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- Unit 1: Complex Numbers (Solutions)
- Unit 10: Trigonometric Identities of Sum and Difference of Angles (Solutions)
- Question 1, Exercise 1.1
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- Question 1, Exercise 1.3
- Question 2, Exercise 1.3
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- Question 1, Review Exercise 1
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- Question 6, Exercise 10.2
- Question 7, Exercise 10.2
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- Question 1, Exercise 10.3
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- Question 4 & 5, Review Exercise 10
- Question 6 & 7, Review Exercise 10
- Question 8 & 9, Review Exercise 10
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 ... eal and imaginary parts of complex numbers. * Solve simultaneous linear equations with complex coef... polynomial as a product of linear factors. * Solve quadratic equation of the form $pz^2+ qz+ r = 0
- Unit 10: Trigonometric Identities of Sum and Difference of Angles (Solutions)
- etric Identities of Sum and Difference of Angles (Solutions) ===== This is a tenth unit of the book Ma... awar, Pakistan. On this page we have provided the solutions of the questions. After reading this unit ... rdion><panel type="default" title="Exercise 10.1 (Solutions)"> * [[fsc-part1-kpk:sol:unit10:ex10-1-p1|Question 1]] * [[fsc-part1-kpk:sol:unit10:ex10-1-
- Question 7, Exercise 10.2 @fsc-part1-kpk:sol:unit10
- ====== Question 7, Exercise 10.2 ====== Solutions of Question 7 of Exercise 10.2 of Unit 10: Trigonomet... \sin }^{4}}\theta =\dfrac{1}{\sec 2\theta }$. ====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(