Let $\vec{a}$ and $\vec{b}$ be two vectors of magnitude 4 inclined at an angle $\frac{\pi}{3}$, then what is the angle between $\vec{a}$ and $\vec{a} - \vec{b}$?

Let $y_1(x)$ and $y_2(x)$ be two solutions of the differential equation $\frac{dy}{dx} = x$. If $y_1(0) = 0$ and $y_2(0) = 4$, then what is the number of points of intersection of the curves $y_1(x)$ and $y_2(x)$?

The differential equation, representing the curve $y = e^{x}(a\cos{x} + b\sin{x})$ where $a$ and $b$ are arbitrary constants, is

If $f(x)=ax-b$ and $g(x)=cx+d$ are such that $f(g(x))=g(f(x))$, then which one of the following holds?