Let $$f(x)$$ be a function satisfying $$f^{\prime }(x)=f(x)$$ with $$f(0)=1$$ and $$g(x)$$ be a function that satisfies $$f\left(x\right)+g\left(x\right)=x^2$$. Then the value of the integral $$\underset{0}{\overset{1}{\int }}f\left(x\right)g\left(x\right)dx,$$ is

If $$f\left(y\right)=e^y,$$ $$g\left(y\right)=y;y>0$$ and $$F\left(t\right)=\underset{0}{\overset{t}{\int }}f\left(t-y\right)g\left(y\right)dy,$$ then :

If $$z$$ and $$\omega$$ are two non-zero complex numbers such that $$\left|z\omega \right|=1$$ and Arg(z) - Arg(\omega ) = {\pi \over 2}, then $$\overline{z}\omega$$ is equal to

Let $$Z_1$$ and $$Z_2$$ be two roots of the equation $$Z^2+aZ+b=0$$, Z being complex. Further , assume that the origin, $$Z_1$$ and $$Z_2$$ form an equilateral triangle. Then :

If {\left( {{{1 + i} \over {1 - i}}} \right)^x} = 1 then :

If the two circles $$(x-1{)}^2+(y-3{)}^2=r^2$$ and $$x^2+y^2-8x+2y+8=0$$ intersect in two distinct point, then :

The lines 2x - 3y = 5 and 3x - 4y = 7 are diameters of a circle having area as 154 sq. units. Then the equation of the circle is :

If \mathop {\lim }\limits_{x \to 0} {{\log \left( {3 + x} \right) - \log \left( {3 - x} \right)} \over x} = k, the value of k is

Let $$f(a)=g(a)=k$$ and their nth derivatives $$f^n(a)$$, $$g^n(a)$$ exist and are not equal for some n. Further if \mathop {\lim }\limits_{x \to a} {{f(a)g(x) - f(a) - g(a)f(x) + f(a)} \over {g(x) - f(x)}} = 4 then the value of k is

\mathop {\lim }\limits_{x \to {\pi \over 2}} {{\left[ {1 - \tan \left( {{x \over 2}} \right)} \right]\left[ {1 - \sin x} \right]} \over {\left[ {1 + \tan \left( {{x \over 2}} \right)} \right]{{\left[ {\pi - 2x} \right]}^3}}} is

If f(x) = \left\{ {\matrix{ {x{e^{ - \left( {{1 \over {\left| x \right|}} + {1 \over x}} \right)}}} & {,x \ne 0} \cr 0 & {,x = 0} \cr } } \right. then $$f(x)$$ is

The trigonometric equation $${\sin }^{-1}x=2{\sin }^{-1}a$$ has a solution for :

A student is to answer 10 out of 13 questions in an examination such that he must choose at least 4 from the first five questions. The number of choices available to him is

The number of ways in which 6 men and 5 women can dine at a round table if no two women are to sit together is given by

If $${}^n{C}_r$$ denotes the number of combination of n things taken r at a time, then the expression $${}^n{C}_{r+1}+{}^n{C}_{r-1}+2\times {}^n{C}_r$$ equals

If the sum of the roots of the quadratic equation $$a{x}^2+bx+c=0$$ is equal to the sum of the squares of their reciprocals, then {a \over c},\,{b \over a} and {c \over b} are in

The value of '$$a$$' for which one root of the quadratic equation $\left( {{a^2} - 5a + 3} \right){x^2} + \left( {3a - 1} \right)x + 2 = 0$ is twice as large as the other is

The number of real solutions of the equation $$x^2-3\left|x\right|+2=0$$ is

The foci of the ellipse {{{x^2}} \over {16}} + {{{y^2}} \over {{b^2}}} = 1 and the hyperbola {{{x^2}} \over {144}} - {{{y^2}} \over {81}} = {1 \over {25}} coincide. Then the value of $$b^2$$ is :

The sum of the radii of inscribed and circumscribed circles for an $$n$$ sided regular polygon of side $$a,$$ is :