If x + iy = ${\left(1 - \mathrm{i}\sqrt{3}\right)}^{100}$, then find (x, y).

• $\left(2^{99}, 2^{99}\sqrt{3}\right)$

• $\left(2^{99}, - 2^{99}\sqrt{3}\right)$

• $\left(- 2^{99}, 2^{99}\sqrt{3}\right)$

• None of these

If $\frac{{\mathrm{x}}^2 + 2\mathrm{x} + 7}{2\mathrm{x} + 3} < 6, \mathrm{x} \in \mathrm{R}$, then

• $\mathrm{x} > 11 \mathrm{or} \mathrm{x} = - \frac{3}{2}$

• x > 11 or x < - 1

• $- \frac{3}{2} < \mathrm{x} < - 1$

• $- 1 < \mathrm{x} < 11 \mathrm{or} \mathrm{x} < - \frac{3}{2}$

133.

If D is the set of all real x such that 1 - e^{(1/x) - 1} is positive, then D is equal to

• $(- \infty , 1]$

• $\left(- \infty , 0\right)$

• $\left(1, \infty \right)$

• $\left(- \infty , 0\right) \cup \left(1, \infty \right)$

If one AM 'A' and two GM p and q are inserted between two given numbers, then. find the value of $\frac{{\mathrm{p}}^2}{\mathrm{q}} + \frac{{\mathrm{q}}^2}{\mathrm{p}}$.

• A

• 2A

• 3A

• 4A

If the roots of the equation x² + ax + b = 0 are c and d, then one of the roots of the equation x² + (2c + a)x + c² + ac + b = 0

• c

• d - c

• 2d

• 2c

If the axes are shifted to the point (1, - 2) without solution, then the equation 2x² + y² - 4x + 4y = 0 becomes

• 2X² + 3Y² = 6

• 2X² + Y²

• X² + 2Y²

• None of the above

In a group ( G, *), then equation x * a = b has a

• unique solution b * a^-1

• unique solution a^-1 * b

• unique solution a^-1 * b^-1

• many solutions

The number of solutions of the equation sin(e^x) = 5^x + 5^-x,  is

• 0

• 1

• 2

• infinitely many

If z satisfies the equation $\left|\mathrm{z}\right|$ - z = 1 + 2i, then z is equal to

• $\frac{3}{2} + 2\mathrm{i}$

• $\frac{3}{2} - 2\mathrm{i}$

• $2 - \frac{3}{2}\mathrm{i}$

• $2 + \frac{3}{2}\mathrm{i}$

If $\mathrm{z} = \frac{1 - \mathrm{i}\sqrt{3}}{1 + \mathrm{i}\sqrt{3}}$, then arg (z) is

• 60°

• 120°

• 240°

• 300°