Let P be a point on the parabola, x2 = 4y. If the distance of P from the center of the circle, x2 + y2 + 6x + 8 = 0 is minimum, then the equation of the tangent to the parabola at P, is :

Tangents drawn from the point ($$-$$8, 0) to the parabola y2 = 8x touch the parabola at $$P$$ and $$Q.$$ If F is the focus of the parabola, then the area of the triangle PFQ (in sq. units) is equal to :

Two parabolas with a common vertex and with axes along x-axis and $$y$$-axis, respectively intersect each other in the first quadrant. If the length of the latus rectum of each parabola is $$3$$, then the equation of the common tangent to the two parabolas is :

Tangent and normal are drawn at P(16, 16) on the parabola y2 = 16x, which intersect the axis of the parabola at A and B, respectively. If C is the centre of the circle through the points P, A and B and $$\angle$$CPB = $$\theta$$, then a value of tan$$\theta$$ is :

The value of integral \int_{{\pi \over 4}}^{{{3\pi } \over 4}} {{x \over {1 + \sin x}}dx} is :

If $$f(x)=\underset{0}{\overset{x}{\int }}t\left(\sin x-\sin t\right)dt$$ then :

If $$I_1={\int }_0^1{e}^{-x}{\cos }^{2}xdx;$$ $$I_2={\int }_0^1{e}^{-x^2}{\cos }^{2}xdx$$ and $$I_3={\int }_0^1{e}^{-x^3}dx;$$ then

The value of the integral \int\limits_{ - {\pi \over 2}}^{{\pi \over 2}} {{{\sin }^4}} x\left( {1 + \log \left( {{{2 + \sin x} \over {2 - \sin x}}} \right)} \right)dx is :

The value of \int\limits_{ - \pi /2}^{\pi /2} {{{{{\sin }^2}x} \over {1 + {2^x}}}} dx is

If $$\alpha ,\beta \in C$$ are the distinct roots of the equation x2 - x + 1 = 0, then $${\alpha }^{101}+{\beta }^{107}$$ is equal to :

The set of all $$\alpha$$ $$\in$$ R, for which w = {{1 + \left( {1 - 8\alpha } \right)z} \over {1 - z}} is purely imaginary number, for all z $$\in$$ C satisfying |z| = 1 and Re z $$\ne$$ 1, is :

If |z $$-$$ 3 + 2i| $$\le$$ 4 then the difference between the greatest value and the least value of |z| is :

The least positive integer n for which {\left( {{{1 + i\sqrt 3 } \over {1 - i\sqrt 3 }}} \right)^n} = 1, is :

If a circle C, whose radius is 3, touches externally the circle, $$x^2+y^2+2x-4y-4=0$$ at the point (2, 2), then the length of the intercept cut by this circle C, on the x-axis is equal to :

The tangent to the circle C1 : x2 + y2 $$-$$ 2x $$-$$ 1 = 0 at the point (2, 1) cuts off a chord of length 4 from a circle C2 whose center is (3, $$-$$2). The radius of C2 is :

A circle passes through the points (2, 3) and (4, 5). If its centre lies on the line, $$y-4x+3=0,$$ then its radius is equal to :

If the tangent at (1, 7) to the curve x2 = y - 6 touches the circle x2 + y2 + 16x + 12y + c = 0, then the value of c is :

If the length of the latus rectum of an ellipse is 4 units and the distance between a focus an its nearest vertex on the major axis is {3 \over 2} units, then its eccentricity is :

Two sets A and B are as under : A = {($$a$$, b) $$\in$$ R $$\times$$ R : |$$a$$ - 5| < 1 and |b - 5| < 1}; B = {($$a$$, b) $$\in$$ R $$\times$$ R : 4($$a$$ - 6)2 + 9(b - 5)2 $$\le$$ 36 }; Then

Let N denote the set of all natural numbers. Define two binary relations on N as R = {(x, y) $$\in$$ N $$\times$$ N : 2x + y = 10} and R2 = {(x, y) $$\in$$ N $$\times$$ N : x + 2y = 10}. Then :