Question 1, Review Exercise 1

i. ${{\left( \dfrac{2i}{1+i} \right)}^{2}}$

• (a) $i$
• (b) $2i$
• (c) $1-i$
• (d) $i+1$
  See Answer

  (B): $2i$

ii. Divide $\dfrac{5+2i}{4-3i}$

• (a) $-\dfrac{7}{25}+\dfrac{26}{25}i$
• (b) $\dfrac{5}{4}-\dfrac{2}{3}i$
• (c) $\dfrac{14}{25}+\dfrac{23}{25}i$
• (d) $\dfrac{26}{7}+\dfrac{23}{7}i$
  See Answer

  (C): $\dfrac{14}{25}+\dfrac{23}{25}i$

iii. ${{i}^{57}}+\frac{1}{{{i}^{25}}}$, when simplified has the value

• (a) $0$
• (b) $2i$
• (c) $-2i$
• (d) $2$
  See Answer

  (A): $0$

iv. 1+{i}^{2}+{i}^{4}+{i}^{6}+…+{i}^{2n}$ is

• (a) positive
• (b) negative
• (c) $0$
• (d) cannot be determined
  See Answer

  (D): cannot be determined

v. If $z=x+iy$ and $|\dfrac{z-5i}{z+5i}|=1$, then $z$ lies on

• (a) $X-axis$
• (b) $Y-axis$
• (c) line $y=5$
• (d) None of these
  See Answer

  (C): $y=5$

vi. The multiplicative inverse of $z=3-2i$, is

• (a) $\dfrac{1}{3}\left( 3+2i \right)$
• (b) $\dfrac{1}{13}\left( 3+2i \right)$
• (c) $\dfrac{1}{13}\left( 3-2i \right)$
• (d) $\dfrac{1}{4}\left( 3-2i \right)$
  See Answer

  (B): $\dfrac{1}{13}\left( 3+2i \right)$

vii. If $\left( x+iy \right)\left( 2-3i \right)=4+i$, then

• (a) $x=-\dfrac{14}{13},y=\dfrac{5}{13}$
• (b) $x=\dfrac{5}{13},y=\dfrac{14}{13}$
• (c) $x=\dfrac{14}{13},y=\dfrac{5}{13}$
• (d) $x=\dfrac{5}{13},y=-\dfrac{14}{13}$
  See Answer

  (B): $x=\dfrac{5}{13},y=\dfrac{14}{13}$