Maths Exam

1. Solution of linear equations

We have the following equations: $$\begin{eqnarray} 2 x -4 y &=& 4 \\ 3 x -y &=& 1 \end{eqnarray}$$
Correct Answer
$$ x=0, y=-1 $$

2. Differentiation A

Differentiate the following equation: $$\sin\left(\ln\left(4x\right)\right)$$
Correct Answer
$$\frac{\cos\left(\ln\left(4x\right)\right)}{x}$$

3. Differentiation A

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

4. Differentiation B

Differentiate the following equation: $${2}^{\left(-2 - x\right)}\left(x + 6 - \exp\left(x\right)\right)$$
Correct Answer
$$(-{2}^{\left(-2 - x\right)}\ln\left(2\right)\left(x + 6 - \exp\left(x\right)\right)) + {2}^{\left(-2 - x\right)}\left(1 - \exp\left(x\right)\right)$$

5. Integration

Please carry out the following integral: $$ \int \ln\left(x + 2\right) dx$$
Correct Answer
$$ \left(x + 2\right)\ln\left(x + 2\right) - \left(x + 2\right) $$

6. Integration

Please carry out the following integral: $$ \int \sqrt{6x} dx$$
Correct Answer
$$ \frac{{\left(6x\right)}^{\frac{3}{2}}}{9} $$

7. Consider the following differential equation: $$ 5\frac{d^2y}{dx^2} + 3\frac{dy}{dx} - 2y = 0 $$

Find the general solution to the above equation.
Correct Answer
$$ A\exp\left(\frac{2x}{5}\right) + B\exp\left((-x)\right) $$

8. Consider the following series: $$ \sum_{j=0}^{10} 2\left(j - 2\right) $$

What does the sum evaluate to?
Correct Answer
$$ 66$$

9. Solution of linear equations

We have the following equations: $$\begin{eqnarray} -2 x +5y &=& 2 \\ x -y &=& 2 \end{eqnarray}$$
Correct Answer
$$ x=4, y=2 $$

10. Differentiation A

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


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