All you have to do is determine what the equation will be in each quadrant when the absolute values are removed.
To do this, note that |A| = A for all A ≥ 0, and |A| = -A for all A < 0.
You should get the following equations for the quadrants.
Q1:...

A) x = sin y + cos x
1 = (cos y) dy/dx - sin x
(cos y) dy/dx = 1 + sin x
dy/dx = (1 + sin x) / cos y
B) xy - y3 = sin x
y + x dy/dx - 3y2 dy/dx = cos x
(x - 3y2) dy/dx = -y +...

Problem 1.
Note that the exact root is: x = - 3√4
Machine precision: -1.5874010519681994
Using seed x0 = -2:
f(x) = x3 + 4 so f'(x) = 3x2.
Thus we iterate: x ← x - (x3 + 4) / (3x2) = 2x/3 - 4/(3x2)
x0 = -2
x1...

Triple both sides and then regroup based on pairs which look like a product rule.
3(x2 - 4xy - 2y2)dx + 3(y2 - 4xy - 2x2) dy = 0
3y2 dy - (6y2 dx + 12xy dy) - (12xy dx + 6x2 dy) + 3x2 dx
x3 - 6x2y - 6xy2 + y3 = C
&n...

f(x) = 4 - 3x - x2
[f(x) - f(a)] / (x - a)
= [(4 - 3x - x2) - (4 - 3a - a2)] / (x - a)
= (-3x - x2 + 3a + a2) / (x - a)
= [-3(x - a) - (x + a)(x - a)] / (x - a)
= -3 - x - a

(z2 - 3z + 2)(z2 + 6z + 8) / (z2 + 3z - 4)(z2 - 4)
= (z-1)(z-2)(z+2)(z+4) / (z+3)(z-1)(z+2)(z-2)
= (z+4) / (z+3)

You can only give bounds for this number.
Let's assume, the n hexagons must form a polyhex (like a polyomino but with regular hexagons).
Maximum:
The first hexagon contributes 6 vertices and each new hexagon after that contributes at most 4 vertices.
MAX...

w(x) = 2 cot x sin2 x = 2 cos x sin x = sin 2x
dw/dx = 2 cos 2x

Coins are memoryless, so it is 1/2.

sin 3x - 1 = 0
sin 3x = 1
3x = 2kπ + π/2 where k ∈ Z
x = 2kπ/3 + π/6 where k ∈ Z
In the [0,2π), the solutions are π/6, 5π/6, and 3π/2.

x2 + y2 - x + y - 12 = 0
x2 + y2 - 4x - 2 = 0
The easiest way to do this is to subtract the second equation from the first, which yields a linear equation through the two intersection points.
3x + y - 10 = 0
So that means
y...

More generally, for odd n,
1n + 2n + 3n + … + nn
is a multiple of n.
To show that, note that nn is a multiple of n and that we can split the sum into pairs of terms (n-k)n + kn summing to a multiple of n.
We...

We can say a couple of things:
1. Since the directrix is vertical and the focus is to it's left, the parabola opens left.
2. We can find the parabola's equation in the form (y - k)2 = 4p(x - h).
The vertex is halfway between the focus and directrix, so...

∂f/∂x = 1/(xy)
∂f/∂y = (-1/y2)ln x

Step 1. Use Mark's response to get 2x2 + 5x - 3 = 0
Then use Viete's formulas to get that the sum of the solutions is -5/2.
Viete's formulas state that the set R={r1, … ,rn} of roots (real and complex) of anxn + an-1xn-1 + … + a1x + a0 = 0 satisfy...

1.
The total area of the squares is 13.
The area of the disk is 4π < 4 × 3.25 = 13.
Thus clearly, some squares must overlap
2. This is actually a Number Theory problem.
If we choose 12 different two digit numbers...

If F = ∇U then we find that:
U = xey + f(y,z)
U = xey + yez + g(x,z)
U = yez + h(x,y)
We can take:
f(y,z) = yez
g(x,z) = 0
h(x,y) = xey
This means that F(x,y,z)...

Let's neaten things up:
f(x) = 2x3 - 6x2 + 3x + 1
f'(x) = 6x2 - 12x + 3
We are iterating:
xn+1 = xn - (2x3 - 6x2 + 3x + 1)/(6x2 - 12x + 3)
Note that convergence is quadratic. That is, there is...

For the question on the mystery number, let x be the number.
x + 1/x = 13/6
6x2 - 13x + 6 = 0
(2x - 3)(3x - 2) = 0
x = 3/2 or 2/3
For the question with the secretary:
lcm(5,6) = 30
In...

rs = 2GM/c2
≈ 2(6.67408 × 10-11 Nm2/kg2)(7.95 × 1028 kg) / (299792458 m/s)2
≈ 118 m