Exercise 1

At the point $P\left(a,a^n\right)$ on the graph of $y=x^n\left(n\in N\right)$ in the first quadrant a normal is drawn. The normal intersects the $Y$-axis at the point $\left(0, b\right)$. If $\underset{a\to 0}{\lim }b=\frac{1}{2},$ then $n$ equal

Equation $\frac{1}{{\left(x+1\right)}^3}-3x+\sin x=0$ has

Suppose $f^{\text{'}}\left(x\right)$ exists for each $x$ and $h\left(x\right)=f\left(x\right)-{\left(f\left(x\right)\right)}^2+{\left(f\left(x\right)\right)}^3 \forall x\in R$, then

Let $f:R\to (0,\infty )$ and $g:R\to R$ be twice differentiable functions such that $f^{"}$ and $g^{"}$ are continuous functions on $R$. Suppose $f^{\text{'}}\left(2\right)=g\left(2\right)=0,f^{"}\left(2\right)\ne 0$ and $g^{\text{'}}\left(2\right)\ne 0$, If $\underset{x\to 2}{\lim }\frac{f\left(x\right)g\left(x\right)}{f^{\text{'}}\left(x\right)g^{\text{'}}\left(x\right)}=1$, then

The number of the tangents that can be drawn to the curve $y^2-2x^3-4y+8=0$ from the point $(1,2)$, is

If the area enclosed by normal to the curve $y=(1+x{)}^y+{\sin }^{-1}\left({\sin }^{2}x\right)$ at $x=0$ with coordinate axes (in sq. units) is $k$, then the value of $2k$ is

If the point of intersection of the tangents drawn to the curve $x^{2}y=1-y$ at the points where it is intersected by the curve $x y=1-y$, is $(a, b)$, then $a+b$ is

A function $y=f\left(x\right)$ is given by $x=\frac{1}{1+t^2} \& y=\frac{1}{t\left(1+t^2\right)}$ for all $t>0$ then $f$ is