Properties of Iota

Q. The value of ${}^{12}{C}_2{+}^{13}{C}_3{+}^{14}{C}_4+........{+}^{999}{C}_{989}$ is

1. 
2. 
3. 
4. 

Q. Evaluate:

Q. Reduce to the standard form.

Q. $1+i^2+i^4+i^6+.....+i^{2n}$ is [EAMCET 1980]

1. Positive

2. Negative
3. Zero
4. Cannot be determined

Q.

What is the value of $\frac{1+i}{1-i}$ (give answer in 'i')

__

Q. Express the given complex number in the form a + ib :

Q.

Let $f\left(x\right)=x{(-1)}^{\left[\frac{1}{x}\right]}.x\ne 0$, where [x] denotes the greatest integer less than or equal to x. then $\underset{x\to 0}{lim}f\left(x\right)$

1. is equal to 2

2. Doesn't exist

3. is equal to 0

4. is equal to -1

Q.

If $S_1$ = ${}^n{C}_0$ + ${}^n{C}_1$ + ${}^n{C}_2$..................${}^n{C}_n$ and $S_2$ = ${}^n{C}_0$ - ${}^n{C}_1$ + ${}^n{C}_2$....................${}^n{C}_n$

Find ${}^n{C}_1$ + ${}^n{C}_3$ + ${}^n{C}_5$..............

1. $S_1+S_2$

2. $\frac{S_1-S_2}{2}$

3. $\frac{S_2-S_1}{2}$

4. $S_1+S_2$

Q. 2. 919

Q.

${\left(\frac{1+i}{1-i}\right)}^2+{\left(\frac{1-i}{1+i}\right)}^2$ is equal to

1. 2i

2. -2i

3. -2

4. 2

Q. The value of $\frac{i^{592}+i^{590}+i^{588}+i^{586}+i^{584}}{i^{582}+i^{580}+i^{578}+i^{576}+i^{574}}-1=$

1. -1
2. -2
3. -3
4. -4

Q. 8. (1-1)4

Q.

The eccentricity of the Ellipse $9x^2+16y^2$ = 576 is

1. 

2. 

3. 

4. 

Q. The value of $\frac{i^{592}+i^{590}+i^{588}+i^{586}+i^{584}}{i^{582}+i^{580}+i^{578}+i^{576}+i^{574}}-1=$

1. -2
2. -1
3. -4
4. -3

Q. $1+i^2+i^4+i^6+.....+i^{2n}$ is [EAMCET 1980]

1. Positive

2. Negative
3. Zero
4. Cannot be determined

Q.

Find \((3\sqrt{3} + 5)^n\) + \((3\sqrt{3} - 5)^n\)

Q.

For natural number n, $2^n$(n-1) ! < $n^n$, if

1. n < 2

2. n > 2

3. 2

4. Never

Q. Evaluate $(96{)}^3$

Q.

$\text{If}{\left(\frac{x}{a}\right)}^n+{\left(\frac{y}{b}\right)}^n=2then\frac{dy}{dx}at(a,b)is$

1. a/b

2. -a/b

3. b/a

4. -b/a

Q. ${\left(\frac{1+i}{1-i}\right)}^2+{\left(\frac{1-i}{1+i}\right)}^2$ is equal to

1. 2i
2. -2i
3. -2
4. 2

Q. If $z_1$ is a complex number other than $-1$ such that $|z_1|=1$ and $z_2=\frac{z_1-1}{z_1+1}$ then show that $z_2$ is purely imaginary.