MaxWong

100% probability means that you can be 100% sure that it will happen.

Example: The Sun rises from the East

0 probability means that it won't happen.

Example: The Sun rises from the West

I will give you some hints for the other problems.

b) What has 2 sides?

c) What has 6 faces?

Using the formula to find the volumes, we have

\(V_{\text{cylinder}} = \pi r^2 \cdot 2r = 2\pi r^3\\ V_{\text{sphere}} = \dfrac43 \pi r^3\\ V_{\text{cone}} = \dfrac13 V_{\text{cylinder}} = \dfrac23 \pi r^3\)

We have \(V_{\text{cylinder}} = 2\pi r^3\) and \(V_{\text{cylinder}} + V_{\text{cone}} = \dfrac43 \pi r^3 + \dfrac23 \pi r^3 = 2\pi r^3\), so the two choices gives the same amount of chocolate.

\(BM = MA\) (why?)

\(BA = \sqrt{6^2 + 8^2} = 10\)

\(\angle BMA = 90^\circ\text{ and } BM = MA \Longleftrightarrow \triangle BMA\text{ is a right isosceles triangle (with angles }90^\circ,45^\circ , 45^\circ )\)

\(BM^2 + MA^2 = BA^2\\ 2BM^2 = 10^2\\ BM = MA = 5\sqrt 2\)

The numbers denote the area of each part. 

In the fullest details:

\(\dfrac{x + 2}{x^2 - 2x} - \dfrac2{x - 2} = -1\\ \dfrac{x + 2}{x(x - 2)} - \dfrac{2}{x - 2} = -1\\ \dfrac{x + 2}{x(x - 2)} - \dfrac{2\color{blue}x\color{black}}{\color{blue}x\color{black}(x - 2)} = -1\\ \dfrac{(x + 2) - 2x}{x(x - 2)} = -1\\ -x + 2 = -x(x - 2)\\ -x + 2 = -x^2 + 2x\\ x^2 -3x + 2 = 0\\ (x - 1)(x - 2) = 0\\ x = 1 \text{ or } x = 2\text{ (rejected)}\)

(d - e)^2 = d^2 - 2de + e^2 = (d + e)^2 - 4de

By Vieta's formula, if p and q are the roots of the quadratic equation ax^2 + bx + c = 0, then p + q = -b/a, pq = c/a.

So, in this case \(\begin{cases}d + e = -\dfrac{7}4\\ de = \dfrac{-1}{4}\end{cases}\).

You can do the rest from here.

Square of binomial <=> Discriminant = 0

\(s^2 - 4(9)(4) = 0\\ s^2 = 144\\ s = \pm12\)

s = -12 is also a solution.

Yeah of course, we don't add to the right :)

heureka:

\(\left(z^2 + \dfrac1{z^2}\right)\left(z^5 + \dfrac1{z^5}\right) = z^{\color{red}7\color{black}} + \dfrac1{z^{\color{red}7\color{black}}} + z^3 + \dfrac1{z^3}\)