Properties of Iota

Q. Characteristic equation for the matrix $A=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]$ is

1. $A^2+5A-2I=O$
2. $A^2-5A+2I=O$
3. $A^2-5A-2I=O$
4. $A^2+5A+2I=O$

Q. If $\alpha ,\beta$ are roots of the equation $x^2+5\left(\sqrt{2}\right)x+10=0,$ $\alpha >\beta$ and $P_n={\alpha }^n-{\beta }^n$ for each positive integer $n$, then the value of $\left(\frac{P_{17}{P}_{20}+5\sqrt{2}{P}_{17}{P}_{19}}{P_{18}{P}_{19}+5\sqrt{2}{P}_{18}^2}\right)$ is equal to

Q.

If $z=\left(\frac{1+i}{1-i}\right)$, then $z^4$ equals

1. 1

2. none of these

3. -1

4. 0

Q. Let $A=\left[\begin{array}{ccc}1& 0& 0\\ 2& 1& 0\\ 3& 2& 1\end{array}\right]$. If $u_1$ and $u_2$ are column matrices such that $A{u}_1=\left[\begin{array}{c}1\\ 0\\ 0\end{array}\right]$ and $A{u}_2=\left[\begin{array}{c}0\\ 1\\ 0\end{array}\right]$, then $u_1+u_2$ is equal to :

1. $\left[\begin{array}{c}-1\\ 1\\ 0\end{array}\right]$
2. $\left[\begin{array}{c}-1\\ 1\\ -1\end{array}\right]$
3. $\left[\begin{array}{c}-1\\ -1\\ 0\end{array}\right]$
4. $\left[\begin{array}{c}1\\ -1\\ -1\end{array}\right]$

Q.

How do you evaluate ${}^{5}C_2+{}^{5}C_1$?

Q. If n is a positive integer, then which of the following relations is false

1. $i^{4n}=1$
2. $i^{4n-1}=i$
3. $i^{4n+1}=i$
4. $i^{4n+2}=-1$

Q. The value of $\sum _{n=0}^{100}{i}^{n!}$ equals (where $i=\sqrt{-1})$

1. $-1$
2. $i$
3. $2i+95$
4. $97+i$

Q.

$\text{The value of}\frac{(i^5+i^6+i^7+i^8+i^9)}{(1+i)}$

1. $\frac{1}{2}(1+i)$

2. $\frac{1}{2}(1-i)$

3. 1

4. $\frac{1}{2}$

Q. The value of the expression $1.(2-w)(2-w^2)+2.(3-w)(3-w^2)+..........+(n-1)(n-w)(n-w^2),$
where ω is an imaginary cube root of unity , 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. -1
2. -2
3. -3
4. -4

Q.

Add : $a+b-3,b-a+3,a-b+3$

Q. If $n$ is even positive integer, then the condition that the greatest term in the expansion of $(1+x{)}^n$ may also have the greatest coefficient is

1. $\frac{n}{n+2}<x<\frac{n+2}{n}$
2. $\frac{n}{n+1}<x<\frac{n+1}{n}$
3. $\frac{n+1}{n+2}<x<\frac{n+2}{n+1}$
4. $\frac{n+2}{n+3}<x<\frac{n+3}{n+2}$

Q.

$\text{The value of}(1+i)(1+i^2)(1+i^3)(1+i^4)\text{is}$

1. 2

2. 0

3. 1

4. i

Q. If $\alpha$ and $\beta$ are the roots of the equation $x^2-2x+2=0$, then least value of $n$ for which ${\left(\frac{\alpha }{\beta }\right)}^n=1$ is:

1. $5$
2. $4$
3. $3$
4. $2$

Q.

If $\frac{1}{{}^{4}C_n}=\frac{1}{{}^{5}C_n}+\frac{1}{{}^{6}C_n}$ then $n$ is equal to?

1. $3$

2. $2$

3. $1$

4. $0$

5. $4$

Q. Which among the following is (are) complex number(s) ?

1. $\pi$
2. $0$
3. $3+2i$
4. $\sqrt{-7}$

Q. How many ordered pairs of positive integers $(x,y)$ satisfy the equation $x\sqrt{y}+y\sqrt{x}+\sqrt{2006xy}-\sqrt{2006x}-\sqrt{2006y}-2006=0$?
(correct answer + 3, wrong answer 0)

Q. Find the sum of all positive integers $x$ such that $\frac{x^3-x+120}{(x-1)(x+1)}$ is an integer.
(correct answer + 2, wrong answer 0)

Q.

If $i^2=-1$, then the sum $i+i^2+i^3+......$ upto 1000 terms is equal to

1. 1

2. -1

3. i

4. 0

Q.

If n is any positive integer, write the value of $\frac{i^{4n+1}-i^{4n-1}}{2}$

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 is

1. -3

2. -4

3. -2

4. -1

Q. If $\int \frac{x^3}{x^4+3x^2+2}dx=\ln \left|\frac{x^2+2}{\sqrt{f\left(x\right)}}\right|+C,$ where $C$ is constant of integration, then the value of $f\left(7\right)$ is

1. $5\sqrt{2}$
2. $50$
3. $29$
4. $15$

Q. If $\alpha \text{and}\beta$ are the roots of the equation $x^2+kx+9=0$ such that $\alpha +2\beta =2\alpha {\beta }^2$ then possible value(s) of [k] is/are (where [•] denotes the greatest integer function)

1. -13
2. -12
3. 12
4. 13

Q. Let $A_n=\left(\frac{3}{4}\right)-{\left(\frac{3}{4}\right)}^2+{\left(\frac{3}{4}\right)}^3-...+(-1{)}^n{\left(\frac{3}{4}\right)}^n$ and $B_n=1-A_n$. Then, the least odd natural number $p$, so that $B_n>A_n$, for all $n\ge p$, is :

1. $9$
2. $11$
3. $7$
4. $5$

Q. The value of determinant $\Delta =\left|\begin{array}{ccc}x& -x^2& -x^3\\ -x^2& x^2& -x\\ -x^3& -x& x^3\end{array}\right|,(\Delta \ne 0,x\ne 0)$ is equal to

1. $-x^3[1+x+x^2+x^3+x^4+x^5]$
2. $-x^3[1+x^2+x^3+x^4+x^5]$
3. $-x^3[1+x^3+x^4+x^5]$
4. $-x^3[1+x+x^2+x^3+x^4]$

Q. Out of 6 books, in how many ways can a set of one or more books be chosen

1. 64
2. 62
3. 65
4. 63

Q. If n is a positive integer, then which of the following relations is false

1. $i^{4n+1}=i$
2. $i^{4n}=1$
3. $i^{4n-1}=i$
4. $i^{4n+2}=-1$

Q. If $n$ is an odd integer, then $(1+i{)}^{6n}+(1-i{)}^{6n}=$

1. $2i$
2. $-2i$
3. $0$
4. $-2$

Q.

Give examples of polynomials $P\left(X\right),G\left(X\right),Q\left(X\right),andR\left(X\right)$, which satisfy the Division Algorithm and $DegP\left(X\right)=DegQ\left(X\right)$.

Q. Solve :-$\frac{i^{592}+i^{590}+i^{588}+i^{586}+i^{584}}{i^{582}+i^{580}+i^{578}+i^{576}+i^{574}}$