The problem asks us to evaluate four expressions and choose the correct answer from a set of options labeled from "a" to "k". The four expressions are: (1) $\sqrt{21} \times \sqrt{35}$ (2) $\sqrt{50} - \sqrt{98} + \sqrt{72}$ (3) $(2\sqrt{3} + \sqrt{5})(2\sqrt{3} - \sqrt{5})$ (4) $(\sqrt{15} - \sqrt{6})^2$

1. Problem Description

The problem asks us to evaluate four expressions and choose the correct answer from a set of options labeled from "a" to "k". The four expressions are:

(1) $\sqrt{21} \times \sqrt{35}$

(2) $\sqrt{50} - \sqrt{98} + \sqrt{72}$

(3) $(2\sqrt{3} + \sqrt{5})(2\sqrt{3} - \sqrt{5})$

(4) $(\sqrt{15} - \sqrt{6})^2$

2. Solution Steps

(1) $\sqrt{21} \times \sqrt{35} = \sqrt{21 \times 35} = \sqrt{3 \times 7 \times 5 \times 7} = \sqrt{3 \times 5 \times 7^2} = 7\sqrt{3 \times 5} = 7\sqrt{15}$

So the answer to (1) is $7\sqrt{15}$, which corresponds to option "u".

(2) $\sqrt{50} - \sqrt{98} + \sqrt{72} = \sqrt{25 \times 2} - \sqrt{49 \times 2} + \sqrt{36 \times 2} = 5\sqrt{2} - 7\sqrt{2} + 6\sqrt{2} = (5 - 7 + 6)\sqrt{2} = 4\sqrt{2}$

So the answer to (2) is $4\sqrt{2}$, which corresponds to option "ka".

(3) $(2\sqrt{3} + \sqrt{5})(2\sqrt{3} - \sqrt{5})$

This is in the form of $(a+b)(a-b) = a^2 - b^2$.

$(2\sqrt{3} + \sqrt{5})(2\sqrt{3} - \sqrt{5}) = (2\sqrt{3})^2 - (\sqrt{5})^2 = 4 \times 3 - 5 = 12 - 5 = 7$

So the answer to (3) is 7, which corresponds to option "a".

(4) $(\sqrt{15} - \sqrt{6})^2 = (\sqrt{15})^2 - 2\sqrt{15}\sqrt{6} + (\sqrt{6})^2 = 15 - 2\sqrt{15 \times 6} + 6 = 21 - 2\sqrt{3 \times 5 \times 2 \times 3} = 21 - 2\sqrt{2 \times 3^2 \times 5} = 21 - 2 \times 3 \sqrt{2 \times 5} = 21 - 6\sqrt{10}$

So the answer to (4) is $21 - 6\sqrt{10}$, which corresponds to option "ke".

3. Final Answer

9: u

10: ka

11: a

12: ke