Iota

Q. If $(a+ib)=\frac{(x+i{)}^2}{(2x^2+1)}$ then prove that $(a^2+b^2)=\frac{(x^2+1{)}^2}{(2x^2+1{)}^2}$.

Q.

If ${\left[\frac{1+i}{1-i}\right]}^m=1$, then find the least positive integral value of m.

Q.

If $(1-i{)}^n=2^n$ , then n is

1. 1

2. 0

3. -1

4. None of these

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

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

Q. If $f\left(x\right)=(a-x^n{)}^{\frac{1}{n}}$ where a > 0 and n is a positive integer, then $f\left[f\right(x\left)\right]$

1. x
2. -x
3. f(x)
4. -f(x)

Q.

$f\left(x\right)=\sqrt{x+1}\&g\left(x\right)=2x-3$

Find the domain of $\frac{f}{g}$

1. $(-1,\infty )$

2. $R-\left\{\frac{3}{2}\right\}$

3. $(-1,\infty )-\left\{\frac{3}{2}\right\}$

4. $[-1,\infty )-\left\{\frac{3}{2}\right\}$

Q.

If n is a natural number then ${\left(\frac{n+1}{2}\right)}^n$ ≥ n ! is true

when

1. n > 1

2. 1

3. n > 2

4. 

Q.

If $a+ib=\frac{(x+i{)}^2}{(2x^2+1)}$, prove that $a^2+b^2=\frac{(x^2+1{)}^2}{(2x^2+1{)}^2}$

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. 8
4. None of these

Q.

Find the simplified product $b-\left(\frac{5}{2b}\times b^2\right)+\frac{3b}{b}-5$.

Q. $\sqrt{-2}\sqrt{-3}=$
[Roorkee 1978]

1. $\sqrt{6}$
2. None of these
3. $-\sqrt{6}$
4. $i\sqrt{6}$

Q.

1 . The sum of the series 1 + 3x + 6$x^2$ + 10 $x^3$ + ....... ∞ will be

1. 

2. 

3. 

4. 

Q. The least integral value of 'a' such that the function $x^2+ax+1$ is strictly increasing on [1, 2] is

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

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

1. 1

2. 0

3. None of these
4. -1

Q. If the inequality $(m-2)x^2+8x+m+4>0$ is satisfied for all $xϵR$, then least integral m is

1. $4$
2. $5$
3. $6$
4. None of these

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

Q. The least integral value of $a$ such that the function $x^2+ax+1$ is strictly increasing on $[1,2]$ is

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

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

1. 1

2. 0

3. -1
4. None of these

Q.

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

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

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

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

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

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

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

Q. Find the modulus of : $\frac{1+i}{1-i}-\frac{1-i}{1+i}$