Question 1Easy4 marks
In this question you must show all stages of your working.
Solutions relying entirely on calculator technology are not acceptable.
Solutions relying entirely on calculator technology are not acceptable.
(a) Simplify fully \(\sqrt{252} + \sqrt{28} - \sqrt{112}\), giving your answer in the form \(k\sqrt{7}\), where \(k\) is an integer.[2]
(b) Expand and simplify \((4 - 4\sqrt{11})(6 - 2\sqrt{11})\), giving your answer in the form \(a + b\sqrt{11}\), where \(a\) and \(b\) are integers.[2]
Show the answer and mark scheme
(a) Answer: \(4\sqrt{7}\)
- M1 for at least two of \(\sqrt{252} = 6\sqrt{7}\), \(\sqrt{28} = 2\sqrt{7}\), \(\sqrt{112} = 4\sqrt{7}\)
- A1 for \(4\sqrt{7}\)
Worked solution: \(\sqrt{252} = \sqrt{36 \times 7} = 6\sqrt{7}\)
\(\sqrt{28} = \sqrt{4 \times 7} = 2\sqrt{7}\)
\(\sqrt{112} = \sqrt{16 \times 7} = 4\sqrt{7}\)
\(\sqrt{252} + \sqrt{28} - \sqrt{112} = 6\sqrt{7} + 2\sqrt{7} - 4\sqrt{7} = 4\sqrt{7}\)
(b) Answer: \(112 - 32\sqrt{11}\)
- M1 for an attempt to expand the brackets with at least three of the four terms correct: \(24 - 8\sqrt{11} - 24\sqrt{11} + 88\)
- A1 for \(112 - 32\sqrt{11}\)
Worked solution: \((4 - 4\sqrt{11})(6 - 2\sqrt{11}) = 24 - 8\sqrt{11} - 24\sqrt{11} + 88 = 112 - 32\sqrt{11}\), using \((\sqrt{11})^2 = 11\).