Question

Which of the following is/are true?

(1) $(1+i{)}^n$ = $2^{\frac{n}{2}}$$(sin\frac{n\pi }{4}+icos\frac{n\pi }{4})$

If $(1+i{)}^n$ = ${}^n{C}_0$ + ${}^n{C}_1$i - ${}^n{C}_2$ - ${}^n{C}_3$i + ${}^n{C}_4$...............

(2) ${}^n{C}_0$ - ${}^n{C}_2$ + ${}^n{C}_4$ - ${}^n{C}_6$................ = $2^{\frac{n}{2}}$ $sin\frac{n\pi }{4}$

(3) ${}^n{C}_1$ - ${}^n{C}_3$ + ${}^n{C}_5$ - ${}^n{C}_7$....................= $2^{\frac{n}{2}}cos\frac{n\pi }{4}$

• A. 2 and 3
• B. 1, 2 and 3
• C. only 1
• D. None of these

Answer 1

The correct option is D

None of these

(1 + i) = $\sqrt{2}$ $e^{\frac{i\pi }{4}}$

⇒ $(1+1{)}^n$ = ${\left(2^{\frac{1}{2}}{e}^{\frac{i\pi }{4}}\right)}^n$ = $2^{\frac{n}{2}}{e}^{\frac{in\pi }{4}}$

= $2^{\frac{n}{2}}$$(cos\frac{n\pi }{4}+isin\frac{n\pi }{4})$

⇒ (1) is wrong

$(1+i{)}^n$ = ${}^n{C}_0$ + ${}^n{C}_1$i - ${}^n{C}_2$ - ${}^n{C}_3$i + ${}^n{C}_4$......................

$2^{\frac{n}{2}}$$cos\frac{n\pi }{4}$ + $i{2}^{\frac{n}{2}}$ sin $\frac{n\pi }{4}$ = (${}^n{C}_0$ - ${}^n{C}_2$ + ${}^n{C}_4$...............)

i(${}^n{C}_1$ - ${}^n{C}_3$ + ${}^n{C}_5$..................)

We can equate the real and imaginary parts separately.

⇒ ${}^n{C}_0$ - ${}^n{C}_2$ + ${}^n{C}_4$ ....................= $2^{\frac{n}{2}}cos\frac{n\pi }{4}$

${}^n{C}_1$ - ${}^n{C}_3$ + ${}^n{C}_5$ .......................=$2^{\frac{n}{2}}sin\frac{n\pi }{4}$

⇒ (2) and (3) are wrong