Maths Exam

1. Solution of linear equations

We have the following equations: $$\begin{eqnarray} 3 x +4y &=& 4 \\ -6 x -6 y &=& 2 \end{eqnarray}$$
Correct Answer
$$ x=\frac{-16}{3}, y=5 $$

2. Differentiation A

Differentiate the following equation: $$-4x - \ln\left(x\right) + \frac{2}{x}$$
Correct Answer
$$-4 - \frac{1}{x} + \frac{-2}{{x}^{2}}$$

3. Differentiation A

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

4. Differentiation B

Differentiate the following equation: $$\left(6 - x + \ln\left(x\right)\right)\ln\left(\frac{-4}{x}\right)$$
Correct Answer
$$\left(\frac{1}{x} - 1\right)\ln\left(\frac{-4}{x}\right) + \frac{(-\left(6 - x + \ln\left(x\right)\right))}{x}$$

5. Integration

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

6. Integration

Please carry out the following integral: $$ \int {\left((-5)x\right)}^{-4} dx$$
Correct Answer
$$ \frac{-1}{1875{x}^{3}} $$

7. Consider the following differential equation: $$ 6\frac{d^2y}{dx^2} - 6\frac{dy}{dx} + 3y = 0 $$

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

8. Consider the following series: $$ \sum_{j=0}^{16} -6\left(j + 4\right) $$

What does the sum evaluate to?
Correct Answer
$$ -1224$$

9. Solution of linear equations

We have the following equations: $$\begin{eqnarray} x +2y &=& 4 \\ -3 x +6y &=& 1 \end{eqnarray}$$
Correct Answer
$$ x=\frac{11}{6}, y=\frac{13}{12} $$

10. Differentiation A

Differentiate the following equation: $$\cos\left(\ln\left(x\right) + \cos\left(x\right)\right)$$
Correct Answer
$$(-\sin\left(\ln\left(x\right) + \cos\left(x\right)\right)\left(\frac{1}{x} - \sin\left(x\right)\right))$$


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