﻿ Maths Exam

### Maths Exam

1. Solution of linear equations

 We have the following equations: $$\begin{eqnarray} -4 x +2y &=& 2 \\ -2 x +4y &=& 4 \end{eqnarray}$$ Correct Answer $$x=0, y=1$$

2. Differentiation A

 Differentiate the following equation: $$\ln\left(\frac{3}{x}\right)\sqrt{\ln\left(x\right)}$$Correct Answer $$\frac{(-\sqrt{\ln\left(x\right)})}{x} + \frac{\ln\left(\frac{3}{x}\right)}{2x\sqrt{\ln\left(x\right)}}$$

3. Differentiation A

 Differentiate the following equation: $$\exp\left(\sqrt{(-x)}\right)$$Correct Answer $$\frac{(-\exp\left(\sqrt{(-x)}\right))}{2i\sqrt{x}}$$

4. Differentiation B

 Differentiate the following equation: $$\frac{\sin\left(x\right)\exp\left(2x\right)}{\sqrt{x}}$$Correct Answer $$\frac{\cos\left(x\right)\exp\left(2x\right) + 2\sin\left(x\right)\exp\left(2x\right)}{\sqrt{x}} - \frac{\sin\left(x\right)\exp\left(2x\right)}{2{x}^{1.5}}$$

5. Integration

 Please carry out the following integral: $$\int \sin\left(2x\right) dx$$Correct Answer $$\frac{(-\cos\left(2x\right))}{2}$$

6. Integration

 Please carry out the following integral: $$\int {\left(x + 6\right)}^{-3} dx$$Correct Answer $$\frac{-1}{2{\left(x + 6\right)}^{2}}$$

7. Consider the following differential equation: $$\frac{d^2y}{dx^2} + 5\frac{dy}{dx} + 4y = 0$$

 Find the general solution to the above equation.Correct Answer $$A\exp\left((-x)\right) + B\exp\left(-4x\right)$$

8. Consider the following series: $$\sum_{j=0}^{18} 2\left(j + 2\right)$$

 What does the sum evaluate to?Correct Answer $$418$$

9. Solution of linear equations

 We have the following equations: $$\begin{eqnarray} -x -4 y &=& 3 \\ -2 x -6 y &=& 3 \end{eqnarray}$$ Correct Answer $$x=3, y=\frac{-3}{2}$$

10. Differentiation A

 Differentiate the following equation: $$\sqrt{\ln\left(x\right)\exp\left(x\right)}$$Correct Answer $$\frac{\frac{\exp\left(x\right)}{x} + \ln\left(x\right)\exp\left(x\right)}{2\sqrt{\ln\left(x\right)}\sqrt{\exp\left(x\right)}}$$

## Moon Formation

A Kotlin N-Body code, and lots of animations of the collision between Earth and a hypothetical Theia that people think created the moon.

## The double slit and observers

A look at the double slit experiment. The first half is meant to be a clear explanation, using simulations. The second half discusses some of the philosophy / interpretations of quantum mecahnics.

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