A particle moves along the parabola $y=x^2$ in the first quadrant in such a way that its x-coordinate (measured in metres) increases at a rate of 10 m/sec. If the angle of inclination $\theta$ of the line joining the particle to the origin change, when x = 3 m, at the rate of k rad/sec., then the value of k is

If the function $f\left(x\right)=x^4+b{x}^2+8x+1$ has a horizontal tangent and a point of inflection for the same value of x then the value of b is equal to

In the curve $y=x^3+ax\text{and}y=b{x}^2+c$ pass through the point $\left(-1,0\right)$ and have a common tangent line at this point then the value of $a+b+c$ is

Let $f\left(x\right)=x^3+x+1$ and let g(x) be its inverse function then equation of the tangent to $y=g\left(x\right)$ at x = 3 is

The equation of the normal to the curve parametrically represented by $x=t^2+3t-8\text{and}y=2t^2-2t-5$ at the point $P\left(2,-1\right)$ is

If the tangent at (1, 1) on $y^2=x{\left(2-x\right)}^2$ meets the curve again at P, then P is

The portion of the tangent at any point on the curve $x=a{t}^3,y=a{t}^4$ between the axes is divided by the abscissa of the point of contact externally in the ratio

Equation of a line which is tangent to both the curve $y=x^2+1\text{and}y=-x^2$ is

From the point (1, 1) tangents are drawn to the curve represented parametrically as $x=2t-t^2\text{and}y=t+t^2$. The distance between the points of contact is

For the functions defined parametrically by the equations
$f\left(t\right)=x=\{\begin{array}{cccc}2t+t^2\sin .\frac{1}{t}& & & t\ne 0\\ 0& & & t=0\end{array}$ and
$g\left(t\right)=y=\{\begin{array}{cc}\frac{1}{t}{\text{sint}}^2& t\ne 0\\ 0& t=0\end{array}$