If \int {{{\cos xdx} \over {{{\sin }^3}x{{\left( {1 + {{\sin }^6}x} \right)}^{2/3}}}}} = f\left( x \right){\left( {1 + {{\sin }^6}x} \right)^{1/\lambda }} + c where c is a constant of integration, then \lambda f\left( {{\pi \over 3}} \right) is equal to

The integral \int {{{dx} \over {{{(x + 4)}^{{8 \over 7}}}{{(x - 3)}^{{6 \over 7}}}}}} is equal to : (where C is a constant of integration)

If ƒ'(x) = tan–1(secx + tanx), - {\pi \over 2} < x < {\pi \over 2}, and ƒ(0) = 0, then ƒ(1) is equal to :

Let a – 2b + c = 1. If f(x)=\left| {\matrix{ {x + a} & {x + 2} & {x + 1} \cr {x + b} & {x + 3} & {x + 2} \cr {x + c} & {x + 4} & {x + 3} \cr } } \right|, then:

For a suitably chosen real constant a, let a function, $$f:R-\left\{-a\right\}\to R$$ be defined by f(x) = {{a - x} \over {a + x}}. Further suppose that for any real number $$x\ne -a$$ and $$f(x)\ne -a$$, (fof)(x) = x. Then f\left( { - {1 \over 2}} \right) is equal to :

If f(x + y) = f(x)f(y) and $$\sum _{x=1}^{\infty }f\left(x\right)=2$$ , x, y $$\in$$ N, where N is the set of all natural number, then the value of {{f\left( 4 \right)} \over {f\left( 2 \right)}} is :

Let f : R $$\to$$ R be a function which satisfies f(x + y) = f(x) + f(y) $$\forall$$ x, y $$\in$$ R. If f(1) = 2 and g(n) = $$\sum _{k=1}^{\left(n-1\right)}f\left(k\right)$$, n $$\in$$ N then the value of n, for which g(n) = 20, is :

Let ƒ : (1, 3) $$\to$$ R be a function defined by f(x) = {{x\left[ x \right]} \over {1 + {x^2}}} , where [x] denotes the greatest integer $$\le$$ x. Then the range of ƒ is

The inverse function of f(x) = {{{8^{2x}} - {8^{ - 2x}}} \over {{8^{2x}} + {8^{ - 2x}}}}, x $$\in$$ (-1, 1), is :

If g(x) = x2 + x - 1 and (goƒ) (x) = 4x2 - 10x + 5, then ƒ\left( {{5 \over 4}} \right) is equal to:

If y = mx + 4 is a tangent to both the parabolas, y2 = 4x and x2 = 2by, then b is equal to :

The locus of a point which divides the line segment joining the point (0, –1) and a point on the parabola, x2 = 4y, internally in the ratio 1 : 2, is :

If one end of a focal chord AB of the parabola y2 = 8x is at A\left( {{1 \over 2}, - 2} \right), then the equation of the tangent to it at B is :

The area (in sq. units) of an equilateral triangle inscribed in the parabola y2 = 8x, with one of its vertices on the vertex of this parabola, is :

Let P be a point on the parabola, y2 = 12x and N be the foot of the perpendicular drawn from P on the axis of the parabola. A line is now drawn through the mid-point M of PN, parallel to its axis which meets the parabola at Q. If the y-intercept of the line NQ is {4 \over 3}, then :

Let the latus ractum of the parabola y2 = 4x be the common chord to the circles C1 and C2 each of them having radius 2$$\sqrt{5}$$. Then, the distance between the centres of the circles C1 and C2 is :

If the common tangent to the parabolas, y2 = 4x and x2 = 4y also touches the circle, x2 + y2 = c2, then c is equal to :

Let L1 be a tangent to the parabola y2 = 4(x + 1) and L2 be a tangent to the parabola y2 = 8(x + 2) such that L1 and L2 intersect at right angles. Then L1 and L2 meet on the straight line :

The centre of the circle passing through the point (0, 1) and touching the parabola y = x2 at the point (2, 4) is :

$$\underset{-\pi }{\overset{\pi }{\int }}\left|\pi -\left|x\right|\right|dx$$ is equal to :