Class 11: Complex Numbers – Exercise 13.1 – ICSE / ISC / CBSE Mathematics Portal for K12 Students

Question 1: Evaluate the following:

i) $i^{457}$ ii) $i^{528}$ iii) $\frac{1}{i^{58}}$ iv) $i^{37}$ $+$ $\frac{1}{i^{67}}$ v) $\Big( i^{41}+ \frac{1}{i^{257}} \Big)^9$

vi) $( i^{77} + i^{70} + i^{87} + i^{414} )^3$ vii) $i^{30} + i^{40} + i^{60}$ viii) $i^{49} + i^{68} + i^{89} + i^{110}$

Answer:

i) $i^{457}$ $=$ $i^{4 \times 114 + 1}$ $=$ ${(i^4)}^{114} \times i$ $=$ $i$

ii) $i^{528}$ $=$ $i^{4 \times 132 }$ $=$ ${(i^4)}^{132}$ $= 1$

iii) $\frac{1}{i^{58}}$ $=$ $\frac{1}{i^{4 \times 14+2}}$ $=$ $\frac{1}{(i^4)^{14} \times i^2}$ $= -1$

iv) $i^{37} +$ $\frac{1}{i^{67}}$

$= i^{4 \times 9 + 1}+$ $\frac{1}{i^{4 \times 15 + 3}}$ $= (i^4)^9 \cdot i +$ $\frac{1}{(i^4)^{16}\cdot i^3}$ $= i -$ $\frac{1}{i}$ $=$ $\frac{i^2 - 1}{i}$ $=$ $\frac{-2}{i}$ $\times$ $\frac{i}{i}$ $=$ $\frac{-2i}{i^2}$ $= 2i$

v) $\Big( i^{41}+$ $\frac{1}{i^{257}}$ $\Big)^9$

$= \Big( i^{4 \times 10 + 1}+$ $\frac{1}{i^{4 \times 64 + 1}}$ $\Big)^9$ $= \Big( i+$ $\frac{1}{i}$ $\Big)^9$ $= \Big($ $\frac{i^2+1}{i}$ $\Big)^9$ $= \Big($ $\frac{-1+1}{i}$ $\Big)^9$ $= 0$

vi) $( i^{77} + i^{70} + i^{87} + i^{414} )^3$

$=$ $( i^{4 \times 19 + 1} + i^{4 \times 17 + 2} + i^{4 \times 21 + 3} + i^{4 \times 103 + 2} )^3$

$=$ $( i + i^{2} + i^{3} + i^{2} )^3$ $=$ $( - 2 + i + i^3)^3$ $= -8$

vii) $i^{30} + i^{40} + i^{60}$

$=$ $i^{4 \times 7 + 2} + i^{4 \times 10} + i^{4 \times 15 }$ $=$ $i^2 + 1 + 1$ $= - 1 + 1 +1 = 1$

viii) $i^{49} + i^{68} + i^{89} + i^{110}$

$= i^{4 \times 12 + 1} + i^{4 \times 17 } + i^{4 \times 22 + 1} + i^{4 \times 27 + 2}$

$= i + 1 + i + i^2 = i + 1 + i - 1 = 2i$

$\\$

Question 2: Show that $1 + i^{10}+ i^{20}+i^{30}$ is a real number

Answer:

$1 + i^{10}+ i^{20}+i^{30}$

$= 1+ i^{4 \times 2 + 2}+ i^{4 \times 5 }+i^{4 \times 7 + 2}$

$= 1 + i^2 + 1 + i^2$

$= 1 - 1 + 1 - 1 = 0$ . This is a real number.

$\\$

Question 3:

i) $i^{49} + i^{68} + i^{89} + i^{110}$ ii) $i^{30} + i^{80} + i^{120}$

iii) $i + i^{2} + i^{3} + i^{4}$ iv) $i^{5} + i^{10} + i^{15}$

v) $\frac{i^{592} + i^{590} + i^{588} + i^{586} + i^{584}}{i^{582} + i^{580} + i^{578} + i^{576} + i^{574}}$ vi) $1 + i^{2} + i^{4} + i^{6}+ \ldots + i^{20}$

vii) $(1+i)^6+(1-i)^3$

Answer:

i) $i^{49} + i^{68} + i^{89} + i^{110}$

$= i^{4 \times 12 + 1} + i^{4 \times 17} + i^{4 \times 22 + 1} + i^{4 \times 27 + 2}$

$= i + 1 + i + i^2 = 2i$

ii) $i^{30} + i^{80} + i^{120}$

$= i^{4 \times 7 + 2} + i^{4 \times 20} + i^{4 \times 30}$

$= i^2 + 1 + 1 = -1 + 2 = 1$

iii) $i + i^{2} + i^{3} + i^{4}$

$= i - 1 - i + 1 = 0$

iv) $i^{5} + i^{10} + i^{15}$

$= i^{4 \times 1 + 1} + i^{4 \times 2 + 2} + i^{4 \times 3 + 3}$

$= i + i^2 + i^3$

$= i - 1 - i = - 1$

v) $\frac{i^{592} + i^{590} + i^{588} + i^{586} + i^{584}}{i^{582} + i^{580} + i^{578} + i^{576} + i^{574}}$

$=$ $\frac{i^{4 \times 148} + i^{4 \times 147 + 2} + i^{4 \times 147} + i^{4 \times 146 + 2} + i^{4 \times 146}}{i^{4 \times 145 + 2} + i^{4 \times 145} + i^{4 \times 144 + 2} + i^{4 \times 144} + i^{4 \times 143 + 2}}$