Let $$f,g:\mathbb{N}-\{1\}\to \mathbb{N}$$ be functions defined by $$f(a)=\alpha$$, where $$\alpha$$ is the maximum of the powers of those primes $$p$$ such that $$p^{\alpha }$$ divides $$a$$, and $$g(a)=a+1$$, for all $$a\in \mathbb{N}-\{1\}$$. Then, the function $$f+g$$ is

Let $$\alpha ,\beta$$ and $$\gamma$$ be three positive real numbers. Let $$f(x)=\alpha x^5+\beta x^3+\gamma x,x\in \mathbf{R}$$ and $$g:\mathbf{R}\to \mathbf{R}$$ be such that $$g(f(x))=x$$ for all $$x\in \mathbf{R}$$. If $${\mathrm{a}}_1,{\mathrm{a}}_2,{\mathrm{a}}_3,\dots ,{\mathrm{a}}_{\mathrm{n}}$$ be in arithmetic progression with mean zero, then the value of $$f\left(g\left(\frac{1}{\mathrm{n}}\sum _{i=1}^{\mathrm{n}}f\left({\mathrm{a}}_i\right)\right)\right)$$ is equal to :

$$\text{ Let }f(x)=a{x}^2+bx+c\text{ be such that }f(1)=3,f(-2)=\lambda \text{ and }$$ $$f(3)=4$$. If $$f(0)+f(1)+f(-2)+f(3)=14$$, then $$\lambda$$ is equal to :

Let PQ be a focal chord of the parabola y2 = 4x such that it subtends an angle of {\pi \over 2} at the point (3, 0). Let the line segment PQ be also a focal chord of the ellipse E:{{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1, $$a^2>b^2$$. If e is the eccentricity of the ellipse E, then the value of {1 \over {{e^2}}} is equal to :

Let P : y2 = 4ax, a > 0 be a parabola with focus S. Let the tangents to the parabola P make an angle of {\pi \over 4} with the line y = 3x + 5 touch the parabola P at A and B. Then the value of a for which A, B and S are collinear is :

If vertex of a parabola is (2, $$-$$1) and the equation of its directrix is 4x $$-$$ 3y = 21, then the length of its latus rectum is :

If the equation of the parabola, whose vertex is at (5, 4) and the directrix is $$3x+y-29=0$$, is $$x^2+a{y}^2+bxy+cx+dy+k=0$$, then $$a+b+c+d+k$$ is equal to :

Let the normal at the point on the parabola y2 = 6x pass through the point (5, $$-$$8). If the tangent at P to the parabola intersects its directrix at the point Q, then the ordinate of the point Q is :

If the line $$y=4+kx,k>0$$, is the tangent to the parabola $$y=x-x^2$$ at the point P and V is the vertex of the parabola, then the slope of the line through P and V is :

If $$y=m_{1}x+c_1$$ and $$y=m_{2}x+c_2$$, $$m_1\ne m_2$$ are two common tangents of circle $$x^2+y^2=2$$ and parabola y2 = x, then the value of $$8|m_1{m}_2|$$ is equal to :

Let $$x=2t$$, y = {{{t^2}} \over 3} be a conic. Let S be the focus and B be the point on the axis of the conic such that $$SA\perp BA$$, where A is any point on the conic. If k is the ordinate of the centroid of the $$\Delta$$SAB, then $$\underset{t\to 1}{\lim }k$$ is equal to :

A particle is moving in the xy-plane along a curve C passing through the point (3, 3). The tangent to the curve C at the point P meets the x-axis at Q. If the y-axis bisects the segment PQ, then C is a parabola with :

Let x2 + y2 + Ax + By + C = 0 be a circle passing through (0, 6) and touching the parabola y = x2 at (2, 4). Then A + C is equal to ___________.

The tangents at the points $$A(1,3)$$ and $$B(1,-1)$$ on the parabola $$y^2-2x-2y=1$$ meet at the point $$P$$. Then the area (in unit $${}^2$$ ) of the triangle $$PAB$$ is :

Let $$\mathrm{P}$$ and $$\mathrm{Q}$$ be any points on the curves $$(x-1{)}^2+(y+1{)}^2=1$$ and $$y=x^2$$, respectively. The distance between $$P$$ and $$Q$$ is minimum for some value of the abscissa of $$P$$ in the interval :

The equation of a common tangent to the parabolas $$y=x^2$$ and $$y=-(x-2{)}^2$$ is

Let $$P(a,b)$$ be a point on the parabola $$y^2=8x$$ such that the tangent at $$P$$ passes through the centre of the circle $$x^2+y^2-10x-14y+65=0$$. Let $$A$$ be the product of all possible values of $$a$$ and $$B$$ be the product of all possible values of $$b$$. Then the value of $$A+B$$ is equal to :

If the length of the latus rectum of a parabola, whose focus is $$(a,a)$$ and the tangent at its vertex is $$x+y=a$$, is 16, then $$|a|$$ is equal to :

If the tangents drawn at the points $$\mathrm{P}$$ and $$\mathrm{Q}$$ on the parabola $$y^2=2x-3$$ intersect at the point $$R(0,1)$$, then the orthocentre of the triangle $$PQR$$ is :

Let the focal chord of the parabola $$\mathrm{P}:y^2=4x$$ along the line $$\mathrm{L}:y=\mathrm{m}x+\mathrm{c},\mathrm{m}>0$$ meet the parabola at the points M and N. Let the line L be a tangent to the hyperbola $$\mathrm{H}:x^2-y^2=4$$. If O is the vertex of P and F is the focus of H on the positive x-axis, then the area of the quadrilateral OMFN is :