The equation of the circle on the common chord of the circles (x - a)² + y² = a² and x² + (y + b)² = b² as diameter is

• x² + y² = 2ab(bx + ay)

• x² + y² = bx + ay

• (a² + b²)(x² + y²) = 2ab(bx - ay)

• (a² + b²)(x² + y²) = 2(bx + ay)

The eccentricity of the hyperbola which passes through (3, 0) and (3$\sqrt{2}$, 2) is

• $\sqrt{\left(13\right)}$

• $\frac{\sqrt{13}}{3}$

• $\sqrt{\frac{13}{4}}$

• None of these

The line x - 1 = 0 is the directrix of the parabola y² - kx + 8 = 0. Then, one of the value of k is

• $\frac{1}{8}$

• 8

• 4

• $\frac{1}{4}$

If $\mathrm{y} = 2^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{${\log }_{\mathrm{x}}\left(8\right)$}\right.}$, then x is equal to

• y

• y²

• y³

• None of these

The region of the complex plane for which $\left|\frac{\mathrm{z} - \mathrm{a}}{\mathrm{z} + \overline{\mathrm{a}}}\right|$ = 1, [Re (a) $\ne$ 0] is

• x - axis

• y - axis

• the straight line x = a

• None of the above

The number of non-zero integral solutions of the equation ${\left|1 - \mathrm{i}\right|}^{\mathrm{x}}$ = 2^x

• infinite

• 1

• 2

• None of these

27.

Let $\mathrm{\alpha }, {\mathrm{\alpha }}^2$ be the roots of x² + x + 1 = 0, then the equation whose roots are ${\mathrm{\alpha }}^{31}, {\mathrm{\alpha }}^{62}$ is

• x² - x + 1 = 0

• x² + x - 1 = 0

• x² + x + 1 = 0

• x⁶⁰ + x³⁰ + 1 = 0

If ${\mathrm{e}}^{\cos \left(\mathrm{x}\right)} - {\mathrm{e}}^{- \cos \left(\mathrm{x}\right)} = 4$, then the value of $\cos \left(\mathrm{x}\right)$ is

• $\log \left(2 + \sqrt{5}\right)$

• $- \log \left(2 + \sqrt{5}\right)$

• $\log \left(- 2 + \sqrt{5}\right)$

• None of these

If ${\tan }^{-1}\left(\frac{\mathrm{a}}{\mathrm{x}}\right) + {\tan }^{-1}\left(\frac{\mathrm{b}}{\mathrm{x}}\right) = \frac{\mathrm{\pi }}{2}$, then x is equal to

• $\sqrt{\mathrm{ab}}$

• $\sqrt{2\mathrm{ab}}$

• 2ab

• ab

A house of height 100 m subtends a right angle at the window of an opposite house. If the height of the window be 64 m, then the distance between the two houses is

• 48 m

• 36 m

• 54 m

• 72 m