If $\int(1-\cos x) \operatorname{cosec}^2 x \, dx = f(x) + c$,then $f(x)$ is equal to

There is a rectangular sheet of dimension $(2m-1) \times (2n-1)$,where $m > 0, n > 0$. It has been divided into squares of unit area by drawing lines perpendicular to the sides. Find the number of rectangles having sides of odd unit length.

If $\vec{a} = \hat{i} - \hat{j} - \hat{k}$ and $\vec{b} = \lambda\hat{i} - 3\hat{j} + \hat{k}$ and the orthogonal projection of $\vec{b}$ on $\vec{a}$ is $\frac{4}{3}(\hat{i} - \hat{j} - \hat{k})$,then $\lambda$ is equal to:

$A$ point charge $q$ is placed at a distance $a/2$ directly above the centre of a square of side $a$. The electric flux through the square is

$A$ particle of mass $4 m$ explodes into three pieces of masses $m, m$ and $2 m$. The equal masses move along $X$-axis and $Y$-axis with velocities $4 ms^{-1}$ and $6 ms^{-1}$ respectively. The magnitude of the velocity of the heavier mass is

Two resistances of $400 \ \Omega$ and $800 \ \Omega$ are connected in series with a $6 \ V$ battery of negligible internal resistance. $A$ voltmeter of resistance $10,000 \ \Omega$ is used to measure the potential difference across the $400 \ \Omega$ resistor. The error in the measurement of the potential difference in volts is approximately: