spain-selectividad

Papers (16)
2025
selectividad__matematicas-II_modelo 7
2024
selectividad__matematicas-II_modelo 7
2023
selectividad__madrid_matematicas-II 8 selectividad__madrid_matematicas-II_extraordinaria 8
2022
selectividad__madrid_matematicas-II 8 selectividad__madrid_matematicas-II_extraordinaria 8
2021
selectividad__madrid_matematicas-II_extraordinaria 8 selectividad__madrid_matematicas-II_ordinaria 8
2020
selectividad__madrid_matematicas-II_extraordinaria 8 selectividad__madrid_matematicas-II_ordinaria 7
2019
selectividad__madrid_matematicas-II_extraordinaria 4 selectividad__madrid_matematicas-II_ordinaria 4
2018
selectividad__madrid_matematicas-II_extraordinaria 4 selectividad__madrid_matematicas-II_ordinaria 4
2017
selectividad__madrid_matematicas-II_extraordinaria 4 selectividad__madrid_matematicas-II_ordinaria 4
2020 selectividad__madrid_matematicas-II_extraordinaria

8 maths questions

QA.1 2 marks Matrices Determinant and Rank Computation View
Let $A$ be a matrix of size $3 \times 4$ such that its first two rows are $(1,1,1,1)$ and $(1,2,3,4)$, and with no zeros in the third row. In each of the following sections, provide an example of matrix $A$ that satisfies the requested condition, justifying it appropriately:\ a) (0.5 points) The third row of $A$ is a linear combination of the first two.\ b) (0.5 points) The three rows of $A$ are linearly independent.\ c) (0.5 points) $A$ is the augmented matrix of a compatible determined system.\ d) (0.5 points) $A$ is the augmented matrix of a compatible indeterminate system.\ e) (0.5 points) $A$ is the augmented matrix of an incompatible system.
QA.2 2 marks Curve Sketching Continuity and Discontinuity Analysis of Piecewise Functions View
Given the function $f(x) = \left\{ \begin{array}{lll} \frac{x-1}{x^{2}-1} & \text{if} & x < 1, x \neq -1 \\ \frac{x^{2}+1}{4x} & \text{if} & x \geq 1 \end{array} \right.$, find:\ a) (0.5 points) Calculate $f(0)$ and $(f \circ f)(0)$.\ b) (1.25 points) Study the continuity and differentiability of $f(x)$ at $x = 1$ and determine if there exists a relative extremum at that point.\ c) (0.75 points) Study its asymptotes.
QA.3 2 marks Vectors 3D & Lines Multi-Part 3D Geometry Problem View
Given the point $P(3,3,0)$ and the line $r \equiv \frac{x-2}{-1} = \frac{y}{1} = \frac{z+1}{0}$, find:\ a) (0.75 points) Write the equation of the plane that contains point $P$ and line $r$.\ b) (1 point) Calculate the point symmetric to $P$ with respect to $r$.\ c) (0.75 points) Find two points $A$ and $B$ on $r$ such that triangle $ABP$ is right-angled, has area $\frac{3}{\sqrt{2}}$ and the right angle is at $A$.
QA.4 2 marks Probability Definitions Conditional Probability and Bayes' Theorem View
There are three urns $A$, $B$ and $C$. Urn $A$ contains 4 red balls and 2 black balls, urn $B$ contains 3 balls of each color and urn $C$ contains 6 black balls. An urn is chosen at random and two balls are drawn from it consecutively and without replacement. Find:\ a) (1 point) Calculate the probability that the first ball drawn is red.\ b) (1 point) Calculate the probability that the first ball drawn is red and the second is black.\ c) (0.5 points) Given that the first ball drawn is red, calculate the probability that the second is black.
QB.1 2 marks 3x3 Matrices Direct Determinant Computation View
Given the matrices $A = \left(\begin{array}{ccc} 0 & -1 & 2 \\ 2 & 1 & -1 \\ 1 & 0 & 1 \end{array}\right)$, $I = \left(\begin{array}{lll} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right)$, $B = \left(\begin{array}{cc} 2 & -1 \\ 1 & 0 \\ 0 & 1 \end{array}\right)$, find:\ a) (1 point) Calculate, if possible, the inverse of matrix $A$.\ b) (0.5 points) Calculate the matrix $C = A^{2} - 2I$.\ c) (1 point) Calculate the determinant of matrix $D = ABB^{t}$ (where $B^{t}$ denotes the transpose of matrix $B$).
QB.2 2 marks Applied differentiation Applied modeling with differentiation View
The power generated by a battery is given by the expression $P(t) = 25te^{-t^{2}/4}$, where $t > 0$ is the operating time.\ a) (0.5 points) Calculate the value towards which the power generated by the battery tends if left operating indefinitely.\ b) (0.75 points) Determine the maximum power generated by the battery and the instant at which it is reached.\ c) (1.25 points) The total energy generated by the battery up to instant $t$, $E(t)$, is related to power by $E'(t) = P(t)$, with $E(0) = 0$. Calculate the energy produced by the battery between instant $t = 0$ and instant $t = 2$.
QB.3 2 marks Vectors 3D & Lines Multi-Part 3D Geometry Problem View
For the parallelogram $ABCD$, the consecutive vertices $A(1,0,-1)$, $B(2,1,0)$ and $C(4,3,-2)$ are known. Find:\ a) (1 point) Calculate an equation of the line that passes through the midpoint of segment $AC$ and is perpendicular to segments $AC$ and $BC$.\ b) (1 point) Find the coordinates of vertex $D$ and the area of the resulting parallelogram.\ c) (0.5 points) Calculate the cosine of the angle formed by vectors $\overrightarrow{AB}$ and $\overrightarrow{AC}$.
QB.4 2 marks Independent Events View
In a random experiment there are two independent events $X$, $Y$. We know that $P(X) = 0.4$ and that $P(X \cap \bar{Y}) = 0.08$ (where $\bar{Y}$ is the complementary event of $Y$). Find:\ a) (1 point) Calculate $P(Y)$.\ b) (0.5 points) Calculate $P(X \cup Y)$.\ c) (1 point) If $X$ is an undesired outcome, so that we consider the experiment a success when $X$ does NOT occur, and we repeat the experiment on 8 occasions, find the probability of having succeeded at least 2 times.