Let f : R $$-$$ {3} $$\to$$ R $$-$$ {1} be defined by f(x) = {{x - 2} \over {x - 3}}. Let g : R $$\to$$ R be given as g(x) = 2x $$-$$ 3. Then, the sum of all the values of x for which f$$-$$1(x) + g$$-$$1(x) = {{13} \over 2} is equal to :

Let [ x ] denote the greatest integer $$\le$$ x, where x $$\in$$ R. If the domain of the real valued function f(x) = \sqrt {{{\left| {[x]} \right| - 2} \over {\left| {[x]} \right| - 3}}} is ($$-$$ $$\infty$$, a) $$]\cup$$ [b, c) $$\cup$$ [4, $$\infty$$), a < b < c, then the value of a + b + c is :

Let f:R - \left\{ {{\alpha \over 6}} \right\} \to R be defined by f(x) = {{5x + 3} \over {6x - \alpha }}. Then the value of $$\alpha$$ for which (fof)(x) = x, for all x \in R - \left\{ {{\alpha \over 6}} \right\}, is :

Let g : N $$\to$$ N be defined as g(3n + 1) = 3n + 2, g(3n + 2) = 3n + 3, g(3n + 3) = 3n + 1, for all n $$\ge$$ 0. Then which of the following statements is true?

Let f : R $$\to$$ R be defined as f(x + y) + f(x - y) = 2f(x)f(y),f\left( {{1 \over 2}} \right) = - 1. Then, the value of \sum\limits_{k = 1}^{20} {{1 \over {\sin (k)\sin (k + f(k))}}} is equal to :

Consider function f : A $$\to$$ B and g : B $$\to$$ C (A, B, C $$\subseteq$$ R) such that (gof)$$-$$1 exists, then :

Let f : N $$\to$$ N be a function such that f(m + n) = f(m) + f(n) for every m, n$$\in$$N. If f(6) = 18, then f(2) . f(3) is equal to :

The range of the function, f(x) = {\log _{\sqrt 5 }}\left( {3 + \cos \left( {{{3\pi } \over 4} + x} \right) + \cos \left( {{\pi \over 4} + x} \right) + \cos \left( {{\pi \over 4} - x} \right) - \cos \left( {{{3\pi } \over 4} - x} \right)} \right) is :

The locus of the mid-point of the line segment joining the focus of the parabola y2 = 4ax to a moving point of the parabola, is another parabola whose directrix is :

If P is a point on the parabola y = x2 + 4 which is closest to the straight line y = 4x $$-$$ 1, then the co-ordinates of P are :

A tangent is drawn to the parabola y2 = 6x which is perpendicular to the line 2x + y = 1. Which of the following points does NOT lie on it?

The shortest distance between the line x $$-$$ y = 1 and the curve x2 = 2y is :

If the three normals drawn to the parabola, y2 = 2x pass through the point (a, 0) a $$\ne$$ 0, then 'a' must be greater than :

Let C be the locus of the mirror image of a point on the parabola y2 = 4x with respect to the line y = x. Then the equation of tangent to C at P(2, 1) is :

Let L be a tangent line to the parabola y2 = 4x $$-$$ 20 at (6, 2). If L is also a tangent to the ellipse {{{x^2}} \over 2} + {{{y^2}} \over b} = 1, then the value of b is equal to :

Let the tangent to the parabola S : y2 = 2x at the point P(2, 2) meet the x-axis at Q and normal at it meet the parabola S at the point R. Then the area (in sq. units) of the triangle PQR is equal to :

Let P be a variable point on the parabola $$y=4x^2+1$$. Then, the locus of the mid-point of the point P and the foot of the perpendicular drawn from the point P to the line y = x is :

Let a parabola b be such that its vertex and focus lie on the positive x-axis at a distance 2 and 4 units from the origin, respectively. If tangents are drawn from O(0, 0) to the parabola P which meet P at S and R, then the area (in sq. units) of $$\Delta$$SOR is equal to :

A tangent and a normal are drawn at the point P(2, $$-$$4) on the parabola y2 = 8x, which meet the directrix of the parabola at the points A and B respectively. If Q(a, b) is a point such that AQBP is a square, then 2a + b is equal to :

If two tangents drawn from a point P to the parabola y2 = 16(x $$-$$ 3) are at right angles, then the locus of point P is :