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A vector $\vec{a}$ has components $3p$ and $1$ with respect to a rectangular cartesian system. This system is rotated through a certain angle about the origin in the counter clockwise sense. If, with respect to new system, $\vec{a}$ has components $p + 1$ and $\sqrt{10}$, then a value of $p$ is equal to:
(1) $1$
(2) $-\frac{5}{4}$
(3) $\frac{4}{5}$
(4) $-1$
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