(a) If \(\frac{\sqrt{5} + 4}{3 - 2\sqrt{5}} - \frac{2 + \sqrt{5}}{4 - 2\sqrt{5}} = a + b\sqrt{5}\), find the values of a and b.
(b)(i) Evaluate : \(\begin{vmatrix} 2 & -1 & 2 \\ 1 & 3 & 4 \\ 1 & 2 & 1 \end{vmatrix}\)
(ii) Using the result in b(i), find, correct to two decimal places, the value of x in the system of equations.
\(2x - y + 2z + 5 = 0\)
\(x + 3y + 4z - 1 = 0\)
\(x + 2y + z + 2 = 0\)
(a) \(\frac{\sqrt{5} + 4}{3 - 2\sqrt{5}} - \frac{2 + \sqrt{5}}{4 - 2\sqrt{5}}\)
= \(\frac{(\sqrt{5} + 4)(4 - 2\sqrt{5}) - (2 + \sqrt{5})(3 - 2\sqrt{5})}{(3 - 2\sqrt{5})(4 - 2\sqrt{5})}\)
= \(\frac{4\sqrt{5} - 10 + 16 - 8\sqrt{5} - 6 + 4\sqrt{5} - 3\sqrt{5} + 10}{12 - 6\sqrt{5} - 8\sqrt{5} + 4(5)}\)
= \(\frac{10 - 3\sqrt{5}}{32 - 14\sqrt{5}}\)
Rationalizing,
\((\frac{10 - 3\sqrt{5}}{32 - 14\sqrt{5}}) (\frac{32 + 14\sqrt{5}}{32 + 14\sqrt{5}})\)
= \(\frac{320 + 140\sqrt{5} - 96\sqrt{5} - 42(5)}{1024 + 448\sqrt{5} - 448\sqrt{5} - 196(5)}\)
= \(\frac{110 + 44\sqrt{5}}{44}\)
= \(\frac{5}{2} + \sqrt{5}\).
You can do the rationalizing before solving if you wish to. I just like mine a little complex.
(b)(i) \(\begin{pmatrix} 2 & -1 & 2 \\ 1 & 3 & 4 \\ 1 & 2 & 1 \end{vmatrix}\)
= \(2(3 - 8) - (-1)(1 - 4) + 2(2 - 3)\)
= \(2(-5) + 1(-3) + 2(-1)\)
= \(-10 - 3 - 2\)
= \(-15\)
(ii) \(2x - y + 2z = -5 ... (1)\)
\(x + 3y + 4z = 1 ..... (2)\)
\(x + 2y + z = -2 ..... (3)\)
\(x = \frac{\begin{vmatrix} -5 & -1 & 2 \\ 1 & 3 & 4 \\ -2 & 2 & 1 \end{vmatrix}}{-15}\)
= \(\frac{-5(3 - 8) + 1(1 + 8) + 2(2 + 6)}{-15}\)
= \(\frac{25 + 9 + 16}{-15}\)
= \(\frac{50}{-15} = -3.33\) (2 d.p)
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