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- Question 1, Exercise 1.3
- hat \begin{align}z-4w&=3i …(i)\\ 2z+3w&=11-5i …(ii)\end{align} Multiply $2$ by (i), we get\\ \begin{align}2z-8w&=6i …(iii)\end{align} Subtract (iii) from (ii), we get\\ \[\begin{array}{cccc} 2z&-8w&=6i \\ \mathop+\limits_{-}2z&\mathop+\limits_{-
- Question 1, Exercise 1.1
- \\ &=i-i\\ &=0.\end{align} GOOD =====Question 1(ii)===== Simplify ${{\left( -i \right)}^{23}}$. GOOD... -1 \right)\\ &=i\end{align} GOOD =====Question 1(iii)===== Simplify ${{\left( -1 \right)}^{\frac{-23}
- Question 2 & 3, Exercise 1.1
- 6 \right)i\\ &=1+12i\end{align} =====Question 3(ii)===== Add the complex numbers $\dfrac{1}{2}-\dfra... \\ &=\dfrac{3}{4}-i\end{align} =====Question 3(iii)===== Add the complex numbers $\left( \sqrt{2},1
- Question 4, Exercise 1.1
- \left( a-2 \right)+bi\end{align} =====Question 4(ii)===== Subtract the second complex number from fir... } \right)i\\ &=-6+0i\end{align} =====Question 4(iii)===== Subtract the second complex number from fi
- Question 5, Exercise 1.1
- 6 \right)i\\ &=-117-i\end{align} =====Question 5(ii)===== Multiply the complex number $3i,2\left( 1-i... 1 \right)+6i\\ &=6+6i\end{align} =====Question 5(iii)===== Multiply the complex number $\sqrt{2}+\sqr
- Question 6, Exercise 1.1
- {1}{2}-\dfrac{1}{2}i\end{align} =====Question 6(ii)===== Perform the indicated division $\dfrac{1}{... }{65}-\dfrac{1}{65}i\end{align} =====Question 6(iii)===== Perform the indicated division $\dfrac{1}{
- Question 7, Exercise 1.1
- +25}\\ &=\sqrt{34}\end{align} =====Question 7(ii)===== If ${{z}_{1}}=1+2i$ and ${{z}_{2}}=2+3i$, e... 6+49}\\ &=\sqrt{65} \end{align} =====Question 7(iii)===== If ${{z}_{1}}=1+2i$ and ${{z}_{2}}=2+3i$,
- Question 8, Exercise 1.1
- }{13}-\dfrac{24i}{13}\end{align} =====Question 8(ii)===== Express the $\dfrac{2+\sqrt{-9}}{-5-\sqrt{-... }{41}-\dfrac{7i}{41}\end{align} =====Question 8(iii)===== Express the $\dfrac{{{\left( 1+i \right)}^
- Question 1, Exercise 1.2
- \begin{align} z_2+z_1&=(1-i)+(2+i)\\ &=3 \ldots (ii)\end{align} From (i) and (ii), we get required result. Now, we prove commutative property under mult
- Question 5, Exercise 1.2
- _1}+\overline{z_2}$$ as required. =====Question 5(ii)===== Let ${{z}_{1}}=2+3i$ and ${{z}_{2}}=2-3i$. ... }=\overline{z_1}\overline{z_2}.$$ =====Question 5(iii)===== Let ${{z}_{1}}=-a-3bi$ and ${{z}_{2}}=2a-3
- Question 7, Exercise 1.2
- Imaginary part $=\dfrac{19}{29}$ =====Question 7(ii)===== Separate into real and imaginary parts $\df... Imaginary part $=-\dfrac{1}{2}$ =====Question 7(iii)===== Separate into real and imaginary parts $\d
- Question 8, Exercise 1.2
- e{Re}\left( z \right)\end{align} =====Question 8(ii)===== Show that $z-\overline{z}=2i\operatorname{... i\operatorname{Im}(z)\end{align} =====Question 8(iii)===== Show that $z\overline{z}={{\left[ \operat
- Question 2, Exercise 1.3
- (z+2)(z-1+3i)(z-1-3i)\end{align} =====Question 2(ii)===== Factorize the polynomial $P(z)$ into linear... }z-\sqrt{7}i \right)\end{align} =====Question 2(iii)===== Factorize the polynomial $P\left( z \right
- Question 5, Exercise 1.3
- {1}{2}\pm\dfrac{\sqrt{11}}{2}i$. =====Question 5(ii)===== Find the solutions of the equation ${{z}^{2... ns are $\dfrac{1\pm\sqrt{5}}{2}$. =====Question 5(iii)===== Find the solutions of the equation ${{z}^{
- Question 6, Exercise 1.3
- 1}{2}\pm \dfrac{\sqrt{3}}{2}i$$ =====Question 6(ii)===== Find the solutions of the equation ${{z}^{3... tions of the required equations. =====Question 6(iii)===== Find the solutions of the equation ${{\lef