Explanation

(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)

Correct Answer: C. {x : 2 ≤ x ≤ 4}

Explanation

P = {x : 2 ≤ x ≤ 8} = {2, 3, 4, 5, 6, 7, 8}

Q = {x : 4 < x ≤ 12} = {5, 6, 7, 8, 9, 10, 11, 12}

Q1 = R - {5, 6, 7, 8, 9, 10, 11, 12} = {..., 2, 3, 4, 13, 14, 15, ...}

∴ P ? Q\(^1\) = {2, 3, 4} = {x : 2 ≤ x ≤ 4}

Correct Answer: C. -4, 1

Correct Answer: D. (x + y)(a - b)(a + b)

Correct Answer: D. y = \(\frac{1}{2}x + 1\)

Explanation

y = mx + c; when x = 0; y = 1

1 = m(0) + c; 1 = 0 + c; c = 1

when x = 2; y = 2

2 = m(2) + c; 2 = 2m + c; but c = 1

2 = 2m + 1

2 - 1 = 2m

2m = 1

m = \(\frac{1}{2}\)

y = \(\frac{1}{2}\)x + 1

Correct Answer: B. 0.08 mm/s

Explanation

Given: \(\frac{\mathrm d A}{\mathrm d t} = 4 mm^{2}/s\)

\(\frac{\mathrm d A}{\mathrm d t} = (\frac{\mathrm d A}{\mathrm d r})(\frac{\mathrm d r}{\mathrm d t})\)

\(A = \pi r^{2} \implies \frac{\mathrm d A}{\mathrm d r} = 2\pi r\)

\(\implies 4 = 2\pi r \times \frac{\mathrm d r}{\mathrm d t}\)

\(\frac{\mathrm d r}{\mathrm d t} = \frac{4}{2\pi r} = \frac{4 \times 7}{2 \times 22 \times 8}\)

= \(0.07954 mm/s \approxeq 0.08 mm/s\)

Correct Answer: C. 86.5%

Explanation

To find the mean of a certain set of numbers we need to add all enteries together and then divide by how many numbers you have. In our case we do: (87+92+77+90)/4 = 86.5

Correct Answer: C. sin 147^o

Correct Answer: C. \(\frac{-33}{65}\)

Explanation

From Pythagoras' theorem

when sin x = \(\frac{12}{13}\), cos x = \(\frac{5}{13}\)

Using cos (x + y) = cosx cosy - sinxsiny

\(\frac{5}{13}\) * \(\frac{3}{5}\) - \(\frac{12}{13}\) * \(\frac{4}{5}\)

= \(\frac{-33}{65}\)

Correct Answer: A. (ax - 2y)(bx - 3)