Ylioppilastutkinto (Finnish Matriculation) · Pitkä matematiikka

Advanced Calculus, Vectors, Geometry and Probability

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

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In advanced mathematics (pitkä matematiikka), what is the limit $lim_{x \to 0} \frac{s i n ( 3 x )}{2 x}$ ?
In advanced mathematics (pitkä matematiikka), what is the derivative of $f (x) = e^{2 x} cos x$ ?
In advanced mathematics (pitkä matematiikka), what is the value of the definite integral $\int_{0}^{1} (3 x^{2} + 2 x - 1) d x$ ?
In advanced mathematics (pitkä matematiikka), what is the scalar product of the vectors $u = 2 i - 3 j + k$ and $v = i + 2 j + 4 k$ ?
In advanced mathematics (pitkä matematiikka), what is the cross product $u \times v$ of $u = (1, 0, 0)$ and $v = (0, 1, 0)$ ?
In advanced mathematics (pitkä matematiikka), what is the sum of the infinite geometric series $\sum_{k = 0}^{\infty} 5 (\frac{2}{3})^{k}$ ?
In advanced mathematics (pitkä matematiikka), what is the modulus of the complex number $z = 3 - 4 i$ ?
In advanced mathematics (pitkä matematiikka), what are the coordinates of the inflection point of $f (x) = x^{3} - 6 x^{2} + 9 x + 2$ ?
In advanced mathematics (pitkä matematiikka), what is the volume of the solid generated by rotating $y = x$ about the $x$ -axis from $x = 0$ to $x = 4$ ?
In advanced mathematics (pitkä matematiikka), what is the derivative of $f (x) = ln (x^{2} + 1)$ ?
In advanced mathematics (pitkä matematiikka), what is the distance from the point $P (1, 2, 3)$ to the plane $2 x - 2 y + z - 7 = 0$ ?
In advanced mathematics (pitkä matematiikka), what is the solution set of the inequality $∣2 x - 5∣ \leq 3$ ?
In advanced mathematics (pitkä matematiikka), what is the value of $\int_{1}^{e} x ln x d x$ evaluated using integration by parts?
In advanced mathematics (pitkä matematiikka), what is the slope of the normal line to the curve $y = x^{2} - 3 x + 2$ at the point where $x = 2$ ?
In advanced mathematics (pitkä matematiikka), what is the value of $lim_{x \to \infty} (1 + \frac{2}{x})^{x}$ ?
In advanced mathematics (pitkä matematiikka), how many real roots does the equation $x^{3} + 3 x + 5 = 0$ possess?
In advanced mathematics (pitkä matematiikka), if $z = 1 + i 3$ , what is $z^{6}$ in standard rectangular form?
In advanced mathematics (pitkä matematiikka), what is the area between the curve $y = 4 - x^{2}$ and the $x$ -axis?
In advanced mathematics (pitkä matematiikka), if a random variable follows $X \sim N (100, 16)$ , what is the standardized value $Z$ corresponding to $X = 108$ ?
In advanced mathematics (pitkä matematiikka), what is the general solution for $x \in [0, 2 π)$ of the equation $sin (2 x) = cos x$ ?
In advanced mathematics (pitkä matematiikka), what is the value of $lim_{x \to 0} \frac{e ^{x} - 1 - x}{x ^{2}}$ ?
In advanced mathematics (pitkä matematiikka), what is the equation of the sphere centered at $(1, - 2, 3)$ with radius $4$ ?
In advanced mathematics (pitkä matematiikka), what is the derivative of $f (x) = x^{x}$ for $x > 0$ ?
In advanced mathematics (pitkä matematiikka), what is the angle between the vectors $a = (1, 1, 0)$ and $b = (0, 1, 1)$ ?
In advanced mathematics (pitkä matematiikka), what is the value of $\int_{0}^{π /2} sin^{3} x cos x d x$ ?
In advanced mathematics (pitkä matematiikka), what is the constant term in the binomial expansion of $(x^{2} + \frac{1}{x})^{6}$ ?
In advanced mathematics (pitkä matematiikka), what is the horizontal asymptote of $f (x) = \frac{4 x ^{2} - 3 x + 1}{2 x ^{2} + 5 x - 7}$ as $x \to \infty$ ?
In advanced mathematics (pitkä matematiikka), what is the maximum volume of an open-top cylindrical can constructed with fixed surface area $12 π$ square units?
In advanced mathematics (pitkä matematiikka), what is the value of $cos (\frac{7 π}{6})$ ?
In advanced mathematics (pitkä matematiikka), for what value of $k$ are the lines $r_{1} = (1, 0, 2) + t (2, k, - 1)$ and $r_{2} = (3, 1, 0) + s (1, 2, 4)$ perpendicular?
In advanced mathematics (pitkä matematiikka), what is the derivative of $f (x) = arctan (2 x)$ ?
In advanced mathematics (pitkä matematiikka), what is the length of the vector $w = (2, - 3, 6)$ ?
In advanced mathematics (pitkä matematiikka), what is the sum of the series $\sum_{n = 1}^{\infty} \frac{1}{n ( n + 1 )}$ using telescoping sums?
In advanced mathematics (pitkä matematiikka), what is the period of the function $f (x) = tan (3 x)$ ?
In advanced mathematics (pitkä matematiikka), what is the acute angle between the planes $x + y = 0$ and $y + z = 0$ ?
In advanced mathematics (pitkä matematiikka), what is the equation of the tangent line to $y = e^{x}$ at $x = 0$ ?
In advanced mathematics (pitkä matematiikka), what is the value of the improper integral $\int_{1}^{\infty} \frac{1}{x ^{2}} d x$ ?
In advanced mathematics (pitkä matematiikka), if two independent events satisfy $P (A) = 0.4$ and $P (B) = 0.5$ , what is $P (A \cup B)$ ?
In advanced mathematics (pitkä matematiikka), what is the derivative of $f (x) = cos^{2} x$ ?
In advanced mathematics (pitkä matematiikka), what is the area of the triangle in $R^{3}$ with vertices $A (0, 0, 0)$ , $B (1, 2, 0)$ , and $C (0, 2, 2)$ ?
In advanced mathematics (pitkä matematiikka), what is the value of $lo g_{2} (32) - lo g_{2} (4)$ ?
In advanced mathematics (pitkä matematiikka), what is the minimum value of $f (x) = e^{x} + e^{- x}$ for $x \in R$ ?
In advanced mathematics (pitkä matematiikka), what is the standard form of the complex number $\frac{1 + i}{1 - i}$ ?

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