In which of the following is incomplete double circulation observed?

If $\int e^x \left( \frac{1-\sin x}{1-\cos x} \right) dx = f(x) + \text{constant}$,then $f(x)$ is equal to

The value of $\cos^{-1}\left(\cos\frac{5\pi}{3}\right) + \sin^{-1}\left(\cos\frac{5\pi}{3}\right)$ is

$A$ bag $X$ contains $2$ white and $3$ black balls and another bag $Y$ contains $4$ white and $2$ black balls. One bag is selected at random and a ball is drawn from it. Then, the probability for the ball chosen to be white is:

The length of the common chord of the two circles $x^2+y^2-4y=0$ and $x^2+y^2-8x-4y+11=0$ is

Acetaldehyde forms a white crystalline precipitate on mixing with a ...... solution of