The domain of the function f\left( x \right) = {{{{\sin }^{ - 1}}\left( {x - 3} \right)} \over {\sqrt {9 - {x^2}} }}

The graph of the function y = f(x) is symmetrical about the line x = 2, then

If $$f:R\to S$$, defined by $$f\left(x\right)=\sin x-\sqrt{3}\cos x+1$$, is onto, then the interval of $$S$$ is

The range of the function f(x) = $${}^{7-x}{P}_{x-3}$$ is

If $$a\ne 0$$ and the line $$2bx+3cy+4d=0$$ passes through the points of intersection of the parabolas $$y^2=4ax$$ and $$x^2=4ay$$, then :

\mathop {Lim}\limits_{n \to \infty } \sum\limits_{r = 1}^n {{1 \over n}{e^{{r \over n}}}} is

The value of I = \int\limits_0^{\pi /2} {{{{{\left( {\sin x + \cos x} \right)}^2}} \over {\sqrt {1 + \sin 2x} }}dx} is

The value of $$\underset{-2}{\overset{3}{\int }}\left|1-x^2\right|dx$$ is

If $$\underset{0}{\overset{\pi }{\int }}xf\left(\sin x\right)dx=A\underset{0}{\overset{\pi /2}{\int }}f\left(\sin x\right)dx,$$ then $$A$$ is

If f\left( x \right) = {{{e^x}} \over {1 + {e^x}}},{I_1} = \int\limits_{f\left( { - a} \right)}^{f\left( a \right)} {xg\left\{ {x\left( {1 - x} \right)} \right\}dx} and $$I_2=\underset{f\left(-a\right)}{\overset{f\left(a\right)}{\int }}g\left\{x\left(1-x\right)\right\}dx,$$ then the value of {{{I_2}} \over {{I_1}}} is

Let z and w be complex numbers such that $$\overline{z}+i\overline{w}=0$$ and arg zw = $$\pi$$. Then arg z equals :

If $$z=x-iy$$ and {z^{{1 \over 3}}} = p + iq, then {{\left( {{x \over p} + {y \over q}} \right)} \over {\left( {{p^2} + {q^2}} \right)}} is equal to :

If $$\left|z^2-1\right|={\left|z\right|}^2+1$$, then z lies on :

A variable circle passes through the fixed point A (p, q) and touches x-axis. The locus of the other end of the diameter through A is :

If the lines 2x + 3y + 1 + 0 and 3x - y - 4 = 0 lie along diameter of a circle of circumference $$10\pi$$, then the equation of the circle is :

If a circle passes through the point (a, b) and cuts the circle $$x^2+y^2=4$$ orthogonally, then the locus of its centre is :

Intercept on the line y = x by the circle $$x^2+y^2-2x=0$$ is AB. Equation of the circle on AB as a diameter is :

The eccentricity of an ellipse, with its centre at the origin, is {1 \over 2}. If one of the directrices is $$x=4$$, then the equation of the ellipse is :

Let $R=\{(1,3),(4,2),(2,4),(2,3),(3,1)\}$ be a relation on the set $A=\{1,2,3,4\}$. The relation $R$ is :

Let f(x) = {{1 - \tan x} \over {4x - \pi }}, x \ne {\pi \over 4}, x \in \left[ {0,{\pi \over 2}} \right]. If $$f(x)$$ is continuous in \left[ {0,{\pi \over 2}} \right], then f\left( {{\pi \over 4}} \right) is