If $$\int$$x5.e$$-$$4x3 dx = {1 \over {48}}e$$-$$4x3 f(x) + C, where C is a constant of inegration, then f(x) is equal to -

Let n $$\ge$$ 2 be a natural number and 0 < \theta < {\pi \over 2}. Then \int {{{{{\left( {{{\sin }^n}\theta - \sin \theta } \right)}^{1/n}}\cos \theta } \over {{{\sin }^{n + 1}}\theta }}} \,d\theta is equal to - (where C is a constant of integration)

If f\left( x \right) = \int {{{5{x^8} + 7{x^6}} \over {{{\left( {{x^2} + 1 + 2{x^7}} \right)}^2}}}} \,dx,\,\left( {x \ge 0} \right), $$f\left(0\right)=0,$$ then the value of $$f(1)$$ is :

For $$x\in R-\left\{0,1\right\}$$, Let f1(x) = 1\over x, f2 (x) = 1 – x and f3 (x) = 1 \over {1 - x} be three given functions. If a function, J(x) satisfies (f2 o J o f1) (x) = f3 (x) then J(x) is equal to :

For x $$\in$$ (0, 3/2), let f(x) = $$\sqrt{x}$$ , g(x) = tan x and h(x) = {{1 - {x^2}} \over {1 + {x^2}}}. If $$\varphi$$ (x) = ((hof)og)(x), then \phi \left( {{\pi \over 3}} \right) is equal to :

Let f(x) = ex – x and g(x) = x2 – x, $$\forall$$ x $$\in$$ R. Then the set of all x $$\in$$ R, where the function h(x) = (fog) (x) is increasing, is :

Let f(x) = x2 , x $$\in$$ R. For any A $$\subseteq$$ R, define g (A) = { x $$\in$$ R : f(x) $$\in$$ A}. If S = [0,4], then which one of the following statements is not true ?

The domain of the definition of the function f(x) = {1 \over {4 - {x^2}}} + {\log _{10}}({x^3} - x) is

If the function ƒ : R – {1, –1} $$\to$$ A defined by ƒ(x) = {{{x^2}} \over {1 - {x^2}}} , is surjective, then A is equal to

Let $$\sum _{k=1}^{10}f(a+k)=16\left(2^{10}-1\right)$$ where the function ƒ satisfies ƒ(x + y) = ƒ(x)ƒ(y) for all natural numbers x, y and ƒ(1) = 2. then the natural number 'a' is

If f(x) = {\log _e}\left( {{{1 - x} \over {1 + x}}} \right), $$\left|x\right|<1$$ then f\left( {{{2x} \over {1 + {x^2}}}} \right) is equal to

Let a function f : (0, $$\infty$$) $$\to$$ (0, $$\infty$$) be defined by f(x) = \left| {1 - {1 \over x}} \right|. Then f is :

The number of functions f from {1, 2, 3, ...., 20} onto {1, 2, 3, ...., 20} such that f(k) is a multiple of 3, whenever k is a multiple of 4, is :

Let fk(x) = {1 \over k}\left( {{{\sin }^k}x + {{\cos }^k}x} \right) for k = 1, 2, 3, ... Then for all x $$\in$$ R, the value of f4(x) $$-$$ f6(x) is equal to

Let f : R $$\to$$ R be defined by f(x) = {x \over {1 + {x^2}}},x \in R. Then the range of f is :

Let N be the set of natural numbers and two functions f and g be defined as f, g : N $$\to$$ N such that f(n) = \left\{ {\matrix{ {{{n + 1} \over 2};} & {if\,\,n\,\,is\,\,odd} \cr {{n \over 2};} & {if\,\,n\,\,is\,\,even} \cr } \,\,} \right.; and g(n) = n $$-$$($$-$$ 1)n. Then fog is -

Let A = {x $$\in$$ R : x is not a positive integer}. Define a function $$f$$ : A $$\to$$ R as $$f(x)$$ = {{2x} \over {x - 1}}, then $$f$$ is :

Let ƒ(x) = ax (a > 0) be written as ƒ(x) = ƒ1 (x) + ƒ2 (x), where ƒ1 (x) is an even function of ƒ2 (x) is an odd function. Then ƒ1 (x + y) + ƒ1 (x – y) equals

The equation of common tangent to the curves y2 = 16x and xy = –4, is :

The tangents to the curve y = (x – 2)2 – 1 at its points of intersection with the line x – y = 3, intersect at the point :