The common chord of the circles x² + y² - 4x - 4y = 0 and 2x²+ 2y² = 32 subtends at the origin an angle equal to

• $\frac{\mathrm{\pi }}{6}$

• $\frac{\mathrm{\pi }}{2}$

The locus of the mid-points of the chords of the circle x² + y² + 2x - 2y - 2= 0, which make an angle of 90° at the centre is

• x² + y² - 2x - 2y = 0

• x² + y² - 2x + 2y = 0

• x² + y² + 2x - 2y = 0

• x² + y² + 2x - 2y - 1 = 0

Let P be the foot of the perpendicular from focus S of hyperbola $\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2} - \frac{{\mathrm{y}}^2}{{\mathrm{b}}^2} = 1$ on the line bx- ay = 0 and let C be the centre of the hyperbola. Then, the area of the rectangle whose sides are equal to that of SP and CP is

• 2ab

• ab

• $\frac{\left({\mathrm{a}}^{2 }+ {\mathrm{b}}^2\right)}{2}$

• $\frac{\mathrm{a}}{\mathrm{b}}$

B is an extremity of the minor axis of an ellipse whose foci are S and S'. If $\angle$SBS' is a right angle, then the eccentricity of the ellipse is

• $\frac{1}{2}$

• $\frac{1}{\sqrt{2}}$

• $\frac{2}{3}$

• $\frac{1}{3}$

The axis of the parabola x² + 2xy + y² - 5x + 5y - 5 = 0 is

• x + y = 0

• x + y - 1 = 0

• x - y + 1 = 0

• $\mathrm{x} - \mathrm{y} = \frac{1}{\sqrt{2}}$

The line segment joining the foci of the hyperbola x^{2 -} y² + 1 = 0 is one of the diameters of a circle. The equation of the circle is

• x² + y² = 4

• x² + y² = $\sqrt{2}$

• x² + y² = 2

• x² + y² = $2\sqrt{2}$

The chord of the curve y = x² + 2ax + b, joining the points where $\mathrm{x} = \mathrm{\alpha } \mathrm{and} \mathrm{y} = \mathrm{\beta }$ is parallel to the tangent to the curve at abscissa x is equal to

• $\frac{\mathrm{a} + \mathrm{b}}{2}$

• $\frac{2\mathrm{a} + \mathrm{b}}{2}$

• $\frac{2\mathrm{\alpha } +\mathrm{\beta }}{2}$

• $\frac{\mathrm{\alpha } + \mathrm{\beta }}{2}$

18.

Let f(x) = x¹³ + x¹¹ + x⁹ + x⁷ + x⁵ + x³ + x + 19. Then , f(x) = 0 has

• 13 real roots

• only one positive and only two negative real roots

• not more than one real root

• has two positive and one negative real root

In a GP series consisting of positive terms, each term is equal to the sum of next two terms. Then, the common ratio of this GP series is

• $\sqrt{5}$

• $\frac{\sqrt{5} - 1}{2}$

• $\frac{\sqrt{5}}{2}$

• $\frac{\sqrt{5} +\hspace{0.17em}1}{2}$

If $\left({\log }_5\left(\mathrm{x}\right)\right){\log }_{\mathrm{x}}\left(3\mathrm{x}\right){\log }_{3\mathrm{x}}\left(\mathrm{y}\right) = {\log }_{\mathrm{x}}\left({\mathrm{x}}^3\right)$, then y equals

• 125

• 25

• 5/3

• 243