If $$f\left(x\right)=x^n,$$ then the value of f\left( 1 \right) - {{f'\left( 1 \right)} \over {1!}} + {{f''\left( 1 \right)} \over {2!}} - {{f'''\left( 1 \right)} \over {3!}} + ..........{{{{\left( { - 1} \right)}^n}{f^n}\left( 1 \right)} \over {n!}} is

Let $$f\left(x\right)$$ be a polynomial function of second degree. If $$f\left(1\right)=f\left(-1\right)$$ and $$a,b,c$$ are in $$A.P,$$ then $$f^{\prime }\left(a\right),f^{\prime }\left(b\right),f^{\prime }\left(c\right)$$ are in

The degree and order of the differential equation of the family of all parabolas whose axis is $$x$$-axis, are respectively.

The solution of the differential equation \left( {1 + {y^2}} \right) + \left( {x - {e^{{{\tan }^{ - 1}}y}}} \right){{dy} \over {dx}} = 0, is :

If the pair of straight lines $$x^2-2pxy-y^2=0$$ and $$x^2-2qxy-y^2=0$$ be such that each pair bisects the angle between the other pair, then :

A square of side a lies above the $$x$$-axis and has one vertex at the origin. The side passing through the origin makes an angle \alpha \left( {0 < \alpha < {\pi \over 4}} \right) with the positive direction of x-axis. The equation of its diagonal not passing through the origin is :

If the equation of the locus of a point equidistant from the point $$\left(a_{1,}{b}_1\right)$$ and $$\left(a_{2,}{b}_2\right)$$ is $$\left(a_1-a_2\right)x+\left(b_1-b_2\right)y+c=0$$ , then the value of $${}^{\prime }{c}^{\prime }$$ is :

If $$x_1,x_2,x_3$$ and $$y_1,y_2,y_3$$ are both in G.P. with the same common ratio, then the points $$\left(x_1,y_1\right),\left(x_2,y_2\right)$$ and $$\left(x_3,y_3\right)$$ :

Locus of centroid of the triangle whose vertices are $$\left(a\cos t,a\sin t\right),\left(b\sin t,-b\cos t\right)$$ and $$\left(1,0\right),$$ where $$t$$ is a parameter, is :

If $$x$$ is positive, the first negative term in the expansion of $${\left(1+x\right)}^{27/5}$$ is

The number of integral terms in the expansion of $${\left(\sqrt{3}+\text{root}8\text{of}5\right)}^{256}$$ is

The function $$f\left(x\right)$$ $$=\log \left(x+\sqrt{x^2+1}\right)$$, is

A function $$f$$ from the set of natural numbers to integers defined by $f\left( n \right) = \left\{ {\matrix{ {{{n - 1} \over 2},\,when\,n\,is\,odd} \cr { - {n \over 2},\,when\,n\,is\,even} \cr } } \right.$ is

If $$f:R\to R$$ satisfies $$f$$(x + y) = $$f$$(x) + $$f$$(y), for all x, y $$\in$$ R and $$f$$(1) = 7, then $$\sum _{r=1}^{n}f\left(r\right)$$ is

Domain of definition of the function f(x) = {3 \over {4 - {x^2}}} + $${\log }_{10}\left(x^3-x\right)$$, is

The normal at the point$$\left(b{t}_1^2,2b{t}_1\right)$$ on a parabola meets the parabola again in the point $$\left(b{t}_2^2,2b{t}_2\right)$$, then :

The value of the integral $$I=\underset{0}{\overset{1}{\int }}x{\left(1-x\right)}^{n}dx$$ is

\mathop {\lim }\limits_{n \to \infty } {{1 + {2^4} + {3^4} + .... + {n^4}} \over {{n^5}}} - \mathop {\lim }\limits_{n \to \infty } {{1 + {2^3} + {3^3} + .... + {n^3}} \over {{n^5}}}

The value of $$\underset{x\to 0}{\lim }\underset{0}{\overset{x^2}{\int }}{\sec }^{2}tdt\frac{x}{s}inx$$ is

If $$f\left(a+b-x\right)=f\left(x\right)$$ then $$\underset{a}{\overset{b}{\int }}xf\left(x\right)dx$$ is equal to