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Mathematics for the Austrian Matura

37 practice questions, each with the answer and an explanation.

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What is the equation of the tangent line to $f (x) = x^{2}$ at the point $x = 3$ ?
Given $a = (23)$ and $b = (- 1 4)$ , what is $a + b$ ?
What is the median of the data 3, 9, 4, 7, 12?
In a right triangle the side opposite the angle $α$ is 3 cm and the hypotenuse is 5 cm. What is $sin α$ ?
What is the area enclosed by the graph of $f (x) = 4 - x^{2}$ and the $x$ -axis?
What is the sum $1 + 2 + 3 + \dots + 20$ ?
What is the length of the vector $v = (34)$ ?
Evaluate $\int_{1}^{3} 2 x d x$ .
What are the solutions of $x^{2} - 4 x - 12 = 0$ ?
What are the coordinates of the vertex of the parabola $f (x) = x^{2} - 6 x + 5$ ?
A price rises by 20 percent and then falls by 20 percent. How does the final price compare with the starting price?
For which value of $k$ are $a = (2 - 1)$ and $b = (1 k)$ perpendicular?
The mean of four numbers is 7. Three of them are 5, 6 and 9. What is the fourth?
A cuboid has edges of 2 cm, 3 cm and 6 cm. How long is its space diagonal?
How many local minima does the function $f (x) = x^{3} - 3 x$ have, and where?
What is the dot product of $123$ and $4 - 5 6$ ?
What is the derivative of $f (x) = 5 x^{2} - 3 x + 1$ ?
Solve $e^{2 x} = 7$ .
Solve $3^{x} = 81$ .
A fair die is rolled 3 times. What is the probability of at least one six?
In a box plot the first quartile is 12 and the third quartile is 20. What is the interquartile range?
A fair coin is tossed 3 times. What is the probability of exactly 2 heads?
A coat costs 80 euros and the price is raised by 15 percent. What is the new price?
What is the area of a circular sector with radius 6 cm and central angle $6 0^{\circ}$ ?
A car drives 150 km in 2 hours and 30 minutes. What is its average speed?
Every value of a data set is increased by 5. Which statement about the standard deviation and the mean is correct?
A machine makes defective parts with probability $0.2$ . In a sample of 50 parts, what is the expected number of defective parts?
A firm's profit in thousand euros is $P (x) = - x^{2} + 12 x - 20$ for $x$ units sold (in hundreds). At which $x$ is the profit largest, and what is it?
The position of a particle is $s (t) = t^{3} - 6 t^{2} + 9 t$ (metres, $t$ in seconds). At which times is its velocity zero?
A substance has a half-life of 5 days. How much of an 80 g sample is left after 15 days?
A line passes through the points $(1, 3)$ and $(3, 11)$ . What is its equation?
A sum of 2000 euros is invested at 3 percent interest per year with compound interest. What is it worth after 2 years?
From a point 40 m from the foot of a tower, the angle of elevation of the top is $3 0^{\circ}$ . How high is the tower (to one decimal place)?
A bag holds 3 red and 7 blue balls. One ball is drawn at random. What is the probability that it is red?
A bacterial culture of 500 cells grows by 20 percent every hour. After how many full hours does it first exceed 1000 cells?
What angle does the line $y = 3 x$ make with the positive $x$ -axis?
Solve the system $3 x + 2 y = 16$ , $x - y = 2$ .

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