Maths Exam

1. Solution of linear equations

We have the following equations: $$\begin{eqnarray} -4 x +3y &=& 3 \\ 4 x -2 y &=& 1 \end{eqnarray}$$
Correct Answer
$$ x=\frac{9}{4}, y=4 $$

2. Differentiation A

Differentiate the following equation: $$1 - \exp\left(x\right)\cos\left(x\right)$$
Correct Answer
$$(-\left(\exp\left(x\right)\cos\left(x\right) - \exp\left(x\right)\sin\left(x\right)\right))$$

3. Differentiation A

Differentiate the following equation: $$\sqrt{\left(x - 5\right)\ln\left(x\right)}$$
Correct Answer
$$\frac{\ln\left(x\right) + \frac{x - 5}{x}}{2\sqrt{x - 5}\sqrt{\ln\left(x\right)}}$$

4. Differentiation B

Differentiate the following equation: $$\left(\frac{-2}{x} - 5x\right)\sin\left(\frac{(-x)}{3}\right)$$
Correct Answer
$$\frac{3\left(\frac{2}{{x}^{2}} - 5\right)\sin\left(\frac{(-x)}{3}\right) - \left(\frac{-2}{x} - 5x\right)\cos\left(\frac{(-x)}{3}\right)}{3}$$

5. Integration

Please carry out the following integral: $$ \int \frac{-6}{2x - 5} dx$$
Correct Answer
$$ null $$

6. Integration

Please carry out the following integral: $$ \int \frac{-3}{-6x} dx$$
Correct Answer
$$ null $$

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

Find the general solution to the above equation.
Correct Answer
$$ A\exp\left((-x)\right) + B\exp\left(\frac{4x}{5}\right) $$

8. Consider the following series: $$ \sum_{j=0}^{10} j + 4 $$

What does the sum evaluate to?
Correct Answer
$$ 99$$

9. Solution of linear equations

We have the following equations: $$\begin{eqnarray} x +5y &=& 3 \\ -3 x -6 y &=& 4 \end{eqnarray}$$
Correct Answer
$$ x=\frac{-38}{9}, y=\frac{13}{9} $$

10. Differentiation A

Differentiate the following equation: $$2{x}^{1.5} + \exp\left(\cos\left(x\right)\right)$$
Correct Answer
$$3\sqrt{x} - \exp\left(\cos\left(x\right)\right)\sin\left(x\right)$$