If the point $\left(2\cos \left(\mathrm{\theta }\right), 2\sin \left(\mathrm{\theta }\right)\right) \mathrm{for} (0, 2\mathrm{\pi })$ lies in the region between the lines x + y = 2 and x - y = 2 containing the origin, then 0 lies in

• $\left(0, \frac{\mathrm{\pi }}{2}\right) \cup \left(\frac{3\mathrm{\pi }}{2}, 2\mathrm{\pi }\right)$

• $\left[0, \mathrm{\pi }\right]$

• $\left(\frac{\mathrm{\pi }}{2}, \frac{3\mathrm{\pi }}{2}\right)$

• $\left[\frac{\mathrm{\pi }}{4}, \frac{\mathrm{\pi }}{2}\right]$

Let 16x² - 3y - 32x - 12y = 44 represents a hyperbola. Then,

• length of the transverse axis is $2\sqrt{3}$

• length of each latusrectum is $32/\sqrt{3}$

• eccentricity is $\sqrt{19/3}$

• equation of a directrix is x = $\frac{\sqrt{19}}{3}$

If the straight line (a - 1)x - by + 4 = 0 is normal to the hyperbola xy = 1, then which of the following does not hold?

• a > 1, b > 0

• a > 1, b < 0

• a < 1, b < 0

• a < 1, b > 0

If y = 4x + 3 is parallel to a tangent to the parabola y² = 12x, then its distance from the normal parallel to the given line is

• $\frac{213}{\sqrt{17}}$

• $\frac{219}{\sqrt{17}}$

• $\frac{211}{\sqrt{17}}$

• $\frac{210}{\sqrt{17}}$

Let the equation of an ellipse be $\frac{{\mathrm{x}}^2}{144} + \frac{{\mathrm{y}}^2}{25} = 1$. Then, the radius of the circle with centre (0, $\sqrt{2}$) and passing through the foci of the ellipse is

• 9

• 7

• 11

• 5

The value of $\mathrm{\lambda }$ for which the curve (7x + 5)² + (7y + 3)² = ${\mathrm{\lambda }}^2$(4x + 3y - 24)² represents a parabola is

• $\pm \frac{6}{5}$

• $\pm \frac{7}{5}$

• $\pm \frac{1}{5}$

• $\pm \frac{2}{5}$

The equation of the common tangent with positive slope to the parabola y² = 8$\sqrt{3}$x and the hyperbola 4x² - y² = 4 is

• $\mathrm{y} = \sqrt{6}\mathrm{x} + \sqrt{2}$

• $\mathrm{y} = \sqrt{6}\mathrm{x} - \sqrt{2}$

• $\mathrm{y} = \sqrt{3}\mathrm{x} + \sqrt{2}$

• $\mathrm{y} = \sqrt{3}\mathrm{x} - \sqrt{2}$

The point on the parabola y² = 64x which is nearest to the line 4x + 3y + 35 = 0 has coordinates

• (9, - 24)

• (1, 81)

• (4, - 16)

• (- 9, - 24)

269.

Let z₁, z₂ be two fixed complex numbers in the argand plane and z be an arbitrary point satisfying $\left|\mathrm{z} - {\mathrm{z}}_1\right| + \left|\mathrm{z} - {\mathrm{z}}_2\right| = 2\left|{\mathrm{z}}_1 - {\mathrm{z}}_2\right|$ Then, the locus of z will be

• an ellipse

• a straight line joining z₁ and z₂

• a parabola

• a bisector of the line segment joining z₁ and z₂

Let z, be a fixed point on the circle of radius 1 centered at the origin in the Argand plane and ${\mathrm{z}}_1 \ne \pm 1$. Consider an equilateral triangle inscribed in the circle with z₁, z₂, z₃ as the vertices taken in the counterclockwise direction. Then, z₁z₂z₃ is equal to

• z₁²

• z₁³

• z₁⁴

• z₁