i\operatorname{Im}\left( z \right)$. ====Solution==== Assume that $z=a+ib$, then $\overline{z... ratorname{Im}\left( z \right) \right]}^{2}}$. ====Solution==== Suppose $z=a+ib$, then $\overline{z}=a-
- Question 1, Exercise 1.3 @fsc-part1-kpk:sol:unit01
- ====== Question 1, Exercise 1.3 ====== Solutions of Question 1 of Exercise 1.3 of Unit 01: Complex Numb... TBB) Peshawar, Pakistan. =====Question 1(i)===== Solve the simultaneous linear equation with complex c... in{align}&z-4w=3i\\ &2z+3w=11-5i\end{align} ====Solution==== Given that \begin{align}z-4w&=3i …(i)... $$z=4-i, \quad w=1-i.$$ =====Question 1(ii)===== Solve the simultaneous linear equation with complex c
- Question 2 & 3, Review Exercise 1 @fsc-part1-kpk:sol:unit01
- ====== Question 2 & 3, Review Exercise 1 ====== Solutions of Question 2 & 3 of Review Exercise 1 of Uni... i}^{n+2}}+{{i}^{n+3}}=0$, $\forall n\in N$ \\ ====Solution==== \begin{align}{{i}^{n}}+{{i}^{n+1}}+{{i}... left( 5+7i \right)$ in the form of $x+iy$.\\ ====Solution==== $\left( 1+3i \right)+\left( 5+7i \right... left( 5+7i \right)$ in the form of $x+iy$.\\ ====Solution==== \begin{align}\left( 1+3i \right)-\left(