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Maths Exam

1. Solution of linear equations

We have the following equations: $$\begin{eqnarray} -2 x -y &=& 4 \\ -5 x -y &=& 4 \end{eqnarray}$$
Correct Answer
$$ x=0, y=-4 $$

2. Differentiation A

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

3. Differentiation A

Differentiate the following equation: $${\cos\left(2x\right)}^{2}$$
Correct Answer
$$-4\cos\left(2x\right)\sin\left(2x\right)$$

4. Differentiation B

Differentiate the following equation: $$\frac{\sqrt{6}\sqrt{x}}{\ln\left(-3 - x\right)}$$
Correct Answer
$$\frac{\sqrt{6}}{2\sqrt{x}\ln\left(-3 - x\right)} - \frac{(-\sqrt{6}\sqrt{x})}{{\ln\left(-3 - x\right)}^{2}\left(-3 - x\right)}$$

5. Integration

Please carry out the following integral: $$ \int \ln\left(-4x\right) dx$$
Correct Answer
$$ \frac{(-\left(-4x\ln\left(-4x\right) + 4x\right))}{4} $$

6. Integration

Please carry out the following integral: $$ \int \sin\left(-6x\right) dx$$
Correct Answer
$$ \frac{\cos\left(-6x\right)}{6} $$

7. Consider the following differential equation: $$ -4\frac{d^2y}{dx^2} + 5\frac{dy}{dx} + 6y = 0 $$

Find the general solution to the above equation.
Correct Answer
$$ A\exp\left(\frac{-3x}{4}\right) + B\exp\left(2x\right) $$

8. Consider the following series: $$ \sum_{j=0}^{11} 6\left(j - 2\right) $$

What does the sum evaluate to?
Correct Answer
$$ 252$$

9. Solution of linear equations

We have the following equations: $$\begin{eqnarray} 6 x +6y &=& 1 \\ -3 x -2 y &=& 2 \end{eqnarray}$$
Correct Answer
$$ x=\frac{-7}{3}, y=\frac{5}{2} $$

10. Differentiation A

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


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