NIMCET 2016 — Mathematics PYQ

Calculations:
Given, Let $\vec{a}$, $\vec{b}$ and $\vec{c}$ be three non-zero vectors, no two of which are collinear.
Let the vector $\vec{a}+2\vec{b}$ is collinear with $\vec{c}$
$\Rightarrow \vec{a}+2\vec{b}=x\vec{c}$ .... (1)

$$\vec{b}+3\vec{c}\text{ is collinear with }\vec{a}$$
$$\Rightarrow \vec{b}+3\vec{c}=y\vec{a}$$
$$\Rightarrow \vec{b}=y\vec{a}-3\vec{c}$$
Put the value of $\vec{b}$ in equation 1st, we get
$$\vec{a}+2(y\vec{a}-3\vec{c})=x\vec{c}$$
$$\Rightarrow (1+2y)\vec{a}-(6+x)\vec{c}=0$$
Compare both sides, we get
$$1+2y=0\text{ and }6+x=0$$
$$\therefore x=-6$$
Put the value of x in equation (1), we get
$$\vec{a}+2\vec{b}=-6\vec{c}$$
$$\vec{a}+2\vec{b}+6\vec{c}=0=\vec{0}$$