Representation of Vectors

Q. A vector $a$ has components $2p$ and $1$ with respect to a rectangular cartesian system. The system is rotated through a certain angle about the origin in the anti-clockwise sense. If $a$ has components $p+1$ and $1$ with respect to the new system, then

1. $p=0$
2. $p=1$ or $-\frac{1}{3}$
3. $p=-1$ or $\frac{1}{3}$
4. $p=1$ or $-1$

Q.

ABCD is a parallelogram and $A_1$ and $B_1$ are the mid-points of the sides BC and CD respectively.
$If\overrightarrow{A{A}_1}+\overrightarrow{A{B}_1}=\lambda \overrightarrow{AC},$ then ${}^{\prime }{\lambda }^{\prime }$ is equal to

1. $\frac{1}{2}$

2. 1

3. $\frac{3}{2}$

4. 2