Iota

Q. If matrix $A$ is non-singular and satisfies $A^2-A+I=O,$ then the inverse of $A$ is equal to

1. $A^{-2}$
2. $A+I$
3. $I-A$
4. $A-I$

Q. If x = 3 + i, then $x^3+3x^2-8x+15$ =
[UPSEAT 2003]

1. 6
2. 10
3. -18
4. -15

Q. If n is a positive integer, then ${\left(\frac{1+i}{1-i}\right)}^{4n+1}$ =

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

Q.

Write the value of $\sqrt{-25}\times \sqrt{-9}.$

Q.

If ${\left(1-i\right)}^n=2^n$, then $n$ is equal to

1. $1$

2. $0$

3. $-1$

4. None of these

Q.

The value of $i^{1+3+5+......+(2n+1)}$ is

1. 1 if n is even, - 1 if n is odd

2. 1 if n is odd, - 1 if n is even

3. i if n is even, - 1 if n is odd

4. i if n is even, - i if n is odd

Q. If ${\left(\frac{1+i}{1-i}\right)}^m=1,$ then the least integral value of m is
[IIT 1982; MNR 1984; UPSEAT 2001; MP PET 2002]

1. 2
2. 4
3. None of these
4. 8

Q. If $z=cos\theta +isin\theta$, then

1. 
2. 
3. 
4. 

Q.

Evaluate ${48}^2$ using suitable identity

Q.

Which of the following statements is true and which is false? justify each false statement with an example.

For any integer a, $a÷0=0÷a=0$

1. True
2. False

Q. If ${\left(\frac{1+i}{1-i}\right)}^m=1,$ then the least positive integral value of $m$ is

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

Q. Find the smallest positive integer n, for which ${\left(\frac{1+i}{1-i}\right)}^n=1.$

Q. If $x=3+i$ then $x^3-3x^2-8x+15=$

1. 6
2. 10
3. -15
4. -18

Q. If $I_n=\int x^n\sqrt{a^2-x^2}dx$ and $(n+k)\cdot I_n=-x^{n-1}(a^2-x^2{)}^p+(n-1)a^2\cdot I_{n-2},$ then $3k-2p=$
(where $m,n\in N;m,n\ge 2$)

Q. If $(1-i{)}^n=2^n,thenn=$
[RPET 1990]

1. 0

2. None of these

3. 1

4. -1

Q.

If ${\left(\frac{1+i}{1-i}\right)}^m$ = 1 then the least integral value of m is

1. 2

2. 4

3. 8

4. None of these

Q. Find the value of: $i^2+i^4+i^6$ +..... upto $(2n+1)$ terms.

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

Q. Let $n$ be a positive integer. Then ${\left(i\right)}^{4n+1}+{(-i)}^{4n+5}=$

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

Q. Multiply $(2\sqrt{-3}+3\sqrt{-2})by(4\sqrt{-3}-5\sqrt{-2})$

Q. The smallest positive integral value of n for which ${\left[\frac{1-i}{1+i}\right]}^n$ is purely imaginary with positive imaginary part is

1. 1
2. 5
3. none of thees
4. 3

Q. Find the smallest positive integer value of $n$ for which $\frac{(1+i{)}^n}{(1-i{)}^{n-2}}$ is a real number.

Q. $\sqrt{-8-6i}$ =

Q. The smallest positive integral value of $n$ for which $(1+i{)}^{2n}=(1-i{)}^{2n}$ is

1. $4$
2. $8$
3. $2$
4. $12$

Q.

Differentiate given problems w.r.t.x.

$(3x^2-9x+5{)}^9$.

Q.

If sin $\theta$ + cos $\theta$ = a, cos $\theta$ - sin $\theta$ = b, then sin $\theta$ (sin $\theta$ - cos $\theta$ ) + $si{n}^2\theta (si{n}^2\theta -co{s}^2\theta )+si{n}^3\theta (si{n}^3\theta -co{s}^3\theta )$+ . . .. . is equal to

1. 

2. 

3. 

4. 

Q. ${\int }_0^{\infty }\left[e^{|-x|}\right]dx$ is equal to (where [.] denotes the greatest integer function)

Q. Find the value of $x^3+7x^2-x+16$, when $x=1+2i$.

Q. Find the value:

Q. Show that $1+i^{10}+i^{20}+i^{30}$ is a real number

Q. Simplify $\sqrt{56-2405}$