Matura (Reifeprüfung Austria) · Mathematics (Matura)

Advanced Analysis, Algebra and Stochastics

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

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In mathematics, what is the nature of the roots of the quadratic equation $2 x^{2} - 4 x + 5 = 0$ ?
In mathematics, what is the derivative of the rational function $f (x) = \frac{2 x - 3}{x ^{2} + 1}$ evaluated at $x = 1$ ?
In mathematics, what is the exact value of the expression $lo g_{2} (32) - lo g_{3} (81) + ln (e^{5})$ ?
In mathematics, what is the exact value of the definite integral $\int_{0}^{1} x e^{x^{2}} d x$ ?
In mathematics, if the 3rd term of an arithmetic progression is 11 and the 7th term is 27, what is the 10th term?
In mathematics, at what value of $x$ does the cubic function $f (x) = x^{3} - 6 x^{2} + 9 x - 2$ have its point of inflection?
In mathematics, what is the sum of the infinite geometric series $8 + 4 + 2 + 1 + \dots$ ?
In mathematics, what is the coefficient of $x^{3}$ in the polynomial expansion of $(2 - x)^{6}$ ?
In mathematics, what is the determinant of the $2 \times 2$ matrix $(54 - 3 2)$ ?
In mathematics, what is the slope of the normal line to the curve $y = 2 x + 1$ at the point where $x = 4$ ?
In mathematics, what is the scalar product of the vectors $u = (3, - 2, 5)$ and $v = (4, 6, 1)$ in $R^{3}$ ?
In mathematics, what is the antiderivative of $\int x cos x d x$ ?
In mathematics, what is the radius of the circle given by the Cartesian equation $x^{2} + y^{2} - 6 x + 8 y = 0$ ?
In mathematics, what is the maximum value attained by the function $f (x) = x e^{- x}$ on the interval $[0, \infty)$ ?
In mathematics, if two independent events $A$ and $B$ have probabilities $P (A) = 0.4$ and $P (B) = 0.5$ , what is $P (A \cup B)$ ?
In mathematics, what is the perpendicular distance from the point $P (2, 3)$ to the line $3 x - 4 y + 1 = 0$ ?
In mathematics, which expression is algebraically equivalent to $\frac{1 - c o s ( 2 x )}{2}$ for all real $x$ ?
In mathematics, what is the value of the limit $lim_{x \to \infty} \frac{4 x ^{3} - 5 x + 1}{7 x ^{3} + 2 x ^{2} - 9}$ ?
In mathematics, what is the modulus (absolute value) of the complex number $z = 5 - 12 i$ ?
In mathematics, what is the area of the finite region enclosed between the parabola $y = 4 - x^{2}$ and the $x$ -axis?
In mathematics, if a discrete random variable has variance $Var (X) = 0.64$ , what is its standard deviation?
In mathematics, if a radioactive isotope decays according to $N (t) = N_{0} e^{- 0.05 t}$ where $t$ is in years, what is its half-life?
In mathematics, what is the slope of any straight line perpendicular to the line with equation $2 x + 5 y = 10$ ?
In mathematics, what is the limit $lim_{x \to 0} \frac{e ^{3 x} - 1}{s i n ( 2 x )}$ ?
In mathematics, in how many different ways can a committee of 3 members be selected from a pool of 8 candidates?
In mathematics, what is the volume of a right circular cone with base radius $r = 6$ cm and vertical height $h = 10$ cm?
In mathematics, what is the fundamental period of the periodic wave function $f (x) = 4 cos (6 x - π)$ ?
In mathematics, on which open interval is the polynomial curve $f (x) = x^{4} - 6 x^{2} + 5$ concave down (inflected downward)?
In mathematics, what is the real solution to the equation $lo g_{5} (2 x - 3) = 2$ ?
In mathematics, what is the magnitude of the cross product $a \times b$ if $∣ a ∣ = 4$ , $∣ b ∣ = 5$ , and the angle between them is $3 0^{\circ}$ ?
In mathematics, what are the coordinates of the vertex of the parabola $y = x^{2} - 8 x + 11$ ?
In mathematics, if a fair die is rolled 180 times, what is the expected number of times a multiple of 3 (3 or 6) appears?
In mathematics, what is the real solution to the exponential equation $3^{2 x - 1} = 27$ ?
In mathematics, what is the value of the definite integral $\int_{0}^{1} \frac{1}{1 + x ^{2}} d x$ ?
In mathematics, what is the average rate of change of $f (x) = 2 x^{2} - 3 x + 1$ over the interval $[1, 4]$ ?
In mathematics, how many local extrema does the polynomial function $f (x) = 3 x^{4} - 4 x^{3} + 1$ have on $R$ ?
In mathematics, what is the sum of the roots of the cubic equation $2 x^{3} - 8 x^{2} + 5 x - 7 = 0$ ?
In mathematics, if a continuous random variable $X$ has probability density function $f (x) = 2 x$ for $0 \leq x \leq 1$ , what is its mean $E (X)$ ?
In mathematics, for which value of $k$ are the vectors $u = (2, - 3)$ and $v = (6, k)$ collinear in $R^{2}$ ?
In mathematics, what is the second-degree Taylor polynomial $T_{2} (x)$ for $f (x) = ln (1 + x)$ centered at $x = 0$ ?
In mathematics, what is the measure of the acute angle between the lines $y = x$ and $y = 0$ in the Cartesian plane?
In mathematics, if $f (x) = x^{3} + x$ and $g$ is the inverse function $f^{- 1}$ , what is the derivative $g^{'} (2)$ ?
In mathematics, if a point is chosen uniformly at random inside a square of side 4, what is the probability it lands inside an inscribed circle?

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