For preparing a $0.1 \, N$ solution of a compound from its impure sample,of which the percentage purity is known,the weight of the substance required will be

If $I_n = \int_0^{\pi / 4} \tan^n \theta \, d\theta$ for $n = 1, 2, 3, \ldots$, then $I_{n-1} + I_{n+1}$ is equal to

Two wires are made of the same material and have the same volume. However,wire $1$ has cross-sectional area $A$ and wire $2$ has cross-sectional area $3A$. If the length of wire $1$ increases by $\Delta x$ on applying force $F$,how much force is needed to stretch wire $2$ by the same amount?

$A$ wire suspended vertically from one of its ends is stretched by attaching a weight of $200 \ N$ to the lower end. The weight stretches the wire by $1 \ mm$. The elastic energy stored in the wire is ......... $J$.

The equation $8 x^2-24 x y+18 y^2-6 x+9 y-5=0$ represents a

The system of equations $3x + 2y + z = 6$,$3x + 4y + 3z = 14$ and $6x + 10y + 8z = a$ has an infinite number of solutions if $a$ is equal to