Given two vectors $\vec{A}$ and $\vec{B}$,such that $\vec{A} + \vec{B} = \vec{C}$ and $A^2 + B^2 = C^2$. Which of the following statements is true?

$A$ vector has a magnitude $x$. If it is rotated by an angle $\theta$, then the magnitude of the change in the vector is $nx$. Match the following two columns:

Column $I$	Column $II$
$(A)$ $\theta=60^{\circ}$	$(p)$ $n=\sqrt{3}$
$(B)$ $\theta=90^{\circ}$	$(q)$ $n=1$
$(C)$ $\theta=120^{\circ}$	$(r)$ $n=\sqrt{2}$
$(D)$ $\theta=180^{\circ}$	$(s)$ $n=2$

What is the wave number of the lowest energy transition associated with the Paschen series?

Find the angle between $\vec{A} = 3\hat{i} - \hat{j} + 4\hat{k}$ and the $Z$-axis.

$\vec{A}$ and $\vec{B}$ are two non-zero vectors inclined at an angle $\theta$. $\hat{a}$ and $\hat{b}$ are unit vectors along $\vec{A}$ and $\vec{B}$ respectively. The component of $\vec{A}$ in the direction of $\vec{B}$ is

Read each statement below carefully and state with reasons,if it is true or false:
$(a)$ The magnitude of a vector is always a scalar.
$(b)$ Each component of a vector is always a scalar.
$(c)$ The total path length is always equal to the magnitude of the displacement vector of a particle.
$(d)$ The average speed of a particle (defined as total path length divided by the time taken to cover the path) is either greater or equal to the magnitude of average velocity of the particle over the same interval of time.
$(e)$ Three vectors not lying in a plane can never add up to give a null vector.