The centre of mass of a system of two particles divides the distance between them

Infinite rods of uniform mass density and lengths $L, L/2, L/4, \dots$ are placed one upon another up to infinity as shown in the figure. Find the $x-$ coordinate of the centre of mass.

The center of mass of a system of particles with masses $1 \, g, 2 \, g$,and $3 \, g$ is at the origin. When a particle of mass $4 \, g$ with position vector $\alpha(\hat{i} + 2\hat{j} + 3\hat{k})$ is added,the center of mass of the system becomes $(1, 2, 3)$. If $\alpha$ is a constant,its value must be:

The figure shows two cylindrical rods whose centers of mass are marked as $A$ and $B$. The line $AB$ divides the region into two parts: one containing point $O$ (region $1$) and the other containing point $O'$ (region $2$). Choose the correct option regarding the center of mass of the combined system.

Two uniform plates of the same thickness and area but of different materials,one shaped like an isosceles triangle and the other shaped like a rectangle,are joined together to form a composite body as shown in the figure. If the centre of mass of the composite body is located at the mid-point of their common side,then the ratio between the masses of the triangle to that of the rectangle is:

Three point masses $m_1 = 1.6 \, kg$,$m_2 = 2.0 \, kg$,and $m_3 = 2.4 \, kg$ are placed at the corners of a thin massless rectangular sheet $(1.2 \, m \times 1.0 \, m)$ as shown. The center of mass will be located at the point ........... $m$.