MaxWong

Let \((x, y) = \left(x, \dfrac1{\sqrt2}(x^2 - 8)\right)\) be a point on the graph.

Then the distance between the origin to that point is \(\sqrt{x^2 + \left(\dfrac1{\sqrt 2}(x^2- 8)\right)^2} = \sqrt{x^2 + \dfrac{(x^2 - 8)^2}2} = \sqrt{\dfrac{x^4 - 14x^2 + 64}2}\)

To minimize the distance, we need to minimize \(x^4 - 14x^2 + 64\). Note that \(x^4 - 14x^2 + 64 = (x^2 - 7)^2 + 15 \geq 15\), with equality attained when x = sqrt(7) or x = -sqrt(7). Then the minimum distance is \(\sqrt{\dfrac{15}2} = \dfrac{\sqrt{30}}2\).

Note that by the Pythagorean theorem, 

\(AC = \sqrt{10^2 + 20^2} = 10\sqrt5\)

Now, consider the interior angle sum of \(\triangle ABM\). We have \(\angle ABM + 90^\circ + \angle BAC = 180^\circ\). That means \(\angle ABM = 90^\circ - \angle BAC\)

Therefore, 

\(\begin{array}{rcl} \cos \angle ABM &=& \sin \angle BAC\\ &=& \dfrac{\boxed{\phantom{\text{aa}}}}{\boxed{\phantom{\text{aa}}}}\\ &=& \boxed{\phantom{\text{aaaa}}} \end{array}\)

Fill in the blanks. You should be able to arrive at the conclusion with the help of SOHCAHTOA.

Two sides are 11 and what? You need to include that number also.

Note that \(\operatorname{lcm}(a, b) = \dfrac{ab}{\gcd(a, b)}\), so if we want the smallest value of lcm, we want to minimize the product a * b.

That means a and b must be as small as possible.

The smallest number with unit digit 2 and divisible by 8 is 32.

The smallest number with unit digit 4 and divisible by 8 is 24.

Can you continue from here?

To get g(8), we must force f(x) = 8 to happen.

When f(x) = 8, 

\(x^2 - 6x - 4 = 0\\ x = 3 \pm \sqrt{13}\)

Then g(8) is either 2(3 + sqrt(13)) + 3 or 2(3 - sqrt(13)) + 3. Now it is your turn to simplify these expressions.

\(2x^2 - 6x + 8 = 2(x^2 - 3x) + 8=2\left(x^2 - 3x + \dfrac94\right) +\dfrac72 = 2\left(x - \dfrac32\right)^2 + \dfrac72\)

Now you should be able to determine the values of a, b, and c.

\(z^2 + z + 1 = 0 \\ (z^2 + z + 1)(z - 1) = 0\\ z^3 - 1 = 0\\ z^3 = 1\)

So, if z^2 + z + 1 = 0, then z^3 = 1. (Note that the reverse is not true: z^3 = 1 does not imply z^2 + z + 1 = 0.)

\(\quad z^{49} + z^{50} + z^{51} + z^{52} + z^{53}\\ = z^{48}(z + z^2 + z^3 + z^4 + z^5)\\ = (z^3)^{16} (z + z^2 + z^3(1 + z + z^2))\\ = 1^{16} (z + z^2 + z^3 (0))\\ = z + z^2 \\ = (z^2 + z + 1) - 1\\ = 0 - 1\\ = -1\)

Just move the 1 to the right-hand side and it's done.

\(z^2 + z + 1 = 0\\ (z^2 + z + 1) - 1 = 0 -1\\ z^2 + z = -1\)

Oh I missed "even". Thanks. But the counting method is still the same.

a) The initial fixed charge is \(t_1 = 8\).

b) Charge per kilometre travelled is 0.75, because for each extra kilometer, the price is increased by 0.75, as described by the recursive formula.

c) This is an arithmetic sequence, since each term is 0.75 more than the previous one.

d) The general term of an arithmetic sequence is \(a + (n - 1)d\), where a is the first term and d is the common difference. Can you find those numbers and substitute into the expression? Don't forget to simplify the result.

e) After you have done d), substitute n = 16 into the expression also.