Let $$f:(-1,1)\to B$$, be a function defined by f\left( x \right) = {\tan ^{ - 1}}{{2x} \over {1 - {x^2}}}, then $$f$$ is both one-one and onto when B is the interval

A real valued function f(x) satisfies the functional equation f(x - y) = f(x)f(y) - f(a - x)f(a + y) where a is given constant and f(0) = 1, f(2a - x) is equal to

Let $$P$$ be the point $$(1,0)$$ and $$Q$$ a point on the parabola $$y^2=8x$$. The locus of mid point of $$PQ$$ is :

Let $$f:R\to R$$ be a differentiable function having $$f\left(2\right)=6$$, f'\left( 2 \right) = \left( {{1 \over {48}}} \right). Then \mathop {\lim }\limits_{x \to 2} \int\limits_6^{f\left( x \right)} {{{4{t^3}} \over {x - 2}}dt} equals :

\mathop {\lim }\limits_{n \to \infty } \left[ {{1 \over {{n^2}}}{{\sec }^2}{1 \over {{n^2}}} + {2 \over {{n^2}}}{{\sec }^2}{4 \over {{n^2}}}.... + {1 \over n}{{\sec }^2}1} \right] equals

If $$I_1=\underset{0}{\overset{1}{\int }}{2}^{x^2}dx,I_2=\underset{0}{\overset{1}{\int }}{2}^{x^3}dx,I_3=\underset{1}{\overset{2}{\int }}{2}^{x^2}dx$$ and $$I_4=\underset{1}{\overset{2}{\int }}{2}^{x^3}dx$$ then

The value of \int\limits_{ - \pi }^\pi {{{{{\cos }^2}} \over {1 + {a^x}}}dx,\,\,a > 0,} is

The value of integral, \int\limits_3^6 {{{\sqrt x } \over {\sqrt {9 - x} + \sqrt x }}} dx is

If the cube roots of unity are 1, $$\omega ,{\omega }^2$$ then the roots of the equation $$(x-1{)}^3$$ + 8 = 0, are :

If \,\omega = {z \over {z - {1 \over 3}i}}\, and $$\left|\omega \right|=1$$, then $$z$$ lies on :

If $$z_1$$ and $$z_2$$ are two non-zero complex numbers such that $$\left|z_1+z_2\right|=\left|z_1\right|+\left|z_2\right|$$, then arg $$z_1$$ - arg $$z_2$$ is equal to :

A circle touches the x-axis and also touches the circle with centre at (0, 3) and radius 2. The locus of the centre of the circle is :

If the pair of lines $$a{x}^2+2\left(a+b\right)xy+b{y}^2=0$$ lie along diameters of a circle and divide the circle into four sectors such that the area of one of the sectors is thrice the area of another sector then :

If the circles $$x^2+y^2+2ax+cy+a=0$$ and $$x^2+y^2-3ax+dy-1=0$$ intersect in two ditinct points P and Q then the line 5x + by - a = 0 passes through P and Q for :

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

An ellipse has $$OB$$ as semi minor axis, $$F$$ and $$F$$' its focii and theangle $$FBF$$' is a right angle. Then the eccentricity of the ellipse is :

Let $R=\{(3,3),(6,6),(9,9),(12,12),(6,12)$, $(3,9),(3,12),(3,6)\}$ be a relation on the set $A=\{3,6,9,12\}$. The relation is :

Let $$\alpha$$ and $$\beta$$ be the distinct roots of $$a{x}^2+bx+c=0$$, then \mathop {\lim }\limits_{x \to \alpha } {{1 - \cos \left( {a{x^2} + bx + c} \right)} \over {{{\left( {x - \alpha } \right)}^2}}} is equal to

Suppose $$f(x)$$ is differentiable at x = 1 and \mathop {\lim }\limits_{h \to 0} {1 \over h}f\left( {1 + h} \right) = 5, then $$f^{\prime }\left(1\right)$$ equals

If $$f$$ is a real valued differentiable function satisfying $$\left|f\left(x\right)-f\left(y\right)\right|$$ $$\le {\left(x-y\right)}^2$$, $$x,y$$ $$\in R$$ and $$f(0)$$ = 0, then $$f(1)$$ equals