Rank of a Matrix

Q.

If $z_1$ and $z_2$ are complex numbers such that $\mathrm{Re}\left(z_1\right)=\left|z_1-1\right|$, $\mathrm{Re}\left(z_2\right)=\left|z_2-1\right|$, and $\arg \left(z_1-z_2\right)=\frac{\mathrm{\pi }}{6}$, then $\mathrm{Im}\left(z_1+z_2\right)$ is equal to:

1. $2\sqrt{3}$

2. $\frac{2}{\sqrt{3}}$

3. $\frac{1}{\sqrt{3}}$

4. $\frac{\sqrt{3}}{2}$

Q.

Which of the following is true if A and C are coefficient and augmented matrices respectively for a system of linear equation. Here n = number of unknowns

1. Rank A = Rank of C=n, Infinite solutions

2. Rank A = Rank of C = r, r < n, Infinite solutions

3. Rank A = Rank of C = n, unique solution

4. Rank A = Rank of C = r< n, unique solution

Q. $\underset{x\to \infty }{lim}\frac{x-\log x}{x+\log x}=$

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

Q.

Given A and C are coefficient and augmented matrices respectively for a system of linear equations. Which of the following cases tells if the equations are consistent?

1. Rank of A < Rank of C

2. Rank of A > Rank of C

3. Rank of A = Rank of C

4. None of these

Q.

The Rank of the following matrix is

$\left[\begin{array}{cccc}1& 0& 0& 3\\ 0& 1& 1& 2\\ 0& 0& 0& 0\\ 0& 0& 1& 1\end{array}\right]-----------$

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