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
$$\frac{\frac{\exp\left(x\right)}{x} + \ln\left(x\right)\exp\left(x\right)}{2\sqrt{\ln\left(x\right)}\sqrt{\exp\left(x\right)}}$$


Other Articles:

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.

Generating Plant-like Structures Using Neural Networks

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.




© Hugo2015. Session @sessionNumber