Vectorial Representation of a Complex Number

Q.

Express the following complex numbers in the standard form a + i b :

$\left(i\right)(1+i)(1+2i)\left(ii\right)\frac{3+2i}{-2+i}\left(iii\right)\frac{1}{(2+i{)}^2}\left(iv\right)\frac{1-i}{1+i}\left(v\right)\frac{(2+i{)}^3}{2+3i}\left(vi\right)\frac{(1+i)(1+\sqrt{3i})}{1-i}\left(vii\right)\frac{2+3i}{4+5i}\left(viii\right)\frac{(1-i{)}^3}{1-i^3}\left(ix\right)(1+2i{)}^{-3}(x\left)\frac{3-4i}{(4-2i)(1+i)}\right(xi\left)(\frac{1}{1-4i}-\frac{2}{1+i})\left(\frac{3-4i}{5+i}\right)\right(xii)\frac{5+\sqrt{2i}}{1-\sqrt{2i}}$

Q. Argument of complex number $\frac{1+\sqrt{3}i}{\sqrt{3}+i}$ is

1. $\frac{5\pi }{6}$
2. $\frac{\pi }{6}$
3. $\frac{\pi }{3}$
4. $-\frac{5\pi }{6}$

Q.

If $z=\frac{4}{1-i}$, then $\overline{z}$ is equal to

1. $2\left(1-i\right)$

2. $1+i$

3. $\frac{2}{1-i}$

4. $\frac{4}{1+i}$

Q.

If $a,b$ and $c$ are different real numbers and $a\hat{i}+b\hat{j}+c\hat{k},b\hat{i}+c\hat{j}+a\hat{k}$ and $c\hat{i}+a\hat{j}+b\hat{k}$ are position vectors of three non-collinear points, then?

1. centroid of $\Delta ABC$ is $\frac{a+b+c}{3}(\hat{i}+\hat{j}+\hat{k})$

2. $(\hat{i}+\hat{j}+\hat{k})$ is not equally inclined to three vectors

3. $∆ABC$ is a scalene triangle

4. perpendicular from the origin to the plane of the triangle does not meet it at the centroid

Q. Match the following with the locus of $z$.
$\begin{array}{cccc}& \text{Column I}& & \text{Column II}\\ \text{(A)}& |z-z_1|=k|z-z_2|,k>0& \text{(p)}& \text{Circle}\\ \text{(B)}& ||z-z_1|-|z-z_2||=k_1,k_1\in R& \text{(q)}& \text{Line}\\ \text{(C)}& |z+z_1|+|z+z_2|=k_2,k_2>0& \text{(r)}& \text{Ellipse}\\ \text{(D)}& \arg \left(\frac{z-z_1}{z-z_2}\right)=\alpha ,\alpha \in R& \text{(s)}& \text{No locus}\\ & & \text{(t)}& \text{Hyperbola}\end{array}$

1. A(P, Q), B(Q, S, T), C(Q, R, S), D(P, Q)
2. Option d
3. Option a
4. Option b

Q. Evaluate:

${lim}_{x\to 2}\frac{x^2-4}{\sqrt{3x-2}-\sqrt{x+2}}$

Q. If $P$ be any point on the plane $lx+my+nz=p$ and $Q$ be a point on the line $OP$ such that $OP.OQ=p^2$. The locus of the point $Q$ is

1. $lx+my+nz=p(x^2+y^2+z^2)$
2. $lx+my+nz=x^2+y^2+z^2$
3. $p(lx+my+nz)=x^2+y^2+z^2$
4. None of these

Q.

$\text{Locus of point}z\text{so that}z,i,\text{and}iz\text{are collinear, is}$

1. A straight line

2. An ellipse

3. A circle

4. A rectangular hyperbola

Q. Argument of complex number $\frac{1+\sqrt{3}i}{\sqrt{3}+i}$ is

1. $\frac{\pi }{3}$
2. $\frac{5\pi }{6}$
3. $-\frac{5\pi }{6}$
4. $\frac{\pi }{6}$

Q.

A man walks a distance of 3 units, from the origin towards the north - east $\left(N{45}^{\circ }E\right)$ direction. From there, he walks a distance of 4 units towards the north - west $\left(N{45}^{\circ }E\right)$ direction to reach a point P. Then the position of P in the Argand plane is:

1. ${3e}^{i\pi /4}+4i$

2. ${(3-4i)e}^{i\pi /4}$

3. ${(4+3i)e}^{i\pi /4}$

4. ${(3+4i)e}^{i\pi /4}$

Q. $P(x_1,y_1)$ and $Q(x_2,y_2)$ are two points on the curve. $y=x^2+3x+1$. If $\overrightarrow{OP}.\hat{i}=1,\overrightarrow{OQ}.\hat{i}=-1$ then $|2\overrightarrow{OP}+3\overrightarrow{OQ}|$

1. $2\sqrt{5}$
2. $5\sqrt{2}$
3. $8$
4. $11$

Q. Express the mentioned below complex number in the standard form $a+ib$
(xii) $\frac{5+\sqrt{2}i}{1-\sqrt{2}i}$

Q. The position vectors of the points $P$ and $Q$ with respect to the origin $O$ are $\overrightarrow{a}=\hat{i}+3\hat{j}-2\hat{k}$ and $\overrightarrow{b}=3\hat{i}-\hat{j}-2\hat{k}$, respectively. If $M$ is a point on $PQ$, such that $OM$ is the bisector of $POQ$, then $\overrightarrow{OM}$ is

1. $2(\hat{i}-\hat{j}+\hat{k})$
2. $2(\hat{i}+\hat{j}+\hat{k})$
3. $2\hat{i}+\hat{j}-2\hat{k}$
4. $2(-\hat{i}+\hat{j}-\hat{k})$

Q. Find the vector $\overline{r}$ perpendicular to $\overline{a}=\hat{i}-2\hat{j}+5\hat{k}$ and $\overline{b}=2\hat{i}+3\hat{j}-\hat{k}$ and $\overline{r}.(2\hat{i}+\hat{j}+\hat{k})+8=0$

1. $-13\hat{i}+11\hat{j}+7\hat{k}$
2. $13\hat{i}-11\hat{j}-7\hat{k}$
3. $7\hat{i}+13\hat{j}+11\hat{k}$
4. $11\hat{i}+13\hat{j}+9\hat{k}$

Q. The real part of ${\left(\frac{1+i}{3-i}\right)}^2=$

1. $1$
2. $16$
3. $16{\omega }^2$
4. $\frac{-3}{25}$

Q.

Locus of point z so that z, i, and iz are collinear, is

1. A straight line

2. A circle

3. An ellipse

4. A rectangular hyperbola

Q. If ${\cos }^{-1}\left(\frac{x^2-y^2}{x^2+y^2}\right)={\tan }^{-1}a$, prove that $\frac{dy}{dx}=\frac{y}{x}$.

Q. The value of ${}^2{P}_1{+}^3{P}_1+.....{+}^n{P}_1$ is equal to

1. $\frac{n^2-n+2}{2}$
2. $\frac{n^2+n+2}{2}$
3. $\frac{n^2-n-1}{2}$
4. $\frac{n^2+n-2}{2}$
5. $\frac{n^2+n-1}{2}$

Q. The 2nd term of $t_n=n^2-2n$ is:

Q. Compute :
$(1+i{)}^{-1}$

Q.

A man walks a distance of 3 units, from the origin towards the north - east $\left(N{45}^{\circ }E\right)$ direction. From there, he walks a distance of 4 units towards the north - west $\left(N{45}^{\circ }E\right)$ direction to reach a point P. Then the position of P in the Argand plane is:

1. 
2. 
3. 
4. 

Q.

POQis a straight line through the origin O, P and Q represent the complex numbers a+ib andc+id respectively and OP=OQ, then

1. None of these

2. 

3. arg(a+ib)=arg(c+id)

4. a+c=b+d

Q. Express the mentioned below complex number in the standard form $a+ib$
(vi) $\frac{(1+i)(1+\sqrt{3}i)}{1-i}$