If $\rm(\vec {a} + \vec{b})$ is perpendicular to $\rm\vec {a}$ and magnitude of $\rm\vec {b}$ is twice that of $\rm\vec {a}$, then what is the value of $\rm(4\vec {a} + \vec{b})\cdot \vec{b}$ equal to?

Let $\rm\vec {a}, \vec{b}$ and $ \rm \vec {c}$ be three vectors such that  $\rm\vec {a}, \vec{b}$ and $ \rm \vec {c}$  are co-planar. Which of the following is/are correct?

1. $\rm(\vec{a}\times \vec{b})\times \vec{c}$ is co-planar with $\rm\vec {a}$ and $\rm\vec {b}$

2. $\rm(\vec{a}\times \vec{b})\times \vec{c}$ is perpendicular to $\rm\vec {a}$ and $\rm\vec {b}$

Select the correct answer using the code given below.

If the position vectors of A and B are (√2 - 1)î - ĵ and î + (√2 + 1)ĵ respectively, then what is the magnitude of $\rm \vec{AB}$?

If y = (1 + x)(1 + x²)(1 + x⁴)(1 + x⁸)(1 + x¹⁶) then what is $\frac{dy}{dx}$ at x = 0 equal to?