List of top Mathematics Questions on types of vectors asked in JEE Main

Let \(\hat a\) and \(\hat b\) be two unit vectors such that the angle between them is \(\frac{\pi}{4}\). If θ is the angle between the vectors (\(\hat a\)+\(\hat b\)) and (\(\hat a\)+2\(\hat b\)+2(\(\hat a\)×\(\hat b\))), then the value of 164 cos2θ is equal to :

Let
\(\vec{a}\) and \(\vec{b}\) be two vectors such that 
\(|\vec{a} + \vec{b}|^2 = |\vec{a}|^2 + 2|\vec{b}|^2,\  \vec{a} \cdot \vec{b} = 3\) and \(|\vec{a} \times \vec{b}|^2 = 75\)
Then \(|\vec{a}|^2\) is equal to _____.