If a and b are real numbers such that $${\left(2+\alpha \right)}^4=a+b\alpha$$ where \alpha = {{ - 1 + i\sqrt 3 } \over 2} then a + b is equal to :

Let u = {{2z + i} \over {z - ki}}, z = x + iy and k > 0. If the curve represented by Re(u) + Im(u) = 1 intersects the y-axis at the points P and Q where PQ = 5, then the value of k is :

If z1 , z2 are complex numbers such that Re(z1) = |z1 – 1|, Re(z2) = |z2 – 1| , and arg(z1 - z2) = {\pi \over 6}, then Im(z1 + z2 ) is equal to :

The imaginary part of {\left( {3 + 2\sqrt { - 54} } \right)^{{1 \over 2}}} - {\left( {3 - 2\sqrt { - 54} } \right)^{{1 \over 2}}} can be :

The value of {\left( {{{1 + \sin {{2\pi } \over 9} + i\cos {{2\pi } \over 9}} \over {1 + \sin {{2\pi } \over 9} - i\cos {{2\pi } \over 9}}}} \right)^3} is :

If z be a complex number satisfying |Re(z)| + |Im(z)| = 4, then |z| cannot be :

Let z be complex number such that \left| {{{z - i} \over {z + 2i}}} \right| = 1 and |z| = {5 \over 2}. Then the value of |z + 3i| is :

If the equation, x2 + bx + 45 = 0 (b $$\in$$ R) has conjugate complex roots and they satisfy |z +1| = 2$$\sqrt{10}$$ , then :

If {{3 + i\sin \theta } \over {4 - i\cos \theta }}, $$\theta$$ $$\in$$ [0, 2$$\theta$$], is a real number, then an argument of sin$$\theta$$ + icos$$\theta$$ is :

If {\mathop{\rm Re}\nolimits} \left( {{{z - 1} \over {2z + i}}} \right) = 1, where z = x + iy, then the point (x, y) lies on a :

If the length of the chord of the circle, x2 + y2 = r2 (r > 0) along the line, y – 2x = 3 is r, then r2 is equal to :

The circle passing through the intersection of the circles, x2 + y2 – 6x = 0 and x2 + y2 – 4y = 0, having its centre on the line, 2x – 3y + 12 = 0, also passes through the point :

A circle touches the y-axis at the point (0, 4) and passes through the point (2, 0). Which of the following lines is not a tangent to this circle?

If a line, y = mx + c is a tangent to the circle, (x – 3)2 + y2 = 1 and it is perpendicular to a line L1, where L1 is the tangent to the circle, x2 + y2 = 1 at the point \left( {{1 \over {\sqrt 2 }},{1 \over {\sqrt 2 }}} \right), then :

Let the tangents drawn from the origin to the circle, x2 + y2 - 8x - 4y + 16 = 0 touch it at the points A and B. The (AB)2 is equal to :

If the distance between the foci of an ellipse is 6 and the distance between its directrices is 12, then the length of its latus rectum is :

If 3x + 4y = 12$$\sqrt{2}$$ is a tangent to the ellipse {{{x^2}} \over {{a^2}}} + {{{y^2}} \over 9} = 1 for some $$a$$ $$\in$$ R, then the distance between the foci of the ellipse is :

Let the line y = mx and the ellipse 2x2 + y2 = 1 intersect at a ponit P in the first quadrant. If the normal to this ellipse at P meets the co-ordinate axes at \left( { - {1 \over {3\sqrt 2 }},0} \right) and (0, $$\beta$$), then $$\beta$$ is equal to :

The length of the minor axis (along y-axis) of an ellipse in the standard form is {4 \over {\sqrt 3 }}. If this ellipse touches the line, x + 6y = 8; then its eccentricity is :

Let {{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1 (a > b) be a given ellipse, length of whose latus rectum is 10. If its eccentricity is the maximum value of the function, \phi \left( t \right) = {5 \over {12}} + t - {t^2}, then a2 + b2 is equal to :