Explanation

(a)
The centre and radius of circle
$x^{2} + y^{2} - x - y - 1 = 0$
are $\left( \frac{1}{2},\frac{1}{2} \right)$ and $\sqrt{\frac{3}{2}}$ respectively and the centre and radius of circle
$x^{2} + y^{2} - 2x + 2y - 7 = 0$
are $(1,\ - 1)$ are 3 respectively
Distance between the centres is $\sqrt{\frac{5}{2}} < 3 - \sqrt{\frac{3}{2}}$
$\left( \because C_{1}C_{2} < r_{1} - r_{2} \right)$
$\therefore$ First circle is completely inside the second circle
The centre and radius of circle
$x^{2} + y^{2} + 14x + 12y + 21 = 0$
are $( - 7,\ 6)$ and 8 respectively and the centre and radius of circle
$x^{2} + y^{2} + 2x - 4y - 4 = 0$
are $( - 1,\ 2)$ and 1 respectively
Distance between the centres is $2\sqrt{13} > 8 + 1\ \ \ (\because C_{1}C_{2} > r_{1} + r_{2})$
These two circles intersect each other, therefore the number of common tangents is 2. Hence, only first statements is correct