Applications of Derivatives

Let two non-collinear unit vectors $\hat a$ and $\hat b$ form an acute angle. A point $P$ moves so that at any time $t$ the position vector $\overrightarrow{OP}$ (where $O$ is the origin) is given by $\hat a\cos t+\hat b\sin t$. When $P$ is farthest from origin $O$, let $M$ be the length of $\overrightarrow{OP}$ and $\hat u$ be the unit vector along $\overrightarrow{OP}$. Then,

Which of the following is true?

Let
$$g(x)=\int_0^{e^x}\frac{f'(t)}{1+t^2}dt.$$
Which of the following is true?

Match the Statements / Expressions in Column I with the Statements / Expressions in Column II and indicate your answer by darkening the appropriate bubbles in the 4 × 4 matrix given in the ORS.

The tangent to the curve $y=e^x$ drawn at the point $(c,e^c)$ intersects the line joining the points $(c-1,e^{c-1})$ and $(c+1,e^{c+1})$

In the following $[x]$ denotes the greatest integer less than or equal to $x$.

Match the functions in Column I with the properties in Column II and indicate your answer by darkening the appropriate bubbles in the 4 × 4 matrix given in the ORS.

Let $f(x)=2+\cos x$ for all real $x$.

STATEMENT-1: For each real $t$, there exists a point $c$ in $[t,t+\pi]$ such that $f'(c)=0$.

because

STATEMENT-2: $f(t)=f(t+2\pi)$ for each real $t$.

The line $y=x$ meets $y=ke^x$ for $k\le0$ at

The positive value of $k$ for which $ke^x-x=0$ has only one root is

For $k>0$, the set of all values of $k$ for which $ke^x-x=0$ has two distinct roots is