Distance Formula Using Complex Numbers

Q. The sum of the series $\frac{1}{x+1}+\frac{2}{x^2+1}+\frac{2^2}{x^4+1}+\cdots +\frac{2^{100}}{x^{2^{100}}+1}$ when $x=2$ is :

1. $1-\frac{2^{101}}{4^{101}-1}$
2. $1-\frac{2^{100}}{4^{100}-1}$
3. $1+\frac{2^{100}}{4^{101}-1}$
4. $1+\frac{2^{101}}{4^{101}-1}$

Q. The sum of $10$ terms of the series $\frac{3}{1^2\times 2^2}+\frac{5}{2^2\times 3^2}+\frac{7}{3^2\times 4^2}+\cdots$ is

1. $\frac{143}{144}$
2. $\frac{99}{100}$
3. $\frac{120}{121}$
4. $1$

Q. If z is a non - real complex number, then the minimum value of $\frac{Im\left(z^5\right)}{{\left(Im\right(z\left)\right)}^5}$ is

1. -4
2. -5
3. -1
4. -2

Q. Let x > 0, y > 0, z > 0 are respectively the $2^{nd},3^{rd},4^{th},$ terms of a G.P and $\Delta =\left|\begin{array}{ccc}{x}^k& x^{k+1}& x^{k+2}\\ y^k& y^{k+1}& y^{k+2}\\ z^k& z^{k+1}& z^{k+2}\end{array}\right|=(r-1{)}^2(1-\frac{1}{r^2})$ (where r is the common ratio) then

1. k=0
2. None of these
3. k=-1
4. k=1

Q. The complex number $z$ satisfying the equation $|z-i|=|z+1|=1$ is

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

Q.

Let the complex numbers z₁, z₂ and z₃ be the vertices of an equilateral triangle. Let z₀ be the circumcentre of the triangle, then $z_1^2+z_2^2+z_3^2$=

1. $z_0^2$

2. $-z_0^2$

3. $3z_0^2$

4. $-3z_0^2$

Q. Let $z_1,z_2$ be two complex numbers represented by points on the circle $|z|=3$ and $|z|=4$ respectively, then

1. $max|3z_1+2z_2|=17$
2. $min|z_1-z_2|=1$
3. $|z_2+\frac{1}{z_1}|\le \frac{13}{3}$
4. $max|3z_1+2z_2|=16$

Q. If A, B, C are represented by 3 + 4i, 5 - 2i , -1 + 16i, then A, B, C are
[RPET 1986]

1. Collinear

2. Vertices of right angled triangle

3. Vertices of equilateral triangle

4. Vertices of isosceles triangle

Q. If $(3-5i{)}^3$ can be expressed in the form $a+ib$ , then the ordered pair $(a,b)$ is

1. $(-198,10)$
2. $(-198,-10)$
3. $(198,-10)$
4. $(198,10)$

Q. If $z=3+2i$, then value of $3z^3-13z^2+9z+65$ is

1. $0$
2. $1$
3. $2$
4. $3$

Q. Let $S$ be the sum of the first $9$ terms of the series: $\{x+ka\}+\{x^2+(k+2\left)a\right\}+\{x^3+(k+4\left)a\right\}+\{x^4+(k+6\left)a\right\}+...$ where $a\ne 0$ and $a\ne 1$. If $S=\frac{x^{10}-x+45a(x-1)}{x-1}$, then $k$ is equal to:

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

Q. Let $f:R\to R$ be defined as $f\left(x\right)=\{\begin{array}{ccc}{x}^5\sin \left(\frac{1}{x}\right)& +5x^2,& x<0\\ 0& & x=0\\ x^5\cos \left(\frac{1}{x}\right)& +\lambda x^2,& x>0\end{array}$ The value of $\lambda$ for which $f^{\prime \prime }\left(0\right)$ exists, is

Q. The smallest value of $\left(\frac{-8p}{7}\right)$ for which $|x^2-5x+7-p|=6+|x^2-5x+1-p|$ for all $x\in [-1,3]$ is

Q. Length of the line segment joining the points -1 -i and 2 + 3i is

1. – 5

2. 15

3. 5

4. 25

Q. If $a$ and $b$ are positive integers such that $N=(a+ib{)}^3-107i$ is a positive integer then $\frac{N}{6}$ is

Q. If the sum of the series $20+19\frac{3}{5}+19\frac{1}{5}+18\frac{4}{5}+\cdots$ upto $n^{th}$ term is $488$ and the $n^{th}$ term is negative, then:

1. $n=60$
2. $n=41$
3. $n^{th}$ term is $-4$
4. $n^{th}$ term is $-4\frac{2}{5}$

Q.

The number of complex numbers z satisfying |z - 2|= 2 and z(1 - i) + $\overline{z}$(1 + i) = 4 is

1. 0

2. 2

3. 3

4. 4

Q. If $b+ic=(1+a)z$ and $a^2+b^2+c^2=1,$ where $a,b,c\in R,$ then $\frac{1+iz}{1-iz}=\frac{a+ib}{k}$ where $k$ is equal to

1. $1+a$
2. $1+b$
3. $1+c$
4. $2+b$

Q. Sum upto $100$ term of the series $i+2i^2+3i^3+\cdots$ is

1. $50(1-i)$
2. $25(1+i)$
3. $100(1-i)$
4. $25i$

Q. Let $z=x+iy$ be a complex number. The equation $arg\left(\frac{z+1}{z}\right)=\frac{\pi }{4}$ represents

1. $x^2+x+y+y^2=0$
2. $x^2-x+y+y^2=0$
3. $x^2+x-y+y^2=0$
4. $x^2+x+y-y^2=0$

Q. Let the complex numbers $z_1,z_2$ and $z_3$ be the vertices of an equilateral triangle . Let $z_0$ be the circumcentre of the triangle , then $z_1^2+z_2^2+z_3^2=$

1. 
2. 
3. 
4. 

Q. If $a^x=b^y=c^z$ and $b^2=ac$, then $\frac{1}{x}+\frac{1}{z}$ is

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

Q. If the expression $\frac{1-i\sin \alpha }{1+2i\sin \alpha }$ is purely real, which of following is/are correct?

1. $Re\left(z\right)=-\frac{1}{5}$
2. $Re\left(z\right)=1$
3. $\alpha =n\pi ,n\in Z$
4. $\alpha =\frac{n\pi }{2},n\in Z$

Q. If $\frac{x+y-8}{2}=\frac{x+2y-14}{3}=\frac{3x-y}{4}$, then the value of $x^2+y^2$ is equal to:

1. $0$
2. $5$
3. $12$
4. $40$

Q. $ifz,=cos\frac{r\alpha }{n^2}+isin\frac{r\alpha }{n^2},wherer=1,2,3,....n,then\stackrel{lim}{n\to \infty }{z}_1{z}_2{z}_3.....z_{n}isequalto$

1. 
2. 
3. 
4. 

Q. The number of different possible values for the sum $x+y+z$, where $x,y,z$ are real numbers such that $x^4+4y^4+16z^4+64=32xyz$ is

1. 1
2. 2
3. 4
4. 8

Q. Let $A$ be the sum of the first $20$ terms and $B$ be the sum of the first $40$ terms of the series
$1^2+2\cdot 2^2+3^2+2\cdot 4^2+5^2+2\cdot 6^2+\dots .$
If $B-2A=100\lambda$, then $\lambda$ is equal to

1. $496$
2. $232$
3. $248$
4. $464$

Q. For all complex numbers $z_1,z_2$ satisfying $|z_1|=12and|z_2-3-4i|$ = 5, the minimum value of $|z_1-z_2|$ is
[IIT Screening 2002]

1. 0

2. 7

3. 2

4. 17

Q. Vertices of a triangle are given by $\hat{i}+3\hat{j}+2\hat{k},2\hat{i}-\hat{j}+\hat{k}$ and $-\hat{i}+2\hat{j}+3\hat{k}$, then area of triangle is (in units)

1. $\frac{\sqrt{107}}{2}$
2. $\sqrt{\frac{107}{6}}$
3. $\sqrt{\frac{107}{2}}$
4. $\frac{\sqrt{207}}{2}$

Q. If t and c are two complex numbers such that $\left|t\right|\ne \left|c\right|,\left|t\right|=1$ and z = (at + b) /(t - c), z = x + iy. Locus of z is
(where a, b are complex numbers)

1. straight line
2. None of these
3. line segment
4. circle