$N$ divisions on the main scale of a vernier calipers coincide with $(N + 1)$ divisions of the vernier scale. If each division of main scale is $a$ units, then the least count of the calipers is

$\int_{-1}^1 \frac{\cosh x}{1+e^{2 x}} d x$ is equal to :

If $k > 1$ and the determinant of the matrix $A^2$,where $A = \begin{bmatrix} k & k\alpha & \alpha \\ 0 & \alpha & k\alpha \\ 0 & 0 & k \end{bmatrix}$,is $k^2$,then $|\alpha|$ is equal to

The maximum possible number of real roots of the equation $x^5 - 6x^2 - 4x + 5 = 0$ is

$A$ particle of mass $10\, g$ is kept on the surface of a uniform sphere of mass $100\, kg$ and a radius of $10\, cm$. Find the work to be done against the gravitational force between them to take the particle far away from the sphere. (Take $G = 6.67 \times 10^{-11} \, Nm^2/kg^2$)

Let $l_1$ and $l_2$ be two lines intersecting at $P$. If $A_1, B_1, C_1$ are points on $l_1$,and $A_2, B_2, C_2, D_2, E_2$ are points on $l_2$,and if none of these coincides with $P$,then the number of triangles formed by these eight points is: